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Given N points distributed at random on [0,1), let np, be the size of the largest number of points clustered within an interval of length p. Previous work finds Pr (np ≥ n), for n> N/2, and for n < N/2, p=1/L, L an integer. The formula for the case p=1/L is in terms of the sum of L × L determinants and is not computationally feasible for large L. The present paper derives such a computational formula. [ABSTRACT FROM AUTHOR]

An earlier paper [3] provides axioms for the seasonal adjustment of economic time series. Seasonally adjusted data should be minimum variance, unbiased, and linear in an appropriate sense. These axioms yield a unique method for seasonal adjustment. The seasonally adjusted data may be obtained by an application of ordinary least squares regression. The problem of seasonal adjustment of economic time series in [3] is not the same as the problem of seasonal adjustment of data for econometric analysis. The first problem is completely resolved by appeal to the axioms of minimum variance, unbiasedness, and linearity. The second problem requires formulation as a standard problem in econometrics: estimation of the parameters of a single equation in a system of simultaneous equations. Lovell [4] has proposed to solve the problem of seasonal adjustment of data for econometric analysis by applying ordinary least squares directly to a structural equation in a system of simultaneous equations. Ordinary least squares estimation of the parameters of a structural equation usually results in "least squares bias." However, under certain special assumptions such a procedure can be justified, as Wold [6] has pointed out. Under these assumptions Lovell's procedure is valid. In this paper the seasonal adjustment of data for econometric analysis is formulated as a general simultaneous equations problem. Conditions for identification of the parameters to be estimated and methods for constructing consistent and asymptotically unbiased and efficient estimates are derived. Extensions to multivariate problems of seasonal adjustment for econometric analysis are sketched. [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 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]

This paper carries on work in [2] and [4] on the asymptotic effects of small specification error in simultaneous equation estimation. The specification errors considered are inherent in the approximate nature of econometric models. It is shown that for any parameter and small enough specification error, the probability limits of two-stage least squares and limited-information maximum-likelihood, while different, lie on the same side of the true parameter value, the difference between them being of the second order of smalls relative to the effects of the specification error on either one of them. A method of consistently estimating the sign of the difference of the effects of specification error on the two estimators is given and a procedure suggested for choosing an estimator to reduce those effects. Two relatively simple examples are worked out. [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]

The experiments to be discussed in this article was designed to investigate empirically the distribution of the sample version of the measure of association (G), &b.gamma;. Professors, William H. Kruskal and Leo A. Goodman have published an article giving interpretive motivation for the coefficient, and have developed the large sample theory for G. In the present research, sampling experiments were performed to test the adequacy of the large sample theory for 5X5 population cross classifications and samples of size ten, twenty-five and fifty. The numerical experiment described in this paper was based on a number of 25 cells multinomial populations, each considered as represented by the probabilities in a 5X5 population cross classification. Gamma is defined by Goodman and Kruskal in terms of probabilities. The 100 population cross classifications were chosen to be representative of practical situations in psychology. The method use to obtain them was to begin with 5X5 cross classification having complete association along the main diagonal and with a variety of marginal distributions.

Abstract Given a population of N units, it is required to draw a sample of n distinct units in such a way that the probability pi[sub i], (i = 1, ..., N) for the ith unit to be in the sample is proportional to its 'size' x[sub i]. One way to achieve this is as follows: The N units in the population are listed in a random order and their x[sub i] are cumulated and a systematic selection of n elements from a "random start" is then made on the cumulation. The mathematical theory associated with this procedure has been presented in [1], where with the help of asymptotic theory, compact expressions for the variance of the estimate of the population total are derived. These expressions contain probabilities P[sub ii]'(i not equal to i'; i, i' = 1, ... , N) that units i and i' both are in the sample. For n = 2, N = 3 and n = 2, N = 4, exact formulae are derived in [1], but for other n and N only approximate formulae are obtained. It is the purpose of this paper to derive an exact formula for P[sub ii]' for any values of n and N. The argument makes use of the solution given in [1] for the case n = 2 and N = 4. The new formula for n = 2 is derived in Section 1 and illustrated in Section 2 by a numerical example for N = 10. An exact formula for general n is derived in Section 3 and examplified in Section 4. [ABSTRACT FROM AUTHOR]

A model is described which leads to an analysis of data on the sequence of sexes in human families, particular attention being given to the detection of a correlation between the sexes of adjacent sibs. Three sources of potential confounding with this correlation are discussed. The method of analysis avoids any confounding with the effects of voluntary limitation of family size, or with the effects of variation in the probability of a male from birth to birth within a family. It is more difficult to separate the effect of correlation between the sexes of adjacent sibs from the effect of variation between families in the probability that a child born to the family will be male. The problem was solved in the present investigation by using an auxiliary analysis to show that families did not, in fact, differ in this probability. If this were not so, a separation of the two factors could still have been made by an additional analysis referred to in the paper. [ABSTRACT FROM AUTHOR]

A general model for the evaluation of partial transfer evidence is developed in this article. This model is less restrictive than were the original models presented in an earlier paper, thus allowing it to be applied to a more general class of situations. A Bayesian approach to the problem of evidence evaluation is explored, and the probability of error calculated in the Bayesian model is shown to be a reasonable approximation to the probability of possible error calculated in the general model when these probabilities are small. [ABSTRACT FROM AUTHOR]

A brief description is given of a population sample of the County Borough of Dudley in which households were sampled with probabilities proportional to the number of adults in the household. 2. The variance of estimates for specific age-sex groups is considered and an empirical regression obtained relating the coefficient of variation to the size of the group. 3. Estimates are given of the variance that would result from a number of alternative schemes of sampling and estimation. 4. The possibility of increased, precision by district stratification is examined and shown to be negligible. 5. The variance to be expected in the age-sex tabulations from the I % sample of the 1951 Population Census is estimated from the results of the previous analysis. [ABSTRACT FROM AUTHOR]

When data are transformed to satisfy a spherical normal linear model, the mean θ of a variate in the original scale is a function of the mean μ and variance σ² of a normal variate. We consider several approximate confidence interval methods for θ, including a new method based on exact confidence intervals for linear functions of μ and σ² Monte Carlo estimates of coverage probabilities demonstrate the suitability of the new method for applications involving a wide range of data transformations, parameter values and sample sizes. [ABSTRACT FROM AUTHOR]

A table of critical values of the Wilcoxon Signed Rank Statistic is presented for N = 4(1)100 pairs of observations at one-tail probability levels of .00005, .0005, .0025, .005(.005).025, .050(.025).150, and .20(.05).45. Probabilities were computed accurately to at least 6 digits, regardless of the location of the decimal point. Therefore, all critical values are correct as tabled. Normal approximation probabilities were found to be biased, too small at the .05 level but too large at the .005 and .0005 probability levels. Approximation errors were less than 10% for all N >35 for the .05, .025, and .005 one-tail probability levels; but this degree of accuracy was not achieved at the .0005 level for N = 100. [ABSTRACT FROM AUTHOR]

How can you use a product-moment correlation coefficient when you do not have interval scale data? This is particularly addressed to uses of such statistics with personality, value, or other 'soft' variables, and to variables that are sometimes considered to be composed of discrete categories. The implicit contrast usually is to a nonparametric or sidtribution free statistic like Maurice G. Kendall's Tau as more appropriate with such data. Since part of the confusion that surrounds the question of the appropriateness of use of various statistics occurs about the type of measurement involved, we may focus on the presentation of Herbert M. Blalock on this topic.

Official life tables are frequently calculated for a period of years, rather than for an individual year, and the question arises, how annual rates are to be combined, in order to give an indication of the average mortality of the period. The author examines this problem, and uses methods based on the binomial probability distribution to suggest a solution of the 'weighting' problem. Taking as his starting-point the work of the Dutch statistician Van Pesch, he modifies the latter's theory so as to make it applicable to the case, where mortality rates have a secular downward trend, and reaches the conclusion that the 'most probable values for the mortality rates are not obtained by applying the weighted mean, but by the application of a weighted mean and a correction term. The inclusion of the correction term means that, practically speaking, the results do not differ from those obtained by the application of the unweighted mean. The unweighted mean, which has the advantage of requiring less computational work, may therefore be given preference over the theoretically more accurate method.' [ABSTRACT FROM AUTHOR]