Estimation for the three-parameter lognormal distribution ...Estimation for the three-parameter lognormal distribution based on progressively censored data

Computational Statistics and Data Analysis 53 (2009) 3580–3592

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Computational Statistics and Data Analysis

journal homepage: www.elsevier.com/locate/csda

Estimation for the three-parameter lognormal distribution based onprogressively censored data

Prasanta Basak a, Indrani Basak a,∗, N. Balakrishnan b

a Penn State Altoona, Altoona, PA, United StatesbMcMaster University, Hamilton, Canada

a r t i c l e i n f o

Article history:Received 27 February 2009Accepted 27 March 2009Available online 1 April 2009

a b s t r a c t

Some work has been done in the past on the estimation of parameters of the three-parameter lognormal distribution based on complete and censored samples. In this article,we develop inferential methods based on progressively Type-II censored samples from athree-parameter lognormal distribution. In particular, we use the EM algorithm as wellas some other numerical methods to determine maximum likelihood estimates (MLEs) ofparameters. The asymptotic variances and covariances of the MLEs from the EM algorithmare computed by using the missing information principle. An alternative estimator, whichis a modification of the MLE, is also proposed. The methodology developed here is thenillustrated with some numerical examples. Finally, we also discuss the interval estimationbased on large-sample theory and examine the actual coverage probabilities of theseconfidence intervals in case of small samples by means of a Monte Carlo simulation study.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Lognormal distribution is one of the distributions commonly used for modeling lifetimes or reaction-times, and isparticularly useful for modeling data which are long-tailed and positively skewed. It has been discussed extensively bymany authors including Cohen (1951, 1988), Hill (1963), Harter and Moore (1966), Crow and Shimizu (1988), Johnson et al.(1994), Munro and Wixley (1970) and Rukhin (1984).It is well-known that there is a close relationship between normal and lognormal distributions. If X = log(Y − γ )

is normally distributed with mean µ and standard deviation σ , then the distribution of Y becomes a three-parameterlognormal distributionwith parameter θ = (γ , µ, σ ). The probability density function of such a three-parameter lognormaldistribution is

f (y; θ) =

1

σ√2π(y− γ )

exp{−[log(y− γ )− µ]2

2σ 2

}, γ < y <∞, σ > 0,−∞ < µ <∞

0, otherwise.(1.1)

In (1.1), σ 2 andµ are the variance andmean of the underlying normal variable X , but become the shape and scale parametersof the lognormal variable Y . It is more convenient to use β = exp(µ) and w = exp(σ 2) as the scale and shape parametersof the lognormal variable Y , respectively. Also, γ is the threshold (location) parameter of the lognormal random variable.

∗ Corresponding author. Tel.: +1 814 9495263; fax: +1 814 949 5456.E-mail address: [email protected] (I. Basak).

0167-9473/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.csda.2009.03.015

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P. Basak et al. / Computational Statistics and Data Analysis 53 (2009) 3580–3592 3581

When the threshold parameter γ is known, the parameter estimation can be done using the well-known results for normaldistribution by making the transformation from Y to X . Estimation methods become more complex when γ is unknown.Adding to that complexity is the fact that quite often the data on lifetimes or reaction-times come with ‘‘censoring.’’Censoring occurs when exact survival times are known only for a portion of the individuals or items under study. Thecomplete survival times may not have been observed by the experimenter either intentionally or unintentionally, and thereare numerous examples for each kind; see, for example, Nelson (1982) and Balakrishnan and Cohen (1991).In this article, we consider a general scheme of progressive Type-II right censoring. Under this scheme, n units are placed

on a life-testing experiment and only m (< n) are completely observed until failure. The censoring occurs progressivelyin m stages. These m stages offer failure times of the m completely observed units. At the time of the first failure (the firststage), R1 of the n − 1 surviving units are randomly withdrawn (censored intentionally) from the experiment, R2 of then− 2− R1 surviving units are withdrawn at the time of the second failure (the second stage), and so on. Finally, at the timeof the m-th failure (the m-th stage), all the remaining Rm = n − m − R1 − · · · − Rm−1 surviving units are withdrawn. Thisscheme (R1, R2, . . . , Rm) is referred to as progressive Type-II right censoring scheme. It is clear that this scheme includes theconventional Type-II right censoring scheme (when R1 = R2 = · · · = Rm−1 = 0 and Rm = n − m) and complete samplingscheme (when n = m and R1 = R2 = · · · = Rm = 0). The ordered lifetime data which arise from such a progressive Type-IIright censoring scheme are called progressively Type-II right censored order statistics. For theory, methods and applicationsof progressive censoring, readers are referred to the book by Balakrishnan and Aggarwala (2000) and the recent discussionpaper by Balakrishnan (2007).Suppose n independent lognormally distributed units are placed on a life-testing experiment. Let Y1:m:n ≤ · · · ≤ Ym:m:n

denote the above mentioned m progressively Type-II right censored order statistics. For ease in notation, let us use Yj (j =1, . . . ,m) to denote these Yj:m:n’s. Note that we observe only Y = (Y1, . . . , Ym). The purpose of this article is to discussdifferent estimation procedures for the parameter θ based on the progressively Type-II right censored order statisticsY. Recently, Ng et al. (2002) considered two-parameter lognormal distribution (which does not include the thresholdparameter) and discussed Newton–Raphson algorithm as well as EM algorithm for finding the MLEs.Inclusion of the third parameter γ introduces an unusual feature in the likelihood function. Hill (1963) has shown that

there exist paths along which the likelihood function of any ordered sample y1, . . . , yn from the three-parameter lognormaldistribution tends to∞ as (γ , µ, σ ) approaches (y1,−∞,∞). Thus, the global maximization of the likelihood leads to theunreasonable estimate (y1,−∞,∞) although, in fact, the likelihood at the point is zero. ButHill (1963) also retained the ideathat reasonable estimates could be obtained by solving the likelihood equations. To get these reasonable estimates, Cohen(1951), Cohen and Whitten (1988) and Harter and Moore (1966) equated partial derivatives of the likelihood function tozero and solved the resulting equations. These estimates are called local maximum likelihood estimates (LMLEs). Harter andMoore (1966) and Calitz (1973) noted that although these LMLEs are not true MLEs according to the usual definition, theyare reasonable estimates and appear to possess most of the desirable properties associated with MLEs. However, it is notedin Harter and Moore (1966) and Cohen and Whitten (1988) that sometimes the likelihood function may have no clearlydefined local maximum for small samples and so LMLEs fail to produce estimates in that case. In the past, some work hasbeen done on estimationmethods for the three-parameter lognormal distribution based on complete and censored samples;see, for example, the books by Cohen and Whitten (1988) and Balakrishnan and Cohen (1991).In Section 2, we describe the Newton–Raphson algorithm for determining the MLEs of the parameter θ based on a

progressively censored sample. The second derivatives of the log-likelihood are required in order to use the algorithm.These computations are complicated when data are progressively censored. Another viable alternative to Newton–Raphsonalgorithm is the well-known EM algorithm and in Section 3 we discuss how that can be used to determine the MLEs inthis case. Asymptotic variances and covariances of the maximum likelihood estimates generated through the EM algorithmare given in Section 4. Section 5 describes some simplified estimation methods which yields simple alternative estimators.All these methods discussed in this article are then illustrated with some numerical examples in Section 6. With theseillustrative examples, we also discuss the interval estimation based on large-sample theory and then examine the actualcoverage probabilities in case of small samples through Monte Carlo simulations.

2. Newton–Raphson algorithm

One of the standard methods of determining the maximum likelihood estimates is the Newton–Raphson algorithm. Inthis section,we describe theNewton–Raphson algorithm for finding theMLEs numericallywhen life-times are distributed asa three-parameter lognormal distributionwith parameter θ. TheseMLEs are localMLEs, asmentioned earlier, correspondingto the Newton–Raphson algorithm, and would be denoted here by LMLE1.The log-likelihood function log L(θ) = log L based on the progressively Type-II right censored order statistics Y, is

log L(θ) = const.−m log σ −m∑j=1

log(yj − γ )−12

m∑j=1

Ψ 20j +

m∑j=1

Rj{log Φ

[Ψ0j]}, (2.1)

where Ψ0j =log(yj−γ )−µ

σ; see Balakrishnan and Aggarwala (2000). In (2.1) and throughout this article, φ and Φ = 1 − Φ

denote the probability density and survival function of the standard normal distribution, respectively. Three likelihood

3582 P. Basak et al. / Computational Statistics and Data Analysis 53 (2009) 3580–3592

equations which need to be solved simultaneously for the required estimate θ = (µ, σ , γ ) are as follows:

∂ log L∂γ

=1σ

m∑j=1

[Ψ0j + RjL0j

] 1yj − γ

+

m∑j=1

1yj − γ

= 0

∂ log L∂µ

=1σ

[m∑j=1

Ψ0j +

m∑j=1

RjL0j

]= 0

∂ log L∂σ

=1σ

[−m+

m∑j=1

Ψ 20j +

m∑j=1

RjΨ0jL0j

]= 0

(2.2)

in which L0j = φ(Ψ0j)/Φ(Ψ0j) is the hazard function of the standard normal distribution at Ψ0j.In theNewton–Raphson algorithm, the simultaneous solution is obtained through an iterative procedure. In each iterative

step, the corrections a, b, c to the previous estimates γ0, µ0, σ0 produce new estimates γ , µ, σ as

γ = γ0 + a, µ = µ0 + b and σ = σ0 + c.

The iteration method is based on Taylor series expansions of the estimating equations in (2.2) in the neighborhood of theprevious simultaneous estimates. Neglecting powers of a, b and c above the first order and using Taylor’s theorem, we getthe following equations which need to be solved for a, b, c:

a∂2 log L∂γ 2

∣∣∣∣0+ b

∂2 log L∂γ ∂µ

∣∣∣∣0+ c

∂2 log L∂γ ∂σ

∣∣∣∣0= −

∂ log L∂γ

∣∣∣∣0

a∂2 log L∂µ∂γ

∣∣∣∣0+ b

∂2 log L∂µ2

∣∣∣∣0+ c

∂2 log L∂µ∂σ

∣∣∣∣0= −

∂ log L∂µ

∣∣∣∣0

a∂2 log L∂σ∂γ

∣∣∣∣0+ b

∂2 log L∂σ∂µ

∣∣∣∣0+ c

∂2 log L∂σ 2

∣∣∣∣0= −

∂ log L∂σ

∣∣∣∣0,

(2.3)

where the notation A|0, for any partial derivative A, means the partial derivative evaluated at (γ0, µ0, σ0).The second derivatives needed in (2.3) are as follows:

∂2 log L∂γ 2

=

m∑j=1

1(yj − γ )2

+1σ

m∑j=1

(Ψ0j + RjL0j)1

(yj − γ )2−1σ 2

m∑j=1

(1− RjΨ0jL0j + RjL20j)1

(yj − γ )2

∂2 log L∂µ2

= −1σ 2

[m+

m∑j=1

RjL0j(L0j − Ψ0j)

]∂2 log L∂σ 2

=1σ 2

[m−

m∑j=1

{3Ψ 20j + Rj(2Ψ0jL0j − Ψ

30jL0j + Ψ

20jL20j)}]

∂2 log L∂γ ∂µ

= −1σ 2

m∑j=1

[1− RjΨ0jL0j + RjL20j

] 1yj − γ

∂2 log L∂γ ∂σ

= −1σ 2

m∑j=1

[2Ψ0j + Rj(L0j − Ψ 20jL0j + Ψ0jL

20j)] 1yj − γ

∂2 log L∂µ∂σ

= −1σ 2

m∑j=1

[2Ψ0j + Rj(L0j − Ψ 20jL0j + Ψ0jL

20j)].

(2.4)

Using the Newton–Raphson method, Ng et al. (2002) discussed methods of finding MLEs while Balakrishnan et al. (2003)discussed the construction of confidence intervals of µ and σ for the two-parameter lognormal distribution. Mi andBalakrishnan (2003) have made use of the fact that the lognormal density is log-concave in order to establish that the MLEsofµ and σ do exist and are unique. Hence, in that case, the EM algorithm and the Newton–Raphson algorithmwill convergeto the same values.In order to obtain LMLE1 for the three-parameter lognormal distribution, we suggest a variation of the above method,

which was used by Cohen (1951). Calitz (1973) found this method of Cohen (1951) produced better convergence in theestimation process. Instead of solving all three equations in (2.2) simultaneously, the suggested method is to start with aγ(0) which is less than y1 and use the second and third equations of (2.2) to get µ(0) = µ(γ(0)) and σ(0) = σ(γ(0)). The firstequation is used as the test equation. If the left hand side of this equation equals zero when (γ(0), µ(0), σ(0)) substitutedthere, then no further iteration is required. Otherwise, another γ(1) is chosen which is less than y1 yielding µ(1) = µ(γ(1))


and σ(1) = σ(γ(1)) and look for the sign change in the left hand side of the first equation of (2.2). Finally, interpolating on γvalues, first find the final estimate γ and then the final estimates µ = µ(γ ) and σ = σ(γ ) are obtained using the secondand third equations of (2.2). The final estimates (γ , µ, σ ) when substituted in the first equation of (2.2) should producezero or close to zero. As noted by Cohen andWhitten (1988), usually a single root γ will be found. In the event that multipleroots of γ occur, Cohen andWhitten (1988) suggested to use the root which results in the closest agreement between y and

E[Y ] = γ + eµ+σ22 .

3. EM Algorithm

In the case of progressively censored samples, an alternative to theNewton–Raphson algorithm is the use of EMalgorithmfor numerically finding theMLEs. One advantage of the EM algorithm is that asymptotic variances and covariances of the EMalgorithm estimates can be computed, which is discussed in Section 4. EM algorithm, introduced by Dempster et al. (1977),is a very popular tool to handle any missing or incomplete data situation; readers are referred to the book by McLachlanand Krishnan (1997) for a detailed discussion on EM algorithm and its applications. This algorithm is an iterative methodwhich has two steps. In the E-step, it replaces any missing data by its expected value and in the M-step the log-likelihoodfunction is maximized with the observed data and expected value of the incomplete data, producing an update of theparameter estimates. The MLEs of the parameters are obtained by repeating the E- and M-steps until convergence occurs.Since progressive censoring model can be viewed as a missing data problem, EM algorithm can be applied to obtain theMLEs of the parameters in this case.Let us denote the censored data vector as Z = (Z1, Z2, . . . , Zm), where the j-th stage censored data vector Zj is a 1 ×Rj

vector, Zj = (Zj1, Zj2, . . . , ZjRj), for j = 1, 2, . . . ,m. The complete data set is then obtained by combining the observed dataY and the censored data Z. E-step of the algorithm requires the computation of the conditional expectation of functions ofcensored data vector Z, conditional on the observed data vector Y and the current value of the parameters. In particular, onecomputes the conditional expectation of the log-likelihood E [log L(Y, Z, θ)|Y = y] as

E [log L(Y, Z, θ)|Y = y] = const.− n log σ −m∑j=1

log(yj − γ )−12

m∑j=1

(log(yj − γ )− µ

σ

)2

−

m∑j=1

Rj∑k=1

E[log(Zjk − γ ) | Zjk > yj

]−12

m∑j=1

Rj∑k=1

E

[(log(Zjk − γ )− µ

σ

)2| Zjk > yj

]. (3.1)

The above conditional expectations are obtained using the result that given Yj = yj, Zj’s have a left-truncated distributionF , truncated at yj. More specifically, the conditional probability density of Z, given Y, is given by (see Balakrishnan andAggarwala (2000))

fZ|Y(z|y; θ) =m∏j=1

Rj∏k=1

fZjk|Yj(zjk|yj; θ), (3.2)

where

fZjk|Yj(zjk|yj; θ) =f (zjk; θ)1− F(yj; θ)

(3.3)

and f (zjk; θ) is given by (1.1) and F denotes the corresponding cumulative distribution function. In the M-step of the(h + 1)-th iteration, we will denote the updated estimates of the parameter θ as θ(h+1). This θ(h+1) maximizes the log-likelihood function involving the observed data Y, conditional expectation of the log-likelihood function of censored datavector Z given the observed data vector Y, and the h-th iteration value of the parameter θ(h). As a starting value θ(0), onecan use a γ(0) < y1 and µ(0) and σ(0) are computed on the basis of the so-called ‘‘pseudo-complete’’ sample which involvesobserved data Y and the censored observations at the j-th step Zj all taken to be yj. Thus, µ(0) = µ(γ(0)) and σ(0) = σ(γ(0))are then given by

µ(0) =1n

m∑j=1

(Rj + 1) log(yj − γ(0)

)σ(0) =

√√√√1n

[m∑j=1

(Rj + 1) log2(yj − γ(0)

)]− µ2(0)

. (3.4)


Starting with the initial estimates in (3.4), the (h+ 1)-th iteration value θ(h+1) is obtained using the h-th iteration value θhas follows:

µ(h+1) =1n

{m∑j=1

log(yj − γ(h)

)+

m∑j=1

RjE[log(Z − γ(h))|Z > yj; θ(h)

]}

σ(h+1) =

√√√√1n

{m∑j=1

log2(yj − γ(h)

)+

m∑j=1

RjE[log2(Z − γ(h))|Z > yj;µ(h+1), σ(h), γ(h)

]}− µ2(h+1)

. (3.5)

The conditional expectations in the above expression (3.5) can be obtained as follows:

E[log(Z − γ(h))|Z > yj; θ(h)

]= σ(h)L0j(h) + µ(h)

E[log2(Z − γ(h))|Z > yj;µ(h+1), σ(h), γ(h)

]= σ 2(h)

[1+ Ψ ∗0jL

∗

0j

]+ 2σ(h)µ(h+1)L∗0j + µ

2(h+1)

}where

Ψ0j(h) = Ψ0j(θ(h)) =log

(yj − γ(h)

)− µ(h)

σ(h),

L0j(h) = L0j(θ(h)) =φ(Ψ0j(h))

Φ(Ψ0j(h)),

Ψ ∗0j =log

(yj − γ(h)

)− µ(h+1)

σ(h),

L∗0j =φ(Ψ ∗0j)

Φ(Ψ ∗0j).

γ(h+1) is obtained by solving the following equation for γ :[µ(h+1) − σ

2(h+1)

] m∑j=1

1yj − γ

−

m∑j=1

log(yj − γ )yj − γ

+[µ(h+1) − σ

2(h+1)

] m∑j=1

RjE[

1Z − γ

∣∣∣∣ Z > yj; γ , θ∗(h+1)]−

m∑j=1

RjE[log(Z − γ )Z − γ

∣∣∣∣ Z > yj; γ , θ∗(h+1)] = 0, (3.6)

where θ∗(h+1) = (µ(h+1), σ(h+1)). The conditional expectations in the above expression (3.6) can be obtained as follows:

E[

1Z − γ

∣∣∣∣ Z > yj; γ , θ∗(h+1)] = e σ2(h+1)2 −µ(h+1)P0j(h+1)(γ )

E[log(Z − γ )Z − γ

∣∣∣∣ Z > yj; γ , θ∗(h+1)] = e σ2(h+1)2 −µ(h+1)[σ(h+1)P0j(h+1)(γ )+ (µ(h+1) − σ 2(h+1))P0j(h+1)(γ )

],

where

Ψ0j(h+1)(γ ) = Ψ0j(γ , θ∗

(h+1)) =log

(yj − γ

)− µ(h+1)

σ(h+1),

P0j(h+1)(γ ) = P0j(γ , θ∗(h+1)) =Φ(Ψ0j(h+1)(γ )+ σ(h+1))

Φ(Ψ0j(h+1)(γ )).

The final MLEs of the parameters are local MLEs corresponding to the EM algorithm, and would be denoted here by LMLE2.In order to obtain LMLE2, we suggest a variation of the above method. As mentioned before, one can start with a γ(0)

which is less than y1 and get µ(0) = µ(γ(0)) and σ(0) = σ(γ(0)) by using (3.4) and then one gets µ(1) and σ(1) by using (3.5).The suggestion is to treat (3.6) as the test equation. If the left hand side of that equation equals zero when (γ(0), µ(1), σ(1))substituted there, thenno further iteration is needed. Otherwise, the suggestedmethod involves choosing anotherγ(1)whichis less than y1 and repeating the procedure looking for the sign change in (3.6). Finally, interpolating on γ values, one getsthe final estimate (γ , µ, σ )which when substituted in (3.6) should produce zero or close to zero.

4. Asymptotic variances and covariances of the EM algorithm estimates

Asymptotic variances and covariances of the MLEs when the EM algorithm is used can be obtained by using the missinginformation principle of Louis (1982) and Tanner (1993). This principle is basically

Observed information = Complete information−Missing information.Based on this principle, Louis (1982) developed the procedure of finding the observed information matrix when EMalgorithm is used to find MLEs in an incomplete data situation. We adopt this principle in the situation of progressive


Type II right censoring.Wewill denote the complete, observed andmissing (expected) information by I(θ), IY (θ) and IZ |Y (θ),respectively. The complete information I(θ) is given by

I(θ) = −E[∂2E[log L(Y, Z, θ)|Y = y]

∂θ2

](4.1)

in which E [log L(Y, Z, θ)|Y = y] is given by (3.1). In (4.1), the expectation is taken with respect to both Y and Z. The Fisherinformation matrix for a single observation which is censored at the time of the j-th failure is given by

I(j)Z |Y (θ) = −E

[∂2 log fZjk|Yj(zjk|yj; θ)

∂θ2

](4.2)

in which fZjk|Yj(zjk|yj; θ) is given by (3.3). In (4.2), the expectation is taken with respect to zjk so that I(j)Z |Y (θ) is a function of yj

and θ . Then, the missing (expected) information is simply

IZ|Y(θ) =m∑j=1

RjI(j)Z |Y (θ), (4.3)

where I(j)Z |Y (θ) is given by (4.2). The observed information IY (θ) is then obtained as follows:

IY (θ) = I(θ)− IZ|Y(θ), (4.4)

where I(θ) and IZ|Y(θ) are given by (4.1) and (4.3), respectively, and are derived below. Finally, upon inverting the observedinformationmatrix IY (θ) in (4.4), one gets the asymptotic variances and covariances of theMLEswhen EM algorithm is used.

4.1. Complete information matrix I(θ)

The log-likelihood function log L∗(θ) based on n uncensored observations yi, i = 1, . . . , n, is given by

log L∗(θ) =n∑i=1

log f (yi; θ), (4.5)

where f (yi; θ) is as given in (1.1). On differentiating the log-likelihood in (4.5) and equating to zero, one obtains theestimating equations given by (2.2) with m = n and Rj = 0; j = 1, 2, . . . ,m. Negative of the second derivatives of thelog-likelihood function log L∗(θ) are obtained by appropriately differentiating the first derivatives in (2.2) with m = n andRj = 0, j = 1, 2, . . . ,m, and are given by (2.4) withm = n and Rj = 0, j = 1, 2, . . . ,m.For the three-parameter lognormal distribution, it can be shown that

E[1

Yi − γ

]= e

σ22 −µ

E[

1(Yi − γ )2

]= e2(σ

2−µ)

E[log(Yi − γ )Yi − γ

]= (µ− σ 2)e

σ22 −µ

E[log(Yi − γ )(Yi − γ )2

]= (µ− 2σ 2)e2(σ

2−µ).

(4.6)

One gets the complete information matrix I(θ) in (4.1), by using (2.4) with m = n, Rj = 0, j = 1, 2, . . . ,m, and (4.6), asfollows:

I(θ) = n

1+ σ 2

σ 2e2(σ

2−µ) 1

σ 2eσ22 −µ −

2σeσ22 −µ

1σ 2eσ22 −µ

1σ 2

0

−2σeσ22 −µ 0

2σ 2

. (4.7)

Expressions for the asymptotic variances and covariances of the MLE of θ in the uncensored case are obtained by invertingthe matrix I(θ) in (4.7). Denoting β = eµ, w = eσ

2and H =

[w(1+ σ 2)− (1+ 2σ 2)

]−1, the asymptotic variances and


covariances are:

V (γ ) =σ 2

nβ2

wH

V (µ) =σ 2

n(1+ H)

V (σ ) =σ 2

2n(1+ 2σ 2H)

Cov(γ , µ) = −σ 2

nβ√wH

Cov(γ , σ ) =σ 3

nβ√wH

Cov(µ, σ ) = −σ 3

nH.

It is worthwhile to mention here that, for the uncensored case, Cohen (1951) has obtained the asymptotic variances andcovariances of theMLEs of the parameters (γ , β, σ ), while those corresponding to the parameters (γ , µ, σ 2)were obtainedby Hill (1963).

4.2. Missing information matrix IZ|Y(θ)

The logarithm of the density function of an observation zjk = z censored at yj, the time of the j-th failure, is given by[see Eq. (3.2)]

log fz|yj(z|yj; θ) = const.− log σ − log(z − γ )− log[1− Φ

{log(yj − γ )− µ

σ

}]−12σ 2

[log(z − γ )− µ]2 . (4.8)

Differentiating (4.8) with respect to γ , µ and σ , one gets

∂ log fz|yj∂γ

=1σ 2

log(z − γ )− µz − γ

+1

z − γ−1σ

1yj − γ

L0j,

∂ log fz|yj∂µ

=1σ

[log(z − γ )− µ

σ− L0j

],

∂ log fz|yj∂σ

=1σ

[(log(z − γ )− µ

σ

)2−(1+ ψ0jL0j

)].

(4.9)

By using the properties of the left-truncated log-normal distribution, it can then be shown that

E[log(Z − γ )− µ|Z > yj; θ

]= σ L0j,

E[{log(Z − γ )− µ}2 |Z > yj; θ

]= σ 2

(1+ Ψ0jL0j

),

E[{log(Z − γ )− µ}3 |Z > yj; θ

]= σ 3

(2+ Ψ 20j

),

E[{log(Z − γ )− µ}4 |Z > yj; θ

]= σ 4

[3(1+ Ψ0jL0j

)+ Ψ 30jL0j

],

E[1

Z − γ|Z > yj; θ

]= e

σ22 −µLL1j,

E

[(1

Z − γ

)2∣∣∣∣∣ Z > yj; θ]= e2(σ

2−µ)LL2j,

E[log(Z − γ )Z − γ

∣∣∣∣ Z > yj; θ] = e σ22 −µ [σ L1j + (µ− σ 2)LL1j] ,E

[(log(Z − γ )Z − γ

)2∣∣∣∣∣ Z > yj; θ]= e2(σ

2−µ)

[{σ 2Ψ2j + 2σ(µ− 2σ 2)

}L2j +

{σ 2 + (µ− 2σ 2)2

}LL2j

],

E[log(Z − γ )

(Z − γ )2

∣∣∣∣ Z > yj; θ] = e2(σ 2−µ) [σ L2j + (µ− 2σ 2)LL2j] ,E[(log(Z − γ ))2

Z − γ

∣∣∣∣ Z > yj; θ] = e σ22 −µ [{σ 2Ψ1j + 2σ(µ− σ 2)} L2j + {σ 2 + (µ− σ 2)2} LL2j]E[(log(Z − γ ))3

Z − γ

∣∣∣∣ Z > yj; θ] = e σ22 −µ [{σ 2Ψ 21j + 2σ 3 − 3σ(µ− σ 2)2} L1j+{3(µ− σ 2)σ 2Ψ1j + 6σ(µ− σ 2)2

}L2j − 2(µ− σ 2)3LL1j +

{3(µ− σ 2)

[σ 2 + (µ− σ 2)2

]}LL2j

],

(4.10)


in which Ψ0j,Ψ1j,Ψ2j, L0j, L1j, L2j, LL1j and LL2j are given by

Ψ0j =log

(yj − γ

)− µ

σ,

Ψ1j =log(yj − γ )− (µ− σ 2)

σ,

Ψ2j =log(yj − γ )− (µ− 2σ 2)

σ,

L0j =φ(Ψ0j)

Φ(Ψ0j), L1j =

φ(Ψ1j)

Φ(Ψ0j), L2j =

φ(Ψ2j)

Φ(Ψ0j),

LL1j =Φ(Ψ1j)

Φ(Ψ0j), LL2j =

Φ(Ψ2j)

Φ(Ψ0j).

(4.11)

The Fisher informationmatrix based on one observation zjk censored at yj, I(j)Z |Y (θ), in (4.2), can be obtained using (4.9)–(4.11)

as follows:

E[∂ log fZ |yj∂γ

]2=

1σ 2(yj − γ )2

L20j −e2(σ

2−µ)

(2+ Ψ2j

σ 2

)σ

L2j + e2(σ2−µ)

(1+

1σ 2

)LL2j −

2eσ22 −µ

σ 2(yj − γ )L0jL1j

E[∂ log fZ |yj∂µ

]2=1σ 2

[1+ Ψ0jL0j − L20j

]E[∂ log fZ |yj∂σ

]2=1σ 2

[2+ Ψ0jL0j

(1− Ψ0jL0j + Ψ 20j

)]E[(∂ log fZ |yj∂γ

)(∂ log fZ |yj∂µ

)]=1σ 3

[eσ22 −µ

{σ(σ + 1)L1j − σ 2(σ + 1)LL1j

}−σ 2L0jLL1j −

σ L0j + 1− σyj − γ

L0j

]E[(∂ log fZ |yj∂γ

)(∂ log fZ |yj∂σ

)]=1σ

[2µ{(σ − 1)L0j + µ

}σ 2(yj − γ )

L0j

+ eσ22 −µ

{−2µ2

σ 3+2µ2 + Ψ 21j

σ 2+1− Ψ0jL0j + 4µ

σ− 3σ

}L1j

+ eσ22 −µ

{2Ψ1j

( µσ 2−µ

σ− 1

)− 4

( µσ 2− 1

) (σ + µ−

µ

σ

)}L2j

+ 2eσ22 −µ

( µσ 2− 1

)(µ2σ−µ2

σ 2− σ 2 + µ

)LL1j + 2e

σ22 −µ

{σ 2 + (µ− σ 2)2

}(µ− σ 2 − µσσ 4

)LL2j

]E[(∂ log fZ |yj∂µ

)(∂ log fZ |yj∂σ

)]=1σ 2

[L0j + Ψ0jL0j(Ψ0j − L0j)

].

(4.12)

Using IZ|Y(θ) from (4.12) and I(θ) from (4.7), the observed information can be obtained from (4.4). Finally, one can get theasymptotic variances and covariances of the MLEs, when the EM algorithm is used, by inverting this observed informationmatrix IY (θ).

5. Alternative estimators

As an alternative estimator, one can use modified MLE (MMLE). MMLEs use first order statistic for estimating γ and aretherefore easier to compute than the MLEs. For small samples when LMLEs fail to converge, these MMLEs (which alwaysexist) produce reasonable estimates. For progressively censored data, one can have two versions of MMLEs. One versioncorresponds to the Newton–Raphson algorithm and the other version corresponds to the EM algorithm, and they will bedenoted by MMLE1 and MMLE2, respectively. In both versions, the likelihood equation ∂ log L

∂γ= 0 is replaced by

γ + eµ+σΦ−1[1n+1 ] = y1 (5.1)

in which n = m+∑mj=1 Rj is the total sample size. Eq. (5.1) is used as a test equation and it replaces the first equation in (2.2)

for the Newton–Raphson algorithm version and Eq. (3.6) for the EM algorithm version. One starts with a first approximationγ1 and make the transformation log(yi − γ1). One then calculates conditional estimates µ1 = µ(γ1) and σ1 = σ(γ1) byusing the second and third equations of (2.2) for the Newton–Raphson algorithm and (3.5) for the EM algorithm. The values(γ1, µ1, σ1) are substituted in Eq. (5.1). If the test equation is satisfied, then no further iteration is required. Otherwise, a


Table 1Progressively censored data for Example 1.

i 1 2 3 4 5 6 7 8 9 10 11 12

Yi 0.265 0.269 0.297 0.315 0.338 0.379 0.392 0.402 0.412 0.416 0.418 0.484Ri 0 0 2 0 0 2 0 0 0 0 0 4

Table 2Estimates and standard deviations of estimates for γ , µ and σ based on data in Table 1.

γ µ σ

Estimator γ σγ µ σµ σ σσLMLE2 0.16 0.09 −1.21 0.19 0.29 0.05


i 1 2 3 4 5 6 7

Yi 152.7 172.0 172.5 173.3 193.0 204.7 234.9Ri 0 0 0 1 0 0 2


γ µ σ

Estimator γ σγ µ σµ σ σσLMLE2 110.30 10.29 3.35 0.89 0.81 0.12

second approximation γ2 is selected and the cycle of calculations as described above is repeated. The iterations are continueduntil two sufficiently close values γi and γi+1 are found such that the following is satisfied:

γi + eµi+σiΦ−1[1n+1 ] < (>)y1 < (>)γi+1 + eµi+1+σi+1Φ

−1[1n+1 ].

Thus, the final MMLE γ of γ is obtained using which the final MMLEs µ, σ of µ and σ are then obtained.

6. Illustrative examples

To illustrate the computational methods presented in this article, we use three examples given in Cohen and Whitten(1988). We modified these original examples to consider progressively censored data. We then assume that each datumcomes from a three-parameter lognormal distribution, and use them to carry out all the estimation procedures discussedin the preceding sections in order to see whether they produce similar results or not. Also, in Section 7, we compare thecoverage probabilities of confidence intervals based on LMLE1 and LMLE2 using a simulation study.We used same γ0 value for obtaining LMLE1 and LMLE2. For the determination of MMLE, as discussed in Section 5,

Φ−1[ 1n+1 ] = −1.67 for Examples 1 and 3 since n = 20 in both these cases. For Example 2,Φ−1[1n+1 ] = −1.34 since n = 10

here. Proceeding as explained in Section 5, we get two versions ofMMLE, viz., MMLE1 andMMLE2 (for the Newton–Raphsonalgorithm and EM algorithm, respectively).

Example 1. Themaximum flood levels (inmillions of cubic feet per second) for 20 four-year periods from1890 to 1969 in theSusquehanna river at Harrisburg, Pennsylvania are given in Cohen andWhitten (1988) and were also used by Dumonceauxand Antle (1973). We modified these data to make it progressively censored with m = 12 stages, and these progressivelycensored data are presented in Table 1.The LMLE2 and their standard deviations are presented in Table 2.For the data in Table 1, the LMLE1 are given by γ = 0.1822, µ = −1.291 and σ = 0.362, the MMLE1 are given by

γ = 0.1581, µ = −1.103 and σ = 0.279, and the MMLE2 are given by γ = 0.1563, µ = −1.1725 and σ = 0.273.

Example 2. This example was used by McCool (1974) and is also given in Cohen and Whitten (1988). The data are fatiguelives (in hours) of 10 bearings of a certain type.Wemodified these data tomake it progressively censoredwithm = 7 stages,and these progressively censored data are presented in Table 3.LMLE1 did not converge for this sample. The LMLE2 and their standard deviations are presented in Table 4.The MMLE1 are given by γ = 112.7815, µ = 3.9231 and σ = 0.8932, while MMLE2 are given by γ = 109.7232, µ =

3.2763 and σ = 0.8036.



i 1 2 3 4 5 6 7 8 9 10 11 12

Yi 127.2 128.7 131.4 133.0 133.1 135.9 144.3 145.8 148.3 153.1 157.3 164.3Ri 0 0 0 2 0 0 2 0 0 0 0 4


γ µ σ


Table 7Progressively censored data for Example 3 with PCS-1.

i 1 2 3 4 5 6 7 8 9 10 11 12

Yi 127.2 128.7 131.4 133.0 133.1 135.9 137.3 144.3 145.8 148.3 153.1 157.2Ri 0 0 0 0 0 0 0 0 0 2 2 4

Table 8Progressively censored data for Example 3 with PCS-2.

i 1 2 3 4 5 6 7 8 9 10 11 12

Yi 127.2 128.7 133.0 135.9 144.3 148.3 153.1 157.2 166.5 174.8 184.1 201.4Ri 4 2 2 0 0 0 0 0 0 0 0 0

Table 9Estimates and standard deviations of estimates for γ , µ and σ for data in Table 7.

γ µ σ


Table 10Estimates and standard deviations of estimates for γ , µ and σ for data in Table 8.

γ µ σ


Example 3. In this example, the complete data are given by Cohen andWhitten (1988) andwere also used by Cohen (1951).The complete data consists of 20 observations from a three-parameter lognormal distributionwith γ = 100, µ = 3.912023and σ = 0.4. We modified these data to make it progressively censored with m = 12 stages, and these progressivelycensored data are presented in Table 5. Although Yi values are reported up to one decimal place in Tables 5, 7 and 8, we usedall three decimal places for these Yi values in Cohen (1951) for our computations. The LMLE2 and their standard deviationsare presented in Table 6.For the data in Table 5, the LMLE1 are given by γ = 110.3112, µ = 3.573 and σ = 0.422, the MMLE1 are given by

γ = 109.1563, µ = 4.419 and σ = 0.295, and the MMLE2 are given by γ = 107.8036, µ = 3.927 and σ = 0.378.In order to examine the effect of delayed censoring, we considered two schemes of progressive censoring. In one scheme,

the censoring was delayed than the other. Keeping the number of stages m = 12 the same as in Table 4, for the delayedcensoring scheme, we considered PCS-1: R1 = 0, R2 = 0, R3 = 0, R4 = 0, R5 = 0, R6 = 0, R7 = 0, R8 = 0, R9 = 0, R10 =2, R11 = 2, R12 = 4. For the other scheme, we took PCS-2: R1 = 4, R2 = 2, R3 = 2, R4 = 0, R5 = 0, R6 = 0, R7 = 0, R8 =0, R9 = 0, R10 = 0, R11 = 0, R12 = 0. The progressively censored data obtained under these two schemes are presented inTables 7 and 8, respectively.The LMLE2 estimates and their standard deviations for the data in Tables 7 and 8 are presented in Tables 9 and 10,

respectively. For the data in Table 7, the LMLE1 are given by γ = 104.4622, µ = 3.992 and σ = 0.373, the MMLE1are given by γ = 103.5689, µ = 4.151 and σ = 0.298, and the MMLE2 are given by γ = 103.8039, µ = 4.107 andσ = 0.315.For the data in Table 8, the LMLE1 are given by γ = 109.3784, µ = 4.041 and σ = 0.413, the MMLE1 are given by

γ = 107.2501, µ = 4.123 and σ = 0.292, and the MMLE2 are given by γ = 107.9231, µ = 4.137 and σ = 0.368. It is


Table 1195% confidence intervals for Example 1.

Estimators γ µ σ

LMLE1 (−.255, .647) (−2.165,−.477) (.085, .673)LMLE2 (−.071, .415) (−1.644,−.816) (.137, .423)MMLE1 (−.250, .574) (−1.855,−.327) (.047, .595)MMLE2 (−.165, .423) (−1.651,−.593) (.085, .437)

Table 1295% confidence intervals for Example 2.

Estimators γ µ σ

LMLE2 (66.682, 152.942) (1.689, 5.535) (.323, 1.303)MMLE1 (69.882, 157.815) (1.284, 6.968) (.423, 1.482)MMLE2 (65.721, 139.061) (1.232, 4.956) (.403, 1.030)

Table 13Coverage probabilities of 95% confidence intervals for γ based on Monte Carlo simulations.

n m R LMLE1 LMLE2

20 12 (7,1,0, . . . ,0,0,0) 90.71 92.2520 12 (0,0,0, . . . ,0,1,7) 90.92 92.8120 14 (5,1,0, . . . ,0,0,0) 91.02 93.0120 14 (0,0,0, . . . ,0,1,5) 91.12 93.8220 16 (3,1,0, . . . ,0,0,0) 93.34 94.7120 16 (0,0,0, . . . ,0,1,3) 93.78 94.2920 18 (1,1,0, . . . ,0,0,0) 93.92 94.3520 18 (0,0,0, . . . ,0,1,1) 94.35 95.7240 24 (14,2,0, . . . ,0,0,0) 92.10 93.0140 24 (0,0,0, . . . ,0,2,14) 92.94 94.1540 28 (10,2,0, . . . ,0,0,0) 93.21 93.9140 28 (0,0,0, . . . ,0,2,10) 94.28 94.2340 32 (6,2,0, . . . ,0,0,0) 94.37 94.4140 32 0,0,0, . . . ,0,2,6) 93.48 94.8740 36 (2,2,0, . . . ,0,0,0) 93.92 94.9140 36 (0,0,0, . . . ,0,2,2) 94.73 94.28

clear from Tables 6, 9 and 10 that the estimates of γ are affected not only by censoring but also by the pattern of censoring,while the estimates of µ and σ remain more or less the same in all the situations.

7. Simulation study

In this section, we report results from two simulation studies that we carried out. In the first simulation study, weconstructed 95% confidence intervals based on 10,000 randomly generated progressively censored samples correspondingto different estimates of γ , µ and σ for each of the estimationmethods discussed in Section 6.We used the same progressivecensoring schemes as given in those examples. We provide the confidence intervals from this simulation study in Tables 11and 12 for Examples 1 and 2, respectively, for different estimators.In the second simulation study, we compared the performance of LMLE1 and LMLE2 in terms of coverage probabilities of

95% confidence intervals for the parameters γ , µ and σ for different sample sizes and different degrees of censoring. 10,000samples were simulated from the lognormal distribution with γ = 100, µ = 3.912023 and σ = 0.4 with sample sizen = 20, 40. For sample size n = 20, we considered m = 12, 14, 16, 18 stages of censoring, and for sample size n = 40 weconsideredm = 24, 28, 32, 36 stages. In each case, we considered two censoring schemeswith one reflecting comparativelydelayed censoring than the other. The coverage probabilities of 95% confidence intervals from this simulation study for theparameters γ , µ and σ are presented in Tables 13–15, respectively.It is observed from these tables that the coverage probabilities are better when the proportion of uncensored data is

larger. The coverage probabilities seem to be almost the same for the two different censoring schemes. The confidenceintervals for the parameter γ seems to be most sensitive to the censoring pattern, while the confidence intervals for theparameters µ and σ seem to be stable and quite satisfactory and close to the nominal level of 95%. In a similar tone, it isknown that different censoring schemes do not change the estimates for the two-parameter exponential distribution (see forexample, Balakrishnan and Sandhu (1996)). The sensitivity of estimation of γ may be due to the fact that γ is the thresholdparameter of the lognormal distribution and the pattern of censoring applied to the sample might have affected the natureof the sample observations as far as the estimation of the threshold parameter is involved. Also, LMLE1 and LMLE2 seem tohave nearly the same coverage probabilities.


Table 14Coverage probabilities of 95% confidence intervals for µ based on Monte Carlo simulations.

n m R LMLE1 LMLE2

20 12 (7,1,0, . . . ,0,0,0) 93.12 93.9420 12 (0,0,0, . . . ,0,1,7) 93.28 93.8220 14 (5,1,0, . . . ,0,0,0) 93.27 93.7320 14 (0,0,0, . . . ,0,1,5) 94.02 93.6120 16 (3,1,0, . . . ,0,0,0) 94.96 95.1120 16 (0,0,0, . . . ,0,1,3) 95.02 95.2520 18 (1,1,0, . . . ,0,0,0) 94.81 95.4220 18 (0,0,0, . . . ,0,1,1) 95.38 95.4640 24 (14,2,0, . . . ,0,0,0) 93.24 94.1240 24 (0,0,0, . . . ,0,2,14) 93.38 94.2240 28 (10,2,0, . . . ,0,0,0) 94.11 94.0940 28 (0,0,0, . . . ,0,2,10) 94.43 94.4240 32 (6,2,0, . . . ,0,0,0) 95.02 95.3640 32 0,0,0, . . . ,0,2,6) 95.15 95.3740 36 (2,2,0, . . . ,0,0,0) 95.39 95.3840 36 (0,0,0, . . . ,0,2,2) 95.58 96.01

Table 15Coverage probabilities of 95% confidence intervals for σ based on Monte Carlo simulations.

n m R PC for LMLE1 PC for LMLE2

20 12 (7,1,0, . . . ,0,0,0) 92.94 93.6620 12 (0,0,0, . . . ,0,1,7) 93.17 93.7920 14 (5,1,0, . . . ,0,0,0) 93.25 93.7320 14 (0,0,0, . . . ,0,1,5) 93.74 93.8520 16 (3,1,0, . . . ,0,0,0) 94.82 94.2120 16 (0,0,0, . . . ,0,1,3) 94.29 94.6220 18 (1,1,0, . . . ,0,0,0) 95.03 95.4320 18 (0,0,0, . . . ,0,1,1) 95.41 95.4440 24 (14,2,0, . . . ,0,0,0) 93.15 93.6240 24 (0,0,0, . . . ,0,2,14) 93.21 93.7940 28 (10,2,0, . . . ,0,0,0) 93.36 93.3640 28 (0,0,0, . . . ,0,2,10) 93.81 93.7640 32 (6,2,0, . . . ,0,0,0) 95.01 95.2140 32 (0,0,0, . . . ,0,2,6) 95.21 95.0140 36 (2,2,0, . . . ,0,0,0) 95.11 95.3540 36 (0,0,0, . . . ,0,2,2) 95.14 95.91

8. Concluding remarks

In this article, we have discussed the EMalgorithm for themaximum likelihood estimation based on progressively Type-IIcensored samples froma three-parameter lognormal distribution.Wehave also considered the traditional Newton–Raphsonmethod for this purpose. Additionally, we have discussed two versions of modifiedmaximum likelihood estimators. The EMalgorithm and the Newton–Raphson method produced similar results in all three examples considered, as well as in termsof coverage probabilities determined from aMonte Carlo simulation study. The coverage probabilities for both methods arebetter and closer to the nominal level of 95% when the proportion of uncensored data is larger.

Acknowledgments

The authors are thankful to the referees for their valuable comments which led to a considerable improvement in thepresentation of this article.

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