Showing papers on "Sampling distribution published in 1983"

Statistics for Research

Abstract: Preface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1. The Role of Statistics. 1.1 The Basic Statistical Procedure. 1.2 The Scientific Method. 1.3 Experimental Data and Survey Data. 1.4 Computer Usage. Review Exercises. Selected Readings. 2. Populations, Samples, and Probability Distributions. 2.1 Populations and Samples. 2.2 Random Sampling. 2.3 Levels of Measurement. 2.4 Random Variables and Probability Distributions. 2.5 Expected Value and Variance of a Probability Distribution. Review Exercises. Selected Readings. 3. Binomial Distributions. 3.1 The Nature of Binomial Distributions. 3.2 Testing Hypotheses. 3.3 Estimation. 3.4 Nonparametric Statistics: Median Test. Review Exercises. Selected Readings. 4. Poisson Distributions. 4.1 The Nature of Poisson Distributions. 4.2 Testing Hypotheses. 4.3 Estimation. 4.4 Poisson Distributions and Binomial Distributions. Review Exercises. Selected Readings. 5. Chi-Square Distributions. 5.1 The Nature of Chi-Square Distributions. 5.2 Goodness-of-Fit Tests. 5.3 Contingency Table Analysis. 5.4 Relative Risks and Odds Ratios. 5.5 Nonparametric Statistics: Median Test for Several Samples. Review Exercises. Selected Readings. 6. Sampling Distribution of Averages. 6.1 Population Mean and Sample Average. 6.2 Population Variance and Sample Variance. 6.3 The Mean and Variance of the Sampling Distribution of Averages. 6.4 Sampling Without Replacement. Review Exercises. 7. Normal Distributions. 7.1 The Standard Normal Distribution. 7.2 Inference From a Single Observation. 7.3 The Central Limit Theorem. 7.4 Inferences About a Population Mean and Variance. 7.5 Using a Normal Distribution to Approximate Other Distributions. 7.6 Nonparametric Statistics: A Test Based on Ranks. Review Exercises. Selected Readings. 8. Student's t Distribution. 8.1 The Nature of t Distributions. 8.2 Inference About a Single Mean. 8.3 Inference About Two Means. 8.4 Inference About Two Variances. 8.5 Nonparametric Statistics: Matched-Pair and Two-Sample Rank Tests. Review Exercises. Selected Readings. 9. Distributions of Two Variables. 9.1 Simple Linear Regression. 9.2 Model Testing. 9.3 Inferences Related to Regression. 9.4 Correlation. 9.5 Nonparametric Statistics: Rank Correlation. 9.6 Computer Usage. 9.7 Estimating Only One Linear Trend Parameter. Review Exercises. Selected Readings. 10. Techniques for One-way Analysis of Variance. 10.1 The Additive Model. 10.2 One-Way Analysis-of-Variance Procedure. 10.3 Multiple-Comparison Procedures. 10.4 One-Degree-of-Freedom Comparisons. 10.5 Estimation. 10.6 Bonferroni Procedures. 10.7 Nonparametric Statistics: Kruskal-Wallis ANOVA for Ranks. Review Exercises. Selected Readings. 11. The Analysis-of-Variance Model. 11.1 Random Effects and Fixed Effects. 11.2 Testing the Assumptions for ANOVA. 11.3 Transformations. Review Exercises. Selected Readings. 12. Other Analysis-of-Variance Designs. 12.1 Nested Design. 12.2 Randomized Complete Block Design. 12.3 Latin Square Design. 12.4 a xb Factorial Design. 12.5 a xb xc Factorial Design. 12.6 Split-Plot Design. 12.7 Split Plot with Repeated Measures. Review Exercises. Selected Readings. 13. Analysis of Covariance. 13.1 Combining Regression with ANOVA. 13.2 One-Way Analysis of Covariance. 13.3 Testing the Assumptions for Analysis of Covariance. 13.4 Multiple-Comparison Procedures. Review Exercises. Selected Readings. 14. Multiple Regression and Correlation. 14.1 Matrix Procedures. 14.2 ANOVA Procedures for Multiple Regression and Correlation. 14.3 Inferences About Effects of Independent Variables. 14.4 Computer Usage. 14.5 Model Fitting. 14.6 Logarithmic Transformations. 14.7 Polynomial Regression. 14.8 Logistic Regression. Review Exercises. Selected Readings. Appendix of Useful Tables. Answers to Most Odd-Numbered Exercises and All Review Exercises. Index.

Handbook of the normal distribution

Abstract: Genesis: an historical background basic properties expansions and algorithms characterizations sampling distributions limit theorems and expansions normal approximations to distributions order statistics from normal samples the bivariate normal distribution bivariate normal sampling distributions point estimation statistical intervals.

Distribution of the Kurtosis Statistic b2 for Normal Samples.

Abstract: SUMMARY D'Agostino & Pearson (1973) gave percentage points of the distribution of b2 for independent observations from a common univariate normal distribution. Their results can be adequately approximated, when the first three moments of the distribution of b2 have been determined, by fitting a linear function of the reciprocal of a X2 variable and then using the Wilson-Hilferty transformation. Evidence is presented suggesting that the same method of approximation is satisfactory for the b2 statistic calculated from any set of linear-least-squares residuals, on the hypothesis of normal homoscedastic errors.

Introduction to Robust and Quasi-Robust Statistical Methods

Abstract: 1. Introduction and Summary.- 1.1. History and main contributions.- 1.2. Why robust estimations?.- 1.3. Summary.- A The Theoretical Background.- 2. Sample spaces, distributions, estimators.- 2.1. Introduction.- 2.2. Example.- 2.3. Metrics for probability distributions.- 2.4. Estimators seen as functionals of distributions.- 3. Robustness, breakdown point and influence function.- 3.1. Definition of robustness.- 3.2. Definition of breakdown point.- 3.3. The influence function.- 4. The jackknife method.- 4.1. Introduction.- 4.2. The jackknife advanced theory.- 4.3. Case study.- 4.4. Comments.- 5. Bootstrap methods, sampling distributions.- 5.1. Bootstrap methods.- 5.2. Sampling distribution of estimators.- B.- 6. Type M estimators.- 6.1. Definition.- 6.2. Influence function and variance.- 6.3. Robust M estimators.- 6.4. Robustness, quasi-robustness and non-robustness.- 6.4.1. Statement of the location problem.- 6.4.2. Least powers.- 6.4.3. Huber's function.- 6.4.4. Modification to Huber's proposal.- 6.4.5. Function "Fair".- 6.4.6. Cauchy-s function.- 6.4.7. Welsch-s function.- 6.4.8. "Bisquare" function.- 6.4.9. Andrews's function.- 6.4.10. Selection of the ?-function.- 7. Type L estimators.- 7.1. Definition.- 7.2. Influence function and variance.- 7.3. The median and related estimators.- 8. Type R estimator.- 8.1. Definition.- 8.2. Influence function and variance.- 9. Type MM estimators.- 9.1. Definition.- 9.2. Influence function and variance.- 9.3. Linear model and robustness - Generalities.- 9.4. Scale of residuals.- 9.5. Robust linear regression.- 9.6. Robust estimation of multivariate location and scatter.- 9.7. Robust non-linear regression.- 9.8. Numerical methods.- 9.8.1. Relaxation methods.- 9.8.2. Simultaneous solutions.- 9.8.3 Solution of fixed-point and non-linear equations.- 10. Quantile estimators and confidence intervals.- 10.1. Quantile estimators.- 10.2. Confidence intervals.- 11. Miscellaneous.- 11.1. Outliers and their treatment.- 11.2. Analysis of variance, constraints on minimization.- 11.3. Adaptive estimators.- 11.4. Recursive estimators.- 11.5. Concluding remark.- 12. References.- 13. Subject index.

Identification and lack of identification

Abstract: THIS PAPER IS INTENDED to stress the distinction between the conditions for lack of identification in models linear with respect to the variables but nonlinear in the parameters in the sense originally defined by Fisher [2], and the less numerous set of conditions required for first order lack of identification. The latter set of conditions involve only the first derivatives of the coefficients as functions of the parameters. It is argued that if the model suffers from first order lack of identification, it will generally be the case that the usual estimators are consistent, although not asymptotically normally distributed. In a leading special case the asymptotic distribution is discussed, and the simulation of a simple model illustrates the extent to which this asymptotic distribution approximates the actual finite sample distribution.

Statistics: An Introduction

Abstract: Part 1 An introduction to statistics: Who Else Uses Statistics? Graphic Presentation of Data. Types of Statistics. Types of Variables. Levels of Measurement. Uses and Abuses of Statistics. A Word of Encouragement. Computer Applications. Part 2 Summarizing data - frequency distributions and graphic presentations: Frequency Distributions. Stem-and-Leaf Charts. Portraying the Frequency Distributions Graphically. Misuses of Charts and Graphs. Part 3 Descriptive statistics - measures of central tendency: The Sample Mean. The Population Mean. Properties of the Arithmetic Mean. The Weighted Arithmetic Mean. The Median. The Mode. Choosing an Average. Estimating the Arithmetic Mean from Grouped Data. Estimating the Median from Grouped Data. Estimating the Mode from Grouped Data. Choosing an Average for Data in a Frequency Distribution. Part 4 Descriptive statistics - measures of dispersion and skewness. Measures of Dispersion for Raw Data. Measures of Dispersion for Grouped Data. Interpreting and Using the Standard Deviation. Box Plots. Relative Dispersion. The Coefficient of Skewness. Software Example. Part 5 An introduction to probability: Concepts of Probability. Types of Probability. Classical Concept of Probablity. Probability Rules. Some Counting Principles. Part 6 Probability distributions: What is a Probability Distribution? Discrete and Continuous Random Variables. The Mean and Variance of a Probability Distribution. The Binomial Probability Distribution. The Poisson Probability Distribution. Part 7 The normal probability distribution: Characteristics of a Normal Probability Distribution. The "Family" of Normal Distributions. The Standard Normal Probability Distribution. The Normal Approximation to the Binomial. The Normal Approximation to the Poisson Distribution. Part 8 Sampling methods and sampling distributions: Designing the Sample Survey or Experiment. Methods of Probability Sampling. The Sampling Error. The Sampling Distribution of the Sample Mean. Part 9 The central limit theorem and confidence intervals: The Central Limit Theorem. Confidence Intervals for Means. The Standard Error of the Sample Mean. The Standard Error of the Sample Proportion. Confidence Intervals for Proportions. The Finite Population Correction Factor. Choosing an Appropriate Sample Size. Part 10 Hypothesis tests - large-sample methods: The General Idea of Hypothesis Testing. A Test Involving the Population Mean (Large Samples.) A Test for Two Population Means (Large Samples.) A Test Involving the Population Proportion (Large Samples). A Test Involving Two Population Proportions (Large Samples). Part 11 Hypothesis tests - small-sample method: Characteristics of the T Distribution. Testing a Hypothesis about a Population Mean. Comparing Two Population Means. Testing with Dependent Observations. Comparing Dependent and Independent Samples.

A test for normality based on the empirical characteristic function

Abstract: Our aim in the present paper is to suggest a simple test of the composite hypothesis of normality against the alternative that the underlying distribution is long tailed. The test is based on the behaviour of the empirical characteristic function in the neighbourhood of the origin, and is very competitive with several well-known tests for normality. Only a table of critical points is required for its implementation. We begin by describing the philosophy behind the test. Let q be the characteristic function of the sampling distribution, and set

Modern business statistics

Abstract: The relationship between sampling and statistics displaying sample data descriptive sample statistics probability, populations and random variables some useful discrete and continuous distributions estimation (one sample) hypothesis testing two related examples (matched pairs) estimation and hypothesis testing with two independent samples contingency tables correlation regression time series analysis formulating general linear models for fitting data multiple linear regression analysis of variance of one-factor experiments analysis of variance of two-factor experiments other useful topics in experiment design decision theory under uncertainty decision theory with sample information tables: binomial distribution, normal distribution, student's distribution, chi-squared distribution and the f distribution

Chapter 8 Exact small sample theory in the simultaneous equations model

Abstract: Publisher Summary This chapter provides a general framework for the distribution problem and details formulae that are frequently useful in the derivation of sampling distributions and moments. It also provides an account of the genesis of the Edgeworth, Nagar, and saddlepoint approximations. The chapter discusses the Wishart distribution and some related issues that are central to modem multivariate analysis and on which much of the present development of exact small-sample theory depends. The chapter discusses the exact theory of single-equation estimators, commencing with a general discussion on the standardizing transformations that provide research economy in the derivation of exact distribution theory in this context and simplify the presentation of final results without loss of generality. The current information on the exact small-sample behavior of structural variance estimators, test statistics, systems methods, reduced-form coefficient estimators, and estimation under misspecification are also discussed in the chapter.

On the Second Order Asymptotic Efficiency of Estimators of Gaussian ARMA Processes

Abstract: In this paper we investigate an optimal property of maximum likelihood and quasi-maximum likelihood estimators of Gaussian autoregressive moving average processes by the second order approximation of the sampling distribution. It is shown that appropriate modifications of these estimators for Gaussian ARMA processes are second order asymptotically efficient if efficiency is measured by the degree of concentration of the sampling distribution up to second order. This concept of efficiency was introduced by Akahira and Takeuchi (1981).

A Regression Test for Exponentiality: Censored and Complete Samples

Abstract: Two new tests for the two-parameter exponential distribution are presented. The test statistics can be used with doubly censored samples, are easy to compute, need no special constants, and have high power compared with several competing tests. The first test statistic is sensitive to monotone hazard functions, and its percentage points can be closely approximated by the standard normal distribution. The second test statistic is sensitive to nonmonotone hazard functions. The chi-squared (2 degrees of freedom) distribution can be used as an approximation to the distribution of this statistic for moderate and large sample sizes. Monte Carlo power estimates and an example are given.

Statistics in research and development

Abstract: What are statistics? describing the sample describing the population testing and estimation - one sample testing and estimation - two samples testing and estimation - qualitative data testing and estimation - assumptions statistical process control detecting process changes investigating the process - an experiment why was the experiment not successful? some simple but effective experiments adapting the simple experiments improving a bad experiment analysis of variance an introduction to Taguchi technique. Appendices: The sigma notation notation and formulae sampling distributions copy of computer print-out from a multiple regression programme partial correlation significance tests on effect estimates from a p-2-n factorial experiment.

Improvements in the Permutation Test for the Spatial Analysis of the Distribution of Artifacts into Classes

Abstract: Refinements and extensions to the permutation test for assessing the intrasite patterning of artifact distributions in an archaeological space are presented. Specifically, a Pearson type III distribution is employed to approximate the sampling distribution of the test statistic, an improved weighting is introduced to increase the efficiency of the test, and a method is presented to permit the inclusion of unclassified artifacts in the spatial analysis.

On the relevance of finite sample distribution theory

Abstract: The implications of finite sample distribution theory for applied econometrics are explored. In general, its relevance is limited by three considerations: (i) sampling errors are of secondary importance in practice, (ii) exact and approximate results depend on the unknown structural parameters of the problem, and (iii) results more accurate than the limiting distribution require further distributional assumptions about some unobservable. Despite these difficulties, some useful information simultaneous equations estimators are sketched.

Predictive distribution of strength under control

Abstract: The joint distribution of strength of materials is derived in terms of a set of conditional distributions to be used in studies on structural reliability. Bayes theorem of probability theory is used to update prior distributions for the parameters of Gaussian sequences by direct observations and/or by compliance tests. Maximum-Likelihood estimators are given for the efficient quantification of prior information. The formulae are applied to concrete production judged by standard tests. It is shown that statistical uncertainties must not be ignored in structural reliability studies.

A Simulation Study of Autoregressive and Window Estimators of the Inverse Correlation Function

Abstract: The results of a simulation study aimed at investigating and comparing the finite sample behaviour of the autoregressive and the window methods of estimating the inverse correlation function are described. Three moving average and one autoregressive processes of second order are considered. The behaviour of these estimators as the order, k, of the fitted autoregression, the bandwidth parameter, m, of the spectral window and the series length, T, are varied is discussed. The usefulness of the asymptotic distribution of these estimators as an approximation to their finite sample distribution is examined.

Sampling Distribution of Consensus Indices when All Bifurcating Trees are Equally Likely

Abstract: Consensus indices (CI) based on a consensus tree (CT) have been increasingly used in recent studies to measure the congruence between hierarchic classifications But their statistical properties have yet not been investigated This study reports on the distribution of CIs examined by a Monte Carlo method The model employed assumes that all possible rooted bifurcating trees are equally likely Other models involving random generation of data, similarity matrices, or other types of trees are under investigation

Structural inference on the parameter of the rayleigh distribution from doubly censored samples

Abstract: We consider the problem of statistical inference on the scale parameter of the Rayleigh Distribution from a type II doubly censored sample, using a Structural Inference approach. We derive the Structural Distribution for the scale parameter. The properties of this distribution are used to obtain different inferential statements about the parameter.

FUN.STAT Quantile Approach to Two Sample Statistical Data Analysis.

Abstract: : FUN.STAT is a name proposed to describe a synthesis of statistical reasoning which combines quantiles and quantile-densities, information and entropy, and functional statistical inference. This paper describes a FUN.STAT approach to the problem of statistical data analysis of two random samples, respectively representing two populations of interest. It is composed of four parts. Part 1 describes how conventional approaches to two sample problems, including representations of linear rank statistics, are equivalent to functionals of a stochastic process. Part 2 motivates the autoregressive density estimation approach to the problem of functional statistical inference of this stochastic process and states several conjectures concerning the properties of the density estimation approach. Part 3 outlines heuristic derivations of the asymptotic distribution theory of the process. Part 4 provides a summary and an example, using TWOSAM which is a computer program for autoregressive two sample statistical data analysis; it has been implemented as a Fortran program and as a SAS procedure.

A note on the accuracy of fisher's approximation to the large sample variance of an intraclass correlationa

Abstract: The adequacy of Fisher's approximation to the large sample variance of an intraclass correlation is investigated in the context of family studies. It is found that the approximation is highly accurate in samples of moderately large size (≧ 30 families), and can also be used for significance-testing under a broad range of circumstances. The exact sampling of distribution of the intraclass correlation coefficient is also derived.

Comparing alternative asymptotically equivalent tests

Abstract: In many econometric inference problems there are a number of alternative statistical procedures available, all having the same asymptotic properties. Because the exact distributions are unknown, choice among the alternatives often is made on the basis of computational convenience. Recent work in theoretical statistics has suggested that second-order asymptotic approximations can lead to a more satisfactory basis for choice. In this chapter I shall discuss some implications of this statistical theory for hypothesis testing in econometric models. My comments are based on current research in progress with my co-workers Chris Cavanagh, Larry Jones, Dennis Sheehan, and Darrell Turkington. This work, in turn, borrows much from earlier studies by the statisticians Chibisov, Efron, and Pfanzagl and the econometricians Durbin, Phillips, and Sargan. Although my comments will concentrate on the problem of hypothesis testing, there are, of course, parallel theories for point and interval estimation. The basic idea underlying second-order asymptotic comparisons of tests is simple. Tests that are asymptotically equivalent often differ in finite samples. Although the exact sampling distributions may be difficult to derive, the first few terms of an Edgeworth-type series expansion for the distribution functions usually are available. The tests can then be compared using the Edgeworth approximations. Of course, it is not very interesting to compare tests unless they have the same significance level. Therefore, we first approximate the distributions of the test statistics under the null hypothesis and, based on that approximation, modify the tests so that they have the same probability of a type I error. Then we use another Edgeworth series, derived under the alternative hypothesis, to approximate the power functions of the modified tests.

On the Asymptotic Distribution of Watson's Statistic

Abstract: An integral representation of the positive stable distributions is used to give a tabulation of the asymptotic test statistic for Watson's optimal invariant test for uniformity of distribution on a circle.

Nonlinear data fusion

Abstract: Consider the following estimation problem: The state trajectory of a random process is observed by K distinct observers, using noise-corrupted observations. Each observer processes his own observation history, to obtain the local conditional distribution of the state, as a function of time. Assume that sufficient statistics representing each local conditional distribution are communicated to a coordinator at a central location at each point in time. The coordinators's estimation problem consists of constructing the overall conditional distribution of the state, conditioned on knowing all of the observations, while using only the sufficient statistics communicated to him.

A sampling strategy useful in full-distribution simulation.

Abstract: Inherent in most simulation processes is a mechanism to sample from known probability distributions. This is most often accomplished with the aid of pseudo-random generation systems. Though, these generators produce sets of numbers which are usually statistically indistinguishable from a uniform distribution, the actual distribution of any individual one of these data sets exhibit peaks and valleys which, when used in simulations, somewhat misrepresent the desired probability distribution. A stratified approach to fulldistribution sampling is presented which represents a marked improvement over random number generated sampling in certain types of simulation procedures.

Computer Simulations to Clarify Key Ideas of Statistics.

Abstract: Hand calculators have come into common use for performing calculations in high school and undergraduate courses in elementary statistics. Computers have also been used to do standard statistical computations and tests. However, computers can be used in elementary statistics courses not only to do standard statistical tests, but to perform simulations which clarify concepts and theorems of statistics. These simulations allow the nonmathematically oriented student in elementary statistics to have inductive experiences with statistical concepts in a time-efficient manner. A basic concept in statistics is that of a sample distribution. The sampling distribution of a sample statistic is defined to be the theoretical probability distribution of values for that sample statistic obtained from all possible samples of a certain fixed size from the population. Typically elementary texts illustrate this concept by using a very small finite population which has a small number of possible samples of a given size. Although large or infinite populations are of interest since they have a large or infinite number of samples of a certain size it is difficult to develop a sample distribution. Computer simulation, however, makes this easy to explore. The computer can simulate taking many samples of a certain size from a population. By taking many samples the law of large numbers guarantees that the observed frequency distribution of the sample statistic is close to the theoretical sample distribution. The sample distribution of the mean is the most important sample distribution. The computer simulations in Figure 1 explore a sample distribution of the mean. These simulations employ an elementary statistical package called MINITAB developed at Pennsylvania State University. The MINITAB command URANDOM simulates random observations from a population where numbers between 0 and 1 all have an equal probability of occurring. The first simulation consists of drawing 200 samples of size 2 and computing the means of the samples. A histogram of the results is given. Continuing with samples of size 2, drawing 750 samples is simulated and then 1500 samples. As the number of samples increases the student can see the isosceles triangle-shaped theoretical sample distribution emerging. Note that, because of the small sample size and the nonnormal population, the sample distribution should not be expected to be normal.

Estimating the Distribution of Spherical Particles from Plane Sections

Abstract: The persistently troublesome problem of solving the Abel integral equation with noisy data is reconsidered. A criterion is formulated for determination of a most probable solution which satisfies the reasonable requirement that frequency estimates be nonnegative. Our prescription, called the constrained minimum variance estimator (COMIVE) results in an optimum (in a sense described in the text) radius distribution for the spherical particles. Numerical examples are given which use input generated by Monte Carlo from various distributions and which use real experimental data. Results demonstrate considerable improvement over those obtained by other methods.

On the asymptotic distribution of cramer-von mises one-sample test statistics under an alternative

Abstract: The asymptotic normality of the Cramer-von Mises one-sample test statistic and one of its variants under an alternative cdf is demonstrated. The derivation herein is unique in that it does not require knowledge of the theory of weak convergence of probability measures defined on metrized function spaces, and thus is accessible to a broader class of students and practitioners.

Bayesian Estimators of the Parameters and Reliability Function from Mixed Exponentially Distributed Time-Censored Life Test Data

Abstract: Bayes estimators of the parameters and the reliability function are obtained for the Mendenhall and Hader (1958) model where the population is made up of a mixture of two exponentially distributed subpopulations and the test is terminated after a predetermined time T hours. Based on a Monte-Carlo study these estimators are compared with their maximum likelihood counterparts. Appropriate Pearson-type curves are fitted to the sampling distributions of Baves estimates and a ‘goodness of fit’ test performed.