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This volume highlights Prof. Hira Koul’s achievements in many areas of Statistics, including Asymptotic theory of statistical inference, Robustness, Weighted empirical processes and their applications, Survival Analysis, Nonlinear time series and Econometrics, among others. Chapters are all original papers that explore the frontiers of these areas and will assist researchers and graduate students working in Statistics, Econometrics and related areas. Prof. Hira Koul was the first Ph.D. student of Prof. Peter Bickel. His distinguished career in Statistics includes the receipt of many prestigious awards, including the Senior Humbolt award (1995), and dedicated service to the profession through editorial work for journals and through leadership roles in professional societies, notably as the past president of the International Indian Statistical Association. Prof. Hira Koul has graduated close to 30 Ph.D. students, and made several seminal contributions in about 125 innovative research papers. The long list of his distinguished collaborators is represented by the contributors to this volume.



1. Professor Hira Lal Koul’s Contribution to Statistics

Professor Hira Koul received his Ph.D. in Statistics from the University of California, Berkeley in 1967 under the supervision of Professor Peter Bickel. He has the unique distinction of being the first doctoral student of Professor Bickel. True to his training at Berkeley, in the initial years of his research career, he focused on developing asymptotic theory of statistical inference. He pioneered the approach of Asymptotic Uniform Linearity (AUL) as a theoretical tool for studying properties of the empirical process based on residuals from a semiparametric model. This approach has been widely employed by several authors in studying the asymptotic properties of tests of composite hyptheses, and has been a particularly powerful tool for deriving limit laws of goodness-of-fit tests. At around the same time, he also developed the theory of weighted empirical processes which played a fundamental role in the study of asymptotic distribution of robust estimators (e.g., Rank-based estimators and M-estimators) in linear regression models. An elegant account of the theory of weighted empirical processes for independent as well as dependent random variables is given in his monographs on the topic
Soumendra Lahiri, Anton Schick, Ashis SenGupta, T.N. Sriram

2. Martingale Estimating Functions for Stochastic Processes: A Review Toward a Unifying Tool

Large sample theory and various estimation methods for stochastic processes are reviewed in a unified framework via martingale estimating functions. Results on asymptotic op¬timality of the estimates are discussed for both ergodic and non-ergodic processes. To illustrate the main results, various parameter estimates for GARCH-type processes, bifur¬cating and explosive autoregressive processes, conditionally linear autoregressive processes, and branching Markov processes are presented.
S. Y. Hwang, I. V. Basawa

3. Asymptotics of $L_\lambda$ -Norms of ARCH(p) Innovation Density Estimators

In this paper we consider, under L\(_{\lambda}\)-norm, the global property for a residual-based kernel estimator of the innovation density estimator in ARCH time series. For any \(1 \leq \lambda <\infty\), we investigate the L\(_{\lambda}\)-norm of the difference between this estimator and a innovation-based kernel density estimator. For \(\lambda>1\), this quantity is shown to be small enough so that the asymptotic distribution for the \(L_\lambda\)-norm of the difference between the innovation-based kernel estimator and the innovation density (which is known from the i.i.d. case) carries over to the difference between the residual-based kernel estimator and the innovation density. For \(\lambda=1\), this is no longer the case as the above quantity is then of the same magnitude as the L 1-norn of the difference between the innovation-based kernel estimator and the innovation density.
Fuxia Cheng

4. Asymptotic Risk and Bayes Risk of Thresholding and Superefficient Estimates and Optimal Thresholding

The classic superefficient estimate of Hodges for a one dimensional normal mean and the modern hard thresholding estimates introduced in the works of David Donoho and Iain Johnstone exhibit some well known risk phenomena. They provide quantifiable improvement over the MLE near zero, but also suffer from risk inflation suitably away from zero. Classic work of Le Cam and Hájek has precisely pinned down certain deep and fundamental aspects of these risk phenomena.
In this article, we study risks and Bayes risks of general thresholding estimates. In particular, we show that reversal to the risk of the MLE occurs at one standard deviation from zero, but the global peak occurs in a small lower neighborhood of the thresholding parameter. We give first order limits and various higher order asymptotic expansions to quantify these phenomena. Separately, we identify those priors in a class under which the thresholding estimate would be Bayesianly preferred to the MLE and use the theory of regular variation to pin down the rate at which the difference of their Bayes risks goes to zero.
We also formulate and answer an optimal thresholding question, which asks for the thresholding estimate that minimizes the global maximum of the risk subject to a specified gain at zero.
Anirban DasGupta, Iain M. Johnstone

5. A Note on Nonparametric Estimation of a Bivariate Survival Function Under Right Censoring

Estimation of bivariate survival function for right censored data has a long history. After providing a brief review of the existing literature, we introduce a class of novel estimators for this problem. We provide numerical evidence of the superiority of our estimators over existing estimators. Applicability of these estimators and related bootstrap inference are illustrated using a real life data set.
Haitao Zheng, Guiping Yang, Sotmnath Data

6. On Equality in Distribution of Ratios $\boldsymbol{X\!{/}(X}{+}\boldsymbol{Y)}$  and  $\boldsymbol{Y\!{/}(X}{+}\boldsymbol{Y)}$

Motivated by a classical result in the independent identically distributed (i.i.d.) case for a pair random variables X, Y, we look for a simple sufficient condition, allowing for possible dependence between \(X \mbox{and} Y\), under which the ratios of the components X,Y to their sum are equal in distribution. Our finding is easily extended to random vectors of higher (\(n \geq 2\)) dimensions to show that exchangeability of a finite sequence \(X_1, \cdots, X_n\) is sufficient to guarantee the desired result. Any Archimedian copula can be used as a generator of such random vectors. Our main result is applicable in many Bayesian contexts, where the observations are conditionally i.i.d. given an environmental variable with a prior.
Sunil K. Dhar, Manish C. Bhattacharjee

7. Nonparametric Distribution-Free Model Checks for Multivariate Dynamic Regressions

This article proposes asymptotic distribution-free speci.cation tests for parametric regres-sion models under time series processes with higher conditional moments of unknown form and multivariate regressors. The proposed test statistics are continuous functionals of a Khmaladze-Rossenblatt.s transform of a function-parametric residual marked process. Thus, our results extend those of Koul and Stute (1999) and Khmaladze and Koul (2004) to the multivariate time-series heteroskedastic case. The asymptotic theory is formally established using new weak convergence theorems for function-parametric processes. Finally, we compare the power prop-erties of bootstrap-based tests and our martingale-transform-based Cramér-von Mises test by a limited Monte Carlo experiment. We conclude that our new test compares very well to bootstrap tests for the alternatives considered and that the asymptotic results are good approximations for .nite sample distributions.
_AMS 2000 subject classi.cation. 62M07, 62G09, 62G10.
Key words and phrases. Function-parametric empirical processes, Khmaladze.s transformation, Rosenblatt.s trans-formation, Omnibus tests.
Miguel A. Delgado, J. Carlos Escanciano

8. Ridge Autoregression R-Estimation: Subspace Restriction

This paper considers the “ridge autoregression R-estimation” of the AR (p)-model when the parameters of the AR(p)-model is suspected to belong to a linear subspace. Accordingly, we introduce ridge autoregression (RARR) modifications to the usual five R-estimators of the parameters of the AR(p)-model. This class of (RARR)-R-estimators, not only alleviates the problem of multicollinearity in the estimated covariance matrix but also retains their asymptotic dominance properties under a quadratic loss function.
A. K. Md. Ehsanes Saleh

9. On Hodges and Lehmann’s “6/π Result”

While the asymptotic relative efficiency (ARE) of Wilcoxon rank-based tests for location and regression with respect to their parametric Student competitors can be arbitrarily large, Hodges and Lehmann (1961) have shown that the ARE of the same Wilcoxon tests with respect to their van der Waerden or normal-score counterparts is bounded from above by \(6/\pi\approx 1.910\). In this chapter, we revisit that result, and investigate similar bounds for statistics based on Student scores. We also consider the serial version of this ARE. More precisely, we study the ARE, under various densities, of the Spearman–Wald–Wolfowitz and Kendall rank-based autocorrelations with respect to the van der Waerden or normal-score ones used to test (ARMA) serial dependence alternatives.
Marc Hallin, Yvik Swan, Thomas Verdebout

10. Fiducial Theory for Free-Knot Splines

We construct the fiducial model for free-knot splines and derive sufficient conditions to show asymptotic consistency of a multivariate fiducial estimator. We show that splines of degree four and higher satisfy those conditions and conduct a simulation study to evaluate quality of the fiducial estimates compared to the competing Bayesian solution. The fiducial confidence intervals achieve the desired confidence level while tending to be shorter than the corresponding Bayesian credible interval using the reference prior. AMS 2000 subject classifications: Primary 62F99, 62G08; secondary 62P10.
Derek L. Sonderegger, Jan Hannig

11. An Empirical Characteristic Function Approach to Selecting a Transformation to Symmetry

Somewhat surprisingly, the empirical characteristic function can provide the basis for selecting a transformation to achieve near symmetry. In this chapter, we propose to estimate the transformation parameter by minimizing a weighted squared distance between the empirical characteristic function of transformed data and the characteristic function of a symmetric distribution. Asymptotic properties are established when a random sample is selected from an unknown distribution. We also consider the selection of weight functions that yield a closed form for the distance function. A small Monte Carlo simulation shows transforming data by our method lead to more symmetry than those by the maximum likelihood method when the population has heavy tails.
In-Kwon Yeo, Richard A. Johnson

12. Averaged Regression Quantiles

We show that weighted averaged regression α-quantile in the linear regression model, with regressor components as weights, is monotone in \(\alpha\in(0,1),\) and is asymptotically equivalent to the α-quantile of the location model. This relation remains true under the local heteroscedasticity of the model errors. As such, the averaged regression quantile provides various scale statistics, used for studentization and standardization in linear model, and an estimate of quantile density based on regression data. The properties are numerically illustrated.
Jana Jurečková, Jan Picek

13. A Study of One Null Array of Random Variables

Denote \(U_1,U_2,\dots,U_n\) a sequence of independent uniformly distributed random variables. We study limit behavior of the sum
$$\sum_{i=1}^m U_i^m$$
when \(m\to\infty\). It is interesting in itself and allows, although not too easily, an explicit form of the limiting infinitely divisible distribution function. It also has interesting application in the theory of large number of rare events (LNRE, see Khmaladze, Statistical Analysis of Large Number of Rare Events, 1989), which will be discussed in adjoint publication.
Estate Khmaladze

14. Frailty, Profile Likelihood, and Medfly Mortality

Unobserved heterogeneity is an increasingly common feature of statistical survival analysis where it is often referred to as frailty. Parametric mixture models are frequently used to capture these effects, but it is sometimes desirable to consider nonparametric mixture models as well. We illustrate the latter approach with a reanalysis of the well-known large scale medfly mortality study of Carey et al. (Science 258:457–61, 1992). Recent developments in convex optimization are exploited to expand the applicability of the Kiefer–Wolfowitz nonparametric maximum likelihood estimator for mixture models. Some ensuing problems of profile likelihood are also addressed.
Roger Koenker, Jiaying Gu

15. Comparison of Autoregressive Curves Through Partial Sums of Quasi-Residuals

This chapter discusses the problem of testing the equality of two nonparametric autoregressive functions against two-sided alternatives. The heteroscedastic error and stationary densities of the two independent strong mixing strictly stationary time series can be possibly different. The chapter adapts the partial sum process idea used in the independent observations settings to construct the tests and derives their asymptotics under both null and alternative hypotheses. Then, a Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at both fixed and local alternatives.
Fang Li

16. Testing for Long Memory Using Penalized Splines and Adaptive Neyman Methods

Testing procedures for the null hypothesis of short memory against long memory alternatives are investigated. Our new test statistic is constructed using penalized splines method and Fan’s (1996) canonical multivariate normal hypothesis testing procedure. Using penalized splines method, we are able to eliminate the effects of nuisance parameters typically induced by short memory autocorrelation. Therefore, under the null hypothesis of any short memory processes, our new test statistic has a known asymptotic distribution. The proposed test statistic is completely data-driven or adaptive, which avoids the need to select any smoothing parameters. Since the convergence of our test statistic toward its asymptotic distribution is relatively slow, Monte Carlo methods are investigated to determine the corresponding critical value. The finite-sample properties of our procedure are compared to other well-known tests in the literature. These show that the empirical size properties of the new statistic can be very robust compared to existing tests and also that it competes well in terms of power.
Linyuan Li, Kewei Lu

17. On the Computation of R-Estimators

In this chapter, we propose a simple iterative algorithm for computing R-estimates of the parameters of the linear regression models. The algorithm can be applied routinely to compute R-estimates based on any score function. We apply this to some well-known datasets and can identify outliers which would not have been detected using least squares.
Kanchan Mukherjee, Yuankun Wang

18. Multiple Change-Point Detection in Piecewise Exponential Hazard Regression Models with Long-Term Survivors and Right Censoring

Change-point detection in hazard rates is an important research topic in survival analysis. In this chapter, we first review the existing methods for a single change-point detection in piecewise exponential hazard models. Then, we propose a new change-point detection algorithm in multiple change-point hazard regression models for fitting failure times that allows the existence of both susceptibles and long-term survivors. For right censored failure time data, the proposed algorithm combines the Kaplan–Meier estimator for the susceptible proportion and weighted least square estimators for the multiple change-points and other model parameters. A simulation study is conducted for various model parameter settings. The results show that the proposed algorithm works superiorly on detecting the number of change-points with almost ignorable misclassification rate and on estimating other model parameters even for small to moderate sample sizes. Last, the proposed method is used to analyze clinical data on breast cancer.
Lianfen Qian, Wei Zhang

19. How to Choose the Number of Gradient Directions for Estimation Problems from Noisy Diffusion Tensor Data

We consider two popular nonparametric models describing measurements obtained from low and high angular resolution diffusion tensor imaging. The balance between the number of distinct directions for measurements and the number of repetitions is investigated from the statistical point of view. We show that designs with multiple independent repetitions using one set of six directions for the low resolution case yield smaller norms of the estimator’s covariance function than designs where a large set of directions with no repetitions is used, assuming that norms of covariances of image components are similar for both types of designs. The difference is inversely proportional to the number of repetitions. Similar result is obtained for the high resolution case. This yields a practical guideline on how to choose the number of gradient directions and the number of repetitions for estimation problems in this imaging context.
Lyudmila Sakhanenko

20. Efficient Estimation in Two-Sided Truncated Location Models

For a family of two-sided truncated location distributions, based on the generalized Neyman–Pearson lemma, an upper bound for the asymptotic distributions of the absolute deviations of all asymptotically median unbiased estimators for the location parameter is established, upon which the asymptotic efficiency is defined. Except for the cases in which the density has the same values at the truncation points, it is shown that there is no asymptotically median unbiased estimator to be two-sided asymptotically efficient. An adaptive asymptotically weak admissible median unbiased estimator of the location parameter is also constructed.
Weixing Song

21. Semiparametric Analysis of Treatment Effect via Failure Probability Ratio and the Ratio of Cumulative Hazards

For clinical trials with time-to-event data, statistical inference often employs the constant hazard ratio assumption. When the hazards are possibly non-proportional, the hazard ratio function is often the focus of analysis and it gives a visual inspection of proportionality assumption or how severe of a deviation there is from it. However, the hazard ratio does not directly reflect the treatment effect on survival or event occurrence. The failure probability ratio and the ratio of cumulative hazards are two measures that relate to the survival experience and supplement the hazard ratio in helping assess the treatment effect. For these ratios, although simple nonparametric estimators are available through the Nelson-Aalen estimator of the cumulative hazard and the Kaplan–Meier estimator of the survival function, often they are not very smooth and can be quite unstable near the beginning of the data range. In this article, point estimates, point-wise confidence intervals and simultaneous confidence intervals of the two ratios are established under a semiparametric model that can be used in a sufficiently wide range of applications. These methods are illustrated for data from two clinical trials.
Song Yang

22. Inference for the Standardized Median

Given a location-scale family that is symmetric about its median, the aim is to robustly estimate an effect size defined as the median divided by an interquantile range (IQR), where the quantile is fixed and to be chosen. It is shown that the sample version of this effect size can be variance stabilized, given information about its density at the median and quantiles defining the IQR. Tests for a significant effect size and confidence intervals for this effect size are derived and assessed.
Robert G. Staudte

23. Efficient Quantile Regression with Auxiliary Information

We discuss efficient estimation in quantile regression models where the quantile regression function is modeled parametrically. In addition, we assume that auxiliary information is available in the form of a conditional constraint. This is, for example, the case if the mean regression function or the variance function can be modeled parametrically, e.g., by a line or a polynomial. In this chapter, we describe efficient estimators of parameters of the quantile regression function for general conditional constraints and for examples of more specific constraints. We do this more generally for a model with responses missing at random, for which an efficient estimator is provided by a complete case statistic. This covers the usual model as a special case. We discuss several examples and illustrate the results with simulations.
Ursula U. Müller, Ingrid Van Keilegom

24. Nonuniform Approximations for Sums of Discrete m-Dependent Random Variables

Nonuniform estimates are obtained for Poisson, compound Poisson, translated Poisson, negative binomial and binomial approximations to sums of of m-dependent integer-valued random variables. Estimates for Wasserstein metric also follow easily from our results. The results are then exemplified by the approximation of Poisson binomial distribution, 2-runs and m-dependent \((k_1,k_2)\)-events.
P. Vellaisamy, V. Čekanavičius


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