An Introduction to Nonparametric Statistics

John E. Kolassa

• Publisher: Chapman and Hall/CRC
• Year: 2020
• ISBN:
  9780429514791
• Edition: 1
• Topic: Mathématiques
• Subtopic: Algèbre linéaire

An Introduction to Nonparametric Statistics presents techniques for statistical analysis in the absence of strong assumptions about the distributions generating the data. Rank-based and resampling techniques are heavily represented, but robust techniques are considered as well. These techniques include one-sample testing and estimation, multi-sample testing and estimation, and regression.

Attention is paid to the intellectual development of the field, with a thorough review of bibliographical references. Computational tools, in R and SAS, are developed and illustrated via examples. Exercises designed to reinforce examples are included.

Features

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• Rank-based techniques including sign, Kruskal-Wallis, Friedman, Mann-Whitney and Wilcoxon tests are presented

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• Tests are inverted to produce estimates and confidence intervals

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• Multivariate tests are explored

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• Techniques reflecting the dependence of a response variable on explanatory variables are presented

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• Density estimation is explored

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• The bootstrap and jackknife are discussed

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This text is intended for a graduate student in applied statistics. The course is best taken after an introductory course in statistical methodology, elementary probability, and regression. Mathematical prerequisites include calculus through multivariate differentiation and integration, and, ideally, a course in matrix algebra.

1

Background

Statistics is the solution to an inverse problem: given the outcome from a random process, the statistician infers aspects of the underlying probabilistic structure that generated the data. This chapter reviews some elementary aspects of probability, and then reviews some classical tools for inference about a distribution’s location parameter.

1.1Probability Background

This section first reviews some important elementary probability distributions, and then reviews a tool for embedding a probability distribution into a larger family that allows for the distribution to be recentered and rescaled. Most statistical techniques described in this volume are best suited to continuous distributions, and so all of these examples of plausible data sources are continuous.

1.1.1Probability Distributions for Observations

Some common probability distributions are shown in Figure 1.1. The continuous distributions described below might plausibly give rise to a data set of independent observations. This volume is intended to direct statistical inference on a data set without knowing the family from which it came. The behavior of various statistical procedures, including both standard parametric analyses, and nonparametric techniques forming the subject of this volume, may depend on the distribution generating the data, and knowledge of these families will be used to explore this behavior.

Figure 1.1: Comparison of Three Densities

1.1.1.1Gaussian Distribution

The normal distribution, or Gaussian distribution, has density

fG(x)=exp⁡(−(x−μ)2/(2σ2))/(σ2π).

The parameter μ is both the expectation and the median, and σ is the standard deviation. The Gaussian cumulative distribution function is

FG(x)=∫−∞xfG(y)dy.

There is no closed form for this integral. This distribution is symmetric about μ; that is, f_G(x) = f_G(2μ − x), and F_G(x) = 1 − F_G(2μ − x). The specific member of this family of distributions with μ = 0 and σ = 1 is cal...