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

Authors: Kaisar Ahmad; S. P. Ahmad; A. Ahmed
Journal: Journal of Probability and Statistics
 (2016)

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## 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.


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## Body

## 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 Figure1:(1)fy;θ,k=2kkΓkθky2k-1exp⁡-ky2θfory>0,k,θ>0,where θ and k are the scale and the shape parameters, respectively.Figure 1
The pdf’s of Nakagami distribution under various values ofk 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(y1,y2,…,yn) be a random sample of size n having pdf (1); then the maximum likelihood estimator of scale parameter θ, when the shape parameter k is known, is given by(2)θ^=∑i=1nyi2n.Proof.
The likelihood function of the pdf (1) is given by(3)Ly;θ,k=2kknΓknθnk∏i=1nyi2k-1exp⁡-kθ∑i=1nyi2.The log likelihood function is given by(4)log⁡Ly;θ,k=nlog⁡2kk-nlog⁡Γk-nklog⁡θ+2k-1∑i=1nlog⁡yi-kθ∑i=1nyi2.Differentiating (4) with respect to θ and equating to zero, we get(5)θ^=∑i=1nyi2n.


### 2.1. Loss Functions Used in This Paper

(i) The quadratic loss function which is given by(6)Lqdθ^,θ=θ^-θθ2;θ>0,which is a symmetric loss function; θ and θ^ represent the true and estimated values of the parameter.(ii) The Al-Bayyati new loss function is of the form(7)Lnlθ^,θ=θc1θ^-θ2;c1εR,which is an asymmetric loss function; θ and θ^ represent the true and estimated values of the parameter.(iii) The entropy loss function is given by(8)Lefθ^,θ=θ^θ-log⁡θ^θ-1;θ>0,where θ and θ^ represent the true and estimated values of the parameter.


## 2.1. Loss Functions Used in This Paper

(i) The quadratic loss function which is given by(6)Lqdθ^,θ=θ^-θθ2;θ>0,which is a symmetric loss function; θ and θ^ represent the true and estimated values of the parameter.(ii) The Al-Bayyati new loss function is of the form(7)Lnlθ^,θ=θc1θ^-θ2;c1εR,which is an asymmetric loss function; θ and θ^ represent the true and estimated values of the parameter.(iii) The entropy loss function is given by(8)Lefθ^,θ=θ^θ-log⁡θ^θ-1;θ>0,where θ and θ^ 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(y1,y2,…,yn) be a random sample of size n having the probability density function (1) and the likelihood function (2).Jeffreys’ prior forθ is given by(9)gθ=1θ;θ>0.By using the Bayes theorem, we have(10)π1θ∣y_∝Ly∣θgθ.Using (2) and (9) in (10),(11)π1θ∣y_∝2kknΓknθnk+1∏i=1nyi2k-1exp⁡-kθ∑i=1nyi2,π1θ∣y_=ρ1θnk+1exp⁡-kθ∑i=1nyi2,where ρ is independent of θ and(12)ρ=k∑i=1nyi2nkΓnk.Using the value of ρ in (11),(13)π1θ∣y_=k∑i=1nyi2nkΓnk1θnk+1exp⁡-kθ∑i=1nyi2.


### 3.2. Posterior Density under Extension of Jeffreys’ Prior

Let(y1,y2,…,yn) be a random sample of size n having the probability density function (1) and the likelihood function (2).The extension of Jeffreys’ forθ is given by(14)g1θ=1θ2c;θ>0.By using the Bayes theorem, we have(15)π2θ∣y_∝Ly∣θg1θ.Using (2) and (14) in (15),(16)π2θ∣y_∝2kknΓknθnk+2c∏i=1nyi2k-1exp⁡-kθ∑i=1nyi2.Thus(17)π2θ∣y_=ρ1θnk+2cexp⁡-kθ∑i=1nyi2,(18)ρ=k∑i=1nyink+2c-1Γnk+2c-1.By using the value of ρ in (17), we have(19)π2θ∣y_=-k/θ∑i=1nyi2nk+2c-1Γnk+2c-11θnk+2cexp⁡-kθ∑i=1nyi2.


### 3.3. Posterior Density under Quasi Prior

Let(y1,y2,…,yn) be a random sample of size n having the probability density function (1) and the likelihood function (2).Quasi prior forθ is given by(20)g2θ=1θd;θ>0,d>0.By using the Bayes theorem, we have(21)π3θ∣y_∝Ly∣θg2θ.Using (2) and (20) in (21),(22)π3θ∣y_∝2kknΓknθnk+d∏i=1nyi2k-1exp⁡-kθ∑i=1nyi2π3θ∣y_=ρ1θnk+dexp⁡-kθ∑i=1nyi2,where ρ is independent of θ and(23)ρ=k∑i=1nyi2nk+d-1Γnk+d-1.Using the value of ρ in (22),(24)π3θ∣y_=k∑i=1nyi2nk+d-1Γnk+d-11θnk+dexp⁡-kθ∑i=1nyi2.


## 3.1. Posterior Density under Jeffreys’ Prior

Let(y1,y2,…,yn) be a random sample of size n having the probability density function (1) and the likelihood function (2).Jeffreys’ prior forθ is given by(9)gθ=1θ;θ>0.By using the Bayes theorem, we have(10)π1θ∣y_∝Ly∣θgθ.Using (2) and (9) in (10),(11)π1θ∣y_∝2kknΓknθnk+1∏i=1nyi2k-1exp⁡-kθ∑i=1nyi2,π1θ∣y_=ρ1θnk+1exp⁡-kθ∑i=1nyi2,where ρ is independent of θ and(12)ρ=k∑i=1nyi2nkΓnk.Using the value of ρ in (11),(13)π1θ∣y_=k∑i=1nyi2nkΓnk1θnk+1exp⁡-kθ∑i=1nyi2.


## 3.2. Posterior Density under Extension of Jeffreys’ Prior

Let(y1,y2,…,yn) be a random sample of size n having the probability density function (1) and the likelihood function (2).The extension of Jeffreys’ forθ is given by(14)g1θ=1θ2c;θ>0.By using the Bayes theorem, we have(15)π2θ∣y_∝Ly∣θg1θ.Using (2) and (14) in (15),(16)π2θ∣y_∝2kknΓknθnk+2c∏i=1nyi2k-1exp⁡-kθ∑i=1nyi2.Thus(17)π2θ∣y_=ρ1θnk+2cexp⁡-kθ∑i=1nyi2,(18)ρ=k∑i=1nyink+2c-1Γnk+2c-1.By using the value of ρ in (17), we have(19)π2θ∣y_=-k/θ∑i=1nyi2nk+2c-1Γnk+2c-11θnk+2cexp⁡-kθ∑i=1nyi2.


## 3.3. Posterior Density under Quasi Prior

Let(y1,y2,…,yn) be a random sample of size n having the probability density function (1) and the likelihood function (2).Quasi prior forθ is given by(20)g2θ=1θd;θ>0,d>0.By using the Bayes theorem, we have(21)π3θ∣y_∝Ly∣θg2θ.Using (2) and (20) in (21),(22)π3θ∣y_∝2kknΓknθnk+d∏i=1nyi2k-1exp⁡-kθ∑i=1nyi2π3θ∣y_=ρ1θnk+dexp⁡-kθ∑i=1nyi2,where ρ is independent of θ and(23)ρ=k∑i=1nyi2nk+d-1Γnk+d-1.Using the value of ρ in (22),(24)π3θ∣y_=k∑i=1nyi2nk+d-1Γnk+d-11θnk+dexp⁡-kθ∑i=1nyi2.


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

Theorem 2.
Assuming the loss functionLqdθ^,θ, the Bayes estimate of the scale parameter θ, if the shape parameter k is known, is of the form(25)θ^qd=k∑i=1nyi2nk+1.Proof.
The risk function of the estimatorθ under the quadratic loss function Lqdθ^,θ is given by the formula (26)Rθ^=∫0∞θ^-θθ2π1θ∣y_dθ.Using (13) in (26), we get(27)Rθ^=∫0∞θ^-θθ2k∑i=1nyi2nkΓnk1θnk+1exp⁡-kθ∑i=1nyi2dθ.On solving (27), we get(28)Rθ^=θ^2Γnk+2Γnkk∑i=1nyi22-2θ^Γnk+1Γnkk∑i=1nyi2+1.Minimization of the risk with respect to θ^ gives us the optimal estimator:(29)θ^qd=k∑i=1nyi2nk+1.Theorem 3.
Assuming the loss functionLnlθ^,θ, the Bayes estimate of the scale parameter θ, if the shape parameter k is known, is of the form(30)θ^nl=k∑i=1nyi2nk-c1-1.Proof.
The risk function of the estimatorθ under the Al-Bayyati loss function Lnlθ^,θ is given by the formula (31)Rθ^=∫0∞θc1θ^-θ2π1θ∣y_dθ.On substituting (13) in (31), we have(32)Rθ^=∫0∞θc1θ^-θ2k∑i=1nyi2nkΓnk1θnk+1exp⁡-kθ∑i=1nyi2dθ.Solving (32), we get(33)Rθ^=θ^2k∑i=1nyi2c1Γnk-c1Γnk+k∑i=1nyi2c1+2Γnk-c1-2Γnk-2θ^k∑i=1nyi2c1+1Γnk-c1-1Γnk.Minimization of the risk with respect to θ^ gives us the optimal estimator:(34)θ^nl=k∑i=1nyi2nk-c1-1.Theorem 4.
Assuming the loss functionLefθ^,θ, the Bayes estimate of the scale parameter θ, if the shape parameter k is known, is of the form(35)θ^ef=k∑i=1nyi2nk.Proof.
The risk function of the estimatorθ under entropy loss function Lefθ^,θ is given by the formula (36)Rθ^=∫0∞θ^θ-log⁡θ^θ-1π1θ∣y_dθ.Using (13) in (36), we get(37)Rθ^=∫0∞θ^θ-log⁡θ^θ-1k∑i=1nyi2nkΓnk1θnk+1exp⁡-kθ∑i=1nyi2dθ.On solving (37), we get(38)Rθ^=θ^Γnk+1Γnkk∑i=1nyi2-log⁡θ^+hθ-1.Minimization of the risk with respect to θ^ gives us the optimal estimator:(39)θ^ef=k∑i=1nyi2nk.


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

Theorem 5.
Assuming the loss functionLqdθ^,θ, the Bayes estimate of the scale parameter θ, if the shape parameter k is known, is of the form(40)θ^qd=k∑i=1nyi2nk+2c.Proof.
The risk function of the estimatorθ under the quadratic loss function Lqdθ^,θ is given by the formula (41)Rθ^=∫0∞θ^-θθ2π2θ∣y_dθ.Using (19) in (41), we get(42)Rθ^=∫0∞θ^-θθ2k∑i=1nyi2nk+2c-1Γnk+2c-11θnk+2cexp⁡-kθ∑i=1nyi2dθ.On solving (42), we get(43)Rθ^=θ^2Γnk+2c+1Γnkk∑i=1nyi22-2θ^Γnk+2cΓnkk∑i=1nyi2+1.Minimization of the risk with respect to θ^ gives us the optimal estimator:(44)θ^qd=k∑i=1nyi2nk+2c.Remark 6.
By replacingc=1/2 in (44), the same Bayes estimate is obtained as in (29).Theorem 7.
Assuming the loss functionLnlθ^,θ, the Bayes estimate of the scale parameter θ, if the shape parameter k is known, is of the form(45)θ^nl=k∑i=1nyi2nk+2c-c1-2.Proof.
The risk function of the estimatorθ under the Al-Bayyati loss function Lnlθ^,θ is given by the formula (46)Rθ^=∫0∞θc1θ^-θ2π2θ∣y_dθ.On substituting (19) in (46), we have(47)Rθ^=∫0∞θc1θ^-θ2k∑i=1nyi2nkΓnk1θnk+2cexp⁡-kθ∑i=1nyi2dθ.Solving (47), we get(48)Rθ^=θ^2k∑i=1nyi2c1Γnk+2c-c1-1Γnk+2c-1+k∑i=1nyi2c1+2Γnk+2c-c1-3Γnk+2c-1-2θ^k∑i=1nyi2c1+1Γnk+2c-c1-2Γnk+2c-1.Minimization of the risk with respect to θ^ gives us the optimal estimator:(49)θ^nl=k∑i=1nyi2nk+2c-c1-2.Remark 8.
By replacingc=1/2 in (49), the same Bayes estimate is obtained as in (34).Theorem 9.
Assuming the loss functionLefθ^,θ, the Bayes estimate of the scale parameter θ, if the shape parameter k is known, is of the form(50)θ^ef=k∑i=1nyi2nk+2c-1.Proof.
The risk function of the estimatorθ under entropy loss function Lefθ^,θ is given by the formula (51)Rθ^=∫0∞θ^θ-log⁡θ^θ-1π2θ∣y_dθ.Using (19) in (51), we get(52)Rθ^=∫0∞θ^θ-log⁡θ^θ-1k∑i=1nyi2nk+2c-1Γnk+2c-11θnk+2cexp⁡-kθ∑i=1nyi2dθ.On solving (52), we get(53)Rθ^=θ^Γnk+2cΓnk+2c-1k∑i=1nyi2-log⁡θ^+hθ-1.Minimization of the risk with respect to θ^ gives us the optimal estimator:(54)θ^ef=k∑i=1nyi2nk+2c-1.Remark 10.
By replacingc=1/2 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 functionLqdθ^,θ, the Bayes estimate of the scale parameter θ, if the shape parameter k is known, is of the form(55)θ^qd=k∑i=1nyi2nk+d.Proof.
The risk function of the estimatorθ under the quadratic loss function Lqdθ^,θ is given by the formula (56)Rθ^=∫0∞θ^-θθ2π3θ∣y_dθ.Using (24) in (56), we get(57)Rθ^=∫0∞θ^-θθ2k∑i=1nyi2nk+dΓnk+d-11θnk+dexp⁡-kθ∑i=1nyi2dθ.On solving (57), we get(58)Rθ^=θ^2Γnk+d+1Γnkk∑i=1nyi22-2θ^Γnk+dΓnkk∑i=1nyi2+1.Minimization of the risk with respect to θ^ gives us the optimal estimator:(59)θ^qd=k∑i=1nyi2nk+d.Remark 12.
By replacingd=1 in (59), the same Bayes estimate is obtained as in (29).Theorem 13.
Assuming the loss functionLnlθ^,θ, the Bayes estimate of the scale parameter θ, if the shape parameter k is known, is of the form(60)θ^nl=∑i=1nyink+d-c1-2.Proof.
The risk function of the estimatorθ under the Al-Bayyati loss function Lnlθ^,θ is given by the formula (61)Rθ^=∫0∞θc1θ^-θ2π3θ∣y_dθ.On substituting (24) in (61), we have(62)Rθ^=∫0∞θc1θ^-θ2k∑i=1nyi2nk+d-1Γnk+d-11θnk+dexp⁡-kθ∑i=1nyi2dθ.Solving (62), we get(63)Rθ^=θ^2k∑i=1nyi2c1Γnk+d-c1-1Γnk+d-1+k∑i=1nyi2c1+2Γnk+d-c1-3Γnk+d-1-2θ^k∑i=1nyi2c1+1Γnk+d-c1-2Γnk+d-1.Minimization of the risk with respect to θ^ gives us the optimal estimator:(64)θ^nl=k∑i=1nyi2nk+d-c1-2.Remark 14.
By replacingd=1 in (64), the same Bayes estimate is obtained as in (34).Theorem 15.
Assuming the loss functionLefθ^,θ, the Bayes estimate of the scale parameter θ, if the shape parameter k is known, is of the form(65)θ^ef=k∑i=1nyi2nk+d-1.Proof.
The risk function of the estimatorθ under the entropy loss function Lefθ^,θ is given by the formula (66)Rθ^=∫0∞θ^θ-log⁡θ^θ-1π3θ∣y_dθ.Using (24) in (66), we get(67)Rθ^=∫0∞θ^θ-log⁡θ^θ-1k∑i=1nyi2nk+d-1Γnk+d-11θnk+dexp⁡-kθ∑i=1nyi2dθ.On solving (67), we get(68)Rθ^=θ^Γnk+dΓnk+d-1k∑i=1nyi2-log⁡θ^+hθ-1.Minimization of the risk with respect to θ^ gives us the optimal estimator:(69)θ^ef=k∑i=1nyi2nk+d-1.Remark 16.
By replacingd=1 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 ofθ,k where k=0.5,1.0 and θ=1.0,1.5. The values of extension were (C=0.5,1.0) and (d=1.0,1.5). The value for the loss parameter was (C1=-1 and 1). 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.
n
k
θ
θ
M
L
θ
q
d
θ
e
f
θ
n
l
C
1
=
-
1
C
1
=
1
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.
n
k
θ
C
θ
M
L
θ
q
d
θ
e
f
θ
n
l
C
1
=
-
1.0
C
1
=
1.0
25
0.5
1.0
0.51.0
221.9361221.9361
205.4964191.3242
221.9361205.4964
221.931205.494
264.2096241.2349
1.0
1.5
0.51.0
20.0598320.05983
19.288318.57392
20.0598319.2883
20.0598319.2883
21.8041620.89565
50
0.5
1.0
0.51.0
354.8246354.8246
341.1775328.5413
354.8246341.1775
354.8246341.1775
385.6789369.6089
1.0
1.5
0.51.0
49.98649.986
49.0058848.06346
49.98649.00588
49.98649.00588
52.0687551.00612
100
0.5
1.0
0.51.0
863.8767863.8767
846.938830.6507
863.8767846.938
863.8767846.938
899.8716881.5069
1.0
1.5
0.51.0
122.1739122.1739
120.9643119.7783
122.1739120.9643
122.1739120.9643
124.6672123.408
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.
n
k
θ
d
θ
M
L
θ
q
d
θ
e
f
θ
n
l
C
1
=
-
1
C
1
=
1.0
25
0.5
1.0
1.01.5
221.9361221.9361
205.4964198.1572
221.9361213.4001
221.9361213.4001
264.2096252.2001
1.0
1.5
1.01.5
20.0598320.05983
19.288318.92437
20.0598319.6665
20.0598319.6665
21.8041621.34024
50
0.5
1.0
1.01.5
354.8246354.8246
341.1775334.7401
354.8246347.8672
354.8246347.8672
385.6789377.4729
1.0
1.5
1.01.5
49.98649.986
49.0058848.5301
49.98649.49109
49.98649.49109
52.0687551.53196
100
0.5
1.0
1.01.5
863.8767863.8767
846.938838.7153
863.8767855.3235
863.8767855.3235
899.8716890.5946
1.0
1.5
1.01.5
122.1739122.1739
120.9643120.3684
122.1739121.5661
122.1739121.5661
124.6672124.0344
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.



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