Multiple-mixed Statistical Model of Random Variables and Optimal Estimation of Distribution Parameters 2003-01-1036

The failure time of automobile and its components is typical random variable. In studying reliability automobile and its components some simple theoretical statistical models, such as Normal distribution, Lognormal distribution and Weibull's distribution, index distribution etc., are used to describe automobile reliability. These distributions can not often represent actual datum well, then more complex theoretical statistical model is needed. If using complex statistical theoretical model the more parameters must be determined. Methods of moment, Maximum likelihood method can not estimate the parameters of complex statistical models yet. With graph method though the type of random variable distribution can be recognized, the precision of estimated parameters is not high. Meanwhile there is not an objectively criterion to judge statistical model reasonability.

Based on the adaptability of Weibull's distribution, multiple-mixed Weibull's distribution suits to describe the complex statistical distribution of random variable. To estimate the parameters of multiple Weibull's distribution correctly, optimization model for estimating parameters of multiple-mixed Weibull's distribution is presented, according to the principle of least-square.

In order to obtain the optimal estimation, the Levebberg-Marquardt method with line-search and Gauss-Newton method are alternatively used. A safeguarded mixed quadratic and cubic polynomial interpolation and extrapolation are chosen as algorithm of line-search. The example shows that multiple Weibull's distribution has quite good adaptability in describing the complex statistical distribution of random variable. The model and algorithm of optimization for estimating multiple Weibull's distribution parameters have steady convergence.