MODIFIED ML ESTIMATION: SIZE-BIASED LOMAX DISTRIBUTION Dr. R. Subba Rao1, A. Naga Durgamamba2, Dr. R.R.L.Kantam3 1

Department of Basic Science, Shri Vishnu Engg. College for Women, Bhimavaram ,Andhra Pradesh, (India)

2

Department of Humanities and Science, Raghu Institute of Technology, Dakamarri,Vizag, Andhra Pradesh, (India) 3

Department of Statistics, Acharya Nagarjuna University, Guntur, Andhra Pradesh, (India)

ABSTRACT Size biased Lomax distribution is considered with a known shape parameter. Some of its distributional characteristics are presented. Estimation of its scale parameter by the well known maximum likelihood method is modified by two different approaches in order to yield linear estimators. The proposed methods are compared with respect to simulated sampling characteristics.

Keywords: Asymptotic Variance, Log Likelihood Function, MLE, Order Statistics, Size Biased Lomax Distribution.

I INTRODUCTION Estimation of parameters of the size biased Lomax model is considered. Let α and σ be the two parameters of the size biased Lomax distribution (SBLD), where α is shape parameter and σ is scale parameter. First we establish the well known maximum likelihood method of estimation of the parameters from complete sample when one of the parameters is known while the other is unknown. Hence we start with general maximum likelihood estimation of α and σ. As the estimating equations are to be solved by numerical iterative techniques we suggest some modifications to maximum likelihood method from complete sample. Such studies based on modified maximum likelihood estimation which is attempted by many researchers with respect to other standard reliability models that include K. Rosaiah et al. derived (i) ML and Modified ML estimation in gamma distribution with known prior relation among the parameters (1993a) and (ii) On modified maximum likelihood estimation of gamma parameters (1993b), R.R.L. Kantam and G. Srinivasa Rao developed Reliability estimation in Rayleigh distribution with censoring some

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approximations to ML Method (1993), R.R.L. Kantam and B. Sriram determined Maximum likelihood estimation from censored samples- Some modifications in length biased version of exponential model (2003), Pareto Distribution – Some methods of Estimation constructed by R. Subba Rao et al. (2010) and Modified Maximum Likelihood Estimation in Pareto-Rayleigh distribution studied by R. Subba Rao et al. (2015). In this paper discussion of complete sample is given in Section 2. Since the process of estimation involves lot of numerical computations all such results are presented in the form of numerical tables towards the end of the Section with appropriate identification and labels.

II ESTIMATION FROM COMPLETE SAMPLE Let f (x) be the probability density function of the two parameter size biased Lomax distribution and is given by: f  x 

 1    1 x  x ; x  0,   1,   0 1     

(2.1)

where α and σ are shape and scale parameters respectively. Let x1  x2  x3  ........  xn be a complete ordered sample of size n drawn from the above distribution. The log likelihood equations to estimate α and σ from the given complete sample are given by  x   x   log L n n  x     log 1  1  . 1  2  ......... 1  n      1          n x   log L n n      log 1  i      1 i 1  

 

(2.2)

xi

n 2  log L 2n     1   x i   i 1 1  





(2.3)

For M.L.Es of α and σ

 log L  0, 

 log L 0 

After simplification, these equations become n

2  1   2 



xi 

 log 1    i 1

n

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n

2  1   i 1  2 

log 1  Zi  (2.4)

n

 xi   2      1  x   i 1  1  i     n

2n

2n





  1 

n

zi

1 z i 1

(2.5) i

where It can be seen that equation (2.5) can be solved only by iterative method for ‘σ’. The M.L.E of α is an analytical expression involving ‘σ’. In order to overcome the iterative techniques that may sometimes lead to convergence problems we approximate the expression h  zi  

zi 1  zi

(2.6)

of the log likelihood equation (2.5) for estimating ‘σ’ by a linear expression say h  zi    i   i zi

(2.7)

in certain admissible ranges of

. Such approximations are not feasible for the log likelihood equation of ‘α’. Hence

we develop our approximate M.L. method for estimation of ‘σ’ with known ‘α’. As per the parametric specifications we take α = 3. After using the linear approximation given by equation (2.7) in the equation (2.5) and solving it for ‘σ’ we get n

ˆ 

  1   i xi

(2.8)

i 1

n

2n    1   i i 1

as an approximate M.L.E. of ‘σ’, which is a linear estimator. Similar approximations are in Tiku (1967) and Balakrishnan (1990). We suggest two methods finding

of equation (2.7).

Method I:

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Let P  i

i , i  1, 2, 3,...........n n 1

Let Zi , Zi be the solutions of the following equations

F  Zi   pi and F  Zi   pi 

where pi  pi 

pi qi , n

pi  pi 

pi qi n

The solutions of Zi , Zi in our size biased Lomax distribution are

F  Zi   Pi   Zi  F 1  Pi   F  Zi   Pi   Zi  F 1  Pi   The slope

of the linear approximation in the equation (2.7) are respectively given by

h  Z i   h  Z i 

i  And

and intercept

(2.9)

Z i  Z i

 i  h  Zi    i Zi

The values of

and

(2.10)

in this method for n = 5, 10, 15, 20 and 25 for α = 3 are given in Table (2.1).

Method II: Consider the Taylor’s expansion of h  zi  

zi in the neighbourhood of ith quantile of size biased Lomax 1  zi

population. We get another linear approximation for h  Z i  is given by

h  Zi    i   i Zi where  i  h1  Zi  with

(2.11) as the ith quantile for the population given by

F  Zi   Pi  Zi  F 1  Pi  ,

Pi 

i n 1

 i  h  Zi    i Zi Substituting these approximations in the equation (2.8) we get another linear estimator of ‘σ' with different values of and

. The values of

and

in this method for n = 5, 10, 15, 20 and 25 for α = 3 are given in Table (2.2).

These two methods are asymptotically as efficient as exact maximum likelihood estimators as can be seen from the following narration (Tiku et al. 1986; Balakrishnan, 1990).

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In the above two modified methods, the basic principle is that certain expressions in the log likelihood equation are linearized in a neighbourhood of the population quantile which depends on the size of the sample also. The larger the size, the narrower is the neighbourhood and hence the closer is the approximation. That is, the exactness of the approximation becomes finer and finer for large values of ‘n’. Hence the approximate log likelihood equation and the exact log likelihood equation tend to each other as n

. Hence the exact and modified MLEs are

asymptotically identical (Tiku et al. 1986). The same may not be true in small samples and these are to be assessed with the help of small sample characteristics of the MMLEs. Because of difficulties in finding the analytical sampling variances, we compared the modified MLEs of two methods (Method I and Method II) of estimates through Monte Carlo simulation. 1,000 random samples of size n = 5 (5) 25 each are generated from size biased Lomax distribution with α = 3. For each sample with α = 3 the

and

of Method I and Method II are as given in Tables 2.1 and 2.2 and are used in

equation 2.8 to get the modified MLE of ‘σ’ by Method I and Method II respectively. The empirical variances of MMLEs by Method-I and Method-II are compared and given in Table 2.3. A comparison of the sampling characteristics namely the bias, variance and M.S.E of Methods I and II, reveal that MMLE of Method I is preferable to that of Method II.

Table – 2.1 Intercept ( ) and Slope ( ) in MMLE by Method I (when α = 3)

n

i

5

1

0.100281

0.454169

5

2

0.115767

0.367651

5

3

0.226304

0.226304

5

4

0.367651

0.115767

5

5

0.454169

0.100281

10

1

0.050617

0.592522

10

2

0.054466

0.539664

10

3

0.100664

0.430874

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10

4

0.151329

0.344394

10

5

0.20775

0.271357

n

i

10

6

0.271357

0.20775

10

7

0.344394

0.151329

10

8

0.430874

0.100664

10

9

0.539664

0.054466

10

10

0.592522

0.050617

15

1

0.033662

0.660326

15

2

0.0354

0.660326

15

3

0.064332

0.528229

15

4

0.09502

0.454645

15

5

0.127878

0.391957

15

6

0.163176

0.336824

15

7

0.201227

0.287377

15

8

0.242441

0.242441

15

9

0.287377

0.201227

15

10

0.336824

0.163176

15

11

0.391957

0.127878

15

12

0.454645

0.09502

15

13

0.528229

0.064332

15

14

0.660326

0.0354

15

15

0.660326

0.033662

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20

1

0.025159

0.702527

20

2

0.026152

0.669137

20

3

0.047119

0.588105

n

i

20

4

0.069009

0.522942

20

5

0.092057

0.467197

20

6

0.116366

0.417954

20

7

0.142028

0.373576

20

8

0.169153

0.333035

20

9

0.19788

0.295632

20

10

0.228379

0.260869

20

11

0.260869

0.228379

20

12

0.295632

0.19788

20

13

0.333035

0.169153

20

14

0.373576

0.142028

20

15

0.417954

0.116366

20

16

0.467197

0.092057

20

17

0.522942

0.069009

20

18

0.588105

0.047119

20

19

0.669137

0.026152

20

20

0.702527

0.025159

25

1

0.02006

0.732041

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25

2

0.020706

0.703045

25

3

0.037107

0.629696

25

4

0.054068

0.570574

25

5

0.071757

0.519872

25

6

0.090227

0.474969

n

i

25

7

0.109517

0.434396

25

8

0.129668

0.397225

25

9

0.150729

0.362829

25

10

0.172764

0.330756

25

11

0.195841

0.300672

25

12

0.22052

0.272322

25

13

0.245502

0.245502

25

14

0.272322

0.22052

25

15

0.300672

0.195841

25

16

0.330756

0.172764

25

17

0.362829

0.150729

25

18

0.397225

0.129668

25

19

0.434396

0.109517

25

20

0.474969

0.090227

25

21

0.519872

0.071757

25

22

0.570574

0.054068

25

23

0.629696

0.037107

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25

24

0.703045

0.020706

25

25

0.732041

0.02006

Table – 2.2 Intercept ( ) and Slope ( ) in MMLE by Method II (when α = 3) n

i

5

1

0.067158

0.54886

5

2

0.047807

0.610511

5

3

0.119761

0.427631

5

4

0.221399

0.280337

5

5

0.375814

0.14974

10

1

0.034592

0.662612

10

2

0.022148

0.724506

10

3

0.05183

0.596505

10

4

0.08787

0.495012

10

5

0.130433

0.408122

10

6

0.180423

0.330898

10

7

0.239605

0.260616

10

8

0.311234

0.195467

10

9

0.40198

0.133942

10

10

0.529845

0.074036

15

1

0.023187

0.71864

15

2

0.014349

0.774776

15

3

0.032882

0.670212

15

4

0.054562

0.587391

15

5

0.079107

0.516588

15

6

0.406506

0.453802

15

7

0.13691

0.396883

15

8

0.170616

0.344502

15

9

0.208085

0.295759

15

10

0.25

0.25

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15

11

0.297394

0.206717

15

12

0.351899

0.165477

15

13

0.416325

0.12586

15

14

0.496279

0.087338

15

15

0.606743

0.048869

20

1

0.017404

0.753558

20

2

0.010593

0.804744

n

i

20

3

0.024019

0.714059

20

4

0.039447

0.642223

20

5

0.056595

0.580801

20

6

0.075357

0.526331

20

7

0.095717

0.476955

20

8

0.117712

0.431528

20

9

0.14143

0.389286

20

10

0.167003

0.349683

20

11

0.194614

0.312313

20

12

0.224509

0.276861

20

13

0.257022

0.243075

20

14

0.292605

0.210746

20

15

0.331893

0.17969

20

16

0.375814

0.14974

20

17

0.425816

0.120725

20

18

0.484366

0.092437

20

19

0.556381

0.064562

20

20

0.654951

0.03637

25

1

0.013915

0.777993

25

2

0.008389

0.825209

25

3

0.018894

0.743984

25

4

0.030839

0.679618

25

5

0.043976

0.624565

25

6

0.058194

0.575727

25

7

0.073441

0.53144

25

8

0.089707

0.490684

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25

9

0.107003

0.452776

25

10

0.125363

0.417231

25

11

0.144836

0.383689

25

12

0.165495

0.351873

25

13

0.187429

0.321567

25

14

0.210755

0.292594

25

15

0.235618

0.264808

25

16

0.262204

0.238087

25

17

0.290749

0.212325

n

i

25

18

0.321567

0.187429

25

19

0.355073

0.163313

25

20

0.391846

0.139894

25

21

0.432728

0.117086

25

22

0.479026

0.094792

25

23

0.532965

0.072876

25

24

0.598965

0.051108

25

25

0.68804

0.028919

Table – 2.3 Sample Characteristics of MMLE of 'σ' from Complete Sample Method I and Method II

BIAS α

VARIANCE

MSE

N MMLE-I

MMLE-II

MMLE-I

MMLE-II

MMLE-I

MMLE-II

3

5

0.25713

0.59366

0.76178

1.135183

0.82789

1.48761

3

10

0.08878

0.29662

0.29284

0.39633

0.30072

0.48431

3

15

0.04531

0.19875

0.13584

0.18001

0.13789

0.21951

3

20

0.03345

0.153892

0.08263

0.1058

0.08374

0.12949

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3

25

0.01304

0.11383

0.05929

0.07359

0.05947

0.08655

REFERENCES [1] K. Rosaiah, R.R.L. Kantam and V.L. Narasimham, ML and Modified ML estimation in gamma distribution with known prior relation among the parameters, Pakistan J. of Statist., Vol.9(3)B, 1993a, 37-48. [2] K. Rosaiah, R.R.L. Kantam and V.L. Narasimham, On modified maximum likelihood estimation of gamma parameters, J. of Statistical Research, Bangladesh, 27, Nos.1 & 2, 1993b, 15-28. [3] R.R.L. Kantam and G. Srinivasa Rao, Reliability estimation in Rayleigh distribution with censoring some approximations to ML Method, Proceedings of II Annual Conference of Society for Development of Statistics, Acharya Nagarjuna University, 1993, 56-63. [4] R.R.L. Kantam and G. Srinivasa Rao, Log-logistic distribution: Modified Maximum likelihood estimation, Gujarat Statistical Review, 29, Nos.1-2, 2002, 25-36. [5] R.R.L. Kantam and B. Sriram, Maximum likelihood estimation from censored samples- Some modifications in length biased version of exponential model, Statistical methods, 5(1), 2003, 63-78. [6] R. Subba Rao, R.R.L. Kantam and G. Prasad, Modified Maximum Likelihood Estimation in Pareto-Rayleigh Distribution, Conference Proceedings of National Seminar on Recent Developments in Applied Statistics – Golden Research Thoughts, 2015, 140-152. [7] R. Subba Rao, R.R.L. Kantam and G. Srinivasa Rao, Pareto Distribution – Some methods of Estimation, International Journal of Computational Mathematical Ideas, Vol.2, No.2, 2010, 82-92. [8] M.L. Tiku, Estimating the Mean and Standard Deviations from a Censored Normal Sample, Biometrika, Vol.4, 1967, 155-165. [9] N. Balakrishnan, Approximate Maximum Likelihood Estimation for a Generalized Logistic Distribution, Journal of the Statistics Planning & inference, Vol.26, 1990, 221-236. [10] M.L. Tiku, W.Y. Tan and N. Balakrishnan, Robust Inference, Marcel Dekker, INC, New York, 1986.

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