How non-normality affects the quadratic discriminant function

The quadratic discriminant function is commonly used for the two group classification problem when the covariance matrices in the two populations are substantially unequal. This procedure is optimal when both populations are multivariate normal with known means and covariance matrices. This study examined the robustness of the QDF to non-normality. Sampling experiments were conducted to estimate expected actual error rates for the QDF when sampling from a variety of non-normal distributions. Results indicated that the QDF was robust to non-normality except when the distributions were highly skewed, in which case relatively large deviations from optimal were observed. In all cases studied the average probabilities of misclassification were relatively stable while the individual population error rates exhibited considerable variability.     

Frequency and timing of intercourse: Its relation to the probability of conception

Abstract This paper describes a simple study of the chance of conception assuming a fixed-length menstrual cycle, a specific probability distribution for the day of ovulation, and a broad and a narrow fertile period. Eight different patterns of intercourse throughout the cycle are considered with frequency of intercourse ranging between 5 and 12 times per cycle. Results are given in terms of chance of conception and in terms of waiting times until conception.     

Two stage multiple comparison procedures based on the studentized range

Stein’s (1945) two sample approach and Tukey’s T-Method of multiple comparisons (see e.g. Miller, 1966, Ch. 2) are combined to obtain fixed width simultaneous confidence intervals and simultaneous test procedures of predetermined Type I and Type II error levels, for all contrasts, in a one way layout. The necessary constants for implementing the two stage procedure are obtained under a least favorable configuration of the parameters. This provides the required protection of the null and alternative hypotheses under any configuration of parameters. A table is provided for some selected designs and error levels and an example is given to illustrate certain features of the new procedure.     

Considerations for Effective Simulation Study Analysis

This article discusses estimation of several percentiles simultaneously, develops a simple test to compare the sizes of two test statistics, and considers the use of logit models to adjust power curves to have the same null hypothesis level.     

Estimates of fecundability from a truncated distribution of conception times

A population consisting of women who have conceived before a time t ignores all women whose conception time exceeds t. Such a population is considered as a truncated population, and the samples are called truncated samples. Under the assumption that fecundability among women varies according to a Beta distribution (with parameters a and b), the distribution of conception times in a truncated population can be considered as truncated Type I geometric. This paper presents an algorithm to obtain the moment and maximum likelihood estimates of a and b from the truncated samples. Large sample properties of the estimators are also studied. Examples using the Hutterite and the Princeton Fertility Survey data are given.     

Some Distributions on the Positive Real Line Which Have No Moments

A sharp upper bound is obtained for the difference of a pair of order statistics in terms of the sample standard deviation.     

《福乐智慧》中所反映的早期苏非

对于一个正在谋求发展的政权,苏非的避世苦修、清心寡欲、淡泊名利、不问世事并不值得提倡,正如11世纪新疆喀喇汗王朝的著名文学作品《福乐智慧》所提到的日出国王统治的国度,伊斯兰教正统派信仰盛行并以积极的态度参与国家治理。当时苏非已经出现,处于发展的初期阶段,属于个人行为,未形成大的苏非教团,但显然在社会和统治阶层中得到了一定程度的认同。究其原因,苏非神秘主义以其虔诚的独特的思想体系和修行方式为穆斯林的宗教生活增加了直觉体验和宗教情感,是对穆斯林宗教生活的合理补充。     

On small sample properties of the generalized signed rank and generalized sign tests

The generalized signed rank (GSR) and generalized sign (GS) tests were recently proposed for matched pair studies with censored observations (Woolson and Lechenbruch, 1980). The results provided in that paper were asymptotic, and no indicatin of small sample behavior was given. In this paper we report on simulation studied of these statistics for a variety of distributions. We find that the GSR is more powerful than the GS, and that censoring does not affect power greatly. In the original paper, we assumed each member of the pair has the same censoring time. We consider a variant of this in which each member of the pair has a censoring time chosen from a uniform distribution, and the minimum of these times is selected as the censoring time for the pair. It is found that the power of the test is slightly reduced because the number of doubly censored pairs is increased.     

A generalized quantile estimator

An alternative to the conventional sample quantlle Is proposed as a nonparametrlc estimator of a continuous population quantlle.The alternative estimator Is a "generalized sample quantlle" obtained by averaging an appropriate subsample quantlle over all subsamples of .a fixed size.Since the resulting statistic is a U-statistic with representation also as a linear combination of order statistics, known results are employed then to establish asymptotic normality.The alternative estimator is shown to be asymptotically efficient in the class of nonparametrlc models specified by Pfanzagl (1975).Analytic results and Monte Carlo studies with a moderate sample size for a variety of distributions Indicate that the proposed estimator usually provides mean square error of estimation less than that of the conventional sample quantile.