Kenneth W. Wachter

(NOTE: Each chapter concludes with Problems and Complements, Notes, and References.) 1. Statistical Models, Goals, and Performance Criteria. Data, Models, Parameters, and Statistics. Bayesian Models. The Decision Theoretic Framework. Prediction. Sufficiency. Exponential Families. 2. Methods of Estimation. Basic Heuristics of Estimation. Minimum Contrast Estimates and Estimating Equations. Maximum Likelihood in Multiparameter Exponential Families. Algorithmic Issues. 3. Measures of Performance. Introduction. Bayes Procedures. Minimax Procedures. Unbiased Estimation and Risk Inequalities. Nondecision Theoretic Criteria. 4. Testing and Confidence Regions. Introduction. Choosing a Test Statistic: The Neyman-Pearson Lemma. Uniformly Most Powerful Tests and Monotone Likelihood Ratio Models. Confidence Bounds, Intervals and Regions. The Duality between Confidence Regions and Tests. Uniformly Most Accurate Confidence Bounds. Frequentist and Bayesian Formulations. Prediction Intervals. Likelihood Ratio Procedures. 5. Asymptotic Approximations. Introduction: The Meaning and Uses of Asymptotics. Consistency. First- and Higher-Order Asymptotics: The Delta Method with Applications. Asymptotic Theory in One Dimension. Asymptotic Behavior and Optimality of the Posterior Distribution. 6. Inference in the Multiparameter Case. Inference for Gaussian Linear Models. Asymptotic Estimation Theory in p Dimensions. Large Sample Tests and Confidence Regions. Large Sample Methods for Discrete Data. Generalized Linear Models. Robustness Properties and Semiparametric Models. Appendix A: A Review of Basic Probability Theory. The Basic Model. Elementary Properties of Probability Models. Discrete Probability Models. Conditional Probability and Independence. Compound Experiments. Bernoulli and Multinomial Trials, Sampling with and without Replacement. Probabilities on Euclidean Space. Random Variables and Vectors: Transformations. Independence of Random Variables and Vectors. The Expectation of a Random Variable. Moments. Moment and Cumulant Generating Functions. Some Classical Discrete and Continuous Distributions. Modes of Convergence of Random Variables and Limit Theorems. Further Limit Theorems and Inequalities. Poisson Process. Appendix B: Additional Topics in Probability and Analysis. Conditioning by a Random Variable or Vector. Distribution Theory for Transformations of Random Vectors. Distribution Theory for Samples from a Normal Population. The Bivariate Normal Distribution. Moments of Random Vectors and Matrices. The Multivariate Normal Distribution. Convergence for Random Vectors: Op and Op Notation. Multivariate Calculus. Convexity and Inequalities. Topics in Matrix Theory and Elementary Hilbert Space Theory. Appendix C: Tables. The Standard Normal Distribution. Auxiliary Table of the Standard Normal Distribution. t Distribution Critical Values. X 2 Distribution Critical Values. F Distribution Critical Values. Index.

There is no generally accepted procedure for identifying ultradian pulsations in hormonal time series. We suggest an approach based on removing long-term trends, such as diurnal rhythms, from the series of observations; identifying peaks in the residual series; and resolving each peak, if appropriate, into overlapping secretory episodes. The first step uses a robust smoothing technique to generate a smoothed series that omits peaks or trends with time constants less than 6--12 h. The smoothed series is subtracted from the original, and in the second step their difference, the residual series, is examined for the presence of peaks. The standard deviation of the assay is calculated at each point, and the residuals are rescaled in terms of this unit. Peaks are identified as individual subseries elevated above the base line of duration n, all the points in which have magnitude at least G(n), where the values of G are cut-off criteria based on the width of the peak. Thus the algorithm selects both narrow high peaks and broader peaks that may be lower. The user selects the G(n) for each hormone based on theoretical considerations or a set of calibration data series. Points that meet these criteria are identified as belonging to peaks and flagged. To assure that the smoothing process is not influenced by runs of closely spaced peaks, these flagged points are then assigned a reduced weight, and the smoothing is repeated; the revised residuals are then reexamined. After these two steps are iterated until there are no further changes, each peak is examined once more to determine whether it can be resolved into more than one overlapping peak. Finally, the process collects statistics on the average frequency and amplitude of the peaks. We have developed computer programs to carry out these algorithms.

In historical accounts of the circumstances of ordinary people's lives, nutrition has been the great unknown. Nearly impossible to measure or assess directly, it has nonetheless been held responsible for the declining mortality rates of the nineteenth century as well as being a major factor in the gap in living standards, morbidity and mortality between rich and poor. The measurement of height is a means of the direct assessment of nutritional status. This important and innovative study uses a wealth of military and philanthropic data to establish the changing heights of Britons during the period of industrialization, and thus establishes an important dimension to the long-standing controversy about living standards during the Industrial Revolution. Sophisticated quantitative analysis enables the authors to present some striking conclusions about the actual physical status of the British people during a period of profound social and economic upheaval, and Height, Health and History will provide an invigorating statistical edge to many debates about the history of the human body itself.

This paper proves almost-sure convergence of the empirical measure of the normalized singular values of increasing rectangular submatrices of an infinite random matrix of independent elements. The limit is the limit as both dimensions grow large in some ratio. The matrix elements are required to have uniformly bounded central $2 + \delta$th moments, and the same means and variances within a row. The first section (relaxing the restriction on variances) proves any limit-in-distribution to be a constant measure rather than a random measure, establishes the existence of subsequences convergent in probability, and gives a criterion for almost-sure convergence. The second section proves the almost-sure limit to exist whenever the distribution of the row variances converges. It identifies the limit as a nonrandom probability measure which may be evaluated as a function of the limiting distribution of row variances and the dimension ratio. These asymptotic formulae underlie recently developed methods of probability plotting for principal components and have applications to multiple discriminant ratios and other linear multivariate statistics.

Theories about biological limits to life span and evolutionary shaping of human longevity depend on facts about mortality at extreme ages, but these facts have remained a matter of debate. Do hazard curves typically level out into high plateaus eventually, as seen in other species, or do exponential increases persist? In this study, we estimated hazard rates from data on all inhabitants of Italy aged 105 and older between 2009 and 2015 (born 1896-1910), a total of 3836 documented cases. We observed level hazard curves, which were essentially constant beyond age 105. Our estimates are free from artifacts of aggregation that limited earlier studies and provide the best evidence to date for the existence of extreme-age mortality plateaus in humans.