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Given N points independently drawn from the uniform distribution on (0, 1), let p[sub n], be the size of the smallest interval that contains n out of the N points; let n[sub p], be the largest number of points to be found in any subinterval of (0, 1) of length p. This paper uses a result of Karlin, McGregor, Barton and Mallows to determine the distribution of n[sub p] for p = 1/k, k an integer. The paper gives simple determinations for the expectations and variances of p[sub n], for all fixed n > (N + 1)/2, and of n[sub 1/2]. The distribution and expectation of n[sub p] are estimated and tabulated for the cases p = 0.1(0.1)0.9, N =2(1)10. [ABSTRACT FROM AUTHOR]

Given a population of N units, it is required to draw a sample of n distinct units in such a way that the probability for the ith unit to be in the sample is proportional to its 'size' x. From the alternative methods of achieving this we consider here only the so-called systematic method which, to the best of our knowledge, was first developed by W. G. Madow (1949): The units in the population are listed in a 'particular' order, their x, accumulated and a systematic selection of n elements from a 'random start' is then made on the accumulation. In a more recent paper (H. O. Hartley and J. N. K. Rao (1962) ) an asymptotic estimation theory (for large N) associated with this procedure was developed for the case when the order of the listed units is random. In this paper we draw attention to certain properties of Madow's estimator: We utilize the fact that with systematic sampling the total number of different samples is N (rather than ([This eq. cannot be change in char.]) as with completely random sampling). This simplification in the definition of the variance of the estimator in repeated sampling enables us to identify the exact variance of Madow's estimator with a 'between sample mean square' in a special analysis of variance (see section 4) and compare it with the variance of the pps estimator in sampling with replacement as well as in other sampling procedures. We also develop two approximate methods of variance estimation (see section 5). We pay particular attention to the case when the units are listed in the order of their size. With this particular arrangement our method can be described as 'systematic with random start' and the gain in precision that we accomplish has of course, analogues in systematic sampling with equal probabilities employing ratio estimators in which there is a relation between the ratio ri =yi/Xi and xi Compared with other methods the present procedure combines the advantage of ease of systematic sample selection with the availability of exact variance formulas for any n and N. Moreover, it usually leads to a more efficient estimate. Its shortcoming resides in the fact that the estimation of the variance is based on certain assumptions. [ABSTRACT FROM AUTHOR]

This paper proposes several methods of estimating parameters in stable distributions. All the methods involve sample characteristic functions. One of the methods which is based upon the method of moments is treated in some detail. Asymptotic normal distributions for the proposed moment estimators are provided. Moreover, all methods provide consistent estimators. The estimation problem is treated for both univariate and multivariate stable distributions. [ABSTRACT FROM AUTHOR]

The article presents information on a nomogram for the "Student"-fisher t test. A nomogram is given for estimating the probability (P) for a given value of the "Student"-Fisher t test. W.S. Gosset, an employee of the Guiness brewing company in Dublin, published papers in 1908 in which he correctly solved three problems: the probable error of a mean, the distribution of the mean divided by its estimated standard deviation and the distribution of the estimated correlation coefficient between independent variates. Later "Student" and economist R.A. Fisher calculated tables of the relevant t distribution and Fisher gives a table of t and probabilities, corresponding to various degrees of freedom. Fisher and F. Yates, scholar provide in addition a column for P. It seemed that presentation of the P, degrees of freedom, t relationship in the form of a nomogram would be advantageous. It makes possible a fairly exact estimate of probabilities less than 0.0001 and makes it possible to get an estimate of P for any value of t from 1 to 65, instead merely of selected values.

This paper deals with certain aspects of variance-component estimation for the unbalanced two-way completely-random classification where the numbers of observations in the subclasses are treated as random variables not necessarily independent of some of the random effects of the model. General results are given on the expectations of two commonly-used estimators of the vector of variance components. Numerical approximations are presented for these expectations for one sub-family of the family of all possible joint distributions of the subclass numbers and the random effects. [ABSTRACT FROM AUTHOR]

The aim of this paper is two-fold: (1) To give a unified treatment of the theory of order statistics when the parent distribution is not necessarily continuous. (2) To assess the effects of grouping on the distribution of order statistics and to indicate the convenience, under suitable conditions, of using order statistics for the estimation of parameters from grouped data with or without censoring. [ABSTRACT FROM AUTHOR]

This paper presents three upper bounds for the variance of an estimator, based on observations selected with varying probabilities from a finite population, the elements of which are ranked with respect to the Y values. Accordingly, the usefulness of these bounds relates to the pre-enumeration analysis where one may well know the intended probabilities and joint probabilities corresponding to the sampling scheme but does not know the Y values. If, however, one can make a conservative guess at the largest Y value, one can use these bounds. Some examples are included to illustrate the theory. [ABSTRACT FROM AUTHOR]

This paper deals with certain aspects of variance component estimation for the unbalanced one-way random classification where the number (N[sub I]) of observations in the ith class is treated as a random variable not necessarily independent of the class effect (A[sub iota]). It is assumed that in general P(N[sub I] = 0) > 0. The conditional expectations (given the number of observations in each class) of all estimators of the between variance component (sigma[sup 2, sub alpha]) belonging to a certain class of estimators are derived. A general expression is found for the expected value of that estimator of sigma[sup 2, sub alpha] yielded by analysis of variance of class means. The limit of this expression (as the number of classes arrow right Infinity) is given; and it is shown that, if the bivariate distribution function of A[sub I], N[sub I] belongs to a certain class of distribution functions, then this limit is less than sigma[sup 2, sub a]. Numerical approximations to the expected values of two estimators of sigma[sup 2, sub a] are presented for one subclass of such distribution functions. [ABSTRACT FROM AUTHOR]

In many linear regression problems, the values of the independent variable or variables may be subject to certain constraints. For example, the independent variables may necessarily be positive; as another example, the variables may not only all be positive but are powers of a single variable (e.g., polynomial regression on time). Previous writers considering the problem of obtaining confidence bands on a regression function for all values of the independent variable have not utilized such constraints; the usual basis for such bands has been the multiple comparison procedure of Scheffe which places no constraints at all upon the independent variables. Any procedure utilizing constraints will necessarily yield a uniform improvement over the method of Scheffe (assuming both methods are applicable) in the sense of yielding narrower bands for a given confidence probability. In the present paper a nontrivial lower bound is obtained for the confidence probability associated with a multiple comparison procedure appropriate to the case where it can be assumed that each independent variable must be of specified sign; this includes, as a subclass, polynomial regression on a non-negative independent variable. This result gives a basis for a multiple comparison procedure less conservative than that of Scheffe when both are applicable. Implementation of the procedure requires the percentage points of a heretofore untabulated distribution. Tables of percentage points of this distribution appropriate to linear combinations of two, three, or four parameters are presented. [ABSTRACT FROM AUTHOR]

Tins paper illustrates, through a special case, an effect of non-independent numerators on F-test's in an analysis of variance. Symmetrical balanced incomplete block designs are considered with "blocks" a random effect and no interaction. The joint density of the adjusted sums of squares is determined under the null hypothesis of no treatment effect. Given the result of testing for the block effect, the conditional probability of a Type I error when testing the null hypothesis concerning the treatment effect is evaluated for a number of designs. [ABSTRACT FROM AUTHOR]