On the Efficiency of Best Asymptotically Normal Estimates of the Poisson Parameter Based on Singly and Doubly Truncated or Censored Samples

Several examples of situations giving rise to truncated and censored samples from Poisson populations have been given by Cohen [1960] and Hartley [1958]. The problem of estimating the Poisson parameter from a sample with the zero class missing has been considered by Cohen [1960], David and Johnson [1952], Moore [1952], Plackett [1953], Rider [1953], and others; and that from various types of truncated and censored samples has been studied by Cohen [1954, 1961b] and Hartley [1958]. However, very little investigation has been made regarding the precision of the various types of sample in the estimation of the parameter. The present paper presents an investigation into the precision of the samples in the estimation of the parameter, mainly in the following two cases: (1) a singly or doubly truncated sample of size N relative to a complete sample of the same size; (2) a singly or doubly censored sample of size N relative to a complete sample of size equal to the expected number of measured (unpooled) observations in the censored sample. The reason for considering only the above two cases is rather obvious. It is well-known that censored samples of a certain size from a population are always less efficient than complete samples of the same size in the estimation of the parameters of the population (see e.g., Raja Rao [1958] and Swamy [1962b]); whereas the precision of a truncated sample of size N relative to a complete sample of the same size depends on the form of the frequency function, the parameter to be estimated, and the points of truncation (see e.g. Swamy [1962a]). Such is the case also when we want to know the precision of a censored sample of size N relative to a complete sample of size E(n), where n is the number of measured observations in the censored sample (see e.g., Cohen [1961] and Doss [1962]). Hence, it is essential to investigate, in the above two cases, the efficiency of the estimates of the parameters