68

Beta distributions of First and Second kind

In this chapter we consider the two kinds of Beta distributions.

5.1. Beta Distribution of First Kind

The Beta distribution of first kind is defined by the following pdf

, 0<x<1

= 0, otherwise

Where a> 0 and b>0 both are shape parameters.

The mean and variance of this distribution are

E (X) = ; V(X) =

The beta distribution has been applied to model the behavior of random variables limited to

intervals of finite length in a wide variety of disciplines. In population genetics it has been

employed for a statistical description of the allele frequencies in the components of a sub-

divided population. It has been utilized in PERT, critical path method (CPM) and other project

management / control systems to describe the statistical distributions of the time to completion

and the cost of a task. In Bayesian inference, the beta distribution is the conjugate prior

probability distribution for the Bernoulli, binomial and geometric distributions

69

5.2. Parameter Estimation

In this chapter we are interested in estimating the parameters of the Beta distribution of first

kind from which the sample comes. Here we present the method of moments as the MLE does

not give a neater expression and the Local frequency ratio method of estimation.

5.2.1 Method of moments

The r th moment about origin is

dx

dx

By using Beta integral, we can simplify as

Replacing Beta by Gamma, we get

Putting r = 1,2

……..(5.1) ( using )

70

………….(5.2)

Equating the sample moments with the corresponding theoretical moments, we get

. and =

Solving these equations, we get the following estimators:

5.2.2 Frequency Ratio Method of Estimation

As explained in the Chapter 3, we now estimate the parameters by considering local frequency

ratio method.

Putting x = x1, x2 and x3 the pdf of Beta distribution of first kind, we get f1, f2 and f3

, 0<x<1

71

In this case we compute theoretically three frequencies as we have two parameters to be

estimated, The Ratio of frequencies is

=

=

Taking logarithms on both sides,

here

Similarly,

=

Taking logarithms on both sides,

here

72

Solving (5.3) and (5.4), we get estimates of a and b as

and

Illustration:

For Estimating parameters, we generate a sample of size 1000 from a Beta (a=2, b=3)

distribution using MATLAB function. For the generated data a frequency distribution is

constructed.

For example, using x= betarnd (2, 3, 1000) the following distribution is obtained.

x

(Mid-value) 0.0484 0.1452 0.2419 0.3387 0.4355 0.5322 0.6290 0.7258 0.8225 0.9193

f 45 116 151 157 168 155 107 63 24 14

Using these frequencies and mid values in the above formulae, we get estimates of a and b as

The above procedure is repeated for 50 samples. The mean, Standard Deviation , of

these 50estimates were computed. The estimated bias was calculated as the mean minus the true

value of the parameter. The Mean Squared Error (MSE) was calculated as the bias squared plus

the variance .

73

Table 5.1

From the above table, we notice that the actual values of (a, b) and the mean estimated values of

(a, b) are almost same. Therefore, it can be taken as a good estimator. Similar procedure is

followed for different sample sizes and different values of (a, b).The results of both the

estimation procedures are listed in the tables 5.2 to 5.6 along with the histograms figures 5.1 to

5.6. The MATLAB programs are listed in the Appendix.

(a=2,b=3)

ns=50

Method of moments Frequency Ratio method

a b a b

Mean 2.0216

3.0248

2.0388

3.0659

Sd 0.1081 0.1562

1.1239

2.2443

0.2720 0.7451 0.3788

0.5050

2.1383 2.8472 2.5355 2.7952

Bias 0.0216 0.0248 0.0388 0.0659

MSE 0.0005 0.0006 0.0028 0.0094

74

5.3 Comparison of Method of moments and Frequency Ratio Method for

Different Sample sizes and Parameters:

Table 5.2: Simulation Statistics for Beta (1, 5)

Table 5.3: Simulation Statistics for Beta (3, 1)

(a=1,b=5)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

a b a b a b a b

Mean 0.9939 4.9999 0.9981 5.0960 1.0066 5.0418 1.0087 5.0742

Sd 0.0547 0.2974 0.2353 2.4049 0.0452 0.2689 0.2457 2.4929

0.1738 0.0788 0.0081 0.0049 0.0411 0.0877 0.109 0.0216

2.5259 2.3274 2.3978 2.2512 2.3523 2.4176 2.7163 2.9372

Bias -0.006 1.7e-005 -0.0019 0.0960 0.0066 0.0418 0.0087 0.0742

MSE 0.003 0.088 0.0001 0.0150 0.002 0.0018 0.0001 0.0117

(a=3,b=1)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

a b a b a b a b

Mean 3.0151 1.0061 3.1538 1.0209 3.0145 1.0095 3.0364 1.0060

Sd 0.1493 0.0451 1.6009 0.2233 0.1766 0.0524 1.6409 0.2286

0.0027 -0.535 0.2121 0.2766 0.9717 0.7116 -0.290 -0.338

2.4480 3.508 2.9718 2.722 5.0290 3.7251 2.5931 2.6515

Bias 0.0151 0.0061 0.1538 0.0209 0.0145 0.0095 0.0364 0.0060

MSE 0.0003 0.002 0.0262 0.0005 0.0002 0.0001 0.0040 0.0001

75

Table 5.4: Simulation Statistics for Beta (4, 3)

Table 5.5: Simulation Statistics for Beta(3,2)

(a=4,b=3)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

a b a b a B a b

Mean 4.0283 3.0102 3.999 2.9986 3.9836 2.9692 4.0362 3.0161

Sd 0.1804 0.1381 2.6090 1.7411 0.1746 0.1306 2.1844 1.3516

0.2512 0.3099 -0.0436 0.0270 0.9974 0.8995 0.3054 0.4703

2.8736 3.266 2.6150 2.7492 4.9974 4.7166 2.9499 2.8962

Bias 0.0283 0.0102 -0.0008 -0.0014 -0.016 -0.030 0.0362 0.0161

MSE 0.0008 0.0001 0.0068 0.0030 0.0003 0.0010 0.0061 0.0021

(a=3,b=2)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

a b a b a B a b

Mean 2.9663 1.9860 3.0435 2.0235 3.0122 1.9965 2.9463 1.9811

Sd 0.1306 0.0923 2.3736 1.2595 0.1374 0.0949 2.6093 1.3951

0.4621 0.0964 0.0938 0.0434 0.2088 0.0861 -0.399 -0.330

2.3359 3.612 2.4233 2.6296 2.1267 2.010 3.0701 3.1817

Bias -0.033 -0.014 0.0435 0.0235 0.0122 -0.003 -0.053 -0.018

MSE 0.0012 0.0002 0.0075 0.0021 0.0002 0.009 0.0097 0.0023

76

Table 5.6: Simulation Statistics for Beta (5,1)

(a=5,b=1)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

a b a b a B a b

Mean 5.0836 1.0096 4.9818 1.0011 4.9791 0.9973 5.1635 1.0265

Sd 0.3212 0.0574 1.6638 0.2332 0.2474 0.0447 1.1614 0.222

0.1310 0.0093 0.4945 0.2647 0.1757 0.1414 -0.149 -0.260

3.122 3.688 3.7453 3.4875 2.9967 2.7970 3.412 3.4515

Bias 0.0839 0.0096 -0.0182 0.0011 -0.021 -0.002 0.1635 0.0265

MSE 0.0071 0.0001 0.0031 0.0001 0.0005 0.002 0.0293 0.0008

77

5.4 GRAPHS FOR DIFFERENT SAMPLE SIZES AND PARAMETERS

FIGURE 5.1 : HISTOGRAM FOR BETA(1,5,50)

FIGURE 5.2 : HISTOGRAM FOR BETA(1,5,100)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

150

200

250

300

Fre

quency

Bins

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

50

100

150

200

250

300

350

Fre

quency

Bins

78

FIGURE 5.3: HISTOGRAM FOR BETA(3,2,50)

FIGURE 5.4: HISTOGRAM FOR BETA(3,2,100)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

Fre

quency

Bins

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

Bins

Fre

quency

79

FIGURE 5.5 :HISTOGRAM FOR BETA(3,1,50)

FIGURE 5.6: HISTOGRAM FOR BETA(3,1,100)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

Fre

quency

Bins

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

Bins

Fre

quency

80

5.5. Beta Distribution of Second Kind

The Beta distribution of Second kind is defined by the following pdf

= 0, otherwise

Where a>0 and b>0 both are shape parameters.

The mean and variance of the distribution are

E (X) = ; V(X) =

5.6. Parameter Estimation

We are interested in estimating the parameters of the Beta distribution of second kind from

which the sample comes. Here we present the method of moments as the MLE is complex in

estimating the parameters and the Local frequency ratio method of estimation.

5.6.1 Method of moments

The rth moment about origin is

Using Beta integral, we get

81

Replacing Beta by Gamma , we get

Putting r= 1, 2, we get

…….. (5.5) ( using )

………….(5.6)

Equating the sample moments with the corresponding theoretical moments,

Solving these equations yields the following estimators:

82

5.6.2 Frequency Ratio Method of Estimation

As usual to estimate the parameters by this method, put x = x1, x2 and x3 in the Beta

distribution, we get the frequencies f1, f2 and f3

The ratio of the frequencies is

Taking logarithms on both sides,

------------------------ (5.7)

Similarly,

Taking logarithms on both sides,

83

------------------------ (5.8)

Solving (5.7) and (5.8), we get

and

We explain the above procedures in the following illustration

Illustration:

As explained earlier, we generate a sample of size 1000 from a Beta (a=2, b=4) distribution

using MATLAB function. For the generated data a frequency distribution is constructed and the

specified parameter values are estimated using above the procedures. This procedure is repeated

50 times. The mean, Standard Deviation, , and bias of these 50 estimates are computed.

The following results were obtained

Table 5.7

(a=2,b=4)

ns=50

Method of moments Frequency Ratio method

a b a b

Mean 2.0651 4.0934 2.0224 4.0010

Sd 0.2173 0.3767 1.4118 5.8261

-1.0008 -0.8184 -0.5162 -0.5124

5.5308 4.7823 3.1181 3.2351

Bias 0.0651 0.0935 0.0224 0.0010

MSE 0.055 0.1507 1.9938 33.9438

84

From the above table, we notice that the actual values of (a, b) and the mean estimated values of

(a, b) are almost same. Therefore, it can be taken as a good estimator. Similar procedure is

followed for different sample sizes and different values of (a, b). The results of both the

estimation procedures are listed in the tables 5.7 to 5.11 along with the histograms figures 5.7 to

5.12. The MATLAB programs are listed in the Appendix.

5.7 Comparison of Method of moments and Frequency Ratio Method for

Different Sample sizes and Parameters

TABLE 5.7: Simulation Statistics for Beta (2, 4)

(a=2;b=4)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

a b a b a b A b

Mean 2.0651 4.0934 2.0224 4.0010 2.022 4.0389 2.0336 4.1199

Sd 0.2173 0.3767 1.4118 5.8261 0.2476 0.4002 1.7219 7.3148

-1.001 -0.818 -0.5162 -0.512 -0.395 -0.272 0.0319 0.0863

5.5308 4.7823 3.1181 3.2351 2.7567 2.5554 2.2265 2.2269

Bias 0.0651 0.0935 0.0224 0.0010 0.0225 0.0386 0.0336 0.1199

MSE 0.055 0.1507 1.9938 33.943 0.0618 0.1616 2.9653 53.5205

85

TABLE 5.8: Simulation Statistics for Beta( 3,3)

TABLE 5.9: Simulation Statistics for Beta( 3,5)

(a=3;b=3)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

a b a b a b a b

Mean 3.2367 3.1924

2.9707 2.9608 3.315 3.2208 3.0410 3.0600

Sd 0.4079 0.3302 2.2200 4.2288 0.3990 0.3213 2.1180 3.8186

0.0448 0.0536 -0.4864 -0.3313 -0.571 -0.473 -0.479 -0.0722

2.2091 2.2750 2.6249 2.2182 3.2900 3.2753 3.4450 2.7652

Bias 0.2367 0.1924 -0.0293 -0.0392 0.3152 0.2208 0.0410 0.0600

MSE 0.2224 0.1460 4.993 17.8839 0.2585 0.1520 4.4879 14.5852

(a=3;b=5)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

a b a b a b a b

Mean 3.0211 5.0276 3.2546 5.1656 3.0432 5.0690 3.3143 5.1878

Sd 0.2623 0.3393 0.4034 0.9808 0.2998 0.4553 0.6754 1.2952

-0.035 0.0960 0.0457 -0.1377 0.0951 0.2226 -0.719 -1.130

2.5159 2.224 2.700 3.1889 2.755 2.8137 7.3415 7.3212

Bias 0.0211 0.0276 0.2546 0.1656 0.0432 0.0690 0.3143 0.1878

MSE 0.0692 0.1583 0.2276 0.9893 0.0918 0.2121 0.5550 1.7129

86

TABLE 5.10 Simulation Statistics for Beta( 5,7)

TABLE 5.11 Simulation Statistics for Beta ( 5,10)

(a=5;b=7)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

a b a b a b a b

Mean

5.0558 7.0746 5.2022 7.1667

5.0623 7.0756 5.1557 7.0974

Sd 0.3771 0.4878 1.8331 3.1447 0.3931 0.5135 1.6845 3.1104

0.0274 0.3004 0.0133 -0.0522 0.1814 0.3812 0.1818 0.0428

2.5045 2.2683 2.3652 2.2117 2.7394 2.9302 3.4553 3.9820

Bias 0.0558

0.0746 0.2022 0.1667 0.062 0.0756 0.155 0.0974

MSE 0.1453 0.2435 3.4012 9.9172 0.1584 0.2684 2.8620 9.6840

(a=5;b=10)

ns=50 ns= 100

Method of

moments

Frequency

Ratio method

Method of

moments

Frequency

Ratio method

A b a b a b a b

Mean 5.0366 10.0662 5.1158 10.0700 5.0202 10.0296 5.224

10.3472

Sd 0.3638 0.6801 2.2092 5.6544 0.3494 0.6534 2.1881 5.3417

0.4713 0.4195 -0.974 -1.0911 0.1172 -0.0142 0.7037 0.6900

4.7670 4.8932 5.0314 5.6287 3.2497 3.0354 4.4808 3.9793

Bias 0.0366 0.0662

0.1158

0.0700 0.0202 0.0296 0.2224 0.3472

MSE 0.1337 0.4670 4.8940 31.9767 0.1225 0.4278 4.8375 28.6545

87

5.8 GRAPHS FOR DIFFERENT SAMPLE SIZES AND PARAMETERS

FIGURE 5.7: HISTOGRAM FOR BETA2(2,4,50)

FIGURE 5.8: HISTOGRAM FOR BETA2(2,4,100)

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

Fre

quency

Bins

0 2 4 6 8 10 12 14 160

20

40

60

80

100

120

140

Bins

Fre

quency

88

FIGURE 5.9: HISTOGRAM FOR BETA2 (3,3,50)

FIGURE 5.10: HISTOGRAM FOR BETA2 (3,3,100)

0 5 10 15 20 250

20

40

60

80

100

120

140

Fre

quency

Bins

0 2 4 6 8 10 12 14 160

50

100

150

Fre

quency

Bins

89

FIGURE 5.11: HISTOGRAM FOR BETA2 (5,7,50)

FIGURE 5.12: HISTOGRAM FOR BETA2 (5,7,100)

0 1 2 3 4 5 60

50

100

150

200

250

300

0 0.5 1 1.5 2 2.5 3 3.5 40

50

100

150

200

250

90

MATLAB PROGRAMS

BETA DISTRIBUTION OF FIRST KIND %PROGRAM TO ESTIMATE THE PARAMETERS BY METHOD OF MOMENTS AND

LOCAL FREQUENCY RATIO METHOD.

%n (sample size);ns(%no of samples);k=no. of classes(bins

created)

%n=1000; ns=100;k=10;a=3;b=1;

for j=1:ns,

ex=betarnd(a,b,n,1);

meanx = mean(ex);

varx = var(ex);

t=((meanx.*(1-meanx))./varx)-1;

aa(j)=meanx* t;

bb(j) = t-aa(j);

maxx=max(ex);

h=maxx/k;

[fr,r]=hist(ex,h/2:h:maxx);

hist(ex,h/2:h:maxx)

[u,v]=sort(fr);

u=fliplr(u);

v=fliplr(v);

y=v(1:3);x=r(y);

f=u(1:3);

91

x1=1-x(1);

x2=1-x(2);

x3 = 1-x(3);

x12=log(x(1)/x(2));

x23=log(x(2)/x(3));

lx12 = log(x1/x2);

lx23=log(x2/x3);

lf12=log(f(1)/f(2));

lf23=log(f(2)/f(3));

em=[x12,lx12;x23,lx23];

rhs=[lf12;lf23];

est=em\rhs;

aest=est(1)+1;

best=est(2)+1;

a1(j)=aest;

b1(j)=best;

clear ex fr r u v ;

end

%LOCAL FREQUENCY METHOD

a1=a1; b1=b1;

ma1=mean (a1); sda1=std (a1); mb1=mean (b1); sdb1=std (b1);

a1z= (a1-ma1)/sda1; b1z= (b1-mb1)/sdb1;

rba1=mean (a1z. ^3); b2a1=mean (a1z. ^4);

rbb1=mean (b1z. ^3); b2b1=mean (b1z. ^4);

92

%bias

ba1=a1-a; mba1=mean (ba1);bb1=b1-b;mbb1=mean(bb1);

%mean square error

msea1=sda1^2+ mba1^2; mseb1=sdb1^2+ mbb1^2;

Local moments = [ma1, sda1, rba1, b2a1, mb1, sdb1,rbb1, b2b1]

locbias=[mba1,msea1,mbb1,mseb1]

%By Method of Moments

a2= aa; b2=bb;

ma2=mean (a2); sda2=std(a2);

mb2=mean (b2); sdb2=std(b2);

a2z= (a2-ma2)/sda2; b2z=(b2-mb2)/sdb2;

rba2=mean (a2z. ^3);b2a2=mean(a2z.^4);

rbb2=mean (b2z.^3);b2b2=mean(b2z.^4);

%bias

ba2=a2-a; mba2= mean (ba2);bb2=b2-b;mbb2=mean(bb2);

%mean square error

msea2=sda2^2+ mba2^2; mseb2=sdb2^2+ mbb2^2;

methodofmoments = [ma2, sda2,rba2,b2a2,mb2, sdb2,rbb2,b2b2]

mombias = [mba2,msea2,mbb2,mseb2]

93

BETA DISTRIBUTION OF SECOND KIND

%PROGRAM TO ESTIMATE THE PARAMETERS BY METHOD OF MOMENTS AND

LOCAL FREQUENCY RATIO METHOD.

n=1000; ns=100;k=19;a=5;b=10;

for j=1:ns,

x=betarnd (a, b, n, 1);

ex=x./(1-x);

Meanx = mean(ex);

varx = var(ex);

t=((meanx*(1+meanx))./varx)+1;

aa(j)=meanx* t;

bb(j) = t+1;

maxx=max(ex);

h=maxx/k;

[fr,r]=hist(ex,h/2:h:maxx);

hist(ex,h/2:h:maxx)

[u,v]=sort(fr);

u=fliplr(u);

v=fliplr(v);

y=v(1:3);x=r(y);

f=u(1:3);

x1=1+x(1);

x2=1+x(2);

94

x3 = 1+x(3);

x12=log(x(1)/x(2));

x23=log(x(2)/x(3));

lx12 = log(x2/x1);

lx23=log(x3/x2);

lf12=log(f(1)/f(2));

lf23=log(f(2)/f(3));

em=[x12,lx12;x23,lx23];

rhs=[lf12;lf23];

est=em\rhs;

aest=est(1)+1;

best=est(2)-aest;

a1(j)=aest;

b1(j)=best;

clear ex fr r u v ;

end

%LOCAL METHOD

a1=a1;b1=b1;

ma1=mean(a1);sda1=std(a1);

mb1=mean(b1);sdb1=std(b1);

a1z=(a1-ma1)/sda1;b1z=(b1-mb1)/sdb1;

rba1=mean(a1z.^3);b2a1=mean(a1z.^4);

rbb1=mean(b1z.^3);b2b1=mean(b1z.^4);

95

%bias

ba1=a1-a;mba1=mean(ba1);

bb1=b1-b;mbb1=mean(bb1);

%mean square error

msea1=sda1^2+ mba1^2;

mseb1=sdb1^2+ mbb1^2;

local moments = [ma1, sda1,rba1,b2a1,mb1,sdb1,rbb1,b2b1]

locbias=[mba1,msea1,mbb1,mseb1]

%By Method of Moments

a2=aa;b2=bb;

ma2=mean(a2);sda2=std(a2); mb2=mean(b2);sdb2=std(b2);

a2z=(a2-ma2)/sda2;b2z=(b2-mb2)/sdb2;

rba2=mean(a2z.^3);b2a2=mean(a2z.^4);

rbb2=mean(b2z.^3);b2b2=mean(b2z.^4);

%bias

ba2=a2-a;mba2=mean(ba2);

bb2=b2-b;mbb2=mean(bb2);

%mean square error

msea2=sda2^2+ mba2^2;

mseb2=sdb2^2+ mbb2^2;

methodofmoments = [ma2, sda2,rba2,b2a2,mb2, sdb2,rbb2,b2b2]

mombias = [mba2,msea2,mbb2,mseb2]