Journal of Probability and Statistics

Volume 2019, Article ID 7519429, 13 pages

https://doi.org/10.1155/2019/7519429

## Exponentiated Inverse Rayleigh Distribution and an Application to Coating Weights of Iron Sheets Data

Department of Statistics, the University of Dodoma, Dodoma, PO. Box: 338, Tanzania

Correspondence should be addressed to Gadde Srinivasa Rao; moc.oohay@oarseddag

Received 6 December 2018; Revised 19 February 2019; Accepted 17 March 2019; Published 1 April 2019

Academic Editor: Alessandro De Gregorio

Copyright © 2019 Gadde Srinivasa Rao and Sauda Mbwambo. 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

This article aims to introduce a generalization of the inverse Rayleigh distribution known as exponentiated inverse Rayleigh distribution (EIRD) which extends a more flexible distribution for modeling life data. Some statistical properties of the EIRD are investigated, such as mode, quantiles, moments, reliability, and hazard function. We describe different methods of parametric estimations of EIRD discussed by using maximum likelihood estimators, percentile based estimators, least squares estimators, and weighted least squares estimators and compare those estimates using extensive numerical simulations. The performances of the proposed methods of estimation are compared by Monte Carlo simulations for both small and large samples. To illustrate these methods in a practical application, a data analysis of real-world coating weights of iron sheets is obtained from the ALAF industry, Tanzania, during January-March, 2018. ALAF industry uses aluminum-zinc galvanization technology in the coating process. This application identifies the EIRD as a better model than other well-known distributions in modeling lifetime data.

#### 1. Introduction

In this research life time distribution known as exponentiated inverse Rayleigh distribution (EIRD) was developed and it can be used in reliability estimation and statistical quality control techniques. The Rayleigh distribution is originated from two parameter Weibull distribution and it is appropriate model for life-testing studies. It can be shown by transformation of random variable that if the random variable (r. v)* T* has Rayleigh distribution, then the r. v. has an inverse Rayleigh distribution (IRD). Reliability sampling plans of IRD are given in Rosaiah and Kantam [1].

Suppose X is a random variable following inverse Rayleigh distribution with scale parameter . Then its pdf, cdf,and reliability function are respectively given byGeneralization of different distribution was discussed in different statistical writings by different authors, mostly applied in reliability estimation, for example, Mudholkar et al. [2], Gupta et al. [3], Nadarajah and Kotz [4], and Mudholkar and Srivastava [5].

Nadarajah and Kotz [4] suggested a method of generating new distribution by using reliability function. Exponentiated Inverse Rayleigh distribution (EIRD) is a generalized form of inverse Rayleigh distribution as suggested by Nadarajah and Kotz [4] as follows: where R(x) in above equation are the reliability function of inverse Rayleigh distribution.

The cumulative density function (CDF) of the EIRD is given by where is the scale parameter and is the shape parameter. The probability density function (PDF) of EIRD isThe inverse Rayleigh distribution is the particular case of (4) for .

Hence for the exponentiated inverse Rayleigh distribution with the scale parameter and shape parameter will be denoted by EIRD .

Reliability and hazard functions of EIRD are as follows:Graphs of PDF, CDF, Reliability, and hazard function of EIRD are depicted in Figures 1–4. From the diagram of the PDF it can be shown that the distribution is left skewed and the CDF shows the increasing pattern as we expected. Also by using reliability function, it can be seen that the distribution can be used in lifetime studies since reliability graph tends to decrease as the time increases. The hazard function curve shows that the first increases and then decreases in shape. The lifetime models that present first increase and then decrease shaped failure rates are very useful in survival analysis. Moreover, the hazard rate will in general converge to a constant value. For an example the infant mortality rises to some extent over time and then mortality decreases after the infants got immunity in the body. For greater details, readers may refer to Kotz and Nadarajah [6].