Abstract
This paper introduces a new four parameters model called the Weibull Generalized Flexible Weibull extension (WGFWE) distribution which exhibits bathtub-shaped hazard rate. Some of it's statistical properties are obtained including ordinary and incomplete moments, quantile and generating functions, reliability and order statistics. The method of maximum likelihood is used for estimating the model parameters and the observed Fisher's information matrix is derived. We illustrate the usefulness of the proposed model by applications to real data.
Highlights
The Weibull distribution is a highly known distribution due to its utility in modelling lifetime data where the hazard rate function is monotone Weibull (1951)
Statistical Properties we study the statistical properties for the Weibull-G Flexible Weibull Extension (WGFWE) distribution, specially quantile function and simulation median, skewness, kurtosis and moments. 3.1 Quantile and simulation The quantile xq of the WGFWE(a, b, α, β) random variable is given by
Let X1:n, X2:n..., Xn:ndenote the order statistics obtained from a random sample X1, X2, ···, Xn which taken from a continuous population with cumulative distribution function cdf F (x;φ) and probability density function pdf f (x; φ), the probability density function of Xr:n is given by where f (x; φ), F (x;φ)are the pdf and cdf of WGFWE(φ) distribution given by Eq [15] and Eq [14] respectively, φ = (a, b, α, β) and B(., .) is the Beta function, we define first order statistics X1:n = min(X1, X2, · · ·, Xn), and the last order statistics as Xn:n = max(X1, X2, · · ·, Xn)
Summary
The Weibull distribution is a highly known distribution due to its utility in modelling lifetime data where the hazard rate function is monotone Weibull (1951). A random variable X is said to have the Flexible Weibull Extension (FWE) distribution with parameters α, β > 0 if it’s probability density function (pdf) is given by while the cumulative distribution function (cdf) is given by. The survival function is given by the equation and the hazard function is Weibull distribution introduced by Weibull (1951), is a popular distribution for modeling phenomenon with monotonic failure rates. This distribution does not provide a good fit to data sets with bathtub shaped or upside-down bathtub shaped (unimodal) failure rates, often encountered in reliability, engineering and biological studies. If G(x) is the baseline cumulative distribution function (cdf) of a random variable, with probability density function (pdf) g(x) and the Weibull cumulative distribution function is with parameters a and b are positive.
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