Journal of Probability and Statistics
Volume 2016 (2016), Article ID 7581918, 8 pages
http://dx.doi.org/10.1155/2016/7581918

Research Article

Classical and Bayesian Approach in Estimation of Scale Parameter of Nakagami Distribution

Kaisar Ahmad,¹ S. P. Ahmad,¹ and A. Ahmed²

¹Department of Statistics, University of Kashmir, Srinagar, Jammu and Kashmir 190006, India
²Department of Statistics and O.R., Aligarh Muslim University, Aligarh, India

Received 19 October 2015; Accepted 17 December 2015

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2016 Kaisar Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Nakagami distribution is considered. The classical maximum likelihood estimator has been obtained. Bayesian method of estimation is employed in order to estimate the scale parameter of Nakagami distribution by using Jeffreys’, Extension of Jeffreys’, and Quasi priors under three different loss functions. Also the simulation study is conducted in R software.

1. Introduction

Nakagami distribution can be considered as a flexible lifetime distribution. It has been used to model attenuation of wireless signals traversing multiple paths (for details see Hoffman [1]), fading of radio signals, data regarding communicational engineering, and so forth. The distribution may also be employed to model failure times of a variety of products (and electrical components) such as ball bearing, vacuum tubes, and electrical insulation. It is also widely considered in biomedical fields, such as to model the time to the occurrence of tumors and appearance of lung cancer. It has the applications in medical imaging studies to model the ultrasounds especially in Echo (heart efficiency test). Shanker et al. [2] and Tsui et al. [3] use the Nakagami distribution to model ultrasound data in medical imaging studies. This distribution is extensively used in reliability theory and reliability engineering and to model the constant hazard rate portion because of its memory less property. Yang and Lin [4] investigated and derived the statistical model of spatial-chromatic distribution of images. Through extensive evaluation of large image databases, they discovered that a two-parameter Nakagami distribution well suits the purpose. Kim and Latchman [5] used the Nakagami distribution in their analysis of multimedia.

The probability density function (pdf) of the Nakagami distribution is given as mentioned in Figure 1:

where and are the scale and the shape parameters, respectively.

Figure 1: The pdf’s of Nakagami distribution under various values of and theta.

2. Materials and Methods

There are two main philosophical approaches to statistics. The first is called the classical approach which was founded by Professor R. A. Fisher in a series of fundamental papers round about 1930. In classical approach we use the same method as obtained by Ahmad et al. [6].

The alternative approach is the Bayesian approach which was first discovered by Reverend Thomas Bayes. In this approach, parameters are treated as random variables and data is treated as fixed. Recently Bayesian estimation approach has received great attention by most researchers among them are Al-Aboud [7] who studied Bayesian estimation for the extreme value distribution using progressive censored data and asymmetric loss. Ahmed et al. [8] considered Bayesian Survival Estimator for Weibull distribution with censored data. An important prerequisite in this approach is the appropriate choice of prior(s) for the parameters. Very often, priors are chosen according to one’s subjective knowledge and beliefs. The other integral part of Bayesian inference is the choice of loss function. A number of symmetric and asymmetric loss functions have been shown to be functional; see Pandey et al. [9], Al-Athari [10], S. P. Ahmad and K. Ahmad [11], Ahmad et al. [12, 13], and so forth.

Theorem 1. Let be a random sample of size n having pdf (1); then the maximum likelihood estimator of scale parameter , when the shape parameter is known, is given by

Proof. The likelihood function of the pdf (1) is given by

The log likelihood function is given by

Differentiating (4) with respect to and equating to zero, we get

2.1. Loss Functions Used in This Paper

(i) The quadratic loss function which is given by

which is a symmetric loss function; and represent the true and estimated values of the parameter.

(ii) The Al-Bayyati new loss function is of the form

which is an asymmetric loss function; and represent the true and estimated values of the parameter.

(iii) The entropy loss function is given by

where and represent the true and estimated values of the parameter.

3. Bayesian Method of Estimation

In this section Bayesian estimation of the scale parameter of Nakagami distribution is obtained by using various priors under different symmetric and asymmetric loss functions.

3.1. Posterior Density under Jeffreys’ Prior

Let be a random sample of size having the probability density function (1) and the likelihood function (2).

Jeffreys’ prior for is given by

By using the Bayes theorem, we have

Using (2) and (9) in (10),

where is independent of and

Using the value of in (11),

3.2. Posterior Density under Extension of Jeffreys’ Prior

Let be a random sample of size having the probability density function (1) and the likelihood function (2).

The extension of Jeffreys’ for is given by

By using the Bayes theorem, we have

Using (2) and (14) in (15),

Thus

By using the value of in (17), we have

3.3. Posterior Density under Quasi Prior

Let be a random sample of size having the probability density function (1) and the likelihood function (2).

Quasi prior for is given by

By using the Bayes theorem, we have

Using (2) and (20) in (21),

where is independent of and

Using the value of in (22),

4. Bayesian Estimation by Using Jeffreys’ Prior under Different Loss Functions

Theorem 2. Assuming the loss function , the Bayes estimate of the scale parameter , if the shape parameter is known, is of the form

Proof. The risk function of the estimator under the quadratic loss function is given by the formula

Using (13) in (26), we get

On solving (27), we get

Minimization of the risk with respect to gives us the optimal estimator:

Theorem 3. Assuming the loss function , the Bayes estimate of the scale parameter , if the shape parameter is known, is of the form

Proof. The risk function of the estimator under the Al-Bayyati loss function is given by the formula

On substituting (13) in (31), we have

Solving (32), we get

Minimization of the risk with respect to gives us the optimal estimator:

Theorem 4. Assuming the loss function , the Bayes estimate of the scale parameter , if the shape parameter is known, is of the form

Proof. The risk function of the estimator under entropy loss function is given by the formula

Using (13) in (36), we get

On solving (37), we get

Minimization of the risk with respect to gives us the optimal estimator:

5. Bayesian Estimation by Using Extension Jeffreys’ Prior under Different Loss Functions

Theorem 5. Assuming the loss function , the Bayes estimate of the scale parameter , if the shape parameter is known, is of the form

Proof. The risk function of the estimator under the quadratic loss function is given by the formula

Using (19) in (41), we get

On solving (42), we get

Minimization of the risk with respect to gives us the optimal estimator:

Remark 6. By replacing in (44), the same Bayes estimate is obtained as in (29).

Theorem 7. Assuming the loss function , the Bayes estimate of the scale parameter , if the shape parameter is known, is of the form

Proof. The risk function of the estimator under the Al-Bayyati loss function is given by the formula

On substituting (19) in (46), we have

Solving (47), we get

Minimization of the risk with respect to gives us the optimal estimator:

Remark 8. By replacing in (49), the same Bayes estimate is obtained as in (34).

Theorem 9. Assuming the loss function , the Bayes estimate of the scale parameter , if the shape parameter is known, is of the form

Proof. The risk function of the estimator under entropy loss function is given by the formula

Using (19) in (51), we get

On solving (52), we get

Minimization of the risk with respect to gives us the optimal estimator:

Remark 10. By replacing in (54), the same Bayes estimate is obtained as in (39).

6. Bayesian Estimation by Using Quasi Prior under Different Loss Functions

Theorem 11. Assuming the loss function , the Bayes estimate of the scale parameter , if the shape parameter is known, is of the form

Proof. The risk function of the estimator under the quadratic loss function is given by the formula

Using (24) in (56), we get

On solving (57), we get

Minimization of the risk with respect to gives us the optimal estimator:

Remark 12. By replacing in (59), the same Bayes estimate is obtained as in (29).

Theorem 13. Assuming the loss function , the Bayes estimate of the scale parameter , if the shape parameter is known, is of the form

Proof. The risk function of the estimator under the Al-Bayyati loss function is given by the formula

On substituting (24) in (61), we have

Solving (62), we get

Minimization of the risk with respect to gives us the optimal estimator:

Remark 14. By replacing in (64), the same Bayes estimate is obtained as in (34).

Theorem 15. Assuming the loss function , the Bayes estimate of the scale parameter , if the shape parameter is known, is of the form

Proof. The risk function of the estimator under the entropy loss function is given by the formula

Using (24) in (66), we get

On solving (67), we get

Minimization of the risk with respect to gives us the optimal estimator:

Remark 16. By replacing in (69), the same Bayes estimate is obtained as in (39).

7. Results and Discussion

We primarily studied the classical maximum likelihood estimation and Bayesian estimation for Nakagami distribution using Jeffreys’, extension of Jeffreys’, and Quasi priors under three different symmetric and asymmetric loss functions. Here our main focus was to find out the estimate of scale parameter for Nakagami distribution. The mathematical derivations were checked by using the different data sets and the estimate was obtained.

For descriptive manner, we generate different random samples of size 25, 50, and 100 to represent small, medium, and large data set for the Nakagami distribution in R Software; a simulation study was carried out 3,000 times for each pairs of where and . The values of extension were () and (). The value for the loss parameter was ( and ). This was iterated 2000 times and the estimates of scale parameter for each method were calculated. The results are presented in (Tables 1, 2, and 3), respectively.

Table 1: Estimates by using Jeffreys’ prior under three different loss functions.

25
0.5
1.0
221.9361
205.4964
221.9361
221.9361
264.2096

1.0
1.5
20.05983
19.2883
20.05983
20.05983
21.80416

50
0.5
1.0
354.8246
341.1775
354.8246
354.8246
385.6789

1.0
1.5
49.986
49.00588
49.986
49.986
52.06875

100
0.5
1.0
863.8767
846.938
863.8767
863.8767
899.8716

1.0
1.5
122.1739
120.9643
122.1739
122.1739
124.6672

ML: maximum likelihood, qd: quadratic loss function, ef: entropy loss function, and nl: Al-Bayyati’s new loss function.

Table 2: Estimates by using Extension Jeffreys’ prior under three different loss functions.

25
0.5
1.0
0.5
1.0
221.9361
221.9361
205.4964
191.3242
221.9361
205.4964
221.931
205.494
264.2096
241.2349

1.0
1.5
0.5
1.0
20.05983
20.05983
19.2883
18.57392
20.05983
19.2883
20.05983
19.2883
21.80416
20.89565

50
0.5
1.0
0.5
1.0
354.8246
354.8246
341.1775
328.5413
354.8246
341.1775
354.8246
341.1775
385.6789
369.6089

1.0
1.5
0.5
1.0
49.986
49.986
49.00588
48.06346
49.986
49.00588
49.986
49.00588
52.06875
51.00612

100
0.5
1.0
0.5
1.0
863.8767
863.8767
846.938
830.6507
863.8767
846.938
863.8767
846.938
899.8716
881.5069

1.0
1.5
0.5
1.0
122.1739
122.1739
120.9643
119.7783
122.1739
120.9643
122.1739
120.9643
124.6672
123.408

ML: maximum likelihood, qd: quadratic loss function, ef: entropy loss function, and nl: Al-Bayyati’s new loss function.

Table 3: Estimates by using Quasi prior under three different loss functions.

25
0.5
1.0
1.0
1.5
221.9361
221.9361
205.4964
198.1572
221.9361
213.4001
221.9361
213.4001
264.2096
252.2001

1.0
1.5
1.0
1.5
20.05983
20.05983
19.2883
18.92437
20.05983
19.6665
20.05983
19.6665
21.80416
21.34024

50
0.5
1.0
1.0
1.5
354.8246
354.8246
341.1775
334.7401
354.8246
347.8672
354.8246
347.8672
385.6789
377.4729

1.0
1.5
1.0
1.5
49.986
49.986
49.00588
48.5301
49.986
49.49109
49.986
49.49109
52.06875
51.53196

100
0.5
1.0
1.0
1.5
863.8767
863.8767
846.938
838.7153
863.8767
855.3235
863.8767
855.3235
899.8716
890.5946

1.0
1.5
1.0
1.5
122.1739
122.1739
120.9643
120.3684
122.1739
121.5661
122.1739
121.5661
124.6672
124.0344

ML: maximum likelihood, qd: quadratic loss function, ef: entropy loss function, and nl: Al-Bayyati’s new loss function.

8. Conclusion

In this paper we have generated three types of data sets with different sample sizes for Nakagami distribution. These data sets were simulated with the help of programs and the behavior of the data was checked in case of parameter estimation for Nakagami distribution in R Software. With these data sets we have obtained the estimate of scale parameter for Nakagami distribution under three different symmetric and asymmetric loss functions by using three different priors. With the help of these results we can also do comparison between loss functions and the priors.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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