Summary
Two tests for variance shift in a sequence of independent normal random variables, when the initial level of variance is unknown, are investigated in this article. The first is a locally most powerful test, and the second is a test based upon cusums of X2 values. Distribution functions of the two test statistics are approximated through the use of Edgeworth expansions and/or the beta distribution by matching the first few moments. Critical points of both test statistics are tabulated for various sample sizes. Powers of the two tests are compared using a Monte Carlo example. An illustration of the application of the tests to stock market price analysis is provided.
In this article we present some empirical evidence which indicates that attempts to represent the probability distribution of the rates of return on common stocks by a member of the stable Paretian family of distributions, with 1 < a < 2, may be misleading and in fact may not produce an adequate fit to observed rates of return. We offer an alternative probability model for describing rates of return based on the hypothesized phenomenon of a changing variance. We test the "goodness of fit" of our model vis-a-vis a stable Paretian model for several series of rates of return. Finally we propose an extension of the stability test of Fama and Roll [6].
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Summary
In this article, a distribution theory is developed to explain a fairly well‐established characteristic of the position errors observed in large navigation systems. Empirical adequacy of the suggested model is compared with other existing models using newly collected data. An application of all these distribution models to the evaluation of aircraft mid‐air collision risk is illustrated, and potential implications of the computed results are sketched. Insights gained from this study may be useful for other areas of statistical application in which long‐tailed distributions have been observed.
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