Abstract
Lognormally distributed variables are found in biological, economic and other systems. Here the sampling distributions of maximum likelihood estimates (MLE) for parameters are developed when data are lognormally distributed and estimation is carried out either by the correct lognormal model or by the mis-specified normal distribution. This is designed as an aid to experimental design when drawing a small sample under an assumption that the population follows a normal distribution while in fact it follows a lognormal distribution. Distributions are derived analytically as far as possible by using a technique for estimator densities and are confirmed by simulations. For an independently and identically distributed lognormal sample, when a normal distribution is used for estimation then the distribution of the MLE of the mean is different to that for the MLE of the lognormal mean. The distribution is not known but can be well enough approximated by another lognormal. An analytic method for the distribution of the mis-specified normal variance uses computational convolution for a sample of size 2. The expected value of the mis-specified normal variance is also found as a way to give information about the effect of the model misspecification on inferences for the mean. The results are demonstrated on an example for a population distribution that is abstracted from a survey.
Highlights
Some analytic expressions are developed for the distributions of maximum likelihood estimators (MLEs) of parameters of samples from the lognormal distribution
Results are described when the data are generated by the lognormal distribution and estimated using the MLEs for the lognormal distribution
The above approaches demonstrate the effect of misspecifying the normal model for estimation on data that were generated by the lognormal distribution
Summary
Some analytic expressions are developed for the distributions of maximum likelihood estimators (MLEs) of parameters of samples from the lognormal distribution. Since this leads to some difficulties even for a sample of size 2, an alternative approach is shown to calculate the expected value of the normal variance estimate. In the former case it turns out that g(θ ) is normal, as is already well known from elementary statistical theory, while in the latter case g(θ ) has a gamma distribution.
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