Estimation of three-parameter Fréchet distribution for the number of days from drug administration to remission in small sample sizes

In medical research, it is common to estimate parameters for each group and then evaluate the estimated parameters for each group without comparing the groups. However, researchers frequently want to determine whether the two distributions using the estimated parameters differ significantly between the two groups. For the Weibull distribution, the two-sample Kolmogorov-Smirnov test (two-sided) was used to examine whether the two distributions were significantly different between the two groups. Based on this, we developed a method to compare the two groups using a three-parameter Fréchet distribution. The number of days from drug administration to remission frequently followed a Fréchet distribution. It is appropriate to use a three-parameter Fréchet distribution with a location parameter because patients typically go into remission after several days of drug administration. We propose a minimum variance linear estimator with a hyperparameter (MVLE-H) method for estimating a three-parameter Fréchet distribution based on the MVLE-H method for estimating a three-parameter Weibull distribution. We verified the effectiveness of the MVLE-H method and the two-sample Kolmogorov-Smirnov test (two-sided) on the three-parameter Fréchet distribution using Monte Carlo simulations and numerical examples.