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The Exponentiated Weibull family is an extension of the Weibull family obtained by adding an additional shape parameter. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates which are quite common in reliability and biological studies. As with any other distribution, many of its interesting characteristics and features can be studied...
A class of procedures is presented for using random samples to test the fit of location-scale families—distributions F(·;θ1,θ2) such that Z=(X−θ1)/θ2 has distribution Working with empirically standardized data, the test statistic is a measure of distance between the empirical characteristic function, and the c.f. of Z under the null hypothesis, ϕ0(t). The closed-form test statistic is derived...
There are many situations in life testing experiment where an item fail instantaneously and hence the observed lifetime is reported as zero. The items that fail prematurely are called early failures. We propose a modified Weibull distribution as a suitable model to represent such situations by mixture of a singular distribution at zero and a two parameter Weibull distribution. We obtain the maximum...
A near-maximum is an observation which falls within a distance a of the maximum observation in an independent and identically distributed sample of size n. Subject to some conditions on the tail thickness of the population distribution, the number Kn(a) of near-maxima is known to converge in probability to one or infinity, or in distribution to a shifted geometric law. In this paper we show that...
Remaining useful life (RUL) is nowadays in fashion, both in theory and applications. Engineers use it mostly when they have to decide whether to do maintenance, or to delay it, due to production requirements. Most often, it is assumed that in later life of an equipment (in wear-out period), the hazard function is increasing, and then the expected RUL, μ(t), is decreasing. We noticed that the standard...
The Weibull distribution plays a central role in modeling duration data. Its maximum likelihood estimator is very sensitive to outliers. We propose three robust and explicit Weibull parameter estimators: the quantile least squares, the repeated median and the median/Qn estimator. We derive their breakdown point, influence function, asymptotic variance and study their finite sample properties in...
The Weibull distribution was discovered by Rosin, Rammler, Sperling and Bennett between 1932 and 1936 in the context of particle measurement. Weibull found the same distribution a little later while investigating the strength of materials. More than 10 years after, in 1951, he finally showed that this distribution has the potential for wide applications in statistics. However, does this justify that...
Using the likelihood depth, new consistent and robust tests for the parameters of the Weibull distribution are developed. Uncensored as well as type-I right-censored data are considered. Tests are given for the shape parameter and also the scale parameter of the Weibull distribution, where in each case the situation that the other parameter is known as well the situation that both parameter are unknown...
In this paper, we compare the largest order statistics arising from independent heterogeneous Weibull random variables based on the likelihood ratio order. Let $$X_{1},\ldots ,X_{n}$$ X 1 , … , X n be independent Weibull random variables with $$X_{i}$$ X i having shape parameter $$0<\alpha \le 1$$ 0 < α ≤ 1 and scale parameter $$\lambda _{i}$$ λ i...
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