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Biometrics & Biostatistics International Journal

Research Article Volume 8 Issue 4

A generalized Rayleigh distribution and its application

Lishamol Tomy,1 Jiju Gillariose2

1Department of Statistics, Deva Matha College, India
2Department of Statistics, St.Thomas College, India

Correspondence: Lishamol Tomy, Department of Statistics, Deva Matha College, Kuravilangad, Kerala-686633, India

Received: May 31, 2019 | Published: July 8, 2019

Citation: Tomy L, Gillariose J. A generalized Rayleigh distribution and its application. Biom Biostat Int J. 2019;8(4):139-143. DOI: 10.15406/bbij.2019.08.00282

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Abstract

This paper studies properties and applications of the generalized Rayleigh-truncated negative binomial distribution established by Jiju & Lishamol.1 For the concerned objective, we applied the model to a real life data set and its performance is compared with that of other four-parameter generalized Rayleigh distributions which are derived using different generators.

Keywords: hazard rate function, marshall-olkin family, maximum likelihood estimation, rayleigh-truncated negative binomial distribution

Introduction

The Rayleigh distribution was introduced by Rayleigh2 and originally proposed in the fields of acoustics and optics. It has emerged as a special case of the Weibull distribution. The Rayleigh distribution has widely used in communication theory to describe hourly median and instantaneous peak power of received radio signals. Moreover, It has received a considerable attention from engineers and physicists for modeling wave propagation, radiation, synthetic aperture radar images, and other related phenomena. There have been many forms for the Rayleigh distribution to provide flexibility for modeling data. Vod3,4 proposed a powerful extension of the Rayleigh distribution and studied its properties. Its probability density function (pdf) is given by

fx,θ,λ2θ2λ2x2λ+1exθ2Γλ+,x,θλ1.   (1)

where θ is the scale parameter, λ is the shape parameter and Γa0ta1etdt is the complete gamma function. Its survival function is

F¯x,θ,λΓλ+xθ2Γλ+γ¯λ+xθ2,x,θλ1.   (2)

where Γa,x0xta1etdt is the incomplete gamma function, hence statistical software"s can be used for various values of and . In literature, there are many studies based on extensions of this generalized Rayleigh (GR) distribution using different generators. Cordeiro et al.,5 derived four-parameter beta-GR distribution, Gomes et al.,6 proposed the four-parameter Kumaraswamy-GR distribution, and MirMostafaee et al.,7 introduced Marshall-Olkin extended GR distribution respectively.

In modern era, the literature has suggested several ways of extending well-known distributions to generate a more flexible of distributions. Recently new generator approach introduced by Nadarajah et al.,8 as a generalization of the family of Marshall-Olkin extended (MOE) distributions by Marshall & Olkin.9 This approach deals with the shape parameter induction in parent (or baseline) distribution to explore tail properties and to improve goodness-of-fits. Let X1,X2,... be a sequence of independent and identically distributed random variables with survival function F¯x and N be a truncated negative binomial random variable, independent of Xi"s, with parameters α<α<1 and β>0 , such that

PNnαβ1αβ(β+n1β1)αnn   (3)

If UNminX1,X2,...XN), then the survival function of UN is

G¯x,α,βαβ1αβ{Fx+αF¯xβ1}xR,α,β   (4)

Similarly, if α>1 and N is a truncated negative binomial random variable with parameters 1α and β>0, then WNmaxX1,X2,...XN) also has the same survival function given in (4). If α1 in (4), then G¯x,α,βF¯x. If β=1, then this family reduces to the Marshall-Olkin family of distributions. The pdf of survival function given in equation (4) is

gx,α,βαβαβfxαβ{Fx+αF¯x}β+1   (5)

Recently, several authors have used this approach to introduce new distributions. Jayakumar & Sankaran10 defined a generalized uniform distribution using the approach of Nadarajah et al.,8 Babu11 introduced Weibull truncated negative binomial distribution. Further, Jayakumar & Sankaran12 introduced generalized exponential truncated negative binomial distribution and studied its properties. Also, Jose & Sivadas13 used the family given by equation (4) to introduce the negative binomial Marshall-Olkin Rayleigh distribution.

The contents of this paper are organized as follows. Section 2 deals with a generalized Rayleigh-truncated negative binomial distribution and its properties. Section 3 gives a real-life application. The concluding remarks are given in Section 4.

Generalized Rayleigh-truncated negative binomial distribution

In this section, we are focused on generalized Rayleigh truncated negative binomial (GR-TNB) distribution introduced by Jiju & Lishamol.1 The distribution is derived by using generator approach of Nadarajah et al.,8 They have examined various statistical properties of this distribution including estimation of parameters and have showed that this distribution is more flexible comparing to other generalizations of the rayleigh distribution. The survival function of the GR-TNB distribution is given by

G¯x,α,β,λ,θαβ1αβ{γλ+xθ2+αγ¯λ+xθ2β1},x,α,β,θλ1   (6)

corresponding pdf is

gx,α,β,λ,θαβαβ2θλ+x2λ+1exθ2Γλ+αβ{γλ+xθ2+αγ¯λ+xθ2}β+1,x,α,β,θλ1   (7)

and the hazard rate function (hrf) of the GR-TNB distribution becomes

hx,α,β,λ,θαβ2θ2λ2x2λ+1exθ2γλ+1,x2θ2+αγ¯λ+1,x2θ21Γλ+γλ+xθ2+αγ¯λ+xθ2β   (8)

The GR-TNB distribution enfolds some sub models such as MOE GR distribution, MOE half normal distribution and MOE Reyleigh distribution. Figure 1 shows the plots for pdf and hrf for GR-TNB distribution with various parameter values. As seen in Figure 1, the pdf and the hrf of the GR-TNB distribution have several different shapes according to the values of the parameters. This shows that GR-TNB distribution is more flexible than Rayleigh distribution. The quantile function of X follows GR-TNB distribution, it can be expressed as

QuθQG[11α[{uαβαβ}1β+1]α]

where u is generated from the uniform (0, 1) distribution and is the (standardized) gamma quantile function with shape parameter and unit scale parameter (Figure 1).

Figure 1 Graphs of pdf and hrf the GR-TNB distribution for different values of α, β, λ and θ.

Simulation study

In this section, we carry out Monte Carlo simulation study to assess the performance of the maximum likelihood estimates (MLE). The results are obtained from generating 1000 samples from the GR-TNB distribution. For each replication, a random sample of size n = 50, 100 and 200 is drawn from the GR-TNB distribution. The GR-TNB random number generation was performed using the quantile function of GR-TNB distribution and the parameters are estimated by using the method of MLE by using package nlm in R, we get MLEs, β^, θ^and λ^ for fixed α=0.2;or α=0.02. The evaluation of the performance is based on the bias and the mean squared errors (MSE) defined as follows:

  1. Average bias of the simulatedestimates of:

1NiNR^iR

  1. Average Mean square error of the simulatedestimates of:

1NiNR^iR2

where is the true value of parameters β, θ and λ and also N is the number of replications. The initial values of parameter are β=1.5,θ=0.1 and λ=0.2. The results of our simulation study are summarized. From this table, we can see that the bias and MSE of the MLEs converge to zero when the sample size is increased (Table 1).

 

α=0.2

α=0.02

n

Parameters

 Bias

 MSE

 Bias

 MSE

n=50

β

0.047

0.2362

0.055

0.99

θ

0.098

0.3756

0.098

0.009

λ

0.018

0.0391

0.092

0.003

n=100

β

0.008

0.0001

0.004

0.099

θ

0.081

0.0098

0.009

0.009

λ

0.005

0.0005

0.039

0.0009

n=200

β

0.004

0.0003

0.001

0.002

θ

0.018

0.009

0.005

0.0001

λ

0.0007

0.00006

0.009

0.000083

Table 1 Simulation Study for GR-TNB(x;α, β, λ, θ) with α=0.2 and α=0.02

Data analysis

In this section, we consider a real data set on breaking stress of carbon fibres of 50 mm length (GPa) to assess the flexibility of the GR-TNB distribution over some well-known generalizations of Rayleigh distribution. The data have been previously used by Nichols & Padgettcit.14 The data are as follows:

0.39, 0.85, 1.08, 1.25, 1.47, 1.57, 1.61, 1.61, 1.69, 1.80, 1.84 ,1.87, 1.89, 2.03, 2.03, 2.05, 2.12, 2.35, 2.41, 2.43, 2.48, 2.50, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.79, 2.81, 2.82, 2.85, 2.87, 2.88, 2.93, 2.95, 2.96, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.56, 3.60, 3.65, 3.68, 3.70, 3.75, 4.20, 4.38, 4.42, 4.70, 4.90

We compare the results of GR-TNB distribution with following four-parameter generalizations of Rayleigh distribution which are generalized by using different generators:

  1. Beta exponential generalized Rayleigh (BExpGR) distribution by Alzaatreh et al.15
  2. Beta extended generalized Rayleigh (BEGR) distribution by Cordeiro et al.16
  3. Exponentiated Kumaraswamy generalized Rayleigh (EKGR) distribution Lemonte et al.17
  4. Extended beta generalized Rayleigh (EBGR) distribution Alexander et al.18

For each distribution, we estimated the unknown parameters (by the maximum likelihood method), the values of the log-likelihood ( logL), AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), the values of the Kolmogorov-Smirnov (K-S) statistic and the corresponding p-values. The result of comparison of the proposed distributions for these data. From these results we can see that GR-TNB distribution provide smallest logL, AIC, BIC and K-S statistics values and highest p-value as compare to other distributions. This strongly claims that the proposed GR-TNB model provides better fit to the concerned data than the other distributions. Therefore, the GR-TNB distribution is an attractive alternative to the other available four-parameter generalized Rayleigh models in the literature (Table 2).

Distribution               

 Estimates (standard error)

-logL

 AIC

 BIC

 K-S

 p-value

GR-TNB (α,β,θ,λ)

 α^= 9.2904333

 84.77286


 177.5457


 186.3043


 0.06376


 0.9513


(17.383550)

 β^= 1.2821518

(1.402232)

θ^= 1.8392590

 (0.259968)

λ^= -0.1287885

 (0.796702)

BExpGR (α,β,θ,λ)

 α^=2.3214738

 88.25908


 184.5182


 193.2768


 0.2121157


 0.00526888


(0.411143989)

 β^= 1.4398055

(0.828164618)

r^= 0.0980958

(0.047475514)

s^= 1.4465704

(0.005754235)

BEGR(α,β,θ,λ)

 α^=2.410055279

 88.47258


 184.9452


 193.7038


 0.1154599


 0.3424505


(0.405482144)

 β^= 6.3950288

(9.68416177)

r^= 4.833964992

(6.914385767)

s^= 0.008840908

(0.004205926)

EKGR(α,β,θ,λ)

 α^= 0.009924694

 85.75583


179.5117


 188.2703


 0.08173829


 0.7699466


(0.02485053)

 β^= 0.205804483

(0.02541743 )

r^= 9.787248978

(7.00657360 )

s^= 1.126722092

(0.04244422)

EBGR (α,β,θ,λ)

 α^= 0.45962398

 188.5795


179.8209


 188.5795


 0.0810696


 0.7785533


(0.47172504)

 β^= 11.07897095

(22.78252971)

r^= 3.5728010

(3.7261659)

s^= 0.05118239

(0.07335831)

Table 2 Estimated values, logL, AIC, BIC, K-S statistics and-value for data set

Conclusions

 In the last two decades, generalization approaches were adopted and practiced by many statisticians. In this study, we concentrated on such a generalization of Rayleigh distribution and presented a simulation study for verifying the validity of its estimates. A data set is used to prove the performance of GR-TNB distribution. The results present that the GR-TNB distribution provides better fits than existing distributions.

Acknowledgments

None.

Conflicts of interest

The second author is grateful to the Department of Science and Technology (DST), Govt. of India for the financial support under the INSPIRE Fellowship.

References

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