Asymptotic Distribution of the Estimated Cumulative Distribution Function of the Bivariate Distribution with Truncated Logistic and Geometric Marginals

Abstract

Let X₁,X₂, . . . be i.i.d. exp(&beta) and N geo(p) independent of X<sub>i<sub>'s. We are interestedin the bivariate distribution of (X,N) where X = max X_i and N. The distribution of (X,N) has truncated logistic and geometric marginals and was developed and described by Kozubowski and Panorska (2008). We derive the limiting distribution of the estimatedcumulative distribution function &Fhat for (X,N). We discuss the limiting behavior of the asymptotic variance of &Fhat and illustrate the convergence using Monte Carlo simulations. We also present examples of estimation of the error probabilities for hydrological and financial data.