Showing papers on "Sampling distribution published in 1973"
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01 Jan 1973TL;DR: In this article, the authors describe a decision-making process for making decisions in an uncertain environment in the context of time series data, and present a series of graphs and tables to describe the relationships between variables.
Abstract: CHAPTER 1 Describing Data: Graphical 1.1 Decision Making in an Uncertain Environment 1.2 Classification of Variables 1.3 Graphs to Describe Categorical Variables 1.4 Graphs to Describe Time-Series Data 1.5 Graphs to Describe Numerical Variables 1.6 Tables and Graphs to Describe Relationships Between Variables 1.7 Data Presentation Errors CHAPTER 2 Describing Data: Numerical 2.1 Measures of Central Tendency 2.2 Measures of Variability 2.3 Weighted Mean and Measures of Grouped Data 2.4 Measures of Relationships Between Variables CHAPTER 3 Probability 3.1 Random Experiment, Outcomes, Events 3.2 Probability and Its Postulates 3.3 Probability Rules 3.4 Bivariate Probabilities 3.5 Bayes' Theorem CHAPTER 4 Discrete Random Variables and Probability Distributions 4.1 Random Variables 4.2 Probability Distributions for Discrete Random Variables 4.3 Properties of Discrete Random Variables 4.4 Binomial Distribution 4.5 Hypergeometric Distribution 4.6 The Poisson Probability Distribution 4.7 Jointly Distributed Discrete Random Variables CHAPTER 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 5.2 Expectations for Continuous Random Variables 5.3 The Normal Distribution 5.4 Normal Distribution Approximation for Binomial Distribution 5.5 The Exponential Distribution 5.6 Jointly Distributed Continuous Random Variables CHAPTER 6 Sampling and Sampling Distributions 6.1 Sampling from a Population 6.2 Sampling Distributions of Sample Means 6.3 Sampling Distributions of Sample Proportions 6.4 Sampling Distributions of Sample Variances CHAPTER 7 Estimation: Single Population 7.1 Properties of Point Estimators 7.2 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Known 7.3 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Unknown 7.4 Confidence Interval Estimation of Population Proportion 7.5 Confidence Interval Estimation of the Variance of a Normal Distribution 7.6 Confidence Interval Estimation: Finite Populations CHAPTER 8 Estimation: Additional Topics 8.1 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Dependent Samples 8.2 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Independent Samples 8.3 Confidence Interval Estimation of the Difference Between Two Population Proportions 8.4 Sample Size Determination: Large Populations 8.5 Sample Size Determination: Finite Populations CHAPTER 9 Hypothesis Testing: Single Population 9.1 Concepts of Hypothesis Testing 9.2 Tests of the Mean of a Normal Distribution: Population Variance Known 9.3 Tests of the Mean of a Normal Distribution: Population Variance Unknown 9.4 Tests of the Population Proportion 9.5 Assessing the Power of a Test 9.6 Tests of the Variance of a Normal Distribution CHAPTER 10 Hypothesis Testing: Additional Topics 10.1 Tests of the Difference Between Two Population Means: Dependent Samples 10.2 Tests of the Difference Between Two Normal Population Means: Independent Samples 10.3 Tests of the Difference Between Two Population Proportions 10.4 Tests of the Equality of the Variances Between Two Normally Distributed Populations 10.5 Some Comments on Hypothesis Testing CHAPTER 11 Simple Regression 11.1 Overview of Linear Models 11.2 Linear Regression Model 11.3 Least Squares Coefficient Estimators 11.4 The Explanatory Power of a Linear Regression Equation 11.5 Statistical Inference: Hypothesis Tests and Confidence Intervals 11.6 Prediction 11.7 Correlation Analysis 11.8 Beta Measure of Financial Risk 11.9 Graphical Analysis CHAPTER 12 Multiple Regression 12.1 The Multiple Regression Model 12.2 Estimation of Coefficients 12.3 Explanatory Power of a Multiple Regression Equation 12.4 Confidence Intervals and Hypothesis Tests for Individual Regression Coefficients 12.5 Tests on Regression Coefficients 12.6 Prediction 12.7 Transformations for Nonlinear Regression Models 12.8 Dummy Variables for Regression Models 12.9 Multiple Regression Analysis Application Procedure CHAPTER 13 Additional Topics in Regression Analysis 13.1 Model-Building Methodology 13.2 Dummy Variables and Experimental Design 13.3 Lagged Values of the Dependent Variables as Regressors 13.4 Specification Bias 13.5 Multicollinearity 13.6 Heteroscedasticity 13.7 Autocorrelated Errors CHAPTER 14 ANALYSIS OF CATEGORICAL DATA 14.1 Goodness-of-Fit Tests: Specified Probabilities 14.2 Goodness-of-Fit Tests: Population Parameters Unknown 14.3 Contingency Tables 14.4 Sign Test and Confidence Interval 14.5 Wilcoxon Signed Rank Test 14.6 Mann--Whitney U Test 14.7 Wilcoxon Rank Sum Test 14.7 Spearman Rank Correlation CHAPTER 15 Analysis of Variance 15.1 Comparison of Several Population Means 15.2 One-Way Analysis of Variance 15.3 The Kruskal--Wallis Test 15.4 Two-Way Analysis of Variance: One Observation per Cell, Randomized Blocks 15.5 Two-Way Analysis of Variance: More Than One Observation per Cell CHAPTER 16 Time-Series Analysis and Forecasting 16.1 Index Numbers 16.2 A Nonparametric Test for Randomness 16.3 Components of a Time Series 16.4 Moving Averages 16.5 Exponential Smoothing 16.6 Autoregressive Models 16.7 Autoregressive Integrated Moving Average Models CHAPTER 17 Sampling: Additional Topics 17.1 Stratified Sampling 17.2 Other Sampling Methods CHAPTER 18 Statistical Decision Theory 18.1 Decision Making Under Uncertainty 18.2 Solutions Not Involving Specification of Probabilities 18.3 Expected Monetary Value TreePlan 18.4 Sample Information: Bayesian Analysis and Value 18.5 Allowing for Risk: Utility Analysis APPENDIX TABLES 1. Cumulative Distribution Function of the Standard Normal Distribution 2. Probability Function of the Binomial Distribution 3. Cumulative Binomial Probabilities 4. Values of e --lambda 5. Individual Poisson Probabilities 6. Cumulative Poisson Probabilities 7. Cutoff Points of the Chi-Square Distribution Function 8. Cutoff Points for the Student's t Distribution 9. Cutoff Points for the F Distribution 10. Cutoff Points for the Distribution of the Wilcoxon Test Statistic 11. Cutoff Points for the Distribution of Spearman Rank Correlation Coefficient 12. Cutoff Points for the Distribution of the Durbin--Watson Test Statistic 13 Critical Values of the Studentized Range Q (page 964 965 Applied Statistical Methods Carlson, Thorne Prentice Hall 1997) 14. Cumulative Distribution Function of the Runs Test Statistic ANSWERS TO SELECTED EVEN-NUMBERED EXERCISES INDEX I-1
991 citations
TL;DR: In this article, the authors introduce Probability One-Dimension Random Variables Functions of One Random Variable and Expectation Joint Probability Distributions Some Important Discrete Distributions some Important Continuous Distributions The Normal Distribution Random Samples and Sampling Distributions Parameter Estimation Tests of Hypotheses Design and Analysis of Single Factor Experiments: The Analysis of Variance Design of Experiments with Several Factors Simple Linear Regression and Correlation Multiple Regression Nonparametric Statistics Statistical Quality Control and Reliability Engineering Stochastic Processes and Queueing Statistical Decision Theory References
Abstract: Introduction and Data Description An Introduction to Probability One-Dimension Random Variables Functions of One Random Variable and Expectation Joint Probability Distributions Some Important Discrete Distributions Some Important Continuous Distributions The Normal Distribution Random Samples and Sampling Distributions Parameter Estimation Tests of Hypotheses Design and Analysis of Single-Factor Experiments: The Analysis of Variance Design of Experiments with Several Factors Simple Linear Regression and Correlation Multiple Regression Nonparametric Statistics Statistical Quality Control and Reliability Engineering Stochastic Processes and Queueing Statistical Decision Theory References Appendix Answers to Selected Exercises Index.
531 citations
TL;DR: In this paper, the weak convergence of the sample df under a given sequence of alternative hypotheses when parameters are estimated from the data is studied under a general class of estimators and it is shown that the sampledf, when normalised, converges weakly to a specified normal process.
Abstract: The weak convergence of the sample df is studied under a given sequence of alternative hypotheses when parameters are estimated from the data. For a general class of estimators it is shown that the sample df, when normalised, converges weakly to a specified normal process. The results are specialised to the case of efficient estimation.
378 citations
148 citations
TL;DR: The theory of rank tests has been extended to include purely discrete random variables under the null hypothesis of randomness (including the two-sample and $k$-sample problems) and under contiguous alternatives for the two methods of assigning scores known as the average scores method and the randomized ranks method as mentioned in this paper.
Abstract: The theory of rank tests has been developed primarily for continuous random variables. Recently the asymptotic theory of linear rank tests has been extended to include purely discrete random variables under the null hypothesis of randomness (including the two-sample and $k$-sample problems) and under contiguous alternatives, for the two methods of assigning scores known as the average scores method and the randomized ranks method. In this paper the theory of rank tests is developed with no assumptions concerning the continuous or discrete nature of the underlying distribution function. Conditional rank tests, given the vector of ties, are shown to be similar, and the locally most powerful conditional rank test is given. The asymptotic distribution of linear rank statistics is given under the null hypotheses of randomness and symmetry (which includes the one-sample problem), and under contiguous alternatives. Three methods of assigning scores, the average scores, midranks, and randomized ranks methods, are discussed and briefly compared.
70 citations
TL;DR: In this article, a sample of size 1,000,000 was used for Monte Carlo analysis, and it was shown that the observed frequencies are not following the expected pattern and that the sampling distribution has marked local maxima approximately at the points -33, 3 0 and 3-6.
Abstract: SUMMARY In a recent Monte Carlo study, a most unsatisfactory sampling distribution was obtained when supposedly standard techniques were employed to simulate a simple random sample from the standard normal distribution A selection of observed and expected frequencies (the latter rounded to the nearest integer) in the tails of the distribution for a sample of size 1,000,000 is given in Table 1 It hardly needs a chi-square test to indicate that the observed frequencies are not following the expected pattern! Note particularly that all the 1,000,000 observations are restricted to the range (-333 36), and that the sampling distribution has marked local maxima approximately at the points - 33, 3 0 and 3-6 This paper investigates such phenomena
66 citations
TL;DR: In this article, the population genetic hypothesis that mating is random with respect to genotype at a specific two allele locus is equivalent to the statistical hypothesis that the genotypic frequency distribution is given by a binomial expansion (p+q)2.
Abstract: The population genetic hypothesis that mating is random with respect to genotype at a specific two allele locus is equivalent to the statistical hypothesis that the genotypic frequency distribution is given by a binomial expansion (p+q)2. An exact,small sample, conditional test of this hypothesis is derived by tabulating critical values of the number of hetero-zygotes in a sample of size n containing given frequencies of each allele. for amy fixed value of relative gene frequency in the sample this conditional distribution rapidly approaches normality, and the conventional chi-square test becomes valid.
48 citations
TL;DR: This paper presents a more general and more accurate procedure, based on correlation transfer, to generate a pseudo-random set with select distribution and correlation, and illustrates the method.
Abstract: Pseudo-random sequences can be computed to approximate a distribution or a sample correlation function. How ever, conventional techniques do not allow the control of both distribution and correlati...
30 citations
TL;DR: In this paper, the density function of the i-th order statistic from a sample with random size is derived for the case that the size has a bionmial distribution, and a simpler derivation is given below.
Abstract: Summary
Raghunandanan and Patil [1] derived the density function of the i-th order statistic from a sample with random size. For the case that the size has a bionmial distribution, a simpler derivation is given below.
23 citations
01 Jan 1973
19 citations
TL;DR: In this article, the methodology of the Zeigler and Tietjen double sample test for testing a hypothesis about the variance of a normal distribution can be used to construct double sample tests for testing hypotheses about the mean of an exponential distribution.
Abstract: In this note it is shown how the methodology of the Zeigler and Tietjen double sample test for testing a hypothesis about the variance of a normal distribution can be used to construct double sample tests for testing hypotheses about the mean of an exponential distribution.
TL;DR: In this article, the effects of non-normality on the sampling distribution of reliability estimates were investigated under a mixed model anova model and both analytical and computer simulation methods were used.
Abstract: Although the calculated reliability of a test based on a small sample of subjects is an estimate of the population reliability and, hence, is subject to sampling fluctuation, little application has been made of statistical inference techniques to the reliability coefficient. The need for such techniques was recognized as early as the 1940s, and a few attempts have been made to introduce analysis of variance (anova) procedures and to relate the sampling distribution of reliability coefficient estimates to the well-known F distribution (e.g. Jackson & Ferguson, 1941; Ebel, 1951). It was, however, Kristof (1963) who first presented a rather complete sampling theory of reliability estimates and a method to apply it. Feldt (1965) derived similar results based on an anova model and applied it to a binary item test case. The above sampling theories were based on normality assumptions regarding the true and error score distributions. It is, however, conceivable that real data will not always meet the rigorous assumption of normality, and yet very little is known about the effects of non-normality on the sampling distribution of reliability estimates. The present paper contains the results of an investigation of the effects of non-normality under a mixed model anova. Both analytical and computer simulation methods are used.
TL;DR: In this article, empirical distributions of Z = (X-bar - ()/(N-1/2)) were obtained for several sample sizes to investigate the approach of the sampling distribution of Z to a normal distribution.
Abstract: From each of two dozen populations varying widely in shape and skewness, empirical distributions of Z = (X-bar - ()/(N-1/2 were obtained for several sample sizes to investigate the approach of the sampling distribution of Z to a normal distribution. For..
TL;DR: In this article, the joint distribution of a certain set of Kn + 1 of the variables Y1(n),… Yn(n) is examined and it is shown that for all asymptotic probability calculations, this joint distribution can be assumed to be normal.
Abstract: For each n, X1(n),…Xn(n) are independent and identically distributed random variables.Y1(n) < … < Yn(n) are the ordered values of X1(n),… Xn(n). Kn is a positive integer, with . The joint distribution of a certain set of Kn +1 of the variables Y1(n),… Yn(n) is examined.It is shown that for all asymptotic probability calculations, this joint distribution can be assumed to be normal. Applications to asymptotic statistical inference are given.
TL;DR: It is hypothesized that subjects, through experience and understanding, acquire an internal representation of the task environment that can be put to use in some desired form only if a proper response-device is sufficiently well understood so that it makes the information accessible.
TL;DR: In this article, an asymptotic expansion of a confluent hypergeometric series is used to approximate the exact finite sample distribution function of a nonconsistent GCL structural variance estimator.
Abstract: An asymptotic expansion of a confluent hypergeometric series is used to approximate the exact finite sample distribution function of a nonconsistent GCL structural variance estimator. A theoretical result is used to motivate the specification of a simple algorithm under which we may accept or reject the use of the asymptotic distribution function of the GCL estimator to approximate the exact distribution function.
TL;DR: In this paper, it was shown that T1∗ and T2∗ are nonincreasing functions of D2, the Mahalanobis sample distance, and that α is a decreasing function of Δ2 = (μ1 − μ2)′Σ−1(μ 1 − μ 2).
TL;DR: In this paper, a Student's test of Ho:? =?o against??? o of the scale parameter? of the Normal, logistic, and Cauchys distributions, when the location parameter? is known is based on a few sample percentiles selected from a large sample.
Abstract: A Student's test of Ho:? = ?o against ? ? ?o of the scale parameter ? of the Normal, logistic, and Cauchys distributions, when the location parameter ? is known is based on a few sample percentiles selected from a large sample. Tables facilitating the computation of the test statistic are provided. The significance level and power of the test statistic for sample sizes ? 200 are simulated by the Monte Carlo method and compared with the significance level and power when it is assumed to have its asymptotic distribution.
TL;DR: In this article, the authors derived a simple way of displaying the Liviatan estimators which makes their nature clear and which allows the small sample distribution of one of them to be easily deduced.
Abstract: THE USE OF the geometric distributed lag in economics is widespread. The Liviatan [6] method for estimating its parameters is simple and provides consistent estimates. Moreover, it can be used to provide initial estimates for more sophisticated techniques [2]. Not much is known about the statistical properties of these estimators, especially their small sample properties. We do know that their asymptotic efficiencies are inferior to most alternatives [1], and recently Nagar and Gupta [7] have provided approximations to the small sample biases. The purpose of this note is to derive a simple way of displaying the Liviatan estimators which makes their nature clear and which allows the small sample distribution of one of them to be easily deduced. The main result is that the estimator of the parameter which defines the geometric distributed lag is a ratio of two ordinary least squares estimators. With this and the assumption that the error terms form a sequence of independent, identically distributed normal variables, it is possible, using the work of Geary [4], to derive the small sample distribution of this estimator. This result forms the main part of this note, which concludes with a brief discussion of the problem of setting confidence limits along the lines suggested by Fieller [3].
TL;DR: In this article, the authors derived asymptotic expansions of the non-null distributions of Hotelling's statistic and Pillai's statistic up to the order n−1 for large sample size n+1, in terms of the normal distribution functions and its derivatives.
Abstract: Asymptotic expansions of the non-null distributions of Hotelling's statistic and Pillai's statistic for testing the independence between two sees of variates in a p‐variate normal population are derived up to the order n−1 for large sample size n+1, in terms of the normal distribution functions and its derivatives. The method is based on differential operators on symmetric matrix by Siotani [8], Ito [6] and etc.
TL;DR: An alternate technique based on confidence intervals of ranks and a rejection procedure finds A(J), which only requires a small working storage and is suited to the case when the elements of A, for largeN, are serially computed.
Abstract: Let vectorA containN elements. When vectorA is sorted the relationA(1) ⩽A(2) ⩽ ⋯ ⩽A(N) holds.A partial sort is a procedure by which theJth, 1 ⩽J ⩽N, element ofA,A(J), is found without completely sorting vectorA.A(J) is the Jth-order statistic ofA. A partial sort technique based on the Quicksort algorithm requires storage ofA. In this article an alternate technique based on confidence intervals of ranks and a rejection procedure findsA(J). This procedure only requires a small working storage and is suited to the case when the elements ofA, for largeN, are serially computed. The minimum number of working storage locations needed for the algorithm to terminate successfully with set probability for specifiedJ andN was computed. Partial sorting, especially this technique, is useful for computing critical values for statistics for which the sampling distributions are analytically known. Error bounds, which can be estimated from a segment of the ordered sample, are given for the estimated critical value. The technique is applied in several examples to estimate the mathematical constant π.
TL;DR: In this paper, a random sample is available from each of the two populations, each of whose probability properties are described by the 2-parameter exponential probability density function, and the two parameters are unknown in each case.
Abstract: Consider two populations, each of whose probability properties are described by the 2-parameter exponential probability density function. The two parameters are unknown in each case. Assuming that a random sample is available from each of the two populations, this paper shows how to calculate an approximation for the probability that one of the populations has a greater reliability than the other. The Bayesian theory of statistical inference is used and we assume a weak prior for the parameters involved.