Showing papers on "Nonparametric statistics published in 1976"

Multivariate binary discrimination by the kernel method

Abstract: SUMMARY An extension of the kernel method of density estimation from continuous to multivariate binary spaces is described. Its simple nonparametric nature together with its consistency properties make it an attractive tool in discrimination problems, with some advantages over already proposed parametric counterparts. The method is illustrated by an application to a particular medical diagnostic problem. Simple extensions of the method to categorical data and to data of mixed binary and continuous form are indicated.

Detection of trends in water quality data from records with dependent observations

Abstract: Classical statistical tests for trend, both parametric and nonparametric, assume independence of observations, a condition rarely encountered in time series obtained by using moderate to high sample frequencies. A method is developed for summarizing the power of the parametric t tests and the nonparametric Spearman's rho test and Mann-Whitney's test against step and linear trends in a dimensionless ‘trend number’ which is a function of trend magnitude, standard deviation of the time series, and sample size. For the case of dependent observations, use of an equivalent independent sample size rather than the actual sample size is shown to enable use of the same trend number developed for the independent case. An important related result is the existence of an upper limit on power (trend detectability) over a fixed time horizon, regardless of the number of samples taken, for a lag 1 Markov process.

Descriptive Statistics for Nonparametric Models. III. Dispersion

Abstract: Measures of dispersion are defined as functionals satisfying certain equivariance and order conditions. In the main part of the paper attention is restricted to symmetric distributions. Different measures are compared in terms of asymptotic relative efficiency, i.e., the inverse ratio of their standardized variances. The efficiency of a trimmed to the untrimmed standard deviation turns out not to have a positive lower bound even over the family of Tukey models. Positive lower bounds for the efficiency (over the family of all symmetric distributions for which the measures are defined) exist if the trimmed standard deviations are replaced by pth power deviations. However, these latter measures are no longer robust, although for p <2 they are more robust than the standard deviation. The results of the paper suggest that a positive bound to the efficiency may be incompatible with robustness but that trimmed standard deviations and pth power deviations for p = 1 or 1.5 are quite satisfactory in practice.

Basic statistical concepts

Abstract: 1. Some Thoughts on Measurement. 2. Frequency Distributions and Graphical Methods. 3. Central Tendency. 4. Variability. 5. The Normal Curve. 6. Sampling Theory for Hypothesis Testing. 7. Correlation. 8. Prediction and Regression. 9. The Significance of the Difference Between Means. 10. Decision Making, Power, and Effect Size. 11. One-Way Analysis of Variance. 12. Two-Way Analysis of Variance. 13. Some Nonparametric Statistical Tests. References. Appendix 1: Formulas. Appendix 2: Basic Mathematics Refresher. Appendix 3: Statistical Tables. Appendix 4: Computer-Assisted Data Analysis. Answers to Odd-Numbered Problems. Index.

Tests of hypotheses based on ranks in the general linear model

Abstract: A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.

A survey of nonparametric tests for scale

Abstract: A surrey of nonparametric tests for scale is presented. Some parametric procedures are also presented for the purpose of comparison. Suggestions on which procedures to use in various situations are given. Both 2-sample and c-sample tests are included

A note on goodness-of-fit statistics for probit and logit models

Abstract: In the case of models designed to explain the choice among a finite set of alternatives, a number of goodness-of-fit statistics have been reported This paper is primarily concerned with the properties of one of these statistics, the likelihood ratio index By comparing the likelihood ratio index with some of the other statistics and by examining its mathematical properties, it is concluded that the index has desirable properties in binary and multinomial situations However, the way in which the likelihood ratio index has been applied in many recent studies has led to results which are possibly unexpected In these cases, the index was a measure of the extent to which a hypothesized model improved upon the explanatory power of a model with all coefficients, including the constant or the coefficients of alternative-specific dummies, equal to zero It is shown that the minimum value of this likelihood ratio index depends on the relative proportions of sampled individuals selecting the various alternatives, contrary to the expectation of a zero minimum value The dependence on the sampled proportions also prevents comparison of indices resulting from different samples A simple adjustment alleviates these difficulties This new definition makes the likelihood ratio index a measure of the extent to which the hypothesized model improves upon the explanatory of a model with only a constant or alternative-specific dummies It is recommended that this index is more appropriate for assessing the value of choice models

Schaum's outline of theory and problems of business statistics

Abstract: Analyzing business data statistical presentations describing business data - measures of location describing business data - measures of variability probability probability distributions for discrete random variables binomial, hypergeometric and Poisson probability distributions for continuous random variables normal and exponential sampling distributions and confidence intervals for the mean other confidence intervals testing hypotheses concerning the value of the population mean testing other hypotheses the chi-square test analysis of variance linear regression and correlation analysis multiple regression and correlation time series analysis and business forecasting index numbers for business and economic data Bayesian decision analysis - payoff tables and decision trees Bayesian decision analysis - the use of sample information Bayesian decision analysis - application of the normal distribution nonparametric statistical tests.

Nonparametric probability density estimation by optimization theoretic techniques

Abstract: Two nonparametric probability density estimators are considered. The first is the kernel estimator. The problem of choosing the kernel scaling factor based solely on a random sample is addressed. An interactive mode is discussed and an algorithm proposed to choose the scaling factor automatically. The second nonparametric probability estimate uses penalty function techniques with the maximum likelihood criterion. A discrete maximum penalized likelihood estimator is proposed and is shown to be consistent in the mean square error. A numerical implementation technique for the discrete solution is discussed and examples displayed. An extensive simulation study compares the integrated mean square error of the discrete and kernel estimators. The robustness of the discrete estimator is demonstrated graphically.

Dead‐zone limiter: An application of conditional tests in nonparametric detection

Abstract: The concept of a conditional statistical test is applied to obtain a nonparametric detector based on the dead‐zone limiter nonlinearity. The resulting detector is easily implemented, and its performance is shown to be superior to that of the sign detector for both large and small samples.Subject Classification: [43]60.20,[43]60.30.

A class of location-scale nonparametric tests

Abstract: SUMMARY A class of two-sample nonparametric tests for location and scale shift, simultaneously, is proposed. The asymptotic distribution of the proposed class is obtained both under the hypothesis and alternative. Using this result it is shown how the asymptotic relative efficiency of tests in the proposed class can be obtained. F2(x) = F(a2x +4f2), respectively. The purpose of this paper is to study a class of two-sample nonparametric test statistics which can be used to test HO: Fl(x) = F2(x) = F(x) for all real x, versus the alternative Ha:acc * a2 or /h t /h' or both. The proposed statistics will be expressed as functions of two nonparametric statistics, one sensitive to scale shift and the other to location shift. Lepage (1971) proposed a statistic which is a combination of the Wilcoxon and Ansari- Bradley statistics. Furthermore, he showed that since the Wilcoxon and Ansari-Bradley statistics are uncorrelated under Ho, his proposed statistic has a limiting central chi- squared distribution with two degrees of freedom. Thus the critical values needed to carry out an approximate test of Ho can be obtained from a standard chi-squared table. The question regarding the asymptotic power behaviour of Lepage's statistic is of interest but is not, however, discussed by Lepage. Thus the asymptotic relative efficiency of the proposed statistics will also be discussed, as a means of comparing such statistics.

Estimation of the Innovation Generalized Variance of a Multivariate Stationary Time Series

Abstract: Recently there has been renewed interest in the nonparametric estimate of the innovation variance of a stationary time series. Empirical evidence has shown that some smoothing of the periodogram improves the estimate. This paper attempts to clarify the smoothing question and extends the results to the estimation of the innovation generalized variance of a multivariate stationary time series.

A Monte Carlo Test of the Robustness of Parametric and Nonparametric Analysis of Covariance Against Unequal Regression Slopes

Abstract: The power of parametric ANCOVA was compared with the power of three nonparametric ANCOVA techniques both when the parametric assumption of equal slopes was violated and when all parametric assumptions were met. The data situations involved two groups with one covariate and one criterion, normal distributions, equal group sizes, and varying slope combinations. Parametric ANCOVA maintained larger empirical power for nearly all of the data situations. Both parametric and nonparametric techniques appeared not to be robust when violation of the parametric assumption of equal slopes was coupled with unequal group sizes and distributions were normal.

Use of values for calcium and protein in serum, and of a derived index obtained from a probability population sample.

Abstract: We measured serum protein and calcium concentrations in 2340 individuals between 10 and 96 years of age from 900 families chosen by probability methods to give a representative population. These values were used to calculate an index, based on a regression analysis of serum protein on calcium, which was then treated as a new variable. Age-sex specific reference values and frequency distributions are presented for this index as well as for protein and calcium calculated by both parametric and nonparametric methods.

Nonparametric Learning Without a Teacher Based on Mode Estimation

Abstract: The present paper discusses a nonsupervised multicategory problem in terms of nonparametric learning. An algorithm for seeking modes of an unknown multidimensional probability density function (pdf) is considered by employing a hypercubic window function. The convergence proof of the algorithm is also presented. The discriminant function for multicategory problems is constructed by using the estimates of the modes of the multimodal pdf. An application of the mode estimation algorithm to nonparametric signal detection is described. The analytical result shows that our machine nearly converges to the optimal machine without supervision.

Schaum's Outline of Business Statistics

Abstract: Schaum's Outline of Business Statistics, 4ed 1. Analyzing Business Data 2. Statistical Presentations and Graphical Displays 3. Describing Business Data: Measures of Location 4. Describing Business Data: Measures of Dispersion 5. Probability 6. Probability Distributions for Discrete Random Variables: Binomial, Hypergeometric, and Poission 7. Probability Distributions for Continuous Random Variables: Normal and Exponential 8. Sampling Distributions and Confidence Intervals for the Mean 9. Other Confidence Intervals 10. Testing Hypotheses Concerning the Value of a Population Mean 11. Testing Other Hypotheses 12. The Chi-Square Test for the Analysis of Qualitative Data 13. Analysis of Variance 14. Linear Regression and Correlation Analysis 15. Multiple Regression and Correlation 16. Time Series Analysis and Business Forecasting 17. Nonparametric Statistics 18. Decision Analysis: Payoff Tables and Decision Trees 19. Statistical Process Control

A Nonparametric Test for the Correspondence Between Two Proximity Matrices

Abstract: A nonparametric test, developed in the biometrics literature for the study of contagious diseases, may be used more generally to compare the patterning of entries in two proximity matrices. The procedure is a simple extension of the randomization strategies appropriate for the standard bivariate correlation context, and provides a relatively easy alternative to some of the more informal analyses commonly used in education.

A new dimension to nonparametric tests

Abstract: It is demonstrated that a statistician can take a preliminary peek- at the data in a special vay in order to select an appropriate nonparametric test statistic without spoiling the distribution-free property. This process (adaptive selection and then testing) generally improves zhe paver over a vide range of underlying distributions and is the nev -dimension to nonparametric tests. Several illustrations are given using standard nonparametric techniques involving the usual one-sample, tvo-sample, k-sample, and independence testing situations. More recent (and not standard) applications to regression and classification problems are then considered. Finally one 'difficulty concerning this adapting process is noted, but a corrective action is suggested.

Asymptotic Comparison of Rank Tests for the Regression Problem When Ties are Present

Abstract: Without assumptions on the underlying distributions we prove the asymptotic normality of averaged scores rank statistics under all product distributions which are contiguous to the null hypothesis, and find a very simple form of the centering constants. In the one-sided two-sample and trend situations this enables us to show that monotonicity of the scores generating function is equivalent to the asymptotic unbiasedness of the corresponding averaged scores rank test. For such asymptotically unbiased tests we prove simple necessary and sufficient conditions for having bounds for their asymptotic relative efficiency under all contiguous alternatives of the model. As a by-product, we get the local asymptotic superiority of averaged scores rank tests to the associated tests with randomized ranks not only for shift but for general alternatives. In addition we prove that every one-sided averaged scores rank test is asymptotically most powerful (asymptotically equivalent to likelihood ratio test) for a suitable nonparametric subclass of alternatives, provided the test and the associated subclass of alternatives are generated by a nondecreasing, square-integrable function defined on the unit interval.

Nonparametric tests for block experiments

Abstract: SUMMARY A method is described for constructing nonparametric tests for experiments arranged in blocks when the underlying distributions are of Lehmann's form. In simple cases the resulting test statistics are those that would be obtained by an analysis of variance following a trans- formation of the data into exponential scores. In the past considerable work has been done on examining the power of various standard nonparametric tests under the so-called Lehmann alternative, where the probability in the upper tail of the alternative distribution is a power of the probability in the upper tail of the null distribution. With the exception of certain two-sample tests little effort seems to have been devoted to deriving tests more appropriate to this underlying situation, which is of considerable importance in life testing. In this note it is shown how the condi- tional likelihood approach adopted by Cox (1972) may be used to derive nonparametric test statistics for a variety of situations. Because of their method of construction the resulting tests necessarily have asymptotically optimal properties under the Lehmann alternative, but their small-sample behaviour still needs investigation. In some cases the test derived by this method is a generalization of an existing well-known test.

A brief survey of nonparametric statistics

Abstract: The area of non-parametrie statistics has briefly been surveyed. The survey is limited to univariate fixed-sample size problems