Showing papers on "Nonparametric statistics published in 1981"

Rank Transformations as a Bridge between Parametric and Nonparametric Statistics

Abstract: Many of the more useful and powerful nonparametric procedures may be presented in a unified manner by treating them as rank transformation procedures. Rank transformation procedures are ones in which the usual parametric procedure is applied to the ranks of the data instead of to the data themselves. This technique should be viewed as a useful tool for developing nonparametric procedures to solve new problems.

Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods

Abstract: SUMMARY We discuss several nonparametric methods for attaching a standard error to a point estimate: the jackknife, the bootstrap, half-sampling, subsampling, balanced repeated replications, the infinitesimal jackknife, influence function techniques and the delta method. The last three methods are shown to be identical. All the methods derive from the same basic idea, which is also the idea underlying the common parametric methods. Extended numerical comparisons are made for the special case of the correlation coefficient.

An Introduction to Mathematical Statistics and Its Applications

Abstract: 1. Introduction 1.1 An Overview 1.2 Some Examples 1.3 A Brief History 1.4 A Chapter Summary 2. Probability 2.1 Introduction 2.2 Sample Spaces and the Algebra of Sets 2.3 The Probability Function 2.4 Conditional Probability 2.5 Independence 2.6 Combinatorics 2.7 Combinatorial Probability 2.8 Taking a Second Look at Statistics (Monte Carlo Techniques) 3. Random Variables 3.1 Introduction 3.2 Binomial and Hypergeometric Probabilities 3.3 Discrete Random Variables 3.4 Continuous Random Variables 3.5 Expected Values 3.6 The Variance 3.7 Joint Densities 3.8 Transforming and Combining Random Variables 3.9 Further Properties of the Mean and Variance 3.10 Order Statistics 3.11 Conditional Densities 3.12 Moment-Generating Functions 3.13 Taking a Second Look at Statistics (Interpreting Means) Appendix 3.A.1 MINITAB Applications 4. Special Distributions 4.1 Introduction 4.2 The Poisson Distribution 4.3 The Normal Distribution 4.4 The Geometric Distribution 4.5 The Negative Binomial Distribution 4.6 The Gamma Distribution 4.7 Taking a Second Look at Statistics (Monte Carlo Simulations) Appendix 4.A.1 MINITAB Applications Appendix 4.A.2 A Proof of the Central Limit Theorem 5. Estimation 5.1 Introduction 5.2 Estimating Parameters: The Method of Maximum Likelihood and the Method of Moments 5.3 Interval Estimation 5.4 Properties of Estimators 5.5 Minimum-Variance Estimators: The Crami?½r-Rao Lower Bound 5.6 Sufficient Estimators 5.7 Consistency 5.8 Bayesian Estimation 5.9 Taking A Second Look at Statistics (Beyond Classical Estimation) Appendix 5.A.1 MINITAB Applications 6. Hypothesis Testing 6.1 Introduction 6.2 The Decision Rule 6.3 Testing Binomial Dataâ H0: p = po 6.4 Type I and Type II Errors 6.5 A Notion of Optimality: The Generalized Likelihood Ratio 6.6 Taking a Second Look at Statistics (Statistical Significance versus â Practicalâ Significance) 7. Inferences Based on the Normal Distribution 7.1 Introduction 7.2 Comparing Y-i?½ s/ vn and Y-i?½ S/ vn 7.3 Deriving the Distribution of Y-i?½ S/ vn 7.4 Drawing Inferences About i?½ 7.5 Drawing Inferences About s2 7.6 Taking a Second Look at Statistics (Type II Error) Appendix 7.A.1 MINITAB Applications Appendix 7.A.2 Some Distribution Results for Y and S2 Appendix 7.A.3 A Proof that the One-Sample t Test is a GLRT Appendix 7.A.4 A Proof of Theorem 7.5.2 8. Types of Data: A Brief Overview 8.1 Introduction 8.2 Classifying Data 8.3 Taking a Second Look at Statistics (Samples Are Not â Validâ !) 9. Two-Sample Inferences 9.1 Introduction 9.2 Testing H0: i?½X =i?½Y 9.3 Testing H0: s2X=s2Yâ The F Test 9.4 Binomial Data: Testing H0: pX = pY 9.5 Confidence Intervals for the Two-Sample Problem 9.6 Taking a Second Look at Statistics (Choosing Samples) Appendix 9.A.1 A Derivation of the Two-Sample t Test (A Proof of Theorem 9.2.2) Appendix 9.A.2 MINITAB Applications 10. Goodness-of-Fit Tests 10.1 Introduction 10.2 The Multinomial Distribution 10.3 Goodness-of-Fit Tests: All Parameters Known 10.4 Goodness-of-Fit Tests: Parameters Unknown 10.5 Contingency Tables 10.6 Taking a Second Look at Statistics (Outliers) Appendix 10.A.1 MINITAB Applications 11. Regression 11.1 Introduction 11.2 The Method of Least Squares 11.3 The Linear Model 11.4 Covariance and Correlation 11.5 The Bivariate Normal Distribution 11.6 Taking a Second Look at Statistics (How Not to Interpret the Sample Correlation Coefficient) Appendix 11.A.1 MINITAB Applications Appendix 11.A.2 A Proof of Theorem 11.3.3 12. The Analysis of Variance 12.1 Introduction 12.2 The F Test 12.3 Multiple Comparisons: Tukeyâ s Method 12.4 Testing Subhypotheses with Contrasts 12.5 Data Transformations 12.6 Taking a Second Look at Statistics (Putting the Subject of Statistics togetherâ the Contributions of Ronald A. Fisher) Appendix 12.A.1 MINITAB Applications Appendix 12.A.2 A Proof of Theorem 12.2.2 Appendix 12.A.3 The Distribution of SSTR/(kâ 1) SSE/(nâ k)When H1 is True 13. Randomized Block Designs 13.1 Introduction 13.2 The F Test for a Randomized Block Design 13.3 The Paired t Test 13.4 Taking a Second Look at Statistics (Choosing between a Two-Sample t Test and a Paired t Test) Appendix 13.A.1 MINITAB Applications 14. Nonparametric Statistics 14.1 Introduction 14.2 The Sign Test 14.3 Wilcoxon Tests 14.4 The Kruskal-Wallis Test 14.5 The Friedman Test 14.6 Testing for Randomness 14.7 Taking a Second Look at Statistics (Comparing Parametric and Nonparametric Procedures) Appendix 14.A.1 MINITAB Applications Appendix: Statistical Tables Answers to Selected Odd-Numbered Questions Bibliography Index

A Comparative Study of Tests for Homogeneity of Variances, with Applications to the Outer Continental Shelf Bidding Data

Abstract: Many of the existing parametric and nonparametric tests for homogeneity of variances, and some variations of these tests, are examined in this paper. Comparisons are made under the null hypothesis (for robustness) and under the alternative (for power). Monte Carlo simulations of various symmetric and asymmetric distributions, for various sample sizes, reveal a few tests that are robust and have good power. These tests are further compared using data from outer continental shelf bidding on oil and gas leases.

Nonparametric standard errors and confidence intervals

Abstract: We investigate several nonparametric methods; the bootstrap, the jackknife, the delta method, and other related techniques. The first and simplest goal is the assignment of nonparametric standard errors to a real-valued statistic. More ambitiously, we consider setting nonparametric confidence intervals for a real-valued parameter. Building on the well understood case of confidence intervals for the median, some hopeful evidence is presented that such a theory may be possible.

The asymptotic properties of nonparametric tests for comparing survival distributions

Abstract: The asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary, and possibly unequal, right censoring. The test statistics include equivalents to the log rank statistic, the modified Wilcoxon statistic and the class of rank invariant test procedures introduced by Peto & Peto. When there are equal censoring distributions and the hazard functions are proportional the sample size formula for the F test used to compare exponential samples is shown to be valid for the log rank test. In certain situations the power of the log rank test falls as the amount of censoring decreases.

Practical Nonparametric Statistics (2nd ed.)

Abstract: (1981). Practical Nonparametric Statistics (2nd ed.) Technometrics: Vol. 23, No. 4, pp. 415-416.

A Bayesian Nonparametric Approach to Reliability

Abstract: : It is suggested that problems in a reliability context may be handled by a Bayesian non-parametric approach. A stochastic process is defined whose sample paths may be assumed to be either increasing hazard rates or decreasing hazard rates by properly choosing the parameter functions of the process. The posterior distribution of the hazard rates are derived for both exact and censored data. Bayes estimates of hazard rates,c.d.f.'s, densities, and means, are found under squared error type loss functions. Some simulation is done and estimates graphed to better understand the estimators. Finally, estimates of the c.d.f. from some data in a paper by Kaplan and Meier are constructed. (Author)

Business Statistics: A Decision Making Approach

Abstract: Chapter 1: The Where, Why and How of Data Collection Chapter 2: Graphs, Charts, and Tables - Describing Your Data Chapter 3: Describing Data Using Numerical Measures Chapter 4: Introduction to Probability Chapter 5: Introduction to Discrete Probability Distributions Chapter 6: Introduction to Continuous Probability Distributions Chapter 7: Introduction to Sampling Distributions Chapter 8: Estimating Single Population Parameters Chapter 9: Introduction to Hypothesis Testing Chapter 10: Estimation and Hypothesis Testing for Two Population Parameters Chapter 11: Hypothesis Tests for One and Two Population Variances Chapter 12: Analysis of Variance Chapter 13: Goodness-of-Fit Tests and Contingency Analysis Chapter 14: Introduction to Linear Regression and Correlation Analysis Chapter 15: Multiple Regression and Model Building Chapter 16: Analyzing and Forecasting Time-Series Data Chapter 17: Introduction to Nonparametric Statistics Chapter 18: Introduction to Quality and Statistical Process Control Chapter 19: Introduction to Decision Analysis

Computer analysis of EEG signals with parametric models

Abstract: Fifty years ago Berger made the first registrations of the electrical activity of the brain with electrodes placed on the intact skull. It immediately became clear that the frequency content of recorded signals plays an important role in describing these signals and also the state of the brain. This paper briefly surveys the main properties the electroencephalogram (EEG), and points out several influential factors. A number of methods have been developed to quantify the EEG in order to complement visual screening; these are conveniently classified as being parametric or nonparametric. The paper emphasizes parametric methods, in which signal analysis is based on a mathematical model of the observed process. The scalar or multivariate model is typically linear, with parameters being either time invariant or time variable. Algorithms to fit the model to observed data are surveyed. Results from the analysis my be used to describe the spectral properties of the EEG, including the way in which characteristic variables change with time. Parametric models have successfully been applied to detect the occurrence of transients with epiliptic origin, so-called spikes and sharp waves. Interesting results have also been obtained by combining parameter estimation with classification algorithms in order to recognize significant functional states of the brain. The paper emphasizes methodology but includes also brief accounts of applications for research and clinical use. These mainly serve to illustrate the progress being made and to indicate the need for further work. The rapid advance of computer technology makes the processed EEG an increasingly viable tool in research and clinical practice.

A class of hypothesis tests for one and two sample censored survival data

Abstract: This paper proposes a class of new non-parametric test statistics useful for goodness-of-fit or two-sample hypothesis testing problems when dealing with randomly right censored survival data. The procedures are especially useful when one desires sensitivity to differences in survival distributions that are particularly evident at at least one point in time. This class is also sufficiently rich to allow certain statistics to be chosen which are yery sensitive to survival differences occurring over a specified period of interest. The asymptotic distribution of each test statistic is obtained and then employed in the formulation of the corresponding test procedure. Size and power of the new procedures are evaluated for small and moderate sample sizes using Monte Carlo simulations. The simulations, generated in the two sample situation, also allow comparisons to be made with the behavior of the Gehan-Wilcoxon and log-rank test procedures.

Simultaneous Confidence Intervals for all Distances from the "Best"

Abstract: In practice, comparisons with the "best" are often the ones of primary interest. In this paper, parametric and nonparametric simultaneous upper confidence intervals for all distances from the "best" are derived under the location model. Their improvement upon the results of Bechhofer (1954), Gupta (1956, 1965), Fabian (1962), and Desu (1970) in the parametric case is discussed. In the nonparametric case, no comparable confidence statements were available previously.

Convergence of Dirichlet Measures and the Interpretation of Their Parameter.

Abstract: The form of the Bayes estimate of the population mean with respect to a Dirichlet prior with parameter a has given rise to the interpretation that a ( X ) is the prior sample size. Furthermore, if a ( X ) is made to tend to zero, then the Bayes estimate mathematically converges to the classical estimator, that is, the sample mean. This has further given rise to the general feeling that allowing a ( X ) to become small not only makes the prior sample size small but also that it corresponds to no prior information. By investigating the limits of prior distributions as the parameter a tends to various values, it is misleading to think of a ( X ) as the prior sample size and the smallness of a ( X ) as no prior information. In fact, very small values of a ( X ) actually mean that the prior has a lot of information concerning the unknown true distribution and is of a form that would be generally unacceptable to a statistician.

Nonparametric bivariate estimation with randomly censored data

Abstract: SUMMARY The estimation of a bivariate distribution function with randomly censored data is considered. It is assumed that the censoring occurs independently of the lifetimes, and that deaths and losses which occur simultaneously can be separated. Two estimators are developed: a reduced-sample estimator and a self-consistent one. It is shown that the latter estimator satisfies a nonparametric likelihood function and is unique up to the final uncensored values in any dimension; it jumps at the points of double deaths in both dimensions. Some key word8: Censored data; Kaplan-Meier estimator; Life table; Product-limit estimator; Survival estimation.

The Analysis of Social Interaction Data: A Nonparametric Technique

Abstract: A nonparametric technique is presented that is appropriate for comparing two social interaction matrices, either when both are obtained empirically, or when one is generated from some given theoret...

A class of nonparametric tests based on multiresponse permutation procedures

Abstract: SUMMARY A class of univariate rank tests based on multiresponse permutation procedures is introduced. Asymptotic equivalence is established between one member of this class and the Kruskal-Wallis test. Simulated power comparisons for location shift detection indicate that another member of the class provides superior detection efficiency for location shifts of bimodal distributions and of heavy tailed unimodal distributions. Furthermore, both tests possess substantial computational advantages over other members of the class.

Linear Functions of Concomitants of Order Statistics with Application to Nonparametric Estimation of a Regression Function

Abstract: Let (Xi , Yi )(i = 1, 2, …, n) be independent identically distributed as (X, Y). Then the rth ordered X variate is denoted by Xr:n and the associated Y variate, the concomitant of the rth order statistic, by Y [r:n]. This paper considers statistics of the form and more generally of the form , where J is a bounded smooth function and may depend on n. Under certain regularity conditions, the asymptotic normality of these statistics is established. These statistics are used to construct consistent estimators of various conditional quantities, for example E(Y | X = x), P(Y ∈ A | X = x) and var(Y | X = x).

Strong consistency properties of nonparametric estimators for randomly censored data, II: Estimation of density and failure rate

Abstract: This article is Part II of a two-part study. Properties of the product-limit estimator established in the previous part [2] are now used to prove the strong consistency of some nonparametric density and failure rate estimators which can be used with randomly censored data.

Nonparametric analysis of an accelerated failure time model

Abstract: SUMMARY Survival distributions can be characterized by and compared through their hazard functions. Tests using a proportional hazards model have good power if the two hazards do not cross, but without time-dependent covariates can have low power if they do. The method contained herein is designed to provide a complement to the proportional hazards model. Differences in survival distributions are parameterized by a scale change and the log rank statistic is used to generate an estimate of the change and a confidence interval based test. This approach is fully efficient for the Weibull family with no censoring. In general it compares favourably with the proportional hazards approach, although neither method dominates.

On nonparametric multivariate binary discrimination

Abstract: Aitchison & Aitken (1976) introduced a novel and ingenious nonparametric method for estimating probabilities in a multidimensional binary space. The technique is designed for use in multivariate binary discrimination. Their estimator depends crucially on an unknown smoothing parameter A, and Aitchison & Aitken proposed a maximum likelihood method for determining A from the sample. Unfortunately this leads to an adaptive estimator which can behave very erratically when there are a number of empty or near empty cells present. We demonstrate this both theoretically and by example. To overcome these difficulties we introduce another method of estimating A which is designed to minimize a global function of the mean squared error.

Linear nonparametric tests for comparison of counting processes, with applications to censored survival data : (preprint)

Abstract: Summary This paper surveys linear nonparametric one- and k-sample tests for counting processes. The necessary probabilistic background is outlined and a master theorem proved, which may be specialized to most known asymptotic results for linear rank tests for censored data as well as to asymptotic results for one- and k-sample tests in more general situations, an important feature being that very general censoring patterns are allowed. A survey is given of existing tests and their relation to the general theory, and we mention examples of applications to Markov processes. We also discuss the relation of the present approach to classical nonparametric hypothesis testing theory based on permutation distributions.

On the Nonconsistency of Maximum Likelihood Nonparametric Density Estimators

Abstract: One criterion proposed in the literature for selecting the smoothing parameter(s) in RosenblattParzen nonparametric constant kernel estimators of a probability density function is a leave-out-one-at-a-time nonparametric maximum likelihood method. Empirical work with this estimator in the univariate case showed that it worked quite well for short tailed distributions. However, it drastically oversmoothed for long tailed distributions. In this paper it is shown that this nonparametric maximum likelihood method will not select consistent estimates of the density for long tailed distributions such as the double exponential and Cauchy distributions. A remedy which was found for estimating long tailed distributions was to apply the nonparametric maximum likelihood procedure to a variable kernel class of estimators. This paper considers one data set, which is a pseudo-random sample of size 100 from a Cauchy distribution, to illustrate the problem with the leave-out-one-at-a-time nonparametric maximum likelihood method and to illustrate a remedy to this problem via a variable kernel class of estimators.

Some Nonparametric Techniques for Estimating the Intensity Function of a Cancer Related Nonstationary Poisson Process

Abstract: An attempt is made to model the appearance times of metastases as a nonstationary Poisson process. Three algorithms are developed for this task. The first follows the kernel approach used in probability density estimation by Parzen and Rosenblatt; the second extends the work of Grenander on mortality measurements to a more general censoring scheme appropriate for the present application; the third employs a discrete maximum penalized likelihood approach. We obtain estimates using both stratification and the proportional hazards model. Contrary to customary belief, it seems that the intensity functions associated with the tumor systems under investigation are nonincreasing.

Laws of the iterated logarithm for nonparametric density estimators

Abstract: We establish a law of the iterated logarithm for a triangular array of independent random variables, and apply it to obtain laws for a large class of nonparametric density estimators. We consider the case of Rosenblatt-Parzen kernel estimators, trigonometric series estimators and orthogonal polynomial estimators in detail, and point out that our technique has wider application.