4.1. Birth and Death Processes - Aqua Phoenixneutrino.aquaphoenix.com/ReactionDiffusion/SERC5chap4.pdfis birth and death processes (BDPs), ... theory of urban development. Queuing

CHAPTER 4

BIRTH AND DEATH PROCESSES AND ORDER STATISTICS

[The first four sections are based on the lectures of Professor P.R. Parthasarathy of theDepartment of Mathematics, Indian Institute of Technology, Madras, Chennai 600 036, India,at the 5th SERC School. These sections will be devoted to birth and death processes and theremaining sections will be devoted to order statistics based on the lectures of Dr. P. YageenThomas, Department of Statistics, University of Kerala, Trivandrum, Kerala, India]

4.1. Birth and Death Processes

4.1.0. Introduction

An important sub-class of Markov chains with continuous time parameter spaceis birth and death processes (BDPs), whose state space is the non-negative inte-gers.These processes are characterized by the property that if a transition occurs,then this transition leads to a neighboring state.

The description of the process is as follows: The process sojourns in a givenstate i for a random length of time following an exponential distribution with pa-rameter λi + µi. When leaving state i, the process enters state (i + 1) or state (i − 1).The motion is analogous to that of a random walk except that transitions occur atrandom times rather than fixed time periods.

BDPs are frequently used as models of the growth of biological populations. Aremarkable variety of dynamic behavior exhibited by many species of plants, insectsand animals has stimulated great interest in the development of both biological ex-periments and mathematical models. In many ecological problems such as animaland cell populations, epidemics, plant tissues, learning processes, competition be-tween species, growth patterns are influenced by population size. Such populationsdo not increase indefinitely, but are limited by, for example, lack of food, cannibal-ism and overcrowding. There are many deterministic models which describe such

109

110 4. BIRTH AND DEATH PROCESSES AND ORDER STATISTICS

density-dependent logistic population growth. They effectively represent the devel-opment of a tumor, the growth of viral plaques or the population of spread in thetheory of urban development.

Queuing theory is an important application area of BDPs. It has proved to beuseful in a wide range of disciplines from computer networks and telecommuni-cations to chemical kinetics and epidemiology. Suppose that customers arrive at asingle server facility in accordance with a Poisson process. That is, times betweensuccessive arrivals are independent exponential variables having mean 1/λ. Uponarrival, each customer goes directly into service if the server is free, and if not, thecustomer joins the queue. When the server finishes serving a customer, the customerleaves the system and the next customer in line, if there are any waiting, enters theservice. The successive service times are assumed to be independent exponentialvariables having mean 1/µ. The number of persons in the system at time t is a BDPwith birth rate λ and death rate µ independent of the number of customers presentin the system at that time.

The presence of only one of the components, viz., birth or death also leads toimportant applications. For example, in the theory of radioactive transformation theradioactive atoms are unstable and disintegrate stochastically. Each of these newatoms is also unstable. By the emission of radioactive particles, these new atomspass through a number of physical states with specified decay rates from one stateto the adjacent state. Thus the radioactive transformations can be modeled as abirth process. Consider the enzyme reaction of blood clotting. In closing a cut, thegelation process of blood clotting is caused by an enzyme known as fibrin, whichis formed by fibrogen. This conversion of fibrogen molecules into fibrin moleculesfollows a birth process.

BDPs on a finite state space cover a large spectrum of operations research andbiological systems. Queuing models with finite state space have applications inproduction and inventory problems, for example, to optimize the size of the stor-age space, to determine the trade-off between throughput and inventory (or waitingtime) and to exhibit the propagation of blockage. The performance of the produce-to-stock manufacturing facility can be obtained from the performance of the fi-nite queuing systems. Network of queues with finite buffers occur widely in com-puter and telecommunications systems. The phenomena of blocking, starvation and

4.1. BIRTH AND DEATH PROCESSES 111

server breakdowns can severely restrict throughput and response time.

There is a steady increase in the applications of BDPs to natural sciences andpractical problems including epidemics, queues and inventories and reliability, pro-duction management, computer-communication systems, neutron propagation, op-tics, chemical reactions, construction and mining and compartmental models.

Therefore, the broad field of applications of birth and death models amply jus-tifies an intensive study of BDPs.

4.1.1. Transient analysis

In the analysis of BDPs, the emphasis is often laid on the steady state solu-tions while the transient or time-dependent analysis has received less attention. Theassumptions required to derive the steady state solutions to queuing systems areseldom satisfied in the design and analysis of real systems. Also, steady state mea-sures of system performance simply do not make sense for systems that never ap-proach equilibrium. Moreover, the steady state results are inappropriate in situationswherein the time horizon of operations is finite. Hence, in many applications, thepractitioner needs a knowledge of the time-dependent behavior of the system ratherthan easily obtainable steady state results. Further, transient solutions are availablefor a wide class of problems and contribute to a more finely tuned analysis of thecosts and benefits of the systems.

BDPs have a huge mathematical literature discussing effective and interestingtechniques to determine numerous important quantities like system size probabili-ties, their stationary behavior, first passage times, etc. The transition probabilitiesof finite BDPs, for example, can be expressed in terms of sums of exponentials.Its complete unified theory can be used for reference in general Markov chains andother stochastic models.

Recurrence relations play an important role in the transient analysis of BDPs.There is hardly a computational task which does not rely on recursive techniques atone time or another. The widespread use of recurrence relations can be ascribed totheir intrinsic constructive quality and the great ease with which they are amenableto mechanization. In particular, “ three-term recurrence relations ” form the nucleusof continued fractions (CFs), orthogonal polynomials (OPs) and BDPs.


The study of the time dependent behavior of BDPs has given rise to many in-tricate and interesting OPs . Several interesting OPs occur in the study of BDPswith linear and quadratic birth and death rates. This orthogonal representation leadsto the spectral measure for BDPs which is important in the study of the transientbehavior.

4.2. Birth and Death Processes

Consider a continuous time Markov chain {x(t), t ≥ 0} with state space S ={0, 1, 2, . . . } with stationary transition probabilities pi j(t), that is,

pi j(t) = P{x(t + s) = j | x(s) = i}.In addition we assume that the pi j(t) satisfy the following postulates:

1. pi,i+1(h) = λih + o(h), as h ↓ 0, i ≥ 0.

2. pi,i−1(h) = µih + o(h), as h ↓ 0, i ≥ 1.

3. pii(h) = 1 − (λi + µi)h + o(h), as h ↓ 0, i ≥ 0.

4. pi j(h) = δi j.

5. µ0 = 0, λ0 > 0, µi, λi > 0, i = 1, 2, . . . .

The o(h) in each case may depend on i. The matrix

Q =

−λ0 λ0 0 0 . . .µ1 −(λ1 + µ1) λ1 0 . . .0 µ2 −(λ2 + µ2) λ2 . . .

0 0 µ3 −(λ3 + µ3) . . ....

......

...

(4.2.1)

is called the infinitesimal generator of the process. The parameters λi and µi arecalled, respectively, the infinitesimal birth and death rates. The process x(t) isknown as a birth and death process. In Postulates 1 and 2 we are assuming thatif the process starts in state i, then in a small interval of time the probabilities of thepopulation increasing or decreasing by 1 are essentially proportional to the lengthof the interval. Sometimes a transition from zero to some ignored state is allowed.

Since the pi j(t) are probabilities, we have pi j(t) ≥ 0 and∞∑

j=0

pi j(t) = 1.


In order to obtain the probability that x(t) = n, we must specify where theprocess starts or more generally the probability distribution for the initial state. Wethen have

P{x(t) = n} =∞∑

i=0

P{x(0) = i}pin(t).

4.2.1. Waiting times

With the aid of the above assumptions we may calculate the distribution of therandom variable ti which is the waiting time of x(t) in state i; that is, given theprocess in state i, we are interested in calculating the distribution of the time ti untilit first leaves state i. Letting

P{ti ≥ t} = Gi(t)

it follows easily by the Markov property that as h ↓ 0

Gi(t + h) = Gi(t)Gi(h) = Gi(t)(pii(h) + o(h)) = Gi(t)[1 − (λi + µi)h] + o(h)

orGi(t + h) −Gi(t)

h= −(λi + µi)Gi(t) + o(1),

so thatG′i(t) = −(λi + µi)Gi(t).

If we use the condition Gi(0) = 1, the solution of this equation is

Gi(t) = e−(λi+µi)t.

That is, ti follows an exponential distribution with mean (λi + µi)−1.

According to Postulates 1 and 2, during a time duration of length h a transitionoccurs from state i to i + 1 with probability λih + o(h) and from state i to i − 1 withprobability µih + o(h). It follows intuitively that, given that a transition occurs attime t, the probability this transition is to state i + 1 is λi(λi + µi)−1 and to state i − 1is µi(λi + µi)−1.

The description of the motion of x(t) is as follows: The process sojourns in agiven state i for a random length of time following an exponential distribution withparameter λi + µi. When leaving state i the process enters either state i + 1 or state

i− 1 with probabilitiesλi

λi + µi, or,

µi

λi + µi, respectively. The motion is analogous to


that of a random walk except that transitions occur at random times rather than atfixed time periods.

4.2.2. The Kolmogorov equations

The transition probabilities pi j(t) satisfy a system of differential equations knownas the backward Kolmogorov differential equations. These are given by

p′0 j(t) = −λ0 p0 j(t) + λ0 p1 j(t), (4.2.2)

p′i j(t) = µ1 pi−1, j(t) − (λi + µi)pi j(t) + λi+1, j(t), i ≥ 1,

and the boundary condition pi j(0) = δi j.

To derive these we have from (4.2.1)

pi j(t + h) =∞∑

k=0

pik(h)pk j(t) = pi,i−1(h)pi−1, j(t) +

pii(h)pi j(t) + pi,i+1 pi+1, j(t) +∑

k

pik(h)pk j(t), (4.2.3)

where the last summation is over all k , i − 1, i, i + 1. Using Postulates 1, 2, and 3we obtain

∑

k

pik(h)pk j(t) ≤∑

k

pik(h)

= 1 − [pii(h) + pi,i−1(h) + pi,i+1(h)]

= 1 − [1 − (λi + µi)h + o(h) + µih + o(h) + λih + o(h)]

= o(h),

so that

pi j(t + h) = µihpi−1, j(t) + (1 − (λi + µi)h)pi j(t) + λihpi+1, j(t) + o(h).

Transposing the terms pi j(t) to the left-hand side and dividing the equation by h, weobtain the equations (4.2.2), after letting h ↓ 0.

The backward equations are deduced by decomposing the time interval (0, t+h),where h is positive and small, into the two periods

(0, h), (h, t + h),


and examining the transitions in each period separately. The equations (4.2.2) fea-ture the initial state as the variable. A different result arises from splitting the timeinterval (0, t + h) into the two periods

(0, t), (t, t + h)

and adapting the preceding analysis. In this viewpoint, under more stringent condi-tions, we can derive a further system of differential equations

p′i0(t) = −λ0 pi0(t) + µ1 pi1(t), (4.2.4)

p′i j(t) = λ j−1 pi, j−1(t) − (λ j + µ j)pi j(t) + µ j+1 pi, j+1(t), j ≥ 1,

with the same initial condition pi j(0) = δi j. These are known as the forward Kol-mogorov differential equations. To do this we interchange t and h in (4.2.3) andunder stronger assumptions in addition to Postulates 1, 2, and 3 it can be shown thatthe last term is again o(h). The remainder of the argument is the same as before.

In general the infinitesimal parameters {λn, µn} may not determine a unique sto-chastic process obeying

∞∑

j=0

pmn(t) = 1 and pmn(t + s) =∞∑

k=0

pmn(t)pkn(s).

A sufficient condition that there exists a unique Markov process with pmn(t)satisfying the above is that

∞∑

n=0

πn

n∑

k=0

1λnπn

= ∞,

where π0 = 1 and

πn = πn−1λn−1

µn=λ0 . . . λn−1

µ1 . . . µn, n ∈ S \ {0}. (4.2.5)

The quantities πn are called the potential coefficients.

We discuss now briefly the behavior of pi j(t) as t becomes large. It can be provedthat the limits

limt→∞

pi j(t) = p j (4.2.6)


exist and are independent of the initial state i and also that they satisfy the equations

0 = −λ0 p0 + µ1 p1 (4.2.7)

0 = λn−1 pn−1 − (λn + µn)pn + µn+1 pn+1, n ≥ 1. (4.2.8)

which is obtained by setting the left-hand side of (4.2.4) equal to zero. The con-vergence of

∑

j p j follows since∑

j pi j(t) = 1. If∑

j p j = 1, then the sequence {p j}is called a stationary distribution (steady state probabilities). The reason for this isthat p j also satisfy

p j =

∞∑

i=0

pi pi j(t), (4.2.9)

which tells us that if the process starts in state i with probability pi, then at any giventime t it will be in state i with same probability pi. The proof of (4.2.9) follows from(4.2.1) and (4.2.6) and if we let t ↑ ∞ and use the fact that

∑∞i=0 pi < ∞. The solution

to (4.2.7) and (4.2.8) is obtained by rewriting them as

µn+1 pn+1 − λn pn = µn pn − λn−1 pn−1

= · · · = µ1 p1 − λ0 p0 = 0.

Therefore,

p j = π j p0, j = 1, 2, . . . . (4.2.10)

In order that the sequence {p j} define a distribution we must have∑∞

j=0 p j = 1.If

∑

πk < ∞, we see in this case that

p j =π j

∑∞k=0 πk

, j = 0, 1, 2, . . . .

If∑

πk = ∞, then necessarily p0 = 0 and the p j are all zero. Hence, we do have alimiting stationary distribution.

Example 4.2.1. Poisson process: For this process, the rates are given by λn = λ

(constant) and µn = 0. This process plays an important role as an arrival process in manycontexts. In this case, the value of j in the function f j of the random variable technique is 1and

f1(x(t)) = λ, independent of x(t).

The probability generating function P(s, t) is given by

∂p∂t= λ(s − 1)P, P(s, 0) = s


which leads to P(s, t) = eλ(s−1)t . Therefore,

Pn(t) = e−λt(λt)n

n!, n = 0, 1, . . .

E[x(t)] = λt and Var[x(t)] = λt, t ≥ 0.

For a Poisson process, the time between two subsequent occurrences of events is expo-nentially distributed with parameter λ. That is, if t1 denotes the time of the first event and tndenotes the time between the (n − 1)th and nth events, the random variables {ti, i = 1, 2, . . . }are independent exponentially distributed with the parameter of the Poisson process.

Example 4.2.2. Simple birth process (Yule process): For this process, the rates aregiven by λn = nλ and µn = 0. This process models a population in which each member actsindependently and gives birth at an exponential rate λ. The simplest example of a pure birthprocess is the Poisson process, which has a constant birth rate λn = λ, n ≥ 0. Here

f1(x(t)) = λx(t).

Starting with a single individual, the probability generating function using random vari-able technique P(s, t) can be given by

∂P∂t= λs(s − 1)

∂P∂s, P(s, 0) = s.

Then,

P(s, t) =

{

1 − e−λt(

1 − 1s

)}−1

.

Therefore, {x(t)} follows a geometric distribution. Hence,

E[x(t)] = eλt and Var[x(t)] = eλt(eλt − 1).

Example 4.2.3. The rates are given by

λn = (N − n)λ, µn+1 = (n + 1)µ, n = 0, 1, . . . ,N − 1.

Thus,

Pn(t) =µm

(λ + µ)N

[

1 − e−(λ+µ)t]m [

µ + λe−(λ+µ)t]N−m

×n

∑

i=0

(

mi

) (

N − mn − i

) [

λ + µe−(λ+µ)t

µ(1 − e−(λ+µ)t)

]i [λ(1 − e−(λ+µ)t)µ + λe−(λ+µ)t

]n−i

,


where x(0) = m. The mean and variance of x(t) at time t are given by

E[x(t)] =1λ + µ

[

λN − e−(λ+µ)t(λN − m(λ + µ))]

,

Var[x(t)] =1

(λ + µ)2

[

λµN − e−(λ+µ)(λN(µ − λ) − m(µ2 − λ2)2)

−e−2(λ+µ)t(λ2N + m(µ2 − λ2))]

.

Exercises 4.2.

4.2.1. If the birth and death rates are given by λn = λ, µn = nµ. Obtain the prob-ability generating function of the number of customers at time t.( M/M/∞, Infiniteserver queue )

4.2.2. For λn = λ, µn = nµ, obtain P00(t). (M/M/1 queueing system)

4.2.3. For a BDP on {0,1} obtain the transition probabilities.

4.2.4. For a BDP with rates λ0 = 1 and other λn = 1/2 and µn = 1/2, show thatP00(t) = e−tI0(t).

4.2.5. Arrivals to a supermarket is a Poisson process with arrival rate 10 per hour.Given that 100 persons arrived up to 1 PM, what is the probability that 2 more per-sons will arrive in 10 minutes? What is the average number of arrivals between 11AM and 1 PM?

4.2.6. Consider a Poisson process. Given that an event has occurred before T findthe pdf of this arrival.

4.2.7. Arrivals at a telephone booth are considered to be Poisson, with an averagetime of 10 minutes between one arrival and the next. The length of a phone call isassumed to be distributed exponentially with mean 3 minutes. Find the probabilitythat a person arriving at the booth will have to wait and his average time spent inthe booth.

4.3. CONTINUED FRACTIONS 119

4.3. Continued Fractions

A CF is denoted by

a1

b1 +a2

b2 +a3

b3 + · · ·or economically by

a1

b1+

a2

b2+

a3

b3+ · · · , (4.3.1)

where an and bn are real or complex numbers. This fraction can be terminated byretaining the terms a1, b1, a2, b2, . . . , an, bn and dropping all the remaining termsan+1, bn+1, . . . . The number obtained by this operation is called the nth convergentor nth approximant and is denoted by An/Bn. Both An and Bn satisfy the recurrencerelation

Un = anUn−2 + bnUn−1 (4.3.2)

with initial values A0 = 0, A1 = a1 and B0 = 1, B1 = b1. A2n

B2n( A2n+1

B2n+1) is called the

even (odd) part of the CF.

A CF is denoted by

a1

b1 +a2

b2 +a3

b3 + · · ·or economically by

a1

b1+

a2

b2+

a3

b3+ · · · , (4.3.3)

where an and bn are real or complex numbers. This fraction can be terminated byretaining the terms a1, b1, a2, b2, . . . , an, bn and dropping all the remaining termsan+1, bn+1, . . . . The number obtained by this operation is called the nth convergentor nth approximant and is denoted by An/Bn. Both An and Bn satisfy the recurrencerelation

Un = anUn−2 + bnUn−1 (4.3.4)


with initial values A0 = 0, A1 = a1 and B0 = 1, B1 = b1. A2n

B2n( A2n+1

B2n+1) is called the

even (odd) part of the CF.

A CF is said to be convergent if at most a finite number of its denominators Bn

vanish and the limit of its sequence of approximants

limn→∞

An

Bn, (4.3.5)

exists and is finite. Otherwise, the CF is said to be divergent. If

an > 0 , bn > 0 ,∞∑

n=1

√

bnbn−1

an+1= ∞,

then the CF is convergent. The value of a CF is defined to be the limit (4.3.5). Novalue is assigned to a divergent CF.

For any convergent CF, the exact value of the fraction is between any two neigh-boring convergents. All even numbered convergents lie to the left of the exact value,that is, they give an approximation to the exact value by defect. All odd numberedconvergents lie to the right of the exact value, that is, they give an approximationto the exact value by excess. For a terminating CF, that is, a fraction with finitenumber of terms, the last convergent coincides with the exact value. If a CF isnon-terminating, the sequence of convergents is infinite. For example,

√2 = 1 + (

√2 − 1) = 1 +

1

1 +√

2= 1 +

12+

12+ · · · .

The successive convergents are

11,

32,

75,

1712,

4129,

9970,

239169, . . . .

That is,1.0, 1.5, 1.4, 1.417, 1.4138, 1.41429, 1.41420, . . . .

Observe that75<√

2 <32,

4129<√

2 <9970, and so on.

Using the power series expansion of

8 tan

(

πh2

)

− tan(πh)

3h= π − π

5

480h4 − 5π7

16 · 7!h6 − · · · ,


we get a CF representation of π as follows:

π = 3 +17+

115+

11+

1292+

11+ · · · .

Its successive convergents are 31 ,

227 ,

333106 ,

355113 , · · · We observe that

∣

∣

∣

∣

∣

π − 355113

∣

∣

∣

∣

∣

< 3 × 10−7.

This shows how fast the CF is converging to the exact value. This result is spectac-ular. However,

ex22

∫ ∞

xe−

t22 dt =

1x+

1x+

2x+

3x+ · · ·

requires about 70 terms to pin the error down to 2.4 × 10−7 when x = 1.

It is well-known that a sequence of OPs {Pn(x)} satisfy a three-term recurrencerelation

Pn+1(x) = (x + bn)Pn(x) − anPn−1(x) , n ≥ 0 (4.3.6)

with P0 = 1, P−1 = 0 and an > 0. This recurrence relation when compared with therecurrence relation (4.3.4) suggests consideration of the Jacobi CF

1x + b1

− a1

x + b2− a2

x + b3− · · · .

In this case the nth convergent is a rational function An−1(x)/Bn(x) whose de-nominator Bn(x) is a polynomial of degree n and numerator An−1(x) is a poly-nomial of degree n − 1. The sequences {Bn(x)} and {An(x)} satisfy (4.3.6) withB0(x) = 1 , B1(x) = x + b1 and A0(x) = 0 , A1(x) = 1. Further, An−1(x) form anassociated OP system and the zeros of An−1(x) and Bn(x) are distinct and interlace.

4.3.1. CFs and BDPs

In this section, approximate transient system size probability values for an infi-nite BDP {x(t), t ≥ 0} is obtained using the CF technique discussed below. Define

Lr−1 =

r−1∏

k=0

λk; Mr =

r∏

k=1

µk, r = 0, 1, 2, . . .


with L−1 = 1 = M0. Define

fr(s) = (−1)r Mr

∫ ∞

0e−stPr(t)dt, r = 0, 1, 2, . . . .

Taking the Laplace transform of the system of equations given by (4.2.4) and as-suming m = 0, f0(s) gives the expression

f0(s) =1

s + λ0 +f1(s)

f0(s)

.

Similarly,

fr(s)fr−1(s)

=− λr−1µr

s + λr + µr +fr+1(s)

fr(s)

, r = 1, 2, 3, . . . .

This leads to

f0(s) =1

s + λ0− λ0µ1

s + λ1 + µ1− λ1µ2

s + λ2 + µ2· · · . (4.3.7)

Let the nth approximant of f0(s) be given byAn−1(s)Bn(s)

where

A0(s) = 1

A1(s) = s + λ1 + µ1

An(s) = (s + λn−1 + µn−1)An−1(s) − λn−2µn−1An−2(s), n = 2, 4, . . . ,

and

B1(s) = s + λ0

B2(s) = (s + λ1 + µ1)B1(s) − λ0µ1

Bn(s) = (s + λn−1 + µn−1)Bn−1(s) − λn−2µn−1Bn−2(s), n = 3, 4, . . . .

Then

f0(s) ≈ An−1(s)Bn(s)

, (4.3.8)


where ≈ signifies “approximately” and Bn(s) can be written in tridiagonal determi-nant form as follows:-

Bn(s) =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

s + λ0 1λ0µ1 s + λ1 + µ1 1

λ1µ2 s + λ2 + µ2 1. . .

λn−2µn−1 s + λn−1 + µn−1

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

n×n

. (4.3.9)

An−1(s) is obtained from Bn(s) by deleting the first row and first column.

We observe that Bn(s) given by (4.3.9) is clearly zero when −s is an eigenvalueof the matrix

Fn =

λ0 µ1

λ0 λ1 + µ1 µ2

λ1 λ2 + µ2 µ3. . .

λn−2 λn−1 + µn−1

n×n

.

This matrix is quasi-symmetric and can be transformed into a real symmetric matrixEn by a similarity transformation En := D−1

n FnDn, where

Dn := Diag{

1,√

L0M1,√

L1M2, . . . ,√

Ln−2 Mn−1

}

.

The matrix thus obtained is given by

En =

λ0√λ0µ1√

λ0µ1 λ1 + µ1√λ1µ2√

λ1µ2 λ2 + µ2√λ2µ3

. . . √λn−2µn−1 λn−1 + µn−1

n×n

.

This is a real symmetric diagonal dominant positive definite tridiagonal matrix withnon-zero subdiagonal elements and therefore the eigenvalues are real and distinct.We denote these eigenvalues by s(n)

1 , s(n)2 , s

(n)3 , . . . , s

(n)n , that is, −s(n)

1 ,−s(n)2 , s

(n)3 , . . . ,−s(n)

n

are the roots of Bn(s). Similarly, the roots of An−1(s) are negative, real and distinctand let −z(n)

1 ,−z(n)2 ,−z(n)

3 , . . . ,−z(n)n−1 be its roots.


Using partial fractions, (4.3.8) can be expressed as

f0(s) ≈n

∑

j=1

n−1∏

r=1

(z(n)r − s(n)

j )

(s + s(n)j )

n∏

r=1,r, j

(s(n)r − s(n)

j )

.

On inverting we get,

P0(t) ≈n

∑

j=1

n−1∏

r=1

(z(n)r − s(n)

j )

n∏

r=1,r, j

(s(n)r − s(n)

j )

e−s(n)j t.

Example 4.3.1. BDP with constant birth and death rates: The rates are given by

λn = λ; µn+1 = µ, n = 0, 1, 2, . . . ,N − 1. (4.3.10)

This process corresponds to an M/M/1/N queuing model with arrival rate λ and service rateµ. We get the roots of sk of BN+1(s) as follows:

s0 = 0, sk = −λ − µ + 2√

λµ coskπ

N + 1, k = 1, 2, ...,N.

By induction, for i = 0, 1, 2, . . . ,N,

Bi(−sk) =

{

λi, if k = 0

4i (λµ)i2 yi,k − 4i−1µ (λµ)

i−12 yi−1,k , if k = 1, 2, 3, . . . ,N.

,

where, for k = 1, 2, 3, . . . ,N and i = 2, 3, 4, . . . ,N + 1,

yi−1,k =

i∏

j=1

sin

( [

(i + 1)k + (N + 1) j]

π

2(N + 1)(i + 1)

)

sin

([

(i + 1)k − (N + 1) j]

π

2(N + 1)(i + 1)

)

.


We get for n = 0, 1, 2, . . . ,N,

Pn(t) = λN−nµn

N∏

j=1

{

λ + µ − 2√

λµ cosjπ

N + 1

}

+4n+m−2µ√ρ

N∑

k=1

{

4√ρYn,k − Yn−1,k

} {

4√ρYm,k − ym−1,k

}

yN−1,ke−t(λ+µ−2√λµ cos kπ

N+1 )

(

1 + ρ − 2√ρ cos

kπN + 1

) N∏

j=1, j,k

sin

(

( j + k)π2(N + 1)

)

sin

(

( j − k)π2(N + 1)

)

,

where ρ = λµ

and m (≥ 0) is the initial state.

Example 4.3.2.

Rates : λn−1 =(γ + n − 1)(β + n)

(γ + 2n − 2)(γ + 2n − 1),

µn =n(γ + n − 1 − β)

(γ + 2n − 1)(γ + 2n), n = 1, 2, 3, . . .

CF : z 2F1(β + 1, 1; γ + 1,−z) =z1+λ0z

1+µ1z

1+λ1z

1+µ2z

1+ · · ·

P00(t) =∞∑

k=0

(−1)k(β + 1)k

(γ + 1)k

tk

k!= 1F1(β + 1, γ + 1,−t).

In particular, if β = 0 and γ = 1, then

P00(t) =1 − e−t

t.

Example 4.3.3.

Rates : λn−1 = (N − n + 1)p , µn = nq , n = 1, 2, · · · ,N

CF :N

∑

j=0

b(N, j; p)z + j

=1z+

N p1+

qz+

(N − 1)p1

+2q

z+ · · · + Nq

z, <(z) > 0

where b(N, j; p) :=(

Nj

)

p jqN− j , j = 0, 1, 2, · · · ,N , q = 1 − p

P00(t) =N

∑

j=0

b(N, j; p)e− jt .


Example 4.3.4.

Rates : λn−1 =1

2n, µn =

12n + 1

, n = 1, 2, 3, · · ·

CF : 1F1(1; z + 1; z) = 1 +2z2+

3z3+

4z4+

5z5+ · · ·

for any complex number z,

P00(t) =∞∑

m=1

e−mmm−2

(m − 1)!e−t/m.

P00(0) = 1

Exercises 4.3.

4.3.1. If the rates are λn−1 =αUn(1/α)

2Un−1(1/α) , µn =αUn−1(1/α)2Un(1/α) , α < 1, n = 1, 2, 3, · · · , show

that P00(t) = 2e−t I1(αt)αt .

4.3.2. For the rates

λn−1 =nr

1 − r, µn =

n1 − r

, r < 1, n = 1, 2, 3, · · · ,

prove that P00(t) = 1−r1−re−t .

4.3.3. For a BDP with rates λn = 1/2 and µn = n/(2n + 1), show that P00(t) = 1−e−t

t .

4.3.4. Obtain a CF expansion of√

5

4.3.5. Express 35511132 as a CF and calculate the first four convergents.

4.3.6. Use recurrence relation to evaluate the tridiagonal determinant with 2cosθ inthe diagonal and 1 in the subdiagonal and superdiagonal.

4.3.7. Give one CF expansion of π.

4.4. ORTHOGONAL POLYNOMIALS 127

4.4. Orthogonal Polynomials

Definition 4.4.1. A sequence of polynomials {Qn(x)}, where Qn(x) is of exactdegree n in x, is said to be orthogonal with respect to a Lebesgue-Stieltjes measuredα(x) if

∫ ∞

−∞Qm(x)Qn(x)dα(x) = 0, m , n. (4.4.1)

Implicit in this definition is the assumption that the moments

mn =

∫ ∞

−∞xndα(x), n = 0, 1, 2, . . . (4.4.2)

exist and are finite. If the non-decreasing, real-valued, bounded function α(x) alsohappens to be absolutely continuous with dα(x) = w(x)dx, w(x) ≥ 0, then (4.4.1)reduces to

∫ ∞

−∞Qm(x)Qn(x)w(x)dx = 0, m , n, (4.4.3)

and the sequence {Qn(x)} is said to be orthogonal with respect to the weight functionw(x). If, on the other hand, α(x) is a step-function with jumps w j at x = x j, j =0, 1, 2, . . . , then (4.4.1) takes the form of a sum:

∞∑

j=0

Qm(x j)Qn(x j)w j = 0, m , n. (4.4.4)

In this case we refer to the sequence {Qn(x)} as OPs of a discrete variable.

General OPs are characterized by the fact that they satisfy a three-term recur-rence relation of the form

xQn(x) = anQn+1(x) + bnQn(x) + cnQn−1(x), (4.4.5)

where Q−1(x) = 0, Q0(x) = 1, an, bn, cn are real and ancn+1 > 0 for n = 0, 1, 2, . . . .This is an important characterization which has been the starting point for much ofthe modern work on general OPs. Monic OPs for a positive Borel measure α on thereal line are polynomials Pn, n = 0, 1, 2, . . . , of degree n and leading coefficient onesuch that

∫

Pn(x)xkdα(x) = 0, k = 0, 1, 2, . . . , n − 1. (4.4.6)


These n orthogonality conditions give n linear equations for the n unknown coeffi-cients ak,n, k = 0, 1, 2, . . . , n−1, of the monic polynomial Pn(x) =

∑nk=0 aknxk, where

ann = 1. This system of n equations for n unknowns always has a unique solutionsince the matrix with entries

∫

xi+ jdα(x), 0 ≤ i, j ≤ n − 1, known as Gram matrix,is positive definite and hence nonsingular. It is well known that such polynomialssatisfy a three-term recurrence relation

Pn+1(x) = (x − bn)Pn(x) − a2nPn−1(x), n ≥ 0 (4.4.7)

with P0 = 1 and P−1 = 0. Often it is more convenient to consider the orthonormalpolynomials pn(x) = γnPn(x) for which

∫

pn(x)pm(x)dα(x) = δmn, m,m ≥ 0 (4.4.8)

so that γn =(∫

P2n(x)dα(x)

)−1/2> 0. The recurrence relation for these orthonormal

polynomials is given by

xpn(x) = an+1 pn+1(x) + bn pn(x) + an pn−1(x), n ≥ 0, (4.4.9)

where

an =

∫

xpn−1(x)pn(x)dα(x) =γn−1

γn, bn =

∫

xp2n(x)dα(x). (4.4.10)

Let us mention at this point the best known examples: the classical OPs ofJacobi, Laguerre and Hermite. These three families of OPs satisfy a second orderlinear differential equation, their derivatives are again OPs of the same family butwith different parameters.

Apart from the three-term recurrence relations, these OPs satisfy a number ofrelationships of the same general form. They satisfy the second order differentialequation

g2(x)p′′n + g1(x)p′1 + dn pn = 0.

Between any two zeros of pn(x) there is a zero of pn+1(x). Many interesting proper-ties of these polynomials depend on their connection with problems of interpolationand mechanical quadrature.

Remarkably for many specific BDPs of practical interest, the weight functiondα(x) has been explicitly determined and general formulas are derived for the OPs.The spectral representation then gives considerable insight into the time-dependentbehavior of the BDPs.


For each system of OPs Rn satisfying the recurrence relation

−xRn(x) = AnRn+1(x) − (An + Cn)Rn(x) + CnRn−1(x),

we can introduce an extra scaling parameter τ , 0 and study the polynomialsS n(x) = Rn(τx), which satisfy the recurrence relation

−xS n(x) =An

τS n+1(x) − (An + Cn)

τS n(x) +

Cn

τS n−1(x).

If the polynomials Rn are orthogonal on the finite set {x0, x1, . . . , xN}, then the poly-nomials S n are orthogonal on the set {x0/τ, x1/τ, . . . , xN/τ}

Example 4.4.1. Chebyshev polynomials of the second kind: Chebyshev polynomialsof the second kind Un satisfy the recurrence relation

2xUn(x) = Un+1(x) + Un−1(x), n ≥ 1, (4.4.11)

with U0(x) = 1 and U1(x) = 2x. They are orthogonal on the interval [−1, 1] with respect tothe weight

√1 − x2.

If we consider only the finite set {U0,U1, . . . ,UN}, then this is also orthogonal on theset {cos jπ/(N + 2), j = 1, 2, . . . ,N + 1}, i.e., the roots of UN+1. Consider orthonormalpolynomials pn satisfying the relation

xpn(x) = an+1 pn+1(x) + bn pn(x) + an pn−1(x),

with p0(x) = 1 and p−1(x) = 0. Then the Chebyshev polynomials of the second kind are theorthonormal polynomials with constant recurrence coefficients an = 1/2 (n = 1, 2, . . .) andbn = 0 (n = 0, 1, 2, . . .). If we change the first recurrence coefficient to b0 = b/2, then weget polynomials Pn which are a special case of co-recursive OPs. Obviously Un and Un−1

are both solutions of the recurrence relation

2xPn(x) = Pn+1(x) + Pn−1(x),

hence the general solution of this second order linear recurrence is Pn(x) = AUn(x) +BUn−1(x), where A and B are determined from the initial conditions P0(x) = 1 and P1(x) =2x − b. This gives

Pn(x) = Un(x) − bUn−1(x).

If we require a finite system of OPs, then we take aN+1 = 0. Finally taking bN = 1/2bgives

PN+1(x) =

(

x − 12b

)

PN(x) − 12

PN−1(x)

= xUN (x) − bxUN−1(x) − 12b

UN(x) +b2

UN−2(x).


Now use UN−2(x) = 2xUN−1(x) − UN(x) to find

PN+1(x) =

(

x − b + 1/b2

)

UN(x). (4.4.12)

Hence this finite system of modified Chebyshev polynomials, with b0 = b/2 and bN = 1/2b,is orthogonal on the zeros of PN+1, i.e., on

{

b + 1/b2, cos

jπN + 1

, j = 1, 2, . . . ,N

}

.

We note that the Chebyshev polynomials of the second kind can be written as

Un(cos(θ)) =sin((n + 1)θ)

sin(θ), n = 0, 1, . . . ; 0 ≤ θ ≤ π. (4.4.13)

The modified Chebyshev polynomials of second kind Vn(x) and Wn(x) are defined asfollows for n = 0, 1, . . . :

Vn(cos(θ)) =cos

(

(n + 12 )θ

)

cos(

θ2

) (4.4.14)

and

Wn(cos(θ)) =sin

(

(n + 12 )θ

)

sin( θ2 ). (4.4.15)

The significance of these observations is that the properties of sines and cosines can be usedto establish many of the properties of Chebyshev polynomials. Chebyshev polynomialshave acquired great practical importance in polynomial approximation methods. Specif-ically, it has been shown that a series of Chebyshev polynomials converges much morerapidly than a power series.

Chebyshev polynomials satisfy the generating function

(1 − 2xt + t2)−1 =

∞∑

n=0

Un(x)tn.

We can write (4.4.13) as

Un(x) =[n/2]∑

k=0

(

n − kk

)

(−1)k(2x)n−2k .

Define

Vn(x) = Un(x) − Un−1(x) and Wn(x) = Un(x) + Un−1(x), n ≥ 1,

where V0(x) = W0(x) = 1. The polynomials Vn(x) and Wn(x) are orthogonal in (−1, 1) with

weight functions√

1+x1−x and

√

1−x1+x , respectively (Szego (1959)).


Example 4.4.2. Krawtchouk Polynomials: These polynomials Kn(x; p,N), with 0 <p < 1, satisfy the recurrence relation

−xKn(x) = p(N − n)Kn+1(x) − [p(N − n) + n(1 − p)]Kn(x) + n(1 − p)Kn−1(x). (4.4.16)

They are orthogonal on the set {0, 1, 2, . . . ,N} with respect to the binomial distribution withparameters N and p.

Example 4.4.3. Dual Hahn Polynomials Hn(x; γ, δ,N): These polynomials satisfy therecurrence relation

−xHn(x) = AnHn+1(x) − (An +Cn)Hn(x) +CnHn−1(x), (4.4.17)

where An = (N − n)(n + d + 1), Cn = n(N − n + δ + 1).They are orthogonal on the set {n(n + d + δ + 1), n = 0, 1, 2, . . . ,N}.

Example 4.4.4. Racah Polynomials Rn(x;α, β, γ, δ): These polynomials with α+ 1 =−N or β + δ + 1 = −N or γ + 1 = −N, satisfy the recurrence relation

−xRn(x) = AnRn+1(x) − (An +Cn)Rn(x) +CnRn−1(x), (4.4.18)

where

An = −(n + α + β + 1)(n + α + 1)(n + β + δ + 1)(n + d + 1)

(2n + α + β + 1)(2n + α + β + 2),

Cn = −n(n + β)(n + α + β − d)(n + α − δ)

(2n + α + β)(2n + α + β + 1).

They are orthogonal on the set {n(n + d + δ + 1), n = 0, 1, 2, . . . ,N}.

4.4.1. OPs and BDPs

For every BDP {x(t), t ≥ 0} with state space S, one has Karlin-McGregor poly-nomials Qn(x) (n = 0, 1, . . . ), satisfying the three-term recurrence relation

Q−1(x) := 0, Q0(x) := 1λnQn+1(x) + (x − λn − µn)Qn(x) + µnQn−1(x) = 0, n ≥ 0

(4.4.19)

which may be written in matrix notation as

Q0(x) = 1, −xQ(x) = Q>Q(x), (4.4.20)


where Q(x) is the column vector with elements Qi(s), i ∈ S and

Q =

−λ0 µ1 0λ0 −λ1 − µ1 µ2

. . .. . .. . .

. . .. . .. . .

. (4.4.21)

We observe that the monic polynomials Pn(x) := (−1)n(λ0λ1 . . . λn)Qn(x), n > 0,satisfy the recurrence formula

xPn(x) = Pn+1(x) + (λn + µn)Pn(x) + λn−1µnPn−1(x), n > 0,P0(x) = 1, P1(x) = x − λ0

(4.4.22)

and that Pn(−x) can be interpreted as the characteristic polynomials of the matrixQT truncated at the nth row and column. Since λn−1µn > 0 for n > 0, the recur-rence relation (4.4.22) and hence (4.4.19) defines a sequence of polynomials whichis orthogonal with respect to a positive measure. Moreover, the parameters in therecurrence relation (4.4.22) are such that {Pn(x)} is orthogonal with respect to a mea-sure on the nonnegative real axis. Let qn(x) denote the nth orthonormal polynomialcorresponding to Qn(x) or Pn(x). It can easily be verified that

qn(x) = π1/2n Qn(x), n ≥ 0, (4.4.23)

where πn are given by (4.2.5).

We now obtain the transition probability functions Pn(t), which satisfy (4.2.4),in terms of the polynomials Qn(x) for the given rates λn and µn. Suppose that thepolynomials {qn(x)} are orthonormal with respect to the measure dα, with supportin [0,∞), that is,

∫ ∞

0qn(x)qm(x)dα(x) = δm,n

and define the functions

fi(x, t) =∞

∑

j=0

P j(t)Q j(x) =∞∑

j=0

π−1/2j P j(t)q j(x), i ∈ S (4.4.24)

or equivalently

f (x, t) = P(t)Q(x), (4.4.25)

where P(t) = (Pmn(t))∞m,n=0. We observe that this matrix satisfies

P′(t) = P(t)Q> (4.4.26)


with

P(0) = I, (4.4.27)

where I = (δi j)∞i, j=0 is the infinite identity matrix. Using (4.4.20) and (4.4.26), wenote that (4.4.25) satisfies the equation

∂

∂tf (x, t) = P′(t)Q(x) = P(t)Q>Q(x) = −x f (x, t)

and by (4.4.27), the initial condition is

f (x, 0) = Q(x).

Hence,

f (x, t) = e−xtQ(x)

or

fi(x, t) = e−xtQi(x), i ∈ S.

By (4.4.24), π−1/2j P j(t) is the jth Fourier coefficient of fi(x, t) with respect to the

orthonormal system {qn(x)} and hence

π−1/2j P j(t) =

∫ ∞

0e−xtQi(x)q j(x)dα(x), i, j ∈ S

or

P j(t) = π j

∫ ∞

0e−xtQi(x)Q j(x)dα(x), i, j ∈ S. (4.4.28)

For every finite BDP {x(t), t ≥ 0} f with finite state spaceS f one has Karlin-McGregorpolynomials Qn, n = 0, 1, 2, . . . ,N, defined by the recurrence relation

−xQn(x) = λnQn+1(x) − (λn + µn)Qn(x) + µnQn−1(x), n = 1, 2, . . . ,N − 1(4.4.29)

with initial values

Q0(x) = 1, Q1(x) =λ0 − xλ0

and the polynomial QN+1 of degree N + 1, defined by

−xQN(x) = QN+1(x) − µN QN(x) + µNQN−1(x). (4.4.30)


Karlin and McGregor have shown that QN+1 has N + 1 distinct, real zeros s0 <

s1 < s2 < · · · < sN. Furthermore, the polynomials {Qn, n = 0, 1, . . . ,N} areorthogonal on the set {s0, s1, . . . , sN}:

N∑

k=0

Qn(sk)Qm(sk)ρk = 0, 0 ≤ n , m ≤ N, (4.4.31)

where ρk are positive weights given by

ρk :=1

N∑

i=0

Q2i (sk)πi

> 0. (4.4.32)

They gave the spectral representation of the Pn(t) in terms of these polynomials, as

Pn(t) = πn

N∑

k=0

e−sk tQm(sk)Qn(sk)ρk, n ∈ S f . (4.4.33)

The monic orthogonal Karlin-McGregor polynomials, corresponding to the poly-nomials Qn(s), are given by Pn(x) := (−1)n(λ0λ1 . . . λn)Qn(x), n = 0, 1, . . . ,N andthey satisfy the recurrence formula

xPn(x) = Pn+1(x) + (λn + µn)Pn(x) + λn−1µnPn−1(x), n = 1, 2, . . . ,N (4.4.34)

with P0(x) = 1 and P−1(x) = 0. So we can also find the monic Karlin-McGregorpolynomial PN+1(x), and its zeros are important because the Karlin-McGregor poly-nomials are orthogonal on the set consisting of these zeros.

Example 4.4.5. Consider the BDP discussed in example 4.3.1. For this BDP, the Karlin-McGregor polynomials Qn(x) satisfy the recurrence relation

−xQn(x) = λQn+1(x) − (λ + µ)Qn(x) + µQn−1(x), n = 1, 2, . . . ,N − 1, (4.4.35)

with Q0(x) = 1 and Q1(x) = (λ − x)/λ. Take the modified Chebyshev polynomials Pn(x) =Un(x) − bUn−1(x) with b =

√

µ/λ, then the recurrence relation (4.4.11) and the change ofvariable

Qn(x) =(

µ

λ

)n2

Pn

(

λ + µ − x

2√λµ

)

give the recurrence (4.4.35) with the initial conditions Q0(x) = 1 and Q1(x) = (λ− x)/λ. Sothe Karlin-McGregor polynomials for this finite BDP turn out to be the modified Chebyshev


polynomials

Un(y) −√

µ/λUn−1(y), y =λ + µ − x

2√µλ,

which are orthogonal on the set {0, λ + µ − 2√λµ cos kπ

N+1 , k = 1, 2, . . . ,N} (see Example4.4.1 for details).

Example 4.4.6. Consider the BDP discussed in example 4.4.2.

The Karlin-McGregor polynomials for this process are the Krawtchouk polyno-mials

Kn

(

xλ + µ

;λ

λ + µ, N

)

,

which are orthogonal on the set

{0, λ + µ, 2(λ + µ), . . . ,N(λ + µ)}.

Example 4.4.7. Consider a finite BDP with quadratic rates

λn = (N − n)(b − (n − 1)c), n = 0, 1, . . . ,N − 1,µn = (N + n + 1)(b + (n + 2)c), n = 1, 2, . . . ,N,

(4.4.36)

with b > 0, −b/(N + 2) < c < 0. Conditions on the parameters are to ensure the positivityof the rates.

The Karlin-McGregor polynomials for this process are the Racah polynomials

Rn

(

− x4c

; −12,

12, −N − 1, −2b + 5c

2c

)

,

which are orthogonal on the set{

−4cn

(

n − 2b + 5c2c

− N

)

, n = 0, 1, 2, . . . ,N

}

.

Example 4.4.8. Consider a finite BDP with quadratic rates

λn = (N − n)(b + (n − 1)d), n = 0, 1, . . . ,N − 1,µn = n(c − (n − 1)d), n = 1, 2, . . . ,N,

(4.4.37)

where b, c > 0 and{

d ≤ 0, if N = 1 or 2,−b

N−2 < d ≤ 0, if N > 2.

The conditions on the parameters are to ensure the positivity of the rates.


The Karlin-McGregor polynomials for this BDP are the dual Hahn polynomials

Hn

(

xd

;b − 2d

d,

c − Ndd,N

)

,

which are orthogonal on the set{

dn

(

n +b + c

d− N − 1

)

, n = 0, 1, 2, . . . ,N

}

.

Exercises 4.4.

4.4.1. Identify the birth and death rates associated with the continued fraction1N

∑N−1j=0

1z+ j =

1z +

λ01 +

µ1

z + · · ·µN

z , z > 0, and show that P00(t) = 1N

∑N−1j=0 e− jt.

4.4.2. Identify the birth and death rates associated with the continued fraction∑N

j=0b(N, j;p)

z+ j =1z +

N p1 +

qz +

(N−1)p1 +

2qz + · · · +

Nqz , <(z) > 0

where b(N, j; p) :=

(

Nj

)

p jqN− j , j = 0, 1, 2, · · · ,N , q = 1 − p and show that

P00(t) =∑N

j=0 b(N, j; p)e− jt.

Reference

Parthasarathy, P.R. and Lenin, R.B. (2004). Birth and Death Process (BDP) Modelswith Applications − Queueing, Communication Systems, Chemical Models, Bio-logical Models: The State-of-the-Art with a Time-Dependent Perspective, Ameri-can Series in Mathematical and Management Sciences 51, Syracuse, NY, AmericanSciences Press, Inc.

4.5. ESTIMATION OF A PARAMETER 137

4.5. Estimation of a Parameter by Ranked-setSampling

[This section is based on the lectures of P. Yageen Thomas, Department of Statistics, Universityof Kerala, Trivandrum 695 581, India.]

4.5.1. Introduction

The concept of ranked-set sampling (RSS) was first introduced by McIntyre(1952) as a process of improving the precision of the sample mean as an estima-tor of the population mean. ranked-set sampling as described in McIntyre (1952)is applicable whenever ranking of a set of sampling units can be done easily bya judgement method (for a detailed discussion on the theory and applications ofranked-set sampling see, Chen et al (2004). Ranking by judgement method is notrecommendable if the judgement method is too crude and is not powerful for rank-ing by discriminating the units of a moderately large sample. In certain situations,one may prefer exact measurement of some easily measurable variable associatedwith the study variable rather than ranking the units by a crude judgement method.Suppose the variable of interest say Y , is difficult or much expensive to measure, butan auxiliary variable X correlated with Y is readily measurable and can be orderedexactly. In this case as an alternative to McIntyre (1952) method of ranked-set sam-pling, Stokes (1977) used an auxiliary variable for the ranking of the sampling units.The procedure of ranked-set sampling described by Stokes (1977) using auxiliaryvariate is as follows: Choose n2 independent units, arrange them randomly into nsets each with n units and observe the value of the auxiliary variable X on each ofthese units. In the first set, that unit for which the measurement on the auxiliaryvariable is the smallest is chosen. In the second set, that unit for which the mea-surement on the auxiliary variable is the second smallest is chosen. The procedureis repeated until in the last set, that unit for which the measurement on the auxiliaryvariable is the largest is chosen. The resulting new set of n units chosen by one fromeach set as described above is called the RSS defined by Stokes (1977). If X(r)r isthe observation measured on the auxiliary variable X from the unit chosen from therth set then we write Y[r]r to denote the corresponding measurement made on thestudy variable Y on this unit, then Y[r]r, r = 1, 2, . . . , n form the ranked-set sample.Clearly Y[r]r is the concomitant of the rth order statistic arising from the rth sample.


A striking example for the application of the ranked-set sampling as proposedby Stokes (1977) is given in Bain (1978, p.99), where the study variate Y representsthe oil pollution of sea water and the auxiliary variable X represents the tar depositin the nearby sea shore. Clearly collecting sea water sample and measuring the oilpollution in it is strenuous and expensive. However the prevalence of pollution inthe sea water is much reflected by the tar deposit in the surrounding terminal seashore. In this example ranking the pollution level of sea water based on the tar de-posit in the sea shore is more natural and scientific than ranking it visually or byjudgement method.

Stokes (1995) has considered the estimation of parameters of location-scalefamily of distributions using RSS. Lam et al (1994, 1996) have obtained the BLUEsof location and scale parameters of exponential distribution and logistic distribu-tion. The Fisher information contained in RSS have been discussed by Chen (2000)and Chen and Bai (2000). Stokes (1980) has considered the method of estimationof correlation coefficient of bivariate normal distribution by using RSS. Modarresand Zheng (2004) have considered the problem of estimation of dependence pa-rameter using RSS. Robust estimate of correlation coefficient for bivariate normaldistribution have been developed by Zheng and Modarres (2006). Stokes (1977)has suggested the ranked-set sample mean as an estimator for the mean of the studyvariate Y , when an auxiliary variable X is used for ranking the sample units, un-der the assumption that (X, Y) follows a bivariate normal distribution. Barnett andMoore (1997) have improved the estimator of Stokes (1977) by deriving the BestLinear Unbiased Estimator (BLUE) of the mean of the study variate Y , based onranked set sample obtained on the study variate Y .

In this paper we are trying to estimate the mean of the population, under asituation where in measurement of observations are strenuous and expensive. Bain(1978, P.99) has proposed an exponential distribution for the study variate Y , theoil pollution of the sea samples. Thus in this paper we assume a Morgenstern typebivariate exponential distribution (MTBED) corresponding to a bivariate randomvariable (X, Y), where X denotes the auxiliary variable (such as tar deposit in the seashore) and Y denotes the study variable (such as the oil pollution in the sea water).A random variable (X, Y) follows MTBED if its probability density function (pdf)is given by (see, Kotz et al, 2000, P.353)


f (x, y) =

1θ1θ2

exp{

− xθ1− y

θ2

} [

1 + α(

1 − 2 exp{

− xθ1

}) (

1 − 2 exp{

− yθ2

})]

,

x > 0, y > 0;−1 ≤ α ≤ 1; θ1 > 0, θ2 > 0

0, otherwise.

(4.5.1)

In Section 4.5 of this note we have derived an estimator θ∗2 of the parameter θ2involved in (4.5.1) using the ranked-set sample mean. It may be noted that if (X, Y)has a MTBED as defined in (4.5.1) then the marginal distributions of both X and Yhave exponential distributions and the pdf of Y is given by

fY(y) =1θ2

e−y/θ2 , y > 0, θ2 > 0. (4.5.2)

The Cramer-Rao Lower Bound (CRLB) of any unbiased estimator of θ2 based

on a random sample of size n drawn from (4.5.2) isθ22n . In Section 4.5, we have also

shown that the variance of the proposed estimator θ∗2 is strictly less than the CRLBθ22n (associated with the marginal pdf (4.5.2)) for all α ∈ A where A = [−1, 1] − {0}.In this section, we have further made an efficiency comparison between θ∗2 and themaximum likelihood estimator (MLE) θ2based on a random sample of size n arisingfrom (4.5.1). In Section 4.5, we have derived the BLUE θ2 of θ2 involved in MTBEDbased on the ranked-set sample and obtained the efficiency of θ2 relative to θ2. InSection 4.5, we have considered the situation where we apply censoring and rankingon each sample and ultimately used a ranked-set sample arising out of this procedureto estimate θ2. In Section 4.5, we have derived the BLUE of θ2 involved in (4.5.1)using unbalanced multistage ranked-set sampling method. In this section, we havefurther analysed the efficiency of the estimator of θ2 based on unbalanced MSRSSwhen compared with the θ2 the MLE of θ2.

4.5.2. ranked-set sample mean as an estimator

Let (X, Y) be a bivariate random variable which follows a MTBED with pdfdefined by (4.5.1). Suppose RSS in the sense of stokes (1977) as explained inSection 4.5 is carried out. Let X(r)r be the observation measured on the auxiliaryvariate X in the rth unit of the RSS and let Y[r]r be the measurement made on theY variate of the same unit, r = 1, 2, · · · , n. Then clearly Y[r]r is distributed as theconcomitant of rth order statistic of a random sample of n arising from (4.5.1). Byusing the expressions for means and variances of concomitants of order statistics


arising from MTBED obtained by Scaria and Nair (1999), the mean and varianceof Y[r]r for 1 ≤ r ≤ n are given below

E[Y[r]r] = θ2

[

1 − αn − 2r + 12(n + 1)

]

, (4.5.3)

Var[Y[r]r] = θ22

[

1 − αn − 2r + 12(n + 1)

− α2 (n − 2r + 1)2

4(n + 1)2

]

. (4.5.4)

Since Y[r]r and Y[s]s for r , s are measurements on Y made from two units involvedin two independent samples we have

Cov[Y[r]r, Y[s]s] = 0, r , s. (4.5.5)

In the following theorem we propose an estimator θ∗2 of θ2 involved in (4.5.1) andprove that it is an unbiased estimator of θ2.

Theorem 4.5.1. Let Y[r]r, r = 1, 2, · · · , n be the ranked set sample observationson a study variate Y obtained out of ranking made on an auxiliary variate X, when(X, Y) follows MTBED as defined in (4.5.1). Then the ranked-set sample mean givenby

θ∗2 =1n

n∑

r=1

Y[r]r,

is an unbiased estimator of θ2 and its variance is given by,

Var[θ∗2] =θ22

n

1 − α2

4n

n∑

r=1

(

n − 2r + 1n + 1

)2

.

Proof 4.5.1.

E[θ∗2] =1n

n∑

r=1

E[Y[r]r].

On using (4.5.3) for E[Y[r]r] in the above equation we get,

E[θ∗2] =1n

n∑

r=1

[1 − αn − 2r + 12(n + 1)

]θ2 (4.5.6)

It is clear to note thatn

∑

r=1

(n − 2r + 1) = 0. (4.5.7)


Applying (4.5.7) in (4.5.6) we get,

E[θ∗2] = θ2.

Thus θ∗2 is an unbiased estimtor of θ2. The variance of θ∗2 is given by

Var[θ∗2] =1n2

n∑

r=1

Var(Y[r]r).

Now using (4.5.4) and (4.5.7) in the above sum we get,

Var[θ∗2] =θ22

n

1 − α2

4n

n∑

r=1

(

n − 2r + 1n + 1

)2

.

Thus the theorem is proved.

Now we compare the variance of θ∗2 with the CRLB θ22/n of any unbiased esti-mator of θ2 involved in (4.5.2) which is the marginal distribution of Y in (4.5.1). Ifwe write e1(θ∗2) to denote the ratio of θ2

2/n with Var(θ∗2) then we have,

e1(θ∗2) =1

[

1 − α2

4n

∑nr=1

(

n−2r+1n+1

)2] . (4.5.8)

It is very trivial to note that

e1(θ∗2) ≥ 1.

Thus we conclude that there is some gain in efficiency on the estimator θ∗2 due toranked-set sampling. The reason for the above conclusion is that a ranked-set sam-ple always provides more information than simple random sample even if ranking isimperfect (see, Chen et al, 2004, P. 58). It is to be noted that Var(θ∗2) is a decreasingfunction of α2 and hence the gain in efficiency of the estimator θ∗2 increases as |α|increases.

Again on simplifying (4.5.8) we get

e1(θ∗2) =1

1 − α2

4 [ 23 ( 2+1/n

1+1/n ) − 1].


Then

limn→∞

e1(θ∗2) = limn→∞

1

1 − α2

4 [ 23 ( 2+1/n

1+1/n ) − 1]

=1

1 − α2

12

.

From the above relation it is clear that the maximum value for e1(θ∗2) is attainedwhen |α| = 1 and in this case e1(θ∗2) tends to 12/11.

Next we obtain the efficiency of θ∗2 by comparing the variance of θ∗2 with theasymptotic variance of MLE of θ2 involved in MTBED. If (X, Y) follows a MTBEDwith pdf defined by (4.5.1), then,

∂ log f (x, y)∂θ1

=1θ1

−1 +xθ1+

2αxθ1

e−xθ1 (2e−

y

θ2 − 1)

1 + α(2e−xθ1 − 1)(2e−

y

θ2 − 1)

and∂ log f (x, y)∂θ2

=1θ2

−1 +y

θ2+

2αyθ2

e−y

θ2 (2e−xθ1 − 1)

1 + α(2e−xθ1 − 1)(2e−

y

θ2 − 1)

.

Then we have,

Iθ1(α) = E

(


)2

,

=1

θ21

{

1 + 4α2

∫ ∞

0

∫ ∞

0

u2e−3u(2e−v − 1)e−v

{1 + α(2e−u − 1)(2e−v − 1)}dvdu

}

.

Iθ2(α) = E

(


)2

,

=1

θ22

{

1 + 4α2

∫ ∞

0

∫ ∞

0

v2e−3v(2e−u − 1)e−u

{1 + α(2e−u − 1)(2e−v − 1)}dvdu

}

and

Iθ1θ2(α) = E

(

∂2 log f (x, y)∂θ1∂θ2

)

,

=1θ1θ2

{

4α∫ ∞

0

∫ ∞

0

uve−2ue−2v

{1 + α(2e−u − 1)(2e−v − 1)}dvdu

}

.


Thus the Fisher information matrix associated with the random variable (X, Y) isgiven by

I(α) =

(

Iθ1 (α) −Iθ1θ2(α)−Iθ1θ2(α) Iθ2(α)

)

. (4.5.9)

We have evaluated the values of θ−21 Iθ1(α) and θ−1

1 θ−12 Iθ1θ2(α) numerically for

α = ±0.25, ±0.50,±0.75,±1 (clearly θ−21 Iθ1(α) = θ−2

2 Iθ2(α)) and are given below.

Table 4.5.1

α θ−21 Iθ1(α) θ−1

1 θ−12 Iθ1θ2(α) α θ−2

1 Iθ1(α) θ−11 θ−12 Iθ1θ2(α)

0.25 1.0062 0.0625 -0.25 1.0062 -0.06280.50 1.0254 0.1258 -0.50 1.0254 -0.12740.75 1.0596 0.1914 -0.75 1.0596 -0.19551.00 1.1148 0.2624 -1.00 1.1148 -0.2712

Thus from (4.5.9) the asymptotic variance of the MLE θ2 of θ2 involved inMTBED based on a bivariate sample of size n is obtained as

Var(θ2) =1n

I(−1)θ2

(α), (4.5.10)

where I(−1)θ2

(α) is the (2, 2)th element of the inverse of I(α) given by (4.5.9).

We have obtained the efficiency e(θ∗2|θ2) = Var(θ2)Var(θ∗2) of θ∗2 relative to θ2 for n =

2(2)10(5)20; α = ±0.25,±0.50,±0.75,±1 and are presented in Table 4.5.1. Fromthe table, one can easily see that θ∗2 is more efficient than θ2 and efficiency increaseswith n and |α| for n ≥ 4.

4.5.3. Best linear unbiased estimator

In this section we provide a better estimator of θ2 than that of θ∗2 by derivingthe BLUE θ2 of θ2 provided the parameter α is known. Let X(r)r be the observationmeasured on the auxiliary variate X in the rth unit of the RSS and let Y[r]r be themeasurement made on the Y variate of the same unit, r = 1, 2, · · · , n. Let

ξr = 1 − αn − 2r + 12(n + 1)

(4.5.11)


and

δr = 1 − αn − 2r + 12(n + 1)

− α2 (n − 2r + 1)2

4(n + 1)2. (4.5.12)

Using (4.5.11) in (4.5.3) and (4.5.12) in (4.5.4) we get E[Y[r]r] = θ2ξr, 1 ≤ r ≤ n,and Var[Y[r]r] = θ22δr, 1 ≤ r ≤ n. Clearly from (4.5.5), we have Cov[Y[r]r, Y[s]s] =0, r, s = 1, 2, . . . , n and r , s. Let Y[n] = (Y[1]1, Y[2]2, · · · , Y[n]n)′ and if the param-eter α involved in ξr and δr is known then proceeding as in David and Nagaraja(2003, p. 185) the BLUE θ2 of θ2 is obtained as

θ2 = (ξ′G−1ξ)−1ξ′G−1Y[n] (4.5.13)

andVar(θ2) = (ξ′G−1ξ)−1θ22, (4.5.14)

where ξ = (ξ1, ξ2, · · · , ξn)′ and G = diag(δ1, δ2, · · · , δn). On substituting the valuesof ξ and G in (4.5.13) and (4.5.14) and simplifying we get

θ2 =

∑nr=1(ξr/δr)Y[r]r∑n

r=1 ξ2r/δr

(4.5.15)

and

Var(θ2) =1

∑nr=1 ξ

2r /δrθ22.

Thus θ2 as given in (4.5.15) can be explicitly written as θ2 =∑n

r=1 arY[r]r, where

ar =ξr/δr

∑nr=1 ξ

2r /δr, r = 1, 2, . . . , n.

Remark 4.5.1. As the association parameter α in (4.5.1) is involved in the BLUEθ2 of θ2 and its variance, an assumption that α is known may sometimes viewedas unrealistic. Hence when α is unknown, our recommendation is to compute thesample correlation coefficient q of the observations (X(r)r, Y[r]r), r = 1, 2, . . . andconsider the model (4.5.1) for a value of α equal to α0 given by

α0 =

−1, if q < − 14

4q, if − 14 ≤ q ≤ 1

4

1, if q > 14 ,


as we know that the correlation coefficient between X and Y involved in the Mor-genstern type bivariate exponential random vector is equal to α4 .

We have computed the the ratio e(θ2|θ2)= Var(θ2)Var(θ2)

as the efficiency of θ2 relative to

θ2 for α = ±0.25,±0.5,±0.75,±1.0 and n = 2(2)10(5)20 and are also given in Table4.5.1. From the table, one can easily see that θ2 is relatively more efficient than θ2.Further we observe from the table that e(θ2|θ2) increases as n and |α| increase.

4.5.4. Estimation based on censored ranked-set sample

In the case of the example of pollution study on sea samples (see, Bain, 1978,P.99), sometimes if there is no tar deposit at the seashore then the correspondinglylocated sea sample will be censored and hence on these units the observations onY is not measured. For ranking on X observations in a sample, the censored unitsare assumed to have distinct and consecutive lower ranks and the remaining unitsare ranked with the next higher ranks in a natural order. If in this censored schemeof ranked set sampling, k units are censored, then we may represent the ranked-setsample observations on the study variate Y as ε1Y[1]1, ε2Y[2]2, . . . , εnY[n]n where,

εi =

0, if the ith unit is censored

1, otherwise

and hence∑n

i=1 εi = n − k. In this case the usual ranked set sample mean is equalto

∑ni=1 εiY[i]i

n−k . It may be noted that εi = 0 need not occur in a natural order fori = 1, 2, . . .n. Hence if we write mi, i = 1, 2, . . . n − k as the integers such that1 ≤ m1 < m2 < · · · < mn−k ≤ n and for which εmi = 1, then,

E

[∑ni=1 εiY[i]i

n − k

]

= θ2

1 − α

2(n + 1)(n − k)

n−k∑

i=1

(n − 2mi + 1)

.

Thus it is clear that the ranked-set sample mean in the censored case is notan unbiased estimator of the population mean θ2. However we can construct anunbiased estimator of θ2 based on this mean. In the following theorem we havegiven the constructed unbiased estimator θ∗2(k) of θ2 based on the ranked-set samplemean under censored situation and its variance.


Theorem 4.5.2. Suppose that the random variable (X, Y) has a MTBED as de-fined in (4.5.1). Let Y[mi]mi , i = 1, 2, . . .n − k be the ranked-set sample observationson the study variate Y resulting out of censoring and ranking applied on the auxil-iary variable X. Then an unbiased estimator of θ2 based on the ranked-set samplemean 1

n−k

∑n−ki=1 Y[mi]mi is given by,

θ∗2(k) =2(n + 1)

[

2(n + 1)(n − k) − α∑n−ki=1 (n − 2mi + 1)

]

n−k∑

i=1

Y[mi]mi

and its variance is given by,

Var[θ∗2(k)] =4(n + 1)2 θ22

[

2(n + 1)(n − k) − α∑n−ki=1 (n − 2mi + 1)

]2

n−k∑

i=1

δmi ,

where δmi is as defined in (4.5.12).

Proof 4.5.2.

E[θ∗2(k)] =2(n + 1)

[

2(n + 1)(n − k) − α∑n−ki=1 (n − 2mi + 1)

]

n−k∑

i=1

E[Y[mi]mi],

=2(n + 1)

[

2(n + 1)(n − k) − α∑n−ki=1 (n − 2mi + 1)

]

n−k∑

i=1

[

1 − αn − 2mi + 12(n + 1)

]

θ2,

=2(n + 1)

[

2(n + 1)(n − k) − α∑n−ki=1 (n − 2mi + 1)

]

n − k − α

2(n + 1)

n−k∑

i=1

(n − 2mi + 1)

θ2,

= θ2.

Thus θ∗2(k) is an unbiased estimator of θ2. The variance of θ∗2(k) is given by,

Var[θ∗2(k)] =4(n + 1)2

[

2(n + 1)(n − k) − α∑n−ki=1 (n − 2mi + 1)

]2

n−k∑

i=1

Var(Y[mi]mi),

=4(n + 1)2 θ22

[

2(n + 1)(n − k) − α∑n−ki=1 (n − 2mi + 1)

]2

n−k∑

i=1

δmi ,

where δmi is as defined in (4.5.12). Hence the theorem.


As a competitor of the estimator θ∗2(k), we now propose the BLUE of θ2 based onthe censored ranked-set sample, resulting out of ranking of observations on X. LetY[n](k) = (Y[m1]m1 , Y[m2]m2 , . . . , Y[mn−k]mn−k)

′, then the mean vector and the dispersionmatrix of Y[n](k) are given by,

E[Y[n](k)] = θ2 ξ(k), (4.5.16)

D[Y[n](k)] = θ22 G(k), (4.5.17)

where ξ(k) = (ξm1 , ξm2, . . . , ξmn−k)′, G(k) = diag(δm1 , δm2 , . . . , δmn−k).

If the parameter α involved in ξ(k) and G(k) is known then (4.5.16) and (4.5.17)together defines a generalized Guass-Markov set up and hence the BLUE θ2(k) ofθ2 is obtained as,

θ2(k) = [(ξ(k))′(G(k))−1ξ(k)]−1(ξ(k))′(G(k))−1Y[n](k) (4.5.18)

and

Var(θ2(k)) = [(ξ(k))′(G(k))−1ξ(k)]−1θ22. (4.5.19)

On substituting the values of ξ(k) and G(k) in (4.5.18) and (4.5.19) and simplifyingwe get

θ2(k) =

∑n−ki=1 (ξmi/δmi)Y[mi]mi∑n−k

i=1 ξ2mi/δmi

(4.5.20)

and

Var(θ2(k)) =1

∑n−ki=1 ξ

2mi/δmi

θ22.

Remark 4.5.2. In the expression for the BLUE’s θ2 given in (4.5.15) and θ2(k)given in (4.5.20) the quantities ξr and δr for 1 ≤ r ≤ n are all non-negative, and con-sequently the coefficients of ranked-set sample observations are also non-negative.Thus the BLUE’s θ2 and θ2(k) of θ2 are always non-negative. Thus unlike in cer-tain situation of BLUE’s where one encounters with inadmissible estimators, theestimators given by θ2 and θ2(k) using ranked set sample are admissible estimators.

Remark 4.5.3. Since both the BLUE θ2(k) and the unbiased estimator θ∗2(k) basedon the censored ranked-set sample utilize the distributional property of the parentdistribution they lose the usual robustness property. Hence in this case the BLUEθ2(k) shall be considered as a more preferable estimators than θ∗2(k).


4.5.5. Unbalanced multistage ranked-set sampling

Al-Saleh and Al-Kadiri (2000) have extended first the usual concept of RSSto double stage ranked-set sampling (DSRSS) with an objective of increasing theprecision of certain estimators of the population when compared with those ob-tained based on usual RSS or using random sampling. Al-Saleh and Al-Omari(2002) have further extended DSRSS to multistage ranked-set sampling (MSRSS)and shown that there is increase in the precision of estimators obtained based onMSRSS when compared with those based on usual RSS and DSRSS. The MSRSS(in r stages)procedure is described as follows:

(1) Randomly select nr+1 sample units from the target population, where r is thenumber of stages of MSRSS.

(2) Allocate the nr+1 selected units randomly into nr−1sets, each of size n2.

(3) For each set in step (2), apply the procedure of ranked-set sampling methodto obtain a (judgment) ranked-set, of size n; this step yields nr−1 (judgment)ranked-sets, of size n each.

(4) Arrange nr−1 ranked-sets of size n each, into nr−2 sets of n2 units each and with-out doing any actual quantification, apply ranked-set sampling method on eachset to yield nr−2 second stage ranked-sets of size n each.

(5) This process is continued, without any actual quantification, until we end upwith the rth stage (judgment) ranked-set of size n.

(6) Finally, the n identified elements in step (5) are now quantified for the variableof interest.

Instead of judgment method of ranking at each stage if there exists an auxiliaryvariate on which one can make measurement very easily and exactly and if the aux-iliary variate is highly correlated with the variable of interest, then we can applyranking based on these measurements to obtain the ranked-set units at each stageof MSRSS. Then on the finally selected units one can make measurement on thevariable of primary interest. In this section we deal with the MSRSS by assumingthat the random variable (X, Y) has a MTBED as defined in (4.5.1), where Y is thevariable of primary interest and X is an auxiliary variable. In Section 4.5, we have


considered a method for estimating θ2 using the Y[r]r measured on the study variateY on the the unit having rth smallest value observed on the auxiliary variable X, ofthe rth sample r = 1, 2, . . . , n and hence the RSS considered there was balanced.Abo-Eleneen and Nagaraja (2002) have shown that in a bivariate sample of size narising from MTBED the concomitant of largest order statistic possesses the max-imum Fisher information on θ2 whenever α > 0 and the concomitant of smallestorder statistic possesses the maximum Fisher information on θ2 whenever α < 0.Hence in this section, first we consider α > 0 and carry out an unbalanced MSRSSwith the help of measurements made on an auxiliary variate to choose the ranked-setand then estimate θ2 involved in MTBED based on the measurement made on thevariable of primary interest. At each stage and from each set we choose an unit ofa sample with the largest value on the auxiliary variable as the units of ranked-setswith an objective of exploiting the maximum Fisher information on the ultimatelychosen ranked-set sample.

Let U(r)i , i = 1, 2, . . . , n be the units chosen by the (r stage) MSRSS. Since

the measurement of auxiliary variable on each unit U (r)i has the largest value, we

may write Y (r)[n]i to denote the value measured on the variable of primary interest on

U(r)i , i = 1, 2, . . . , n. Then it is easy to see that each Y (r)

[n]i is the concomitant of thelargest order statistic of nr independently and identically distributed bivariate ran-dom variables with MTBED. Moreover Y (r)

[n]i , i = 1, 2, . . . , n are also independentlydistributed with pdf given by (see, Scaria and Nair, 1999)

f (r)[n]i(y;α) =

1θ2

e−y

θ2

{

1 + α

(

nr − 1nr + 1

)

(

1 − 2e−y

θ2

)

}

. (4.5.21)

Thus the mean and variance of Y (r)[n]i for i = 1, 2, . . . , n are given below.

E[Y (r)[n]i] = θ2

[

1 +α

2

(

nr − 1nr + 1

)]

, (4.5.22)

Var[Y (r)[n]i] = θ

22

[

1 +α

2

(

nr − 1nr + 1

)

− α2

4

(

nr − 1nr + 1

)]

. (4.5.23)

If we denote

ξnr = 1 +α

2

(

nr − 1nr + 1

)

(4.5.24)

and

δnr = 1 +α

2

(

nr − 1nr + 1

)

− α2

4

(

nr − 1nr + 1

)

(4.5.25)


then (4.5.22) and (4.5.23) can be written as

E[Y (r)[n]i] = θ2ξnr (4.5.26)

andVar[Y (r)

[n]i] = θ22δnr . (4.5.27)

Let Y(r)[n] = (Y (r)

[n]1, Y(r)[n]2, · · · , Y

(r)[n]n)′, then by using (4.5.26) and (4.5.27) we get the

mean vector and dispersion matrix of Y(r)[n] as,

E[Y(r)[n]] = θ2ξnr1 (4.5.28)

andD[Y(r)

[n]] = θ22δnr I, (4.5.29)

where 1 is the column vector of n ones and I is an identity matrix of order n. Ifα > 0 involved in ξnr and δnr is known then (4.5.28) and (4.5.29) together define ageneralized Gauss-Markov setup and hence the BLUE of θ2 is obtained as

θn(r)2 =

1nξnr

n∑

i=1

Y (r)[n]i, (4.5.30)

with variance given by

Var(θn(r)2 ) =

δnr

n(ξnr )2θ22. (4.5.31)

If we take r = 1 in the MSRSS method described above, then we get the usual singlestage unbalanced RSS. Then the BLUE θn(1)

2 of θ2 is given by

θn(1)2 =

1nξn

n∑

i=1

Y[n]i,

with variance

Var(θn(1)2 ) =

δn

n(ξn)2θ22,

where we write Y[n]r instead of Y (1)[n]r and it represent the measurement on the variable

of primary interest of the unit selected in the RSS. Also ξn and δn are obtained byputting r = 1 in (4.5.24) and (4.5.25) respectively.

We have evaluated the ratio e(θn(1)2 |θ2) = Var(θ2)

Var(θn(1)2 )

for α = 0.25(0.25)1; n =

2(2)10(5)20 as a measure of efficiency of our estimator θn(1)2 relative to the MLE

θ2 of θ2 based on n observations and are also provided in Table 4.5.2. From the ta-ble, one can see that the efficiency increases with increase in α and n. Moreover the


efficiency of the estimator θn(1)2 is larger than the estimator θ∗2 based on RSS mean

and the BLUE θ2 based on usual RSS.

Al-Saleh (2004) has considered the steady-state RSS by letting r to +∞. Ifwe apply the steady-state RSS to the problem considered in this paper then theasymptotic distribution of Y (r)

[n]i is given by the pdf

f (∞)[n]i (y;α) =

1θ2

e−y

θ2

{

1 + α(

1 − 2e−y

θ2

)}

. (4.5.32)

From the definition of our unbalanced MSRSS it follows that Y (∞)[n]i , i = 1, 2, . . . , n are

independently and identically distributed random variables each with pdf as definedin (4.5.32). Then Y (∞)

[n]i , i = 1, 2, . . . , n may be regarded as unbalanced steady-state

ranked-set sample of size n. Then the mean and variance of Y (∞)[n]i for i = 1, 2, . . . , n

are given below.

E[Y (∞)[n]i ] = θ2

[

1 +α

2

]

and

Var[Y (∞)[n]i ] = θ22

[

1 +α

2− α

2

4

]

.

Let Y(∞)[n] = (Y (∞)

[n]1, Y(∞)[n]2, · · · , Y

(∞)[n]n)′. Then the BLUE θn(∞)

2 based on Y(∞)[n] and the

variance of θn(∞)2 is obtained by taking the limit as r → ∞ in (4.5.30) and (4.5.31)

respectively and are given by

θn(∞)2 =

1

n[

1 + α2]

n∑

i=1

Y (∞)[n]i ,

and

Var(θn(∞)2 ) =

[

1 + α2 −α2

4

]

n[

1 + α2]2θ22. (4.5.33)


From (4.5.10) and (4.5.33) we get the efficiency of θn(∞)2 relative to θ2 by taking the

ratio of Var(θ2) with Var(θn(∞)2 ) and is given by

e(θn(∞)2 |θ2) =

Var(θ2)

Var(θn(∞)2 ),

=I(−1)θ2

(α)[

1 + α2]2

[

1 + α2 − α2

4

] .

Thus the efficiency e(θn(∞)2 |θ2) is free of the sample size n. That is, for a fixed

α, e(θn(∞)2 |θ2) is a constant for all n. We have evaluated the value e(θn(∞)

2 |θ2) forα = 0.25(0.25)1 and are presented in Table 4.5.3. From the table one can seethat the efficiency of θn(∞)

2 increases as α increases. Moreover, the estimator θn(∞)2

possesses the highest efficiency amoung the other estimators of θ2 proposed in thispaper and the value of the efficiencies ranges from 1.1382 to 1.7093.

As mentioned earlier, for MTBED the concomitant of smallest order statisticpossesses the maximum Fisher information on θ2 whenever α < 0. Therefore whenα < 0 we consider an unbalanced MSRSS in which at each stage and from each setwe choose an unit of a sample with the smallest value on the auxiliary variable asthe units of ranked-sets with an objective of exploiting the maximum Fisher infor-mation on the ultimately chosen ranked-set sample.

Let Y (r)[1]i, i = 1, 2, . . . , n be the value measured on the variable of primary interest

on the units selected at the rth stage of the unbalanced MSRSS. Then it is easy to seethat each Y (r)

[1]i is the concomitant of the smallest order statistic of nr , independently

and identically distributed bivariate random variables with MTBED. Moreover Y (r)[1]i,

i = 1, 2, . . . , n are also independently distributed with pdf given by

f (r)[1]i(y;α) =

1θ2

e−y

θ2

{

1 − α(

nr − 1nr + 1

)

(

1 − 2e−y

θ2

)

}

(4.5.34)

Clearly from (4.5.21) and (4.5.34) we have

f (r)[1]i(y;α) = f (r)

[n]i(y;−α). (4.5.35)


and hence E(Y (r)[n]i) for α > 0 and E(Y (r)

[1]i) for α < 0 are identically equal. Similarly

Var(Y (r)[n]i) for α > 0 and Var(Y (r)

[1]i) for α < 0 are identically equal. Consequently if

θ1(r)2 is the BLUE of θ2, involved in MTBED for α < 0, based on the unbalanced

MSRSS observations Y (r)[1]i, i = 1, 2, . . . , n, then the coefficients of Y (r)

[1]i, i = 1, 2, . . . , n

in the BLUE θ1(r)2 for α < 0 are same as the coefficients of Y (r)

[n]i, i = 1, 2, . . . , n in the

BLUE θn(r)2 for α > 0. Further, we have Var(θ1(r)

2 ) = Var(θn(r)2 ) and hence Var(θ1(1)

2 ) =Var(θn(1)

2 ) and Var(θ1(∞)2 ) = Var(θn(∞)

2 ), where θ1(1)2 is the BLUE of θ2, for α < 0

based on the usual unbalanced single stage RSS observations Y[1]i, i = 1, 2, . . . , nand θ1(∞)

2 is the BLUE of θ2, for α < 0 based on the unbalanced steady-state RSSobservations Y (∞)

[1]i , i = 1, 2, . . . , n. We have obtained the efficiency e(θ1(1)2 |θ2) of

the BLUE θ1(r)2 relative to θ2, the MLE of θ2 for α = −0.25,−0.5,−0.75,−1; n =

2(2)10(5)20 and are incorporated in Table 4.5.2. Similarly as in the case of θn(∞)2 for

a fixed α the efficiency e(θ1(∞)2 |θ2) is same for all n. We have evaluated e(θ1(∞)

2 |θ2) forα = −0.25,−0.5,−0.75,−1 and are incorporated in Table 4.5.3. From the table, onecan see that efficiency increases as |α| increases and the value of efficiency rangesfrom 1.1382 to 1.7161.

Remark 4.5.4. If (X, Y) follows an MTBED with pdf defined in (4.5.1), then thecorrelation coefficient between X and Y is given by,

Corr(X, Y) =α

4,−1 ≤ α ≤ 1.

Clearly when |α| goes to 1, correlation coefficient between X and Y is high. Thus,using the ranks of Y to induce the ranks of X becomes more accurate. Thus when|α| is large (that is α tends to ±1) we see that the ranked-set sample obtained basedon the ranking made on X becomes more informative for making inference on θ2

than the case with small values of |α|. From the table we notice that for a givensample size the efficiencies of all estimators increase as |α| increases. Consequentlywe note that more information on θ2 can be extracted from the ranked-set samplewhen |α| is large subject to |α| ≤ 1. Thus we conclude that concomitant ranking ismore effective in estimating θ2 when the absolute value of the association parameterα is large (that is when α tends to ±1).


Table 4.5.2Efficiencies of the estimators θ∗2, θ2, θn(1)

2 and θ1(1)2 relative to θ2 of θ2

involved in Morgenstern type bivariate exponential distribution

n α e(θ∗2|θ2) e(θ2|θ2) e(θn(1)2 |θ2) α e(θ∗2|θ2) e(θ2|θ2) e(θ1(1)

2 |θ2)

0.25 0.9994 0.9994 1.0410 -0.25 0.9994 0.9994 1.04102 0.50 0.9970 0.9970 1.0795 -0.50 0.9974 0.9974 1.0800

0.75 0.9911 0.9911 1.1130 -0.75 0.9926 0.9926 1.11471.00 0.9767 0.9768 1.1349 -1.00 0.9806 0.9807 1.13940.25 1.0008 1.0008 1.0782 -0.25 1.0008 1.0008 1.0782

4 0.50 1.0026 1.0027 1.1613 -0.50 1.0030 1.0031 1.16180.75 1.0038 1.0044 1.2466 -0.75 1.0054 1.0060 1.24851.00 0.9996 1.0019 1.3263 -1.00 1.0036 1.0059 1.33160.25 1.0014 1.0014 1.0948 -0.25 1.0014 1.0014 1.0948

6 0.50 1.0051 1.0052 1.1994 -0.50 1.0055 1.0056 1.19980.75 1.0094 1.0105 1.3111 -0.75 1.0109 1.0120 1.31311.00 1.0097 1.0140 1.4224 -1.00 1.0137 1.0180 1.42810.25 1.0018 1.0018 1.1042 -0.25 1.0018 1.0018 1.1042

8 0.50 1.0064 1.0066 1.2213 -0.50 1.0068 1.0070 1.22180.75 1.0125 1.0139 1.3490 -0.75 1.0141 1.0154 1.35111.00 1.0154 1.0211 1.4800 -1.00 1.0195 1.0252 1.48600.25 1.0020 1.0020 1.1103 -0.25 1.0020 1.0020 1.1103

10 0.50 1.0073 1.0075 1.2355 -0.50 1.0077 1.0079 1.23600.75 1.0145 1.0161 1.3739 -0.75 1.0161 1.0176 1.37601.00 1.0191 1.0259 1.5184 -1.00 1.0232 1.0300 1.52450.25 1.0023 1.0023 1.1189 -0.25 1.0023 1.0023 1.1189

15 0.50 1.0085 1.0088 1.2560 -0.50 1.0089 1.0092 1.25650.75 1.0173 1.0192 1.4100 -0.75 1.0189 1.0208 1.41221.00 1.0243 1.0328 1.5747 -1.00 1.0284 1.0370 1.58100.25 1.0024 1.0024 1.1234 -0.25 1.0024 1.0024 1.1234

20 0.50 1.0091 1.0095 1.2669 -0.50 1.0095 1.0099 1.26740.75 1.0188 1.0209 1.4295 -0.75 1.0204 1.0225 1.43171.00 1.0270 1.0366 1.6054 -1.00 1.0311 1.0408 1.6118


Table 4.5.3Efficiencies of the estimators θn(∞)

2 and θ1(∞)2 relative to θ2

α e(θn(∞)2 |θ2) α e(θ1(∞)

2 |θ2)

0.25 1.1382 -0.25 1.13820.50 1.3028 -0.50 1.30330.75 1.4943 -0.75 1.49601.00 1.7093 -1.00 1.7161

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