Exponentiated New Modified Weighted Rayleigh Distribution with Application

Badmus, Nofiu Idowu^1* , Faweya, Olanrewaju²

^1*University of Lagos, Faculty of Science, Department of Statistics, Akoka, Lagos, Nigeria

²Ekiti State University, Faculty of Science, Department of Statistics, Ekiti State, Nigeria

Corresponding Author Email: nibadmus@unilag.edu.ng

DOI : http://dx.doi.org/10.46890/SL.2022.v03i05.001

Abstract

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INTRODUCTION

In Statistics, Rayleigh distribution is one of the continuous and important distributions which can be applied to many areas like: biology, engineering, health, agriculture and other related sciences. It has two principal parameters namely scale and shape (  and  in density function of Rayleigh( ), where  and emanated from Gaussian distribution. While, weighted distributions are useful to adjust the probabilities of the events as observed and recorded Ramadan (2013). Weighted distributions have been in existence for decades for instance, Patil and Rao (1977) developed it and used for survey of their application, Mahfoud and Patil (1982) worked and written essay in honour of C. R. Rao while, Azzalini (1985) expanded his research on a class of distributions which includes the normal ones.

Some years later in literature, it was discovered that many researchers have done tremendous studies on the extension of Azzalini’s work to various levels such as weighted Weibull, modified weighted Weibull and beta weighted Weibull includes Shahbaz et al. (2010), Ramadan (2013), Aleem et al. (2013), Badmus and Ikegwu (2013) and Nasiru (2015). Recently, Badmus and Bamiduro (2014) deeply examined on exponentiated weighted Weibull that has better representation than weighted Weibull and Weibull itself. Oguntunde (2015) proposed exponentiated weighted exponential distribution and studied some of its basic statistical properties. Elsherpieny et al. (2017) and Umar et al. (2018) developed exponentiated new weighted Weibull distributions with applications.

There are two motivational factor for the study: the proposed model is better than some of the existing models in literature and can be fitted to various real life data sets since Rayleigh distribution is useful in several areas like: health, biology, Agriculture and other related sciences. By and large, the main aim of the work is to propose a distribution with high flexibility and wide range applicability from existing distribution in literature. Finally, the rest of the article is organized as follows: In section 2, modified new weighted Rayleigh is defined. Exponentiated new modified weighted Rayleigh (ENMWR) distribution is defined and discussed in section 3. In section 4, various properties are derive. The proposed model is illustrated using a real data set to show its flexibility in section 5 and we conclude in section 6.

MATERIAL AND METHODS

The New Modified weighted Rayleigh (NMWR) Distribution

This section presents the density, distribution, reliability and hazard function of the NMWR distribution generated from a new weighted Weibull (NWW) distribution investigated by Nasiru (2015). The modification is based on the definition given in weighted Weibull and modified weighted Weibull distribution (i.e for any  and , this implies that both the variable and the parameters can take any integer value greater than zero) Mahdy and Aleem et al. (2013). Badmus et al. (2016) introduced a generalized modified weighted Weibull (GMWW) distribution and Badmus et al. (2017) upgraded modified weighted Weibull (MWW) distribution to a beta modified weighted Rayleigh (BMWR) distribution where two more additional shape parameters were introduced for flexibility of the model.

Suppose  is the distribution function (cdf),  is the density function (pdf),  is the reliability function and  is the hazard function exist as the MNWR distribution is defined as

for any  and  Mahdy (2013), Aleem et al. (2013), and Elsherpieny et al. (2017)

                                        (1)

Then, the associating pdf is given by

                                   (2)

 and  Where and are scale and shape parameters. Hence, the reliability function is given by

                             (3)

and the hazard function of the NMWR is given by

                                     (4)

 and

THE EXPONENTIATED NEW MODIFIED WEIGHTED RAYLEIGH (ENMWR) DISTRIBUTION

The pdf of ENMWR distribution is derived using (1) and (2) to obtain (5) as follows:

 and                                                   (5)

where, c is an additional shape parameter. Meanwhile, the associating cdf of the ENMWR is given by

 and                                                   (6)

Therefore, it can be noticed that taking the limit in (6) when  and  as

and

Then,
 

, ;

 and                                                   (7)

Some Known and New Reduced Distributions Emanate from the Proposed Distribution

There are some known distributions in literature and new distributions emanate as special cases from ENMWR distribution such as: Modified weighted Rayleigh (MWR) distribution when  ((Aleem et al. (2013) and Badmus et al. (2017)), Modified weighted Exponential (MWE) distribution when  ((Aleem et al. (2013) and Badmus et al. (2016)), Modified Exponentiated Weibull Rayleigh (MEWR) distribution when  New and Modified Exponentiated weighted Rayleigh (MEWR) distribution when  New

 and                                                   (8)

Quantile Function and Second Quantile (2nd Qut)

The quantile function of the ENMWR distribution is denoted by  and is given by

. Therefore, using the expression (6) above, we get

                                                    (9)

Then, second quantile can be obtained by setting  in (9) and is given (10) below

                                                       (10)

Moment and Generating Function

If a random variable (r.v) K has the ENMWR distribution and the r-th non moment is given by

                                  (11)

Elsherpiency et al. (2017). Also, by letting  we have the following:

Similarly, the

Skewness (SC) and Kurtosis Coefficient (KC)

The measures of skewness and kurtosis coefficient are given by

and the kurtosis

Order Statistics (OS)

Suppose the R.Vs  are called the OS of the sample  with pdf of the  order statistics, is given as:

 for      (12)

Furthermore, the density function of the  OS is written as:

                                        (13)

Again, the pdf for both smallest and largest OS  and  are given by

  (14)

and

         (15)

Maximum Likelihood Estimation (MLE)

Here, we are able to established the estimation of the parameters of the proposed distribution with the help of the method of MLEs. Moreover, suppose a random sample of size n consists  from the ENMWR distribution, then

hence, the likelihood function of the density function above is given by

                   (16)

By taking the logarithm in (16), we obtain

                                                      (17)

Also, taking first and second order partial derivatives with respect to the parameters  and equating to 0 as follows:

and

Fisher Information (FI) Matrix Derive on  and

FI matrix is an instrument/tool to prepare for variances. In this work, the FI matrix used is given by 

The variances of , i.e var  provides from FI matrix based on  is

where,  stands for sample size,  the integral for partial derivatives of  taken over pdf , parameters  and  in the subscript of E are the second partial derivatives. Villa and Cankaya (2020), pp 24.

RESULTS AND DISCUSSION

Application

Here, we validate the application of the exponentiated new modified weighted Rayleigh distribution with lifetime data of 20 electronic components studied which used by Elsherpieny et al. (2017), page 50 (four-parameter exponentiated new weighted Weibull distribution).  Furthermore, several known distributions are fitted to this data as earlier mentioned above and the results compared in Table 1 below:

 Discussion and Conclusion

In this study, we were able to modified the existing four parameters (the exponentiated new weighted Weibull distribution) by equating one of the parameters i.e (  to became exponentiated new modified weighted Rayleigh distribution. In addition, we also examined on the statistical properties of the MENWR distribution for instance, quantile, moments and generating function, order statistics, estimation of parameters and model selection criteria such as AIC, BIC, CAIC, HQIC and their p-values using R-programming for data analysis presented in this paper.

Table 1: Estimates, Standard errors (.), and Statistics

Models	Estimates
ENMWR (Proposed)	0.45390 (0.11769) 0.96411 (12.14493) 2 0.05232 (0.63478)	0.04163	0.25540	0.11743 0.91580P
EMWR (New)	1.06811 (0.23884) 1.52934e-05 (0.07317) 2 1	0.02244	0.17203	0.50670 2.779e-05P
EME (New)	0.45394 (0.11770) 0 2 0.10095 (0.03760)	0.04162	0.25540	0.11745 0.91590P
MWR (Developed by Aleem et al. 2013, Mahdy 2013 and Badmus et al. 2017)	1 1.00893 (13.76033) 2 0.08686 (1.19508)	0.04065	0.25103	0.19648 0.37420P
ENWW (Developed by Elsherpieny et al. (2017))	0.23624 (0.14121) 0.00229 (55.16131) 3.41971 (1.59281) 0.00932 (0.02500)	0.03223	0.21390	0.10319 0.96860P

Table 2: Model Performance

	Model Selection Criteria	Ranks
Model
ENMWR	31.80722	67.61443	68.32031	69.60589	68.00318	1st
EMWR	96.75673	197.5135	198.2193	199.5049	197.9022	5th
EME	31.80722	69.61443	71.11443	72.60163	70.19756	2nd
MWR	37.51925	79.03849	79.74437	81.02996	79.42725	4th
ENWW	31.47063	70.94125	73.60792	74.92418	71.71876	3rd

CONCLUSION

However, results from the estimated parameters, goodness of fit and model performance in Tables 1 and 2 above show that the exponentiated new modified Weighted Rayleigh distribution has appeared to be superior than other distributions considered here.

ACKNOWLEDGMENTS

The authors gratefully acknowledge those who helped in proof-reading and with their valuable comments to the manuscript a success.

AUTHOR’S CONTRIBUTION

This work was carried out in collaboration between both authors. Badmus initiated the study, performed the statistical analysis, wrote the protocol and first draft of the manuscript. Authors Badmus and Faweya searched for the literatures used and did the analyses of the study. Then, both authors read, revised and approved the final manuscript submitted for publication.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest regarding the publication of this manuscript.

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