The Statistical Analysis Of Geometric Distribution And Weibull Distribution

This article is composed of two parts: the statistical analysis for geometric distribution and the statistical analysis for Weibull distribution as well.Part one: The statistical analysis for geometric distributionThe geometric distribution is one of the better known discrete probability distributions and has many useful applications. Its applications include in the fields of information engineering, electronics industry, theory of controls and economic, etc.For example, we use geometric distribution to describe the life distribution of runs of a species in transect surveys of plant populations and inventory demand distributions. In the theory of reliability, geometric distribution is one of the most important discrete probability distributions because of its loss of memory. This article gives the key results for geometric distribution with respect to the following three aspects:1. The statistical character for geometric distribution.(1) The distributions of the order statistics for discrete distribution are studied in this section. Therefore, the probability distributions of the 1st, 2nd and 3rd order statistics are derived, respectively.(2) We point out the similarity and difference between the properties of the order statistics from the geometric distribution population and exponential distribution population.(3) We make some progress in proving the Arnolds's guestrimate.2. The statistical analysis of geometric distribution.(1) After giving the various methods for getting the point estimations of parameters based on the type-â…¡ censored data. We do the comparison of the accuracy of all these point estimations. In the portion, the pivotal quantity used to get the approximate interval estimation of parameters is also derived. Afterwards, we show the feasibility of our method by using the Monte-Carlo simulation.(2) The point estimation and interval estimation of parameters under type-â…  censored case and the point estimation of parameters based on the missing data and grouped data are obtained, respectively.(3) The point estimations of parameters under the constant stress and step-stress accelerated life testing are derived in this part, respectively.3. The approach property and statistical approach property of discrete lifedistribution class and geometric distribution.(l)We carry out the research on the characteristic property of geometric distribution. Hence, we propose the two new concepts: statistical closed property and statistical approach property.(2) Make a detailed study of the discrete new better than used in expectation class and the approach property of geometric distribution. Furthermore, we narrow the upper bound by adding certain constraints.Part Two: The statistical analysis for Weibull distributionWeibull distribution is one of the widely used continuous life distributions, it can be used to describe the failure distribution of quite a lot product. For example, the weakness in metals etc. caused by repeated stress. This article gives the point estimation of parameters based on tampered failure rate model for Weibull distribution under step-stress accelerated life testing. The main results are listed as follows:(l)We present the inverse moment estimations and approximate maximum likelihood estimations of parameters, and then we examine the accuracy of the AMLE, MLE and interval estimations. Moreover we derive the method to get interval estimation of the shape parameter.(2)Focus on the point estimation of parameters under Type- â…¡ life testing basedon step-increasing data.(3) Derive a way to get the point estimations of parameters under normal working condition stress.