Showing papers on "Sampling distribution published in 1991"

Statistics for the Life Sciences

Abstract: 1. Introduction. Statistics and the Life Sciences. Examples and Overview. 2. Description of Populations and Samples. Introduction. Frequency Distributions: Techniques for Data. Frequency Distributions: Shapes and Examples. Descriptive Statistics: Measures of Center. Boxplots. Measures of Dispersion. Effect of Transformation of Variables (Optional). Samples and Populations: Statistical Inference. Perspective. 3. Random Sampling, Probability, and the Binomial Distribution. Probability and the Life Sciences. Random Sampling. Introduction to Probability. Probability Trees. Probability Rules (Optional). Density Curves. Random Variables. The Binomial Distribution. Fitting a Binomial Distribution to Data (Optional). 4. The Normal Distribution. Introduction. The Normal Curves. Areas Under a Normal Curve. Assessing Normality. The Continuity Correction (Optional). Perspective. 5. Sampling Distributions. Basic Ideas. Dichotomous Observations. Quantitative Observations. Illustration of the Central Limit Theorem (Optional). The Normal Approximation to the Binomial Distribution (Optional). Perspective. 6. Confidence Intervals. Statistical Estimation. Standard Error of the Mean. Confidence Interval. Planning a Study to Estimate. Conditions for Validity of Estimation Methods. Confidence Interval for a Population Proportion. Perspective and Summary. 7. Comparison of Two Independent Samples. Introduction. Standard Error of (y1 - y2). Confidence Interval. Hypothesis Testing: The t-test. Further Discussion of the t-test. One-Tailed Tests. More on Interpretation of Statistical Significance. Planning for Adequate Power (Optional). Student's t: Conditions and Summary. More on Principles of Testing Hypotheses. The Wilcoxon-Mann-Whitney Test. Perspective. 8. Statistical Principles of Design. Introduction. Observational Studies. Experiments. Restricted Randomization: Blocking and Stratification. Levels of Replication. Sampling Concerns (Optional). Perspective. 9. Comparison of Paired Samples. Introduction. The Paired-Sample t-Test and Confidence Interval. The Paired Design. The Sign Test. The Wilcoxon Signed-Rank Test. Further Considerations in Paired Experiments. Perspective. 10. Analysis of Categorical Data. Inference for Proportions: The Chi-Squared Goodness-of-Fit Test. The Chi-Squared Test for a 2 X 2 Contingency Table. Independence and Association in a 2 X 2 Contingency Table. Fisher's Exact Test (Optional). The r x k Contingency Table. Applicability of Methods. Confidence Interval for a Difference Between Proportions. Paired Data and 2 X 2 Tables (Optional). Relative Risk and the Odds Ratio (Optional). Summary of Chi-Squared Tests. 11. Comparing the Means of Many Independent Samples. Introduction. The Basic Analysis of Variance. The Analysis of Variance Model (Optional). The Global F-Test. Applicability of Methods. Two-Way ANOVA (Optional). Linear Combinations of Means (Optional). Multiple Comparisons (Optional). Perspective. 12. Linear Regression and Correlation. Introduction. The Fitted Regression Line. Parametric Interpretation of Regression: The Linear Model. Statistical Inference Concerning B1. The Correlation Coefficient. Guidelines for Interpreting Regression and Correlation. Perspective. Summary of Formulas. 13. A Summary of Inference Methods. Introduction. Data Analysis Samples. Appendices. Chapter Notes. Statistical Tables. Answers to Selected Exercises. Index. Index of Examples.

A modified score test statistic having chi-squared distribution to order n−1

Abstract: SUMMARY We give a general modified score test statistic whose nuil distribution is chi-squared to order n-1, where n is the sample size. The modified statistic depends on the joint cumulants of log likelihood derivatives for the full data. Some applications are discussed. Following Cox & Reid (1987a) we also derive a general formula for Bartlett-type corrections to improve test statistics whose null asymptotic distributions are chi-squared.

Sensitivity in Bayesian Statistics: The Prior and the Likelihood

Abstract: One paradigm for sensitivity analyses in Bayesian statistics is to specify Γ, a reasonable class of priors, and to compute the corresponding class of posterior inferences. The class Γ is chosen to represent uncertainty about the prior. There is often additional uncertainty, however, about the family of sampling distributions. This article introduces a method for computing ranges of posterior expectations over reasonable classes of sampling distributions that lie “close to” a given parametric family. By treating the prior as a probability measure on the space of sampling distributions this article also gives a unified treatment to what are usually considered two separate problems—sensitivity to the prior and sensitivity to the sampling model. First the notion of “close to” is made explicit. Then, an algorithm is given for turning ratio-linear problems into sequences of linear problems. In addition to solving the problem at hand, the algorithm simplifies many other robust Bayesian computational pro...

Asymptotic expansions of the information matrix test statistic

Abstract: The information matrix (IM) test is shown to have a finite sample distribution which is poorly approximated by its asymptotic X 2 distribution in models and sample sizes commonly encountered in applied econometric research The quality of the x2 approximation depends upon the method chosen to compute the test Failure to exploit restrictions on the covariance matrix of the test can lead to a test with appalling finite sample properties Order O(n -1) approximations to the exact distribution of an efficient form of the IM test are reported These are developed from asymptotic expansions of the Edgeworth and Cornish-Fisher types They are compared with Monte Carlo estimates of the finite sample distribution of the test and are found to be superior to the usual x2 approximations in sample sizes of the magnitude found in applied micro-econometric work The methods developed in the paper are applied to normal and exponential models and to normal regression models Results are provided for the full IM test and for heteroskedasticity and nonnormality diagnostic tests which are special cases of the IM test In geieral the quality of alternative approximations is sensitive to covariate design However commonly used nonnormality tests are found to have distributions which, to order O(n-1), are invariant under changes in covariate design This leads to simple design and parameter invariant size corrections for nonnormality tests

A continuous time approximation to the unstable first-order autoregressive process: the case without an intercept

Abstract: We consider a first-order autoregression with i.i.d. errors and a fixed initial condition. The asymptotic distribution of the normalized least-squares estimator as the sampling interval converges to zero is shown to be the same as the exact distribution of the continuous-time estimator in an Ornstein-Uhlenbeck process. This asymptotic distribution permits explicit consideration of the effect of the initial condition. The appropriate moment-generating function is derived and used to tabulate the limiting distribution and probability density functions, the moments and some power functions. The adequacy of this asymptotic approximation is found to be excellent for values of the autoregressive parameter near one and any fixed initial condition. Copyright 1991 by The Econometric Society.

Approximations of marginal tail probabilities for a class of smooth functions with applications to Bayesian and conditional inference

Abstract: SUMMARY This paper presents an asymptotic approximation of marginal tail probabilities for a real-valued function of a random vector, where the function has continuous gradient that does not vanish at the mode of the joint density of the random vector. This approximation has error 0(n-312) and improves upon a related standard normal approximation which has error 0(n-1). Derivation involves the application of a tail probability formula given by DiCiccio, Field & Fraser (1990) to an approximation of a marginal density derived by Tierney, Kass & Kadane (1989). The approximation can be applied for Bayesian and conditional inference as well as for approximating sampling distributions, and the accuracy of the approximation is illustrated through several numerical examples related to such applications. In the context of conditional inference, we develop refinements of the standard normal approximation to the distribution of two different signed root likelihood ratio statistics for a component of the natural parameter in exponential families.

Smoothing Methods for Estimating Test Score Distributions

Abstract: Frequency distributions of test scores may appear irregular and, as estimates of a population distribution, contain a substantial amount of sampling error. Techniques for smoothing score distributions are available that have the capacity to improve estimation. In this article, estimation/smoothing methods that are flexible enough to fit a wide variety of test score distributions are reviewed. The methods are a kernel method, a strong true-score model-based method, and a method that uses polynomial log-linear models. The use of these methods is then reviewed, and applications of the methods are presented that include describing and comparing test score distributions, estimating norms, and estimating equipercentile equivalents in test score equating. Suggestions forfurther research are also provided. Entire test score distributions need to be considered for many measurement purposes that include describing and comparing score distributions, estimating percentile norms, and estimating equipercentile equating relationships. The sample score distribution often appears very irregular when graphed, and it may contain a substantial amount of sampling error. Statistical methods can be used to address these problems by producing smoothed estimates of the population score distribution that are intended to have less estimation error than the sample score distribution. Statistical methods for estimating univariate test score distributions are described and evaluated in this article. It focuses on estimating distributions of discrete number-correct scores on educational tests. For this reason, the methods that are considered are those that produce discrete distributions. These three methods are flexible enough to adequately fit a wide variety of potential test score distributions. They are: (a) a kernel method, (b) a method based on Lord's (1965) strong true-score model, and (c) a method that uses a polynomial log-linear model. The description of each method includes a discussion of procedures for choosing the degree of smoothing and/or evaluating the fit of the smoothed to the sample distribution as well as a discussion of other related methods. The kernel method for discrete test score distributions has not been treated elsewhere in published sources, and for this reason it is presented here in detail. Because the other two methods are treated elsewhere, they will only be summarized here. Research on these methods is reviewed and suggestions are made for areas of future research. Selected applications of

Strategies for sampling spatially heterogeneous phenomena; the example of river gravels

Abstract: The paucity of critical results in the field sciences is partly the result of ineffective data collection. Multiple samples must be collected to take advantage of the power of inferential statistics, but the number of samples needed may be minimized by using efficient sampling methods. This study shows that for gravel river bars characterized by the presence of several unevenly distributed facies, the grid technique for sample selection is one of the best and easiest to use in the field. A special combination of grid and random techniques, known as systematic unaligned sampling, performs slightly better, but site selection and navigation in the field are more complex. Random sampling performs less well than either of the other methods. If the sampling question requires estimation of a statistic to represent the entire area or volume, rather than estimates of the variability within and between different zones of the area or volume, then the use of composite sampling further reduces the sampling effort required. A few replications of these composite samples will provide an estimate of the mean and standard deviation of a population statistic which can be used to construct confidence limits about the estimated statistic.

On counting the number of data pairs for semivariogram estimation

Abstract: In planning spatial sampling studies for the purpose of estimating the semivariogram, the number of data pairs separated by a given distance is sometimes used as a comparative index of the precision which can be expected from a given sampling design. Because spatial data are correlated, this index can be unreliable. An alternative index which partially corrects for this correlation, themaximum equivalent uncorrelated pairs, is proposed for comparing spatial designs. The index is developed under the assumption that the underlying stochastic process is Gaussian and is appropriate when the (population) semivariogram is to be estimated by the sample semivariogram.

On maximally selected Chi-square statistics

Abstract: Two samples can be compared by selecting a cutpoint and then forming a 2 x 2 table of the numbers of observations above and below the cutpoint in each sample. Miller and Siegmund (1982, Biometrics 38, 1011-1016) investigated asymptotic theory relating to the distribution of the "standard" chisquare statistic when the cutpoint is selected to maximize its value; in a companion article, Halpern (1982, Biometrics 38, 1017-1023) studied the finite-sample distribution of this maximally selected chi-square statistic via simulation. Exact finite-sample distribution theory, derived from Durbin's (1971, Journal of Applied Probability 8, 431-453; 1973, Distribution Theory for Tests Based on the Sample Distribution Function, Philadelphia: SIAM) combinatorial approach, is presented here.

Random dendrograms and null hypotheses in cladistic biogeography

Abstract: Simberloff et al. (1981, pages 40-63 in Vicariance biogeography: A critique (G. Nelson and D. E. Rosen, eds.), Columbia Univ. Press, New York) proposed three different sampling distributions of trees for testing cladistic biogeographic hypotheses. Two of these distributions involved sampling from a uniform distribution of cladograms (either labeled or unlabeled), whereas the third distribution was calculated by a simple Markovian model. This paper presents a proof that the third distribution of cladograms is identical to the uniform distribution of dendrograms (rooted trees with internal nodes ranked). Hence, all three sampling distributions are uniform distributions that differ solely in the kind of tree being sampled. The paper concludes by suggesting ways of constructing nonuniform tree distributions. (Biogeography; cladistics; dendrograms; null hypotheses; random trees.)

Consistent ordered sampling distributions : characterization and convergence

Abstract: This paper is concerned with models for sampling from populations in which there exists a total order on the collection of types, but only the relative ordering of types which actually appear in the sample is known. The need for consistency between different sample sizes limits the possible models to what are here called ‘consistent ordered sampling distributions'. We give conditions under which weak convergence of population distributions implies convergence of sampling distributions and conversely those under which population convergence may be inferred from convergence of sampling distributions. A central result exhibits a collection of ‘ordered sampling functions', none of which is continuous, which separates measures in a certain class. More generally, we characterize all consistent ordered sampling distributions, proving an analogue of de Finetti's theorem in this context. These results are applied to an unsolved problem in genetics where it is shown that equilibrium age-ordered population allele frequencies for a wide class of exchangeable reproductive models converge weakly, as the population size becomes large, to the so-called GEM distribution. This provides an alternative characterization which is more informative and often more convenient than Kingman's (1977) characterization in terms of the Poisson–Dirichlet distribution.

Bayesian residual analysis in the presence of censoring

Abstract: SUMMARY Use of the posterior distribution of the realized error terms for residual analysis in a linear model was advocated by Zellner (1975) and Zellner & Moulton (1985). The same idea was used by Chaloner & Brant (1988) to define outliers and calculate posterior probabilities of observations being outliers. This paper extends the same concept to other models, including regression models for lifetime data, by using an approach similar to that of Cox & Snell (1968). Residual plots are proposed where realized errors are represented by interval estimates. Incorporating censored observations into this framework is straightforward. 1. THE REALIZED ERROR TERMS Cox & Snell (1968) proposed a general definition of residuals for models where each observation yi can be written as yi = gi(0, Ei) with 0 a vector of unknown parameters and the Ei, for i 1,..., n, a sample of independent and identically distributed random variables from a known distribution. Suppose that the equation yi = gi(0, ?,) has a unique solution Ei = hi(y,, 0). Cox & Snell define Ei = hi(yi, 0) to be the residuals where 0 are the maximum likelihood estimates of 0. In the Bayesian approach used here, define ?i = hi(yi, 0), for i = 1, . . ., n, to be the residuals. Each E, is just a function of the unknown parameters and the posterior distribution is therefore straightforward to calculate. The posterior distribution can be examined for indications of possible departures from the assumed model and the presence of outliers. The posterior distribution of the realized errors is very different from the sampling distribution of their estimates. The posterior distribution represents the uncertainty about functions of the parameters; if the parameters, 0, and observations, yi, are known then so are the realized errors ei = h,(y,, 0). With a large sample size, the posterior distribution of the ri will be, approximately, multivariate normal centred at the posterior mean and with covariance matrix the posterior covariance matrix. An alternative approximation to the posterior mean of the realized

Bayesian bootstrap clones and a biometry function

Abstract: A class of random-weighted empirical distributions is introduced. These random-weighted distributions can be used to play the role of the random-weighted distribution of Rubin (1981) and, in fact, the random-weighted distribution of Rubin is a member of this family. The BBC approximations to posterior distributions and sampling distributions are discussed. It is shown that approximations based on these random-weighted distributions and the bootstrap of Efron (1979) are first-order asymptotically equivalent. An example of a life expectancy function (Yang, 1968) is given.

A handbook of introductory statistical methods

Abstract: Definitions Sample Mean and Variance: Coefficient of Variation Sample and Population Relative Frequency Distributions Probability (Continuous Variates) Population Mean and Variance: Sample Estimates The Simple Linear Transformation of a Variate The Addition and Subtraction of Independent Variates The Distribution of the Sample Mean Statistical Models Normal (Gaussian) Distributions The Standard Normal (Gaussian) Reference Distribution T-Distributions Confidence Intervals for the Population Mean of a Normal Distribution and a General Formulation Chi-Squared (X2) Reference Distributions Confidence Intervals for the Population Variance of a Normal Distribution Hypothesis Testing (Significance Testing) Hypothesis Testing of the Mean of a Normal Distribution: The Two-Tailed T-Test Hypothesis Testing for the Variance of a Normal Distribution: The Two Tailed X2 Test Confidence Intervals and Hypothesis Tests Related Testing One-Sided Alternative Hypotheses Type I and Type II Errors and Power.

Statistical Inference for the Bounds of Random Variables

Abstract: This chapter describes and studies statistical procedures having a significant place in the theory of global random search (these procedures are included into some of the methods of Chapter 4). Most attention is paid to linear statistical procedures that are simple to realize.

Univariate Data Analysis

Abstract: As outlined in Chapter 1 the techniques of statistical inference usually require that assumptions be made regarding the sample data. Such assumptions usually include the type of sampling process that produced the data and in some cases the nature of the population distribution from which the sample was drawn. When assumptions are violated the techniques employed can lead to misleading results. Good statistical practice therefore requires that the data be studied in detail before statistical inference procedures are applied. The techniques of exploratory data analysis are designed to provide such a preliminary view.

A Continuous Time Approximation to the Stationary First-Order Autoregressive Model

Abstract: We consider the least-squares estimator in a strictly stationary first-order autoregression without an estimated intercept. We study its continuous time asymptotic distribution based on an asymptotic framework where the sampling interval converges to zero as the sample size increases. We derive a momentgenerating function which permits the calculation of percentage points and moments of this asymptotic distribution and assess the adequacy of the approximation to the finite sample distribution. In general, the approximation is excellent for values of the autoregressive parameter near one. We also consider the behavior of the power function of tests based on the normalized leastsquares estimator. Interesting nonmonotonic properties are uncovered. This analysis extends the study of Perron [15] and helps to provide explanations for the finite sample results established by Nankervis and Savin [13].

Some Lagrange multiplier tests for seasonal differencing

Abstract: SUMMARY Determining whether a time series has a unit root is an important problem in many time series analyses. For seasonal time series the problem is more complicated as one has to decide whether both regular and seasonal differencing or just one of them would suffice to transform a series into stationarity. This important problem is addressed via the Lagrange multiplier test approach. The large sample representations of the test statistics in terms of integrals of Wiener processes are obtained. These facilitate the tabulation of the large sample distribution of the statistics. Some empirical results are reported.

Test efficiency analysis of random self-test of sequential circuits

Abstract: Motivated by the work of K. Kim et al. (1988) and A. Krasniewski and S. Pilarski (1989), the problem of test efficiency in random testing of sequential circuits using built-in self-test (BIST) techniques is addressed. It is shown that, given a circuit with n primary inputs and the goal of maximizing expected pattern coverage, different pattern-sampling distributions for its 2/sup n/ possible patterns can be partially ordered. The exact distributions for pattern coverage for both equiprobable and nonequiprobable pattern-sampling distributions are derived. Approximations for pattern-coverage distributions under equiprobable pattern-sampling conditions and corresponding numerical results are presented. A limiting distribution function for pattern-coverage distribution is derived. The authors also present numerical results on confidence levels for obtaining a specified pattern coverage. The distribution for the number of test cycles (R) required to achieve a specified pattern coverage is also derived. The authors derive and use the expression for the expected value of R to illustrate the increase in the effect of achieving a specified coverage j as j increases. >

Effective procedures for estimating beta distribution's parameters and Their confidence intervals

Abstract: First, we compare several methods for computing the ML estimators of the two-parameter beta distribution; the most effective one is identified. Second, a simple way is found to characterize the sampling distribution of the ML estimators; this characterization leads to a practical way of establishing confidence intervals for the ML estimators.

One-step Bootstrapping for Smooth Iterative Procedures

Abstract: SUMMARY Resampling techniques have the potential to provide useful information about the sampling distribution of estimators of many population characteristics. Ambitious schemes such as the bootstrap and iterated bootstrap imply a substantial increase in computational effort. For some iterative procedures, such as generalized least squares or the EM algorithm, it is possible to avoid fully iterating each bootstrap replication to convergence. By analysing expansions of the defining equation, we can extract asymptotically correct bootstrap estimates from a single step for each replication. In this paper we demonstrate the large sample validity of this computationally efficient approach and illustrate its small sample applicability. Whether or not the adjustment represents an adequate replacement for full iteration depends on the nature of the problem and the desired accuracy for the bootstrap quantiles. If subsequent iterations are adjusted, then greater enhancement of the rate is achieved and the practical increase in accuracy is significant.

Testing Equality of Means in the Presence of Correlation and Missing Data

Abstract: A statistic, derived from the combination of two dependent tests, is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one or both variates. The null distribution of the statistic is approximated by a well-known distribution. The empirical powers of the statistic are computed and compared with some of the known statistics. The comparisons support the use of the proposed test.

Topics in statistical methodology

Abstract: Probabilistic Foundation of Mathematical Statistics. Functions of Random Variables, Expectations and Limit Theorems. Discrete Probability Distributions. Absolutely Continuous Distributions. Basic Statistical Inference and the Censoring Techniques. Bivariate Distribution, Regression and Correlation Analysis. Multinomial Distribution, Pearsonian chi2 and Transformation of Statistics. Sampling Distribution. Appendices. References. Index.

Two-Step and Related Estimators in Contemporary Rational-Expectations Models: An Analysis of Small-Sample Properties

Abstract: This article examines the performance of ordinary least squares, generalized least squares, and Pagan's (1986) double-length estimator (DLE) in several rational-expectations models. The three approaches are equivalent in the simplest of models but may differ appreciably in models typically encountered in applied work. Small-sample properties of the estimators are examined in several contemporary macroeconomic models. The following conclusions are reached: (a) All estimators exhibit similar sampling distributions in a monetary-neutrality framework, (b) the least squares procedures maintain smaller sampling variance and deliver more reliable tests in a permanent-income model in very small samples, (c) DLE generally delivers superior performance in a nonlinear aggregate-supply model with unanticipated “shock” regressors, and (d) overall, DLE outperforms the LS alternatives except in the smallest of samples.

An Inductive Proof of the Sampling Distributions for the MLE's of the Parameters in an Inverse Gaussian Distribution

Abstract: Tweedie showed that the sampling distributions of the MLE's for the parameters of an inverse Gaussian distribution were independent inverse Gaussian and chi-squared distributions. His proof, however, requires substantial mathematical machinery, for example, Laplace transforms, Lerch's theorem, or conditional moment generating functions. We present an inductive proof that only requires multivariable transformations that are presented in intermediate mathematical statistics classes.