The Annals of Mathematical Statistics

Estimation of Parameters in Truncated Pearson Frequency Distributions

A. C. Cohen, Jr.

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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].

Article information

Source
Ann. Math. Statist., Volume 22, Number 2 (1951), 256-265.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177729645

Digital Object Identifier
doi:10.1214/aoms/1177729645

Mathematical Reviews number (MathSciNet)
MR41394

Zentralblatt MATH identifier
0043.13705

JSTOR
links.jstor.org

Citation

Cohen, A. C. Estimation of Parameters in Truncated Pearson Frequency Distributions. Ann. Math. Statist. 22 (1951), no. 2, 256--265. doi:10.1214/aoms/1177729645. https://projecteuclid.org/euclid.aoms/1177729645


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