A Theorem on Products of Random Variables, With Application to Regression

It is shown that the product of two independent random variables in the domain of attraction of the normal distribution is also in the domain of attraction of the normal distribution, while if the product is in the domain of attraction of the normal distribution and one of the variables has finite variance, the other is in the domain of attraction of the normal distribution. This result is applied to prove the asymptotic normality of the regression coefficient in a linear regression when the error variance is not necessarily finite.     

Evaluations of Women and Men Among Black and White College Students1

Rainwater's assertion that blacks have low self-evaluations because they receive more negative evaluations from other blacks than whites receive from other whites is challenged here by reference to Heiss and Owens evidence that negative self evaluation among blacks is limited to work-related traits. Substantial support was found for our major hypotheses that, compared to whites, (1) blacks would report more negative evaluation of “most men” but not of “most women” (since the provider role is traditionally ascribed primarily to men), and (2) the more negative evaluation of “most men” by blacks would be limited to work-related traits.     

Forecasting births in post-transition population: stochastic renewal with serially correlated fertility

Time series analysis of fertility can improve demographic forecasts. The optimal forecast and its variance for births to an age-structured population subject to serially correlated random fertility are developed. The general case in which the fertility process had arbitrary autoregressive structure is dealt with and then the 4 special cases of white noise, 1st-order autoregressive, 2nd-order autoregressive, and random walk are considered. Consequently, it is determined that the predictions and their variances are highly sensitive to the autoregressive structure of fertility and, therefore, if stochastic models are to be used for prediction, they must emphasize this aspect of the problem. Preliminary empirical efforts to model the time series of U.S. fertility from 1917 to 1972 proved unsuccessful, but it is obvious that at least a 2nd-order autoregressive scheme is require d. The analysis proveded should be helpful in: 1) any application of the procedures requires a successful parameterization of the fertility process; 2) fertility variations could be decomposed into the effects of nuptiality and marital fertility and then births and marriages could be jointly predicted; and 3) the simplifying approximations should be dropped and each age-specific rate could be analyzed and predicted.     

Earnings,relative income,and family formation

The probability of first marriage for men who graduated from Wisconsin high schools in 1957 was analyzed with respect to their Social Security earnings records, Wisconsin income tax reports for parents, and other variables. The findings provide no support for Easterlin's hypothesis that marriage will occur earlier when young men judge their economic prospects favorably with respect to their parents' income. However, young men's earnings and time spent in schooling to increase them were found to be important influences on marriage timing. Additional schooling had little effect net of the time it absorbed.     

Health professionals' attitudes toward abortion

Attitudes toward abortion, as obtained in a 1971 nationwidesurvey, are presented for students and faculty in nursing, medicine,and social work, and are compared to attitudes of the generalpopulation. Attitudinal differences are examined among the professions,between students and faculty, and in terms of religion. Theresults reveal implications for the abortion-related servicesprovided by health professionals.     

THE FUNDAMENTAL FORMULA OF BLASCHKE IN INTEGRAL GEOMETRY AND GEOMETRICAL PROBABILITY, AND ITS ITERATION, FOR DOMAINS WITH FIXED ORIENTATIONS

The so-called “principal formulae” of planar integral geometry are conventionally couched in terms of the “kinematic density”dxdydθ. Here a corresponding theory with respect to the “Lebesgue density”dxdy, that is with rotations suppressed, is developed. The only real difference is that the new “fundamental formula of Blaschke”contains a term depending upon the relative orientations of the two domains involved. In particular, the remarkable iteration property of these formulae carries over. The usual principal formulae follow as a corollary of the formulae given here, upon, averaging over orientations.