Left-censored samples from Weibull distribution – statistical inference and applications

The problem of statistical analysis of left-censored samples from a skewed distribution using the maximum likelihood method is solved only partially, see e.g. Mitra and Kundu [J. Stat. Comput. Sim. (2008)], Fusek and Michalek [J. Stat. Comput. Sim. (2015)]. On one hand, the maximum likelihood approach can certainly be used when data are not normally distributed and the obtained estimates have good asymptotic properties. On the other hand, likelihood equations can be nonlinear, rather complicated, and it is often impossible to find their analytic solution. Moreover, it is difficult to determine Fisher information matrix which is necessary for description of asymptotic variability of obtained estimates.
This contribution focuses on type I left-censored skewed samples from Weibull distribution. A computational procedure for determination of maximum likelihood estimates of parameters of Weibull distribution will be described and the analyti form of the Fisher information matrix will be presented. Bias of the ML estimates of expected value and variance considering various sample sizes and levels of censoring is examined using simulations. Simulations are focused primarily on the properties of estimators for small sample sizes. Parameters of the simulated samples are chosen with respect to real environmental data. The asymptotic Wald's test for comparison of two independent multiply left-censored Weibull samples is given and its power function considering various sample sizes is analyzed. Derived methods are used for assessing the effectiveness of the wastewater treatment plant with respect to presence of musk compounds in fish tissue samples.