A study of the number of solutions of the system of the log-likelihood equations for the 3-parameter Weibull distribution. (Q1928179)

In the case of the three parameter Weibull distribution, which plays an important role in life testing and reliability theory, the author considers the system of the log-likelihood equations of the parameters. First, using a solution of the log-likelihood equation on a scale parameter, the shape parameter is derived as a certain form of a function of a location parameter from the log-likelihood equations on a shape and a location parameter. Next, the form of the function of a location parameter and the expression of the form as the location parameter tends to \(-\infty\) are given. Substituting the form into the log-likelihood equation on a shape parameter, the determination of the number of solutions is discussed. Some concrete examples are also given.