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The Reflected Weibull Distribution

A. Clifford Cohen
Technometrics
Vol. 15, No. 4 (Nov., 1973), pp. 867-873
DOI: 10.2307/1267396
Stable URL: http://www.jstor.org/stable/1267396
Page Count: 7
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The Reflected Weibull Distribution
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Abstract

For combinations of parameter values ordinarily encountered in life testing and reliability data, the similarity of the Weibull with the Pearson Type III (gamma) distribution is well known. When the Weibull distribution is reflected about the vertical axis x = γ, we obtain a version with cumulative probability function $F(x;\beta,\gamma,\delta)={\rm exp}-[(\gamma -x)/\beta]^{\delta};-\infty 0,\delta >0$. When $\delta >\delta _{0}$, where to 13 decimals δ 0=3.6023494257197, we have a distribution with positive skewness (i. e., $\alpha _{3}>0$), which for certain combinations of parameter values, closely resembles the Pearson Type VI distribution. It is to be noted that when δ =δ 0, then α 3=0. The β 1,β 2 curves (Pearson's Betas) for the case in which $\delta >\delta _{0}$ and for the case in which $0<\delta <\delta _{0}$, are presented. An illustrative example is included to demonstrate use of the reflected Weibull in fitting sample data and to provide comparisons with results obtained from fitting the Type VI, Type III, Type V, and lognormal distributions.

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