Estimation

Abstract

We have seen briefly that the principal parameters μ and λ are estimated by their maximum likelihood estimators. The moment estimate for the variance ${m^3}/\lambda$ is ${s^2} = \sum _{i = 1}^n{\left( {{x_i} - \overline x } \right)^2}/\left( {n - 1} \right).$ As a result it is possible to show that as a consistent estimator of ${\lambda ^{ - 1}}$ ${s^2}/{\overline x ^3}$ has an asymptotic efficiency of ф/ (ф + 3). For small ф, the estimator is not reliable. The reciprocal of X can be used to estimate 1/μ, but has a bias equal to 1/λ, and a mean squared error equal to (Ф + 3)/λ². We reproduce in Table 2.1 uniformly minimum variance ${m^3}/\lambda is{s^2} = \sum _{i = 1}^n{\left( {{x_i} - \overline x } \right)^2}/\left( {n - 1} \right).$ unbiased estimators (UMVUE) of several parametric functions of μ and λ as given by Iwase and Setô (1983) as well as Korwar (1980). Uniformly minimum variance unbiased estimators for cumulants and the density itself were given by Park et al. (1988). We state two propositions in the next section that provide these estimators. Also presented is Table 2.2 indicating estimates of the reliability R(t) = 1 — F(t).