Estimating the Right-Tail Probability in Log-Gamma and Log-Weibull Distributions

We consider estimation of the right-tail probability in a log-gamma distribution and a log-Weibull distribution having each known shape parameter, especially we derive a UMVUE of the right-tail probability in the log-gamma and a log-Weibull distributions having each known shape parameter. And we derive densities of product, quotient, and the ratio of two independent log-gamma random variables. And we obtain the several facts through the numerical performance: (a) The log-Weibull density is skewed to the right side when alpha is greater than 1, but skewed to the left side when alpha is 1/2 and 1/3. (b) The kurtosis of the density is larger as alpha goes to a number(≥1/2). (c) The UMVUE performs better than the estimator tilde{R}(t) when t is larger time, and a proposed estimator tilde{R}(t) performs better than the UMVUE when t is small time in a sense of MSE.