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A commonly used density model for radar clutter is chi-square for power, or, equivalently, Rayleigh for amplitude. However, for many modern high resolution radar systems, this density underestimates the tails of the measured clutter density. Log-normal and Weibull distributions have proved to be better suited for the clutter in these high resolution radars. Generalizing the chi-square density by replacing it with the noncentral chi-square density and allowing the mean power level (the noncentrality parameter) to vary, we can both suitably shape the clutter density to produce larger tails and model the fluctuation of the average clutter power, commonly referred to as speckle. The resulting form, although appearing cumbersome, readily allows for efficient and accurate computations of the probability of detection in clutter.

The prevalence of eye-tracking dysfunction (ETD) is significantly elevated in individuals with a diagnosis of schizophrenia and in their nonschizophrenic relatives, suggesting that ETD marks a familial (most likely genetic) risk factor for schizophrenia. Birth in a season with intemperate weather is also a widely reported risk factor for schizophrenia and is particularly marked for the subgroup with no family history of the disorder. This study examined how these two risk factors covaried in 78 patients with a Diagnostic and Statistical Manual of Mental Disorders (3rd ed., rev.; American Psychiatric Association, 1987) diagnosis of schizophrenia. Eye tracking and birth-month weather were independently assessed. As hypothesized, patients without ETD were significantly more likely to be born in months with intemperate weather (both hot and cold) than either patients with ETD or people in the general population. Etiologic factors associated with severe weather near birth may be important sources of nonfamilial schizophrenia.

The two chi-square statistics which are associated with theta, lambda and eta functions are used in the selection of the autoregressive (AR) moving average (MA) orders of an ARMA (p, q). They are visually processed to statistically determine the ARMA process. A Quenouille-Walker's goodness-of-fit test, which is a Lagrange multiplier test statistic, is equivalent to the two chi-square statistics in an asymptotic manner.

A chi-square analysis of the resignations of 65 US Supreme Court Justices indicates that there is a political influence which makes the pattern non-stochastic. All judges studied were appointed after 1872, when the modern Republican Party was in existence. The affiliation of the President has no effect, but the party majority in the Senate has an influence. The judges are seen to retire when the Senate is dominated by their party, thus guaranteeing that the successor judge will continue the former's politics.

This paper studies the problem of model selection based on Pearson chi-square type statistics. Such goodness-of-fit statistics have been considered by Moore (1978) and diagnostic tests based on them have recently been extended to general econometric models by Andrews (1988a, b). Here we consider another important use of these statistics. Specifically, we propose some convenient asymptotically standard normal tests for model selection based on chi-square type statistics. Following Vuong (1989), the null hypothesis is that the competing models are equally close to the data-generating process (DGP) vs. the alternative hypotheses that one model is closer to the DGP where closeness of a model is measured according to the discrepancy implicit in the Pearson type statistic used. Our model selection tests have the desirable feature that neither model needs to be correctly specified nor nested in each other. As a prerequisite, we study the corresponding class of minimum chi-square estimators.

The maximum likelihood estimation of the power of a future study based on the results of a previous study provides a good example in which maximum likelihood estimators perform poorly. In particular, they are inconsistent and, for higher dimensional treatment outcomes, are badly biased. This is easily demonstrated using the noncentral chi-squared distribution functions available in the computer package SAS.