Some Results on the Distribution of a Statistic based on the Pareto and Log- normal Distributions and the Power of an Associated Test*
نویسنده
چکیده
In the October 1975 edition of this Review, Harrison and Nolan examine the "residual wealth" assumption which Lyons (1972 and 1975) made when using the mortality multiplier method to estimate the distribution of personal wealth in Ireland. The assumption was that all of the unexamined estates of adults who died in 1966 had a net value of zero. Scrutiny of such assumptions is valuable because, as Atkinson (1974) has shown, wealth distribution estimates seem to be highly sensitive to the values assumed for residual wealth. Amongst other things, Harrison and Nolan devise a statistical procedure for testing the validity of an arbitrarily small assumed value of average residual wealth. On the maintained hypothesis that wealth is log-normally distributed, and using a result due to Cramer (1971), they argue that if the null hypothesis is true —that is, if the assumed value of average residual wealth is correct in the sense that it produces a variance for the estimated distribution of the natural logarithm of wealth diat is in accord with the log-normal variance suggested by the Cramer result—then their test statistic is F distributed. However, Chesher and McMahon (1976), while not demonstrating that the statistic is not F distributed, have recently drawn attention to certain flaws in the Harrison-Nolan argument. Consequently the precise nature of the statistic, and the status of the conclusion reached by Harrison and Nolan on the basis of their use of the statistic, are in doubt. For this reason Harrison (1976), in his reply to Chesher and McMahon, undertook to investigate the statistic further using the method of simulation. It is the purpose of this paper to briefly describe the study that was carried out and to report and comment on the results. Section II is concerned with the
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