Abstract
In this paper, a new three-parameter distribution called the Alpha Logarithm Transformed Fr\'{e}chet (ALTF) distribution is introduced which offers a more flexible distribution for modeling lifetime data. Various properties of the proposed distribution, including explicit expressions for the quantiles, moments, incomplete moments, conditional moments, moment generating function R\'{e}nyi and $\delta$-entropies, stochastic ordering, stress-strength reliability and order statistics are derived. The new distribution can have decreasing, reversed J-shaped and upside-down bathtub failure rate functions depending on its parameter values. The maximum likelihood method is used to estimate the distribution parameters. A simulation study is conducted to evaluate the performance of the maximum likelihood estimates. Finally, the proposed extended model is applied on real data sets and the results are given which illustrate the superior performance of the ALTF distribution compared to some other well-known distributions.
Highlights
The Frechet distribution occupies an important place in describing the lifetime of components and analyzing several extreme events including earthquakes, floods, rain fall, queues in supermarkets, wind speeds and sea waves etc
In this paper we have proposed a new three-parameter family of distributions, so-called the Alpha Logarithm Transformed Frechet (ALTF) distribution
The proposed ALTF model has two shape parameters and one scale parameter. It includes as special sub-models: the one parameter Frechet, two parameter Frechet, inverse exponential and inverse Rayleigh distributions
Summary
The Frechet distribution occupies an important place in describing the lifetime of components and analyzing several extreme events including earthquakes, floods, rain fall, queues in supermarkets, wind speeds and sea waves etc. Frechet distribution provides a reasonable parametric fit for modeling phenomenon with non-monotone failure rates, such as the upside-down bathtub failure rates, which are common in reliability and biological studies. (iii) It is shown in Section 2 that the ALTF distribution can be viewed as a mixture of Frechet distribution; (iv) it can be viewed as a suitable model for fitting skewed data which may not be properly fitted by other common distributions and can be used in a variety of problems in different areas such as public health, biomedical studies and industrial reliability and survival analysis; and (v) The ALTF distribution outperforms several well-known lifetime distributions with respect to two real data examples.
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