Showing papers on "Sampling distribution published in 1984"

Statistics: A Tool for Social Research

Abstract: 1. Introduction. PART I: DESCRIPTIVE STATISTICS. 2. Basic Descriptive Statistics: Tables, Percentages, Ratios and Rates, and Graphs. 3. Measures of Central Tendency. 4. Measures of Dispersion. 5. The Normal Curve. PART II INFERENTIAL STATISTICS. 6. Introduction to Inferential Statistics: Sampling and the Sampling Distribution. 7. Estimation Procedures. 8. Hypothesis Testing I: The One-Sample Case. 9. Hypothesis Testing II: The Two-Sample Case. 10. Hypothesis Testing III: The Analysis of Variance. 11. Hypothesis Testing IV: Chi Square. PART III BIVARIATE MEASURES OF ASSOCIATION. 12. Bivariate Association for Nominal- and Ordinal-Level Variables. 13. Association Between Variables Measured at the Interval-Ratio Level. PART IV: MULTIVARIATE TECHNIQUES. 14. Elaborating Bivariate Tables. 15. Partial Correlation and Multiple Regression and Correlation. Appendix A: Area Under the Normal Curve. Appendix B: Distribution of t. Appendix C: Distribution of Chi Square. Appendix D: Distribution of F.

Introductory Statistics for Management and Economics

Abstract: 1. What Statistics Is All About 2. Data Collection and Sampling Theory 3. Summarizing Data in Tables and Graphs 4. Summary Statistics: Measures of Location and Dispersion 5. Introduction to Probability 6. Discrete Probability Distributions 7. Some Important Discrete Distributions 8. Some Useful Continuous Probability Distributions 9. Sampling Theory and Some Important Sampling Distributions 10. Estimating and Constructing Confidence Intervals 11. Hypothesis Testing 12. Tests of Hypotheses Involving Two Populations 13. Chi-Square Tests 14. Analysis of Variance 15. Regression and Correlation 16. Multiple Regression Models 17. Special Topics in Multiple Regression Analysis 18. Residual Analysis and Violations of the Basic Assumptions 19. Time Series Analysis I: Estimation of the Trend Component 20. Time Series Analysis II: Estimation of the Seasonal Component 21. Some Nonparametric Tests 22. Introduction to Statistical Decision Theory 23. Quality Control

The Exact Distribution of LIML: I

Abstract: It is shown that the exact finite sample distribution of the limited information maximum likelihood (LIML) estimator in a general and leading single equation case is multivariate Cauchy. When the LIML estimator utilizes a known error covariance matrix (LIMLK) it is proved that the same Cauchy distribution still applies. The corresponding result for the instrumental variable (IV) estimator is a form of multivariate t density where the degrees of freedom depend on the number of instruments.(This abstract was borrowed from another version of this item.)

Alternative approach to maximum-entropy inference

Abstract: A consistent approach to the inference of a probability distribution given a limited number of expectation values of relevant variables is discussed. There are two key assumptions: that the experiment can be independently repeated a finite number (not necessarily large) of times and that the theoretical expectation values of the relevant observables are to be estimated from their measured sample averages. Three independent but complementary routes for deriving the form of the distribution from these two assumptions are reviewed. All three lead to a unique distribution which is identical with the one obtained by the maximum-entropy formalism. The present derivation thus provides an alternative approach to the inference problem which does not invoke Shannon's notion of missing information or entropy. The approach is more limited in scope than the one proposed by Jaynes, but has the advantage that it is objective and that the operational origin of the "given" expectation values is specified.

The asymptotic distributions of incomplete U-statistics

Abstract: An incomplete U-statistic is obtained by sampling the terms of an U-statistic. This paper derives the asymptotic distribution (if the variance is finite). Depending on the number of sampled terms, the resulting distribution is either the same as for the U-statistic, a normal distribution, or something intermediate. Also the case of a non-random sampling of the terms is treated. As an example, a non-parametric test of the independence of two circular random variables is studied. The results are generalized to generalized U-statistics.

Bounds and expansions for Fisher information when the moments are known

Abstract: SUMMARY An expansion of the score statistic as a polynomial in the random variable can be used to provide a nondecreasing sequence of lower bounds for the Fisher information when the moments are known as functions of the parameter. If the random variable is asymptotically normal, the first three bounds provide the first three terms in the asymptotic expansion for the Fisher information. This leads to a definition of efficiency which may be used to advantage with small sample sizes, providing lower bounds for the efficiency of any statistic whose moments are known but whose exact distribution is intractable. Fisher (1925) suggested that the efficiency of a statistic be measured by its squared correlation with the maximum likelihood estimate. This idea was extended by Rao (1962) who defined first- and second-order efficiency by considering how well a linear or quadratic function of the statistic correlated with the derivative of the log likelihood. In ? 2 of the present paper, we develop this further by looking at polynomial expansions of the score statistic to obtain a nondecreasing sequence of lower bounds for the Fisher information 4y(0) of a statistic Y = y(X) which is some function of the data X. These bounds may be expressed in terms of the moments of the statistic and their first derivatives with respect to the unknown parameter 0. If we return to Fisher's (1925) definition of efficiency as the ratio of fy(0)/Ox(0), whre Ox(0) is the Fisher information in the whole sample, we can use this to obtain an efficiency for Y which is invariant under one-one transformations and which does not need to appeal to asymptotic arguments. The results are particularly useful for random variables whose moments are known but whose exact sampling distribution is either unknown or intractable. In ? 3, we consider other implications, such as the possibility of using different sequences of functions and the question of convergence. A recent application of these results by Brockwell & Brown (1981) is given as an example. Rao (1962) pointed out that the limit of the ratio fy(0)/fx(O) in large samples was the asymptotic squared correlation between the statistic Y and the derivative of the log likelihood function, lx(O) = alx(0)/aO, for the full data set. When this ratio tends to one, both Fisher and Rao suggested using lim {fx(0) -fy(0)} as n -+ oo as a means of discriminating between alternative statistics. This asymptotic concept of second-order

Statistical Inference in Life-Table Experiments: The Finite Rate of Increase

Abstract: Life-table experiments are frequently used to examine the effects of food level, toxicants, and other experimental treatments on a population's finite rate of increase λ. Although methods for computing the variance of λ have been suggested, the sampling distribution of λ, which is needed for statistical inference, has not been described. We used Monte Carlo procedures to simulate sampling distributions of λ for a variety of assumptions regarding survivorship and fecundity schedules and initial cohort sizes. The distribution of λ can be bimodal when cohort size is small and when juvenile mortality is large. Under these circumstances the probability that none of the initial cohort members reproduces is high enough to produce a significant frequency of zero values for λ. Zero therefore becomes the lower mode of the distribution. Many commonly observed mortality schedules and commonly used cohort sizes yield distributions of λ that are skewed toward low values. Although the skewness and variance of distributi...

Theory of Spatial Statistics

Abstract: Classical statistics is based upon sampling theory. This theory involves articulations of the concepts of statistical population, sample, sample space and probability. Meanwhile, spatial statistics is concerned with the application of sampling theory to geographic situations. It involves a translation of these four notions into a geographic context. The primary objective of this paper is to discuss these translations.

Some data-analytic modifications to Bayes-Stein estimation

Abstract: The usual Bayes-Stein shrinkages of maximum likelihood estimates towards a common value may be refined by taking fuller account of the locations of the individual observations. Under a Bayesian formulation, the types of shrinkages depend critically upon the nature of the common distribution assumed for the parameters at the second stage of the prior model. In the present paper this distribution is estimated empirically from the data, permitting the data to determine the nature of the shrinkages. For example, when the observations are located in two or more clearly distinct groups, the maximum likelihood estimates are roughly speaking constrained towards common values within each group. The method also detects outliers; an extreme observation will either the regarded as an outlier and not substantially adjusted towards the other observations, or it will be rejected as an outlier, in which case a more radical adjustment takes place. The method is appropriate for a wide range of sampling distributions and may also be viewed as an alternative to standard multiple comparisons, cluster analysis, and nonparametric kernel methods.

Fitting the Log-Normal Distribution to Loss Data Subject to Multiple Deductibles

Abstract: The application of deductibles to insurance claims yields a statistical sample of truncated loss amounts. The problem is compounded when a different deductible value may have been applied to each policy in the sample. This article develops a statistical procedure for fitting the Log-normal distribution to claims experience in which a different deductible amount may have been applied to each loss. Maximum likelihood estimators for the parameters and their asymptotic sampling distribution are derived. The method is illustrated by applying it to a sample of industrial fire insurance claims.

A characterization of the logarithmic series distribution and its application

Abstract: This paper gives a characterization of the logarithmic series distribution with probability function . Our characterization follows that of Poisson and positive Poisson given by Bol'shev (1965) and Singh (1978), respectively, A statistic is suggested as a consequence of the characterization to test whether a random sample X1,X2..Xnfollows a logarithmic series probability law. The desirability of the test statistic over the usual goodness-of-fit test is discussed, A numerical example is considered for illustration.

A small sample test for an absolutely continuous bivariate exponential model

Abstract: The finite sample distribution of the likelihood ratio sta-tistic is obtained for testing independence, given marginal homo-geneity, in the absolutely continuous bivariate exponential distri-bution of Block and Basu (1974). This test is discussed in light of the analysis of Gross and Lam (1981) on times to relief of head-aches for standard and new treatments on ten subjects.

The Performance of Variable Sampling Plans When the Normal Distribution is Truncated

Abstract: The robustness of standard variable sampling plans due to Lieberman and Resnikoff is considered with respect to truncation of the normal distribution. It is shown how variable sampling plans can be designed if the truncation point and the normal populat..

Some aspects of parameter inference for nearly nonstationary and nearly non invertible ARMA models (II).

Abstract: This article will extend the discussion in Ahtola and Tiao (1984a) of the finite sample distribution of the score function in nearly nonstationary first order autoregressions to nearly noninvertible first order moving average models. This distribution theory can be used to appreciate the behavior of the score function in situations where the asymptotic normal theory is known to give poor approximations in finite samples. The approximate distributions suggested here can be used to test for the value of the moving average parameter when it is close to unity. In particular, a test for noninvertibility can be obtained with an exact finite sample distribution of the test statistic under the null hypothesis

Inner tolerance limits controlling tail proportions for the normal distribution

Abstract: Inner tolerance limits are defined to be statistics based on a random sample which lie between certain quantiles of the distribution sampled with a certain confidence. Assuming an underlying normal distribution, tables are presented which allow inner limits to be implemented over a range of sample sizes, quantiles, and confidences.

Distribution Results for Positive Definite Quadratic Forms with Repeated Roots.

Abstract: : The distribution of a statistic which is a positive linear combination of independent chi-square random variables is evaluated in certain cases where some of the degrees of freedom are larger than one. Such statistics arise in positive definite quadratic forms of normal random vectors and as the trace of a Wishart matrix. They also arise in the asymptotic distribution for chi-squared goodness-of-fit tests with estimated parameters and for the average Kendall tau statistic. (Author)

19 Empirical distribution function

Abstract: Publisher Summary This chapter describes the empirical distribution function. A statistical estimation of F(x) based on a random sample (X 1 . . . X n ,) is the so-called empirical or sample distribution function. F(x) is considered also a (random) function of x . To apply statistical methods based on empirical distribution, such as goodness of fit or two-sample tests, confidence intervals, one needs the exact or limiting distributions of statistics concerned.

The Robustness of some Statistical Test. The Multivariate Case

Abstract: In this work some results will be shown, that were obtained by means of Monte Carlo experiments and that concern the robustness of the following statistical parameters: Quadratic form of the mean; Determinant of the variance and covariance matrix ǀSǀ I; Hotelling T2; Correlation coefficient; Covariance; for bidimensional statistical variables

A test for homogeneity when the sample is a positive lagrangian poisson distribution

Abstract: In this note, we obtain, based on the sample sum, a statistic to test the homogeneity of a random sample from a positive (zero truncated) Lagrangian Poisson distribution given in Consul and Jain (1973). This test statistic conforms, in a special case, to Singh (1978). A goodness-of-fit test statistic for the Borel-Tanner distribution is obtained as a particular case cf our results.

Sufficient statistics for families of distributions with densities of variable support. II

Abstract: Let , be independent random vectors with density . The support of depends on ≡. We set . Then is a sufficient statistic if , where andh are measurable functions. The random vectors and T form together a sufficient statistic if We call a singular sufficient statistic. In the paper one investigates the behavior of a singular sufficient statistic with a minimal number . Under wide assumptions it is proved that is bounded in probability for n→∞ (Theorem 3.1). One investigates the limit distribution of and (Theorems 3.4, 3.5). Some weak analogues of Dynkin's theorems are proved for the statistic T.