Estimation of the parameters in the truncated normal distribution when the truncation point is known

Abstract

In various fields of science, such as biology, economics and medicine, scientific data frequently follow a truncated normal distribution. Measurement of variables in some parts of the population present difficulties. Because of the importance of this distribution, many statisticians have been involved with the estimation of the relevant parameters.

The problem with the estimation of the parameters is that the method of maximum likelihood gives rise to two equations which cannot be explicitly solved and, further, the results obtained are not acceptable due to the biases being large. Cox & Hinkley (1974) have presented an approximation formula based on a Taylor expansion, which can be used to find the expected value and variance of the maximum likelihood estimators. An alternative approach for estimating the parameters is by application of Shenton & Bowman's formula (1977).

In this thesis the method of Shenton & Bowman is extended to the two-parameters case to give the means, variances and covariances of the maximum likelihood estimators of the truncated normal distribution simultaneously.

The maximum product spacing method, which is asymptotically as efficient as the maximum likelihood and in some cases hyper-efficient, is used for the truncated normal distribution.

Finally, a comparison is made between the above methods and also with the method of estimation by means of simulation.

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