Estimation of Parameters in Truncated Pearson Frequency Distributions

Abstract

A method based on higher moments is presented in this paper by which the type of a univariate Pearson frequency distribution (population) can be determined and its parameters estimated from truncated samples with known points of truncation and an unknown number of missing observations. Estimating equations applicable to the four-parameter distributions involve the first six moments of a doubly truncated sample or the first five moments of a singly truncated sample. When the number of parameters to be estimated is reduced, there is a corresponding reduction in the order of the sample moments required. A sample is described as singly or doubly truncated according to whether one or both "tails" are missing. Estimates obtained by the method of this paper enjoy the property of being consistent and they are relatively simple to calculate in practice. They should be satisfactory for (a) rough estimation, (b) graduation, and (c) first approximations on which to base iterations to maximum likelihood estimates. Previous investigations of truncated univariate distributions include studies of truncated normal distributions by Pearson and Lee [1], [2], Fisher [3], Stevens [4], Cochran [5], Ipsen [6], Hald [7], and this writer [8], [9]. In addition, the truncated binomial distribution has been studied by Finney [10], and the truncated Type III distribution by this writer [11].