Estimating the Mean and Variance of Normal Populations from Singly Truncated and Doubly Truncated Samples

This paper is concerned with the problem of estimating the mean and variance of normal populations from singly and doubly truncated samples having known truncation points. Maximum likelihood estimating equations are derived which, with the aid of standard tables of areas and ordinates of the normal frequency function, can be readily solved by simple iterative processes. Asymptotic variances and covariances of these estimates are obtained from the information matrices. Numerical examples are given which illustrate the practical application of these results. In Sections 3 to 8 inclusive, the following cases of doubly truncated samples are considered: I, number of unmeasured observations unknown; II, number of unmeasured observations in each `tail' known; and III, total number of unmeasured observations known, but not the number in each `tail'. In Section 9, singly truncated samples are treated as special cases of I and II above.