22 Weibull, log-weibull and gamma order statistics

Abstract

Publisher Summary This chapter focuses on the Weibull, Log–Weibull, and Gamma order statistics. The Weibull distribution, which is also known as the third asymptotic distribution of smallest values, was introduced by Weibull in connection with a study of material strength. In its most general form the Weibull distribution has three parameters: a scale parameter (characteristic value) θ, a shape parameter K, and a location (threshold) parameter c. If the random variable X has the two-parameter Weibull distribution with cdf, then the random variable Y = InX has the Log–Weibull distribution, which is also known as the first asymptotic distribution of smallest values. Thus, a logarithmic transformation converts a scale-and-shape-parameter distribution (two-parameter Weibull) into a more tractable location-and-scale-parameter distribution (log-Weibull). The Gamma distribution, which is also known as the Pearson Type III distribution, was derived by Laplace (1820). It has been used extensively in reliability and life testing and in various other applications. In its most general form the Gamma distribution has three parameters: a scale parameter θ, a shape parameter α, and a location parameter c. The chapter discusses the Weibull order statistics, Log–Weibull order statistics, Gamma order statistics, and the applications of these three.