This volume is a collection of papers in celebration of the 80th birthday of Yuan-Shih Chow, whose influential work in probability and mathematical statistics has contributed greatly to mathematics education and the development of statistics research and application in Taiwan and mainland China.
The twenty-two papers cover a wide range of problems reflecting both the broad scope of areas where Professor Chow has made major contributions and recent advances in probability theory and statistics.
Sample Chapter(s)
Chapter 1: Maximizing Expected Value with Two Stage Stopping Rules (642 KB)
https://doi.org/10.1142/9789812772558_fmatter
A Conversation with Yuan Shih Chow.
Publications of Yuan Shih Chow.
PREFACE.
CONTENTS.
https://doi.org/10.1142/9789812772558_0001
Let Xn,…,X1 be i.i.d. random variables with distribution function F and finite expectation. A statistician, knowing F, observes the X values sequentially and is given two chances to choose X's using stopping rules. The statistician's goal is to select a value of X as large as possible. Let equal the expectation of the larger of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behavior of the sequence
for a large class of F's belonging to the domain of attraction (for the maximum)
, where
with α > 1. The results are compared with those for the asymptotic behavior of the classical one choice value sequence
, as well as with the "prophet value" sequence E(max{Xn,…,X1}), and indicate that substantial improvement is obtained when given two chances to stop, rather than one.
https://doi.org/10.1142/9789812772558_0002
No abstract received.
https://doi.org/10.1142/9789812772558_0003
When the variance is known, a level 1 – α confidence interval of specified width 2h > 0 for the mean of a normal distribution requires a sample of size at least η = c2σ2/h2, where c is the upper quantile of the standard normal distribution. If the variance is unknown, then such an interval may be constructed using Stein's double sampling procedure in which an initial sample of size m ≥ 2 is drawn and used to estimate η. Here is it shown that if the experimenter can specify a prior guess, η0 say, for η, then
is an approximately minimax choice for the initial sample size. The formulation is, in fact, more general and includes point estimation with equivariant loss as well as interval estimation.
https://doi.org/10.1142/9789812772558_0004
We propose methods for incorporating overrunning data, either balanced or unbalanced, into the final analysis of a sequential clinical trial comparing an experimental arm with a control arm. We consider inference on the primary endpoint for which the sequential test is designed and a correlated secondary endpoint. By separating the monitoring process into arm-specific processes, we derive the sufficient statistics and show how the independent overrunning data can be combined with the trial data at stopping and thus allow the likelihood-based inference to be conducted.
https://doi.org/10.1142/9789812772558_0005
This note considers the stationary sequence generated by applying an instantaneous filter to a linear random field in Z2. The class of filters under consideration includes polynomials and indicator functions. Instead of imposing certain mixing conditions on the random fields, it is assumed that the weights of the innovations satisfy a summability property. By building a martingale decomposition based on a suitable filtration, asymptotic normality is proved for the partial sums of the stationary sequence.
https://doi.org/10.1142/9789812772558_0006
We study properties of thinning and Markow chain thinning of renewal processes. Among others, for some special renewal processes we investigate whether these processes can be obtained through Markov chain thinning.
https://doi.org/10.1142/9789812772558_0007
We study three stochastic differential games. In each game, two players control a process X = {Xt, 0 ≤ t < ∞} which takes values in the interval I = (0,1), is absorbed at the endpoints of I, and satisfies a stochastic differential equation
In the first of our games, which is zero-sum, player 𝔄 has a continuous reward function u : [0,1] → ℝ. In addition to α(·), player 𝔄 chooses a stopping rule τ and seeks to maximize the expectation of u(Xτ); whereas player 𝔅 chooses β(·) and aims to minimize this expectation.
In the second game, players 𝔄 and 𝔅 each have continuous reward functions u(·) and v(·), choose stopping rules τ and ρ, and seek to maximize the expectations of u(Xτ) and v(Xρ), respectively.
In the third game the two players again have continuous reward functions u(·) and v(·), now assumed to be unimodal, and choose stopping rules τ and ρ. This game terminates at the minimum τ∧ρ the stopping rules τ and ρ, and players 𝔄, 𝔅 want to maximize the expectations of u(Xτ∧ρ) and ν(Xτ∧ρ), respectively.
Under mild technical assumptions we show that the first game has a value, and find a saddle point of optimal strategies for the players. The other two games are not zero-sum, in general, and for them we construct Nash equilibria.
https://doi.org/10.1142/9789812772558_0008
Let Y have a symmetric Dirichlet multinomial distributions in ℝm, and let Sm = h(Y1) + ⋯ + h(Ym). We derive a central limit theorem for Sm as the sample size n and the number of cells m tend to infinity at the same rate. The rate of convergence is shown to be of order m1/6. The approach is based on approximation of marginal distributions for the Dirichlet multinomial distribution by negative binomial distributions, and a blocking technique similar to that used to study renormalization groups in statistical physics. These theorems generalize and refine results for the classical occupancy problem.
https://doi.org/10.1142/9789812772558_0009
Optimal stopping has been one of Y.S. Chow's major research areas in probability. This paper reviews the seminal work of Chow and Robbins on the optimal stopping problem for Sn/n and subsequent developments and intercrosses with other areas of probability theory. It also uses certain ideas and techniques from these developments to devise relatively simple but accurate approximations to the optimal stopping boundary.
https://doi.org/10.1142/9789812772558_0010
In Dubins and Savage's casino, there are only limited gambles available and the optimal win probability, attained by bold play, is a singular function of the intial fortune, f. Vardi asked the question of how-to-gamble-if-you-must in a casino where any bet is available as long as its expected value is less than a fixed constant c < 0 times the amount staked, and one cannot stake more than one has at any moment. Vardi's casino has the optimal probability, 1 – (1 – f)1+c, though it is only attainable within ∊, and the ∊-optimal betting strategy is far from bold play in some aspects.
https://doi.org/10.1142/9789812772558_0011
We consider the upper limit of Sn/n1/p where Sn are partial sums of iid random variables. Under the assumption of E(X+)p < ∞, we provide an integral test which determines the upper limit up to certain universal constant factors depending on p only. The problem is closely related to moment properties of ladder variables. We prove our theorem by considering the lower limit of Tk/kp where Tk is the k-th epoch of the random walk.
https://doi.org/10.1142/9789812772558_0012
A sequence of dependent 2 × 2 contingency tables often arises in epidemiologic cohort studies, controlled clinical trials, and other follow-up studies. Due to dependence, it is unclear, however, whether and how the conditional approach for a single 2 × 2 table can be extended to analyze a sequence of dependent 2 × 2 tables. We show that distributional properties can be derived by considering a "tangent" sequence of independent 2 × 2 tables, with each 2 × 2 table being represented by a sum of independent, yet not identically distributed, Bernoulli trials, whose success probabilities can easily be computed (Kou and Ying, 1996). The method has four applications: (1) We provide a characterization of the validity of a weighted log-rank test. (2) The method leads to a simple algorithm to compute the maximum partial likelihood estimator of the common odds ratio, as well as its variance estimator. The efficiency over the traditional Mantel-Haenszel estimator is also demonstrated. (3) We show how to use the method to provide estimation in Breslow-Zelen model for regression with nonhomogeneous odd ratios. (4) The method is applied to analysis of a proportional hazards model with tied observations. The computation is straightforward by using a link with the roots of Jacobi polynomials.
https://doi.org/10.1142/9789812772558_0013
Many robust estimators of location, e.g. trimmed means, implicitly assume that the data come from a symmetric distribution. Consequently, it is important to check this assumption with an appropriate statistical test that does not assume a known value of the median or location parameter. This article replaces the mean and standard deviation in the classic Hotelling-Solomons measure of asymmetry by corresponding robust estimators; the median and mean deviation from the median. The asymptotic distribution theory of the test statistic is developed and the new procedure is compared to tests recently proposed by Cabilio and Masaro (1996) and Mira (1999). Using their approach to approximating the variance of this class of statistics, it is shown that the new test has greater power than the existing tests to detect the asymmetry of skewed contaminated normal data as well as a majority of skewed distributions belonging to the lambda family. The increased power of the new test suggests that the use of robust estimators in goodness of fit type tests deserves further study.
https://doi.org/10.1142/9789812772558_0014
The Cox proportional hazards regression model has been widely used in the analysis of survival/duration data. It is semiparametric because the model includes a baseline hazard function that is completely unspecified. We study here the statistical inference of the Cox model where some information about the baseline hazard function is available, but it still remains as an infinite dimensional nuisance parameter. We incorporate the information about the baseline hazard into the inference for regression coefficient by using the empirical likelihood method (Owen 2001) and obtained the modified test/estimator and their asymptotic distributions. The modified estimator is shown to be better than the regular Cox partial likelihood estimator in theory and in several simulations.
https://doi.org/10.1142/9789812772558_0015
A circularity statistic, based upon pairwise Mann-Whitney statistics, and measuring the non-transitivity effect A > B > C > A, was introduced in Brown & Hettmansperger (2002). In the present paper, its large sample null distribution is shown to be logistic. To test for non-transitivity, possibly indicating the presence of mixture terms, one of the components of the logistic limit variable is used as a regulator to prevent the circularity statistic being inflated by large transitive rather than non-transitive effects. An example is discussed.
https://doi.org/10.1142/9789812772558_0016
We describe a Bayesian framework for shape-restricted regression in which the prior is given by Bernstein polynomials. We present consistency theorems concerning the posterior distribution in this Bayesian approach. This study includes monotone regression and a few other shape-restricted regressions.
https://doi.org/10.1142/9789812772558_0017
By using the theory of spherical distributions and some properties of invariant statistics, we develop a stabilized uniform Q-Q plot for checking the multi-normality assumption in high-dimensional data analysis. Acceptance regions associated with the plot are given. Empirical performance of the acceptance regions is studied by Monte Carlo simulation. Application of the Q-Q plot is illustrated by a real data set.
https://doi.org/10.1142/9789812772558_0018
In this paper we consider the problem of finding the maximum of a regression function using the Bayesian approach. We first express the solution as a function of unknown parameters of a model. We then invoke the plug-in principle to substitute Bayes estimates for unknown parameters at each stage. The resulting scheme differs from the classic Kiefer-Wolfowitz scheme (Kiefer and Wolfowitz, 1952) in several important aspects. In particular it does not depend on a sequence of constants an → 0 to dampen the effect of random errors as in Robbins and Monro (1951). The convergence of the scheme to the desired value with probability one is established under mild conditions. In the case of several independent variables, our model is the same as the second order model considered in response surface methods. Hence the convergence result of this paper would be of interest to response surface methods.
https://doi.org/10.1142/9789812772558_0019
We consider L1-norm kernel estimator of the conditional median θ(x), for a wide class of stationary processes. Asymptotic normality of the resulting estimator θn(x) is established under different regularity conditions on bandwidths. Applications of our main results are also given.
https://doi.org/10.1142/9789812772558_0020
We discuss some statistical aspects in materials science that involve microstructures and materials properties. Materials scientists have applied homogenization theory and the finite element method (FEM) for justification and computation of certain effective properties, which are considered as expectations with respect to the distributions of underlying microstructures. However, many important issues remain unresolved, including the implementation of more efficient Monte Carlo computational methods, quantification of statistical variability of FEM, etc. Through several examples, we illustrate stochastic geometry models for microstructural features, Markov chain Monte Carlo (MCMC) methods in computation of elastic moduli and thermal (or electrical) conductivity, and MCMC based confidence intervals related to FEM.
https://doi.org/10.1142/9789812772558_0021
Population-based association study using unrelated individuals is a powerful strategy (Risch and Merikangas, 1996; Risch, 2000) for detecting association between markers and quantitative trait loci (QTLs). However, association test using unrelated individuals may suffer from confounding due to population structure. In this paper, we examine the impact of confounding due to population substructure on commonly used statistical procedures. Two study designs for genetic association study are considered: 1) retrospective sampling of cases and controls according to two cutoff points of the quantitative trait values (high or low) with allele frequency based test statistic; 2) random sample of individuals with regression analysis. For the first design, we consider the impact of confounder on association analysis between markers and QTLs. It is found that the false positive rate (or type I error) could be inflated substantially in the presence of both population stratification and trait heterogeneity; under other situations (e.g., in the presence of population stratification without trait heterogeneity), the inflated false positive rate is relatively minor. For the second design, we consider candidate marker association analysis. It is found that the inflated false positive rate could also be considerable in the presence of both population stratification and trait heterogeneity. Simulation studies confirm the theoretical results.
https://doi.org/10.1142/9789812772558_0022
The paper surveys almost two decades of progress by me and colleagues in three psychometric research areas involving the probability modeling and statistical analysis of standardized ability test data: nonparametric multidimensional latent ability structure modeling and assessment, test fairness modeling and assessment, and modeling and assessment of skills diagnosis via educational testing. In the process, it is suggested that the unidimensional scoring testing paradigm that has driven standardized ability testing research for over half a century is giving way to a new multidimensional latent ability modeling and multiple scoring paradigm that in particular explains and allows the effective detection of test bias and embraces skills-level formative assessment, opening up a plethora of challenging, exciting, and societally important research problems for psychometricians. It is hoped that this light-stepping history will interest probabilists and statisticians in exploring the field of psychometrics.
Informally, test bias occurs when an examinee is under or over evaluated by his test score in terms of the purpose of the test. Also informally, skills diagnosis refers to evaluating examinee levels of mastery (usually done dichotomously as master versus nonmaster of each skill) on a moderate number of carefully selected skills for which having student skills profiles can greatly help individual student learning and classroom level.
My strategy, strongly influenced by his probabilistic background, for producing interesting and effective psychometric research is to choose psychometric research questions from practical challenges facing educational testing. Then, I and colleagues bring to bear sophisticated probability modeling and modern statistical thought to solve these questions, making effectiveness of the resulting research in meeting the educational testing challenges the ultimate criterion for judging its worth.
It is somebody's ancient proverb that the acorn sometimes falls far from the oak tree. Y. S. Chow taught me the tools of probability limit theorem research, taught me to approach research with enthusiasm and tenacity, and provided a very supportive environment for me and his other graduate students. Although psychometrics/educational measurement is far from the probabilistic oak tree, whatever success I've had as a psychometrician has been strongly influenced by the supportive, demanding, and creative environment Y. S. creates for his students. By now I have had many Ph.D. students in psychometrics: it was the just described model of Y. S.'s for mentoring Ph.D. students that I followed with all of them.