Abstract

This article addresses the various properties and different methods of estimation of the unknown parameters of the Transmuted Rayleigh (TR) distribution from the frequentist point of view. Although, our main focus is on estimation from frequentist point of view, yet, various mathematical and statistical properties of the TR distribution (such as quantiles, moments, moment generating function, conditional moments, hazard rate, mean residual lifetime, mean past lifetime, mean deviation about mean and median, the stochastic ordering, various entropies, stress-strength parameter and order statistics) are derived. We briefly describe different frequentist methods of estimation approaches, namely, maximum likelihood estimators, moments estimators, L-moment estimators, percentile based estimators, least squares estimators, method of maximum product of spacings, method of Cram\'er-von-Mises, methods of Anderson-Darling and right-tail Anderson-Darling and compare them using extensive numerical simulations. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. Finally, the potentiality of the model is analyzed by means of two real data sets which is further illustrated by obtaining bias and standard error of the estimates and the bootstrap percentile confidence intervals using bootstrap resampling.

Highlights

  • Rayleigh distribution was introduced by Rayleigh (1880) and relates to a problem in the field of acoustics

  • In terms of performance of the methods of estimation, we found that maximum product spacing (MPS)estimators is the best method as it produces the least estimate biases with the least RMSE for most of the configurations considered in our studies

  • We present fit statistics of Rayleigh, Weibull, Gamma, and transmuted Rayleigh distribution in Table 13. these show that the transmuted Rayleigh distribution best fits the data ans has the largest log−likelihood (−105.39), the smallest AIC (214.77), and the smallest BIC (219.99) among all four candidate distributions

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Summary

Introduction

Rayleigh distribution was introduced by Rayleigh (1880) and relates to a problem in the field of acoustics. The transmuted Rayleigh (TR) distribution was introduced by Merovci (2013c) He only studied moments and the maximum likelihood estimation of the unknown parameters. This distribution has a unimodal pdf and increasing hazard rates. The principal motivation of the paper is two fold: one is empirical and shows that the studied transmuted Rayleigh distribution outperforms at least two twoparameter distributions with respect to a real data set; the other is to show how different frequentist estimators of this distribution perform for different sample sizes and different parameter values, and to develop a guideline to choose the best estimation method for the transmuted Rayleigh distribution, which we think would be of interest to applied statisticians.

Statistical and Mathematical Properties
Moment Generating Function
Stochastic Ordering
Hazard Function
Mean Residual Life Function
Mean Deviation
Entropies
Order Statistics
2.11. Stress Strength Parameter
Method of Maximum Likelihood Estimation
Method of Least-Square Estimation
Method of Percentile Estimation
Method of L-Moments Estimation
Method of Maximum Product of Spacings
Method of Cramér-Von-Mises
Methods of Anderson-Darling and Right-tail Anderson-Darling
Simulation
Real Data Analysis
Example 1
Finding a suitable distribution for guinea pig data
Example 2
Finding a suitable distribution for fibre data
Conclusion
Full Text
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