Showing papers on "Nonparametric statistics published in 1977"

On the Nonparametric Estimation of Regression Functions

Abstract: AN important statistical problem is the estimation of a regression function g(x) = E(y I x). Typically, g(x) has a specified functional form and parameter estimates are obtained according to certain desirable criteria, such as least squares. Assuming normal errors the investigator can test the appropriateness of the hypothesized model. One may wish, however, to have an estimation technique applicable for an arbitrary g(x). In a recent paper, Priestley and Chao (1972) considered the problem of estimating an unknown regression function g(x) given observations at a fixed set of points. Their estimate, referred to here as the Priestley-Chao (PC) estimate, is nonparametric in the sense that g(x) is restricted only by certain smoothing requirements. It can be viewed as a moving average of sample Y's whose weights are based on a class of kernels suggested by Rosenblatt (1956) and Parzen (1962). These weights are similar to those used in nonparametric density estimation. In this paper, the results of Priestley and Chao will be reviewed and further properties of their estimate considered.

How to apply the method of stochastic approximation in the non-parametric estimation of a regression function 1

Abstract: Let (Xi,Yi), (X2,Y2),… be independent, identically distributed pairs of r.v.'s such that .Let r0(x)≡0 and where αn=n-α(0<α<1) and K(.) is a suitable ensity function. Theorems are proved stating that r n (x) is a good estimation of r(x).

Tests of equality of dependent correlation coefficients

Abstract: SUMMARY A parametric test given known variance ratios, and a nonparametric test, both for the equality of correlation coefficients of one variable with a set of other variables, are proposed and their power functions are compared

On the theory of the evaluation of joint orientation measurements

Abstract: On the Theory of the Evaluation of Joint Orientation Measurements The current procedure for the evaluation of sets of joint-orientation measurements in terms of regional stresses is critically reviewed. It is shown that the counting circles on a sphere for constructing density diagrams are not circles on an equal-area (Lambert) projection of such a sphere; the correct curves are presented. It is shown that the search for modal maxima in the density diagrams corresponds to a nonparametric statistic for which it is difficult to give confidence limits. General considerations about the statistics of experiments yield the result that 12–15 joint measurements in an outcrop are sufficient to define 3 maxima, if such maxima exist at all. In order to give confidence limits, it is best to introduce a parametric model for the distribution of joint orientations. For a single cluster of orientations the probability density function is chosen as proportional to exp (k2 cos2θ) whereθ is the polar deviation angle from the “mean” direction; since the basic joint-orientations havethree fundamental directions, three probability density functions of the above type have to be superposed. These are determined by giving 11 parameters which can be determined by a function-minimization procedure from a given set of measurements. This procedure is best carried out on a computer. In this fashion, confidence limits for the preferred joint directions can be obtained. For the determination of the stresses, only two preferred sets of joints are required reducing the necessary number of parameters to 7. According to the usual theory of the statistics of trials, it is seen that 21 measurements are required to determine them. Thus, using a parametric statistic, it is seen that 21 measurements of joint orientations in an outcrop should suffice to determine the stress field that produced them. If the sharpness of the distributions is assumed as a priori known, this reduces again to the 15 or 20 measurements required for a non-parametric evaluation.

On a Class of Partially Sequential Two-Sample Test Procedures

Abstract: A class of test procedures is proposed for the two-sample setting when one of the samples has a fixed sample size, and the second is obtained by an inverse binomial sequential sampling scheme. These procedures allow either a classical approach or a completely nonparametric attack to a wide variety of two-sample problems, Properties of the tests are discussed, Including limiting distributions as the fixed sample size tends to infinity, and the capability of obtaining asymptotic power restrictions against specified alternatives of interest. Criteria for selecting a particular procedure are discussed.

On the Asymptotic Equivalence of Two Ranking Methods for $K$-Sample Linear Rank Statistics

Abstract: Two methods of ranking $K$ samples for rank tests comparing $K$ populations are considered. The first method ranks the $K$ samples jointly; the second ranks the $K$ samples pairwise. These procedures were first suggested by Dunn (1964), and Steel (1960), respectively. It is shown that both ranking procedures are asymptotically equivalent for rank-sum tests satisfying certain nonrestrictive conditions. The problem is formulated in terms of multiple comparisons, but is applicable to other nonparametric procedures based on $K$-sample rank statistics.

An evaluation of some non-parametric and parametric tests for location equality

Abstract: The Kruskal-Wallis and normal scores non-parametric tests of location equality are compared to the parametric analysis of variance F test. In addition to quantifying and manipulating degrees of variance heterogeneity due to combining heterogeneous variances with unequal sample sizes, the tests are compared for their sensitivity, as well as rates of Type I error, under varying conditions of mean variability when sampling from Gaussian and exponential distributions. Even though the asymptotic literature (Pratt, 1964; Puri, 1964) favours the normal scores test, it is found that the Kruskal-Wallis test is preferable to the normal scores test while the choice between the Kruskal-Wallis and analysis of variance tests depends upon population shape.

Tied-Down Wiener Process Approximations for Aligned Rank Order Processes and Some Applications

Abstract: For independent random variables distributed symmetrically around an unknown location parameter, aligned rank order statistics are constructed by using an estimator of the location parameter based on suitable rank statistics. The sequence of these aligned rank order statistics is then incorporated in the construction of suitable stochastic processes which converge weakly to some Gaussian functions, and, in particular, to tied-down Wiener processes in the most typical cases. The results are extended for contiguous alternatives and then applied in two specific problems in nonparametric inference. First, the problem of testing for shift at an unknown time point is treated, and then, some sequential type asymptotic nonparametric tests for symmetry around an unknown origin are considered.

Nonparametric Methods for Quantitative Analysis

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Robust Univariate Test of Symmetry

Abstract: This paper presents a test of symmetry based on the gaps of the order statistics. This test is sensitive to small skewed transformations of symmetric random variables and is adjusted for the apparent long-tailedness of the sample. A comparison of this adjusted test, the usual skewness test g 1, and Gupta's asymptotically nonparametric procedure shows that the procedure presented here has the best performance as a test of symmetry.

A nonparametric test of total independence based on Kendall's tau

Abstract: compute multivariate generalizations of tau. Under the hypothesis of total independence in K dimensions, the computed quantities can be shown to have asymptotic normal distributions, thereby permitting a test of the total independence hypothesis. Significant taus will provide information about nonindependence.

A class of distributions useful in life testing and reliability with applications to nonparametric testing

Abstract: . A new class of distributions having a finite range that are useful in life testing and reliability is proposed. Methods of estimating the unknown parameters are studied. Explicit expressions for lower moments of order statistics in random samples of any size are derived. A uniformly most powerful test is derived for the ‘shape’ parameter and it is illustrated by the data on times to failure of refrigerator motors. This class of distributions and their generalization constitute augmented Lehmann alternatives and locally most powerful rank tests with respect to these alternatives are derived and their Pitman efficiencies are evaluated. For a certain hypothesis - testing problem, an asymptotically distribution - free test is obtained.

Tests for the exponential distribution with censored data using Cramer-von Mises statistics

Abstract: SUMMARY This paper considers modifications of Cramer-von Mfises type statistics so that the statistics can be used to test the goodness of fit of censored data to the exponential distribution. Asymptotic percentage points are given for the statistics. In this paper we give the asymptotic distributions of two Cramer-von Mises type statistics used to test for the exponential distribution with censored data when the scale parameter must be estimated. The distributions given are those of the statistics

Asymptotic expansions for the distribution functions of linear combinations of order statistics

Abstract: Publisher Summary This chapter discusses asymptotic expansions and explains their need. It reviews the classical theory of Edgeworth expansions for sums of independent and identically distributed random variables, and indicates the two main techniques for extending this theory to more general statistics. The chapter presents an account of as yet unpublished results of Bjerve and Helmers who establish Berry–Esseen type bounds for linear combinations of order statistics. For many years, mathematical statisticians have spent a great deal of effort and ingenuity toward applying the central limit theorem in statistics. The estimators and test statistics that interest statisticians are as a rule not sums of independent random variables, and much work went into showing that they can often be approximated sufficiently well by such sums to ensure asymptotic normality. This work can be traced throughout the development of mathematical statistics from the proof of the asymptotic normality of the maximum likelihood estimator to much of the recent work in nonparametric statistics.

Nonparametric percentile estimate of clinical normal ranges.

Abstract: In this study, the nonparametric percentile estimate (PE), a statistical procedure requiring no previous assumption regarding the distribution of the underlying population, was used to determine adult normal limits for fasting plasma glucose, serum alkaline phosphatase, and urine amylase. Comparisons to the gaussian distribution were performed with histograms, symmetry calculations, plots on probability paper, and the chi-square test. The nonparametric and traditional PEs agreed perfectly when the glucose data fit the gaussian and log-gaussian curves. The nonparametric PE varied from the gaussian PE but was identical to the log-gaussian PE when the alkaline phosphatase activities followed the log-gaussian form. It differed sharply from the gaussian PE and was similar to the log-gaussian PE when the amylase values did not follow either the gaussian or log-gaussian model. The non-parametric PE was as efficient as its gaussian theory competitors when the assumed distributions were correct and, in most instances, was more accurate when the assumed distributions were incorrect.

A Nonparametric Analogy of Analysis of Covariance

Abstract: A nonparametric test of the hypothesis of no treatment effect is suggested for a situation where measures of the severity of the condition treated can be obtained and ranked both pre- and post-treatment. The test allows use to be made of the pre-treatment rank as a "concomitant variable," and is based on the nature and degree of permutation of the post-treatment ranks relative to the pre-treatment ranks. Evidence is given which shows that the distribution of the suggested test statistic rapidly approaches the F distribution as the number of subjects is increased. For small samples, a randomization test may be performed.

Comparison of two nonparametric methods for estimating the reliability of motor performance tests.

Abstract: The purpose of this study was to compare the power and sampling distributions of the coefficient of concordance (W) and the weighted average tau (W a ) for the selected sets of parameters m, n, and α. The values of m and n were selected to represent situations in physical education for which test reliability and judge objectivity might be estimated. The comparison of W and Wa required the generation of data matrices with varying amounts of error (α) for each set of values m and n. The obtained distributions were used to make the necessary comparisons. Several conclusions were drawn: First, the sampling distributions of W and Wa were similar for the sets of parameters investigated in this study. Second, W and Wa were equally powerful for the sets of parameters investigated in this study. And third, the comparisons did not indicate that either statistic is more appropriate for estimating the reliability of motor performance tests. On the basis of the practical considerations of ease of computation and famil...

A class of nonparametric tests for location and scale parameters

Abstract: A class of nonparametric two-sample tests for testing identity of distributions versus alternatives containing both location and scale parameters is proposed and some properties are derived. A recursion formula for the exact distribution under the hypothesis is presented and, the asymptotic distribution is given under both the hypothesis and a contiguous sequence of alternatives. Some asymptotic optimality properties are deduced for particular tests of the class and finally, the asymptotic efficiency is found.

Technical Note—A Note on the Nonparametric Analysis of the Stollmack-Harris Recidivism Data

Abstract: We show how quantitative methodology from the field of medical trials can be applied to the analysis of recidivism data from correctional release programs. The methods used are nonparametric and so do not depend on any assumptions concerning an underlying model. The data of Stollmack and Harris are used for illustration.