Search

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 develops two models, each of which is designed to estimate the probability of surviving from birth to selected exact ages of early childhood: namely ages two, three and five. The models are designed for use in areas with deficient registration systems. They require, as input, statistics which can be derived from retrospective data supplied by census or survey respondents. The first model, the age model, converts statistics on the proportion dead of children ever born to women in age groups 20-24, 25-29 and 30-34 into estimates of q2, q3 and q5. The second model, the marriage model, converts statistics on the proportion dead of children ever born to women of five-year marriage duration intervals into these estimates. The models can be used independently or simultaneously. These models were developed from data generated by a large number of empirical fertility and mortality schedules. Regression analysis was used to determine the parameter values of the relationships specified, and several sets of equations for estimating values of qa for a =2, 3 and 5 comprise the final product of the paper. It should be noted that the conceptual basis for the models was first derived by William Brass. The data generated for the regression analysis provided an opportunity to test the original Brass estimated model. We are able to report that the model performed well over the wide range of fertility and mortality conditions included in the test. [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]

While distribution-free procedures are often appropriate when testing statistical hypotheses, they may become complicated or involve loss of power when the data are grouped. For rank tests the ties caused by grouping are generally broken either by using a randomization procedure or averaging the tied ranks. In this paper the power loss due to equi-spaced grouping (in terms of Pitman asymptotic relative efficiency) is investigated for some commonly used tests, for each method of tie-breaking. The tests considered are Wilcoxon's and Mood's tests for the two-sample problem, Mann's test for randomness, and Pitman's independence test. It is shown how the power loss depends on the width of the grouping intervals and the distribution of the data, and some numerical studies are given. The results seem to indicate that the power loss is small even for a sizable group interval, and that it may be preferable to break ties by randomization than by averaging ranks. [ABSTRACT FROM AUTHOR]

Regression upon principal components of the percentage points of the income and education distributions for 1950 census tracts in the city of Chicago led to the estimation of "beta coefficient profiles" for television receiver and refrigerator ownership, for central heating system usage, and for a measure of dwelling unit overcrowding. The betas are standardized coefficients of regression of a dependent variable upon the proportions of families in the classes of the marginal income and education distributions. They measure the relative contribution of families in these classes to the over-all per cent saturation of the dependent variable in the tract. The coefficients were estimated by techniques developed in the first portion of the paper; estimation by classical regression methods would have been impossible because of multicollinearity. The empirical results are in substantial agreement with findings from regressions of the dependent variables upon the mean values of income and education, and their squares. The statistical devices appear to be useful in exploratory empirical research. [ABSTRACT FROM AUTHOR]

The need for data compression, a consequence of the demands made on the telemetry system of a space vehicle, prompts consideration of the use of sample quantiles in estimating population parameters and obtaining tests of goodness of fit for large samples. In this paper optimal unbiased estimators of the mean and standard deviation are given using up to twenty quantiles when the parent population is normal. Moreover, the estimators are relatively insensitive to deviations from normality. A distribution-free goodness-of-fit test is presented based on the sum of the squares of four quantiles after an orthogonal transformation to independent normal deviates. If a frequency function is of the form f(x; p) = pf[sub 1](x) + (1 - p) f[sub 2](x), 0 < p < 1, where f, and f[sub 2] are normal frequency functions, the distribution is likely to be bimodal. Another goodness-of-fit test is obtained using four quantiles, which is likely to have considerable power with a null hypothesis of normality and the alternative hypothesis of bimodality. The "data compression ratios" obtained with the use of a quantile system can be on the order of 100 to 1. [ABSTRACT FROM AUTHOR]