Abstract
In this paper, a new three-parameter lifetime model called the Topp-Leone odd log-logistic exponential distribution is proposed. Its density function can be expressed as a linear mixture of exponentiated exponential densities and can be reversed-J shaped, skewed to the left and to the right. Further, the hazard rate function of the new model can be monotone, unimodal, constant, J-shaped, constant-increasing-decreasing and decreasing-increasing-decreasing and bathtub-shaped. Our main focus is on estimation from a frequentist point of view, yet, some statistical and reliability characteristics for the proposed model are derived. We briefly describe different estimators namely, the maximum likelihood estimators, ordinary least-squares estimators, weighted least-squares estimators, percentile estimators, maximum product of spacings estimators, Cramér-von-Mises minimum distance estimators, Anderson-Darling estimators and right-tail Anderson-Darling estimators. Monte Carlo simulations are performed to compare the performance of the proposed methods of estimation for both small and large samples. We illustrate the performance of the proposed distribution by means of two real data sets and both the data sets show the new distribution is more appropriate as compared to some other well-known distributions.
Highlights
Statisticians have been interested in defining new classes of univariate distributions by adding one or more shape parameters to a baseline model to generate new extended distributions that provide greater flexibility in modeling real data in many applied fields
We have introduced a new three-parameter lifetime model, called Topp-Leone odd log-logistic exponential (TLOLLEx) distribution, which extends the exponential (Ex) distribution
The Topp–Leone odd log-logistic exponential (TLOLLEx) probability density function (PDF) can be expressed as a linear mixture of Ex densities
Summary
Statisticians have been interested in defining new classes of univariate distributions by adding one or more shape parameters to a baseline model to generate new extended distributions that provide greater flexibility in modeling real data in many applied fields. We estimate the unknown parameters of the TLOLLEx distribution using eight frequentist estimators These estimators are: the maximum likelihood estimators, least squares and weighted least-squares estimators, percentile based estimators, the maximum product of spacing estimators, Cramér–von Mises estimators, Anderson–Darling and Right-tail Anderson–Darling estimators. The MLE of the unknown parameters a, b and λ of the TLOLLEx distribution can be obtained by maximizing the above equation. The MLE of a and λ are denoted by a and λ, respectively These estimates can be obtained by numerically by solving the following non-linear equations. The PCE of the parameters of TLOLLEx distribution can be obtained by minimizing the following function.
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