| We examine the problem of finding the exact distributions of linear functions of k independent generalized gamma variables, X 1, X2,..., Xk. Special cases of generalized gamma distributions include the exponential, gamma and Weibull distributions. A linear function of such variables is often a quantity of interest in the analysis of survival data, reliability of certain systems and stochastic processes and hence we present this problem in the context of life testing. The exact distributions of these linear functions are needed to compute survival functions, hazard functions and other important functions in practical problems. Stacy (1962) obtained some exact results involving generalized gamma variables and Huzurbazar and Huzurbazar (1999) used saddlepoint approximations where the input variables are gamma or Weibull. We examine this problem where the k independent real scalar random variables, X1, X 2,..., Xk, are of gamma type with general parameters. For this case, various exact distributions are obtained and it is shown that most of these representations are easily computable. These exact results are compared with the usual saddlepoint approximations. We also examine numerically inverting the Laplace transform in this context, showing that it is one of the most efficient and accurate ways of estimating the exact distribution for certain cases. Results of this thesis are being published and presented in co-authorship with A. M. Mathai in Koulis and Mathai (2000). |
