A NOTE ON THE DISTRIBUTION OF THE DOUBLY NON-CENTRAL F DISTRIBUTION1

A three-moment central-F approximation is examined for its usefulness in giving accurate values of the probability integral of the doubly non-central F distribution.     

A SIMPLE PROCEDURE FOR SAMPLING πpswor1

The author's 1963 procedure for selecting two units per stratum πpswor is generalized to any feasible fixed number of sample units. Simple closed expressions are given for the working probabilities at each draw but the formula for the joint probabilities of inclusion of pairs of units is recursive and tedious to apply. The stability of the variance estimator is believed to be close to the optimum that can be obtained using the Horvitz-Thompson estimator of total in situations where that estimator itself is optimal. Some suggestions are made for rotation of the sample.     

ON POINT MULTISERIAL CORRELATION1

Abstract
We present a simple form for the estimator of the point multiserial correlation coefficient between a quantitative variate X and a qualitative variate 7. Given a bivariate sample grouped in the form of an r × c contingency table the estimator is based on finding the optimum Y -scores which maximize the correlation coefficient. The resulting estimator is equivalent to Das Gupta's (1960) for ungrouped X -values, with the advantage of simplicity in its calculation. Under the assumption of conditional normality, the significance of point multiserial correlation may be studied by an F -test.     

SAMPLING ON SEVERAL SUCCESSIVE OCCASIONS WITH EQUAL AND UNEQUAL PROBABILITIES AND WITHOUT REPLACEMENT1

The applicability of the technique of controlled selection with equal probabilities proposed by the authors, Avadhani and Sukhatme (1965, 1968), and that of unequal probability sampling suggested by Rao et al. (1962) to sampling on several successive occasions is investigated in this paper. Appropriate recurrence formulae for the optimum replacement fraction and the corresponding weight are also derived.     

TWO-STAGE SAMPLING OVER TWO OCCASIONS

We propose a general sampling strategy for estimating a finite population total for the current occasion based on a two-stage sampling scheme adopted on the current and a previous occasion. This is in generalization of sampling schemes considered earlier by, among others, Singh (1968), Abraham, Khosla & Kathuria (1969), Singh & Kathuria (1969) and Kathuria (1975). Applications of the general strategy in specific situations are illustrated.     

QUADRAT SAMPLING AND THE ESTIMATION OF NEYMAN TYPE A AND THOMAS DISTRIBUTIONAL PARAMETERS1

Computer simulation is employed to investigate the effects of quadrat size, sampling and estimation method, and cluster spread, density and size when estimating Neyman Type A and Thomas distributional parameters from quadrat samples. Results obtained indicate that quadrat size and cluster spread affect the estimates. Use of first order standard errors in selecting suitable quadrat sizes is discussed. Where clusters have non-zero spread, the usual situation in practice, cluster density is generally overestimated and cluster size underestimated, which suggests the need for development of modified estimation procedures.     

ON THE DISTRIBUTIONS OF ORDER STATISTICS1

Approximations to the distributions of order statistics based on x²t are obtained. These are easy to compute and provide reasonably accurate values for the percentage points and probability integrals of the distributions.     

ON THE BEHRENS–FISHER PROBLEM1

In this article the Behrens-Fisher problem is reformulated in terms of a structural model of inference. For this version of the problem a solution is obtained which is valid for arbitrary absolutely continuous error distributions. These results are further discussed for the standard normal distribution and for some other special cases with not normally distributed populations.     

A DISTRIBUTION-FREE TEST FOR TWO-WAY LAYOUT1

Many nonparametric tests have been proposed for the hypothesis of no row (treatment) effect in a one-way layout design. Examples of such tests are Kruskal-Wallis H-test, Bhapkar's (1961) V-test and Deshpande's (1965) L-test. However not many tests are available for testing the same hypothesis in a two-way layout design without interaction. Perhaps the only “established” test is the one due to Friedman (1937). However, it applies to the case of one observation per cell only. In this paper, a new distribution-free test is proposed for the hypothesis of row effect in a two-way layout design. It applies to the case of several observations per cell, not necessarily equal. The asymptotic efficiency of the proposed test relative to other tests is studied.     

A NOTE ON OPTIMUM STRATIFICATION IN SAMPLING WITH VARYING PROBABILITIES1

Singh and Sukhatme [4] have considered the problem of optimum stratification on an auxiliary variable x when the units from the different strata are selected with probability proportional to the value of the auxiliary variable and the sample sizes for the different strata are determined by using Neyman allocation method. The present paper considers the same problem for the proportional and equal allocation methods. The rules for finding approximately optimum strata boundaries for these two allocation methods have been given. An investigation into the relative efficiency of these allocation methods with respect to the Neyman allocation has also been made. The performance of equal allocation is found to be better than that of proportional allocation and practically equivalent to the Neyman allocation.     

ON SOME PRODUCT TYPE ESTIMATORS IN VARYING PROBABILITY1

It is shown that varying probability in product method estimation increases the efficiency of estimate under some usual conditions. Further, a necessary condition, for the product estimate from a sample drawn with pps and without replacement to be more efficient than the product estimate from a sample drawn with pps and replacement, has been obtained; incidentally, it is the same as obtained by Narain (1951) for varying probability without replacement to be more efficient than varying probability with replacement. The use of the R.H.C. sampling scheme has also been suggested, and its variance and bias has been shown to be always less than that of pps with replacement.     

A LINEAR MODEL WITH ERRORS LACKING A VARIANCE1

The problem discussed is that of estimating β= (β¹, …, β_k) in the model Y=βX +ε when X has a specified multivariate distribution and the error ε does not necessarily have a finite second moment, for example, ε symmetric stable. We construct a moment estimator based on the empirical characteristic function and establish asymptotic unbiassedness and normality. Most of the paper is concerned with the case when X is normal. Forms of the suggested estimator are given in (2.5), (4.6) and (5.5).     

A BIVARIATE BINOMIAL DISTRIBUTION AND SOME APPLICATIONS1

This study is concerned with the joint distribution of the total numbers of occurrences of binary characters A and B, given three independent samples in which both characters, A but not B, and B but not A, are observed. The distribution function is given; its conditional distributions and regression functions are found; bounds on certain joint probabilities are established; and conditions for bivariate Poisson and Gaussian limits are studied. An application yields the joint distribution of sign statistics for the pair-wise comparison of treatments with a control.