Soumendra N. Lahiri

A Two-Sample Test for Equality of Means in High Dimension

We develop a test statistic for testing the equality of two population mean vectors in the “large-p-small-n” setting. Such a test must surmount the rank-deficiency of the sample covariance matrix, which breaks down the classic Hotelling T2 test. The proposed procedure, called the generalized component test, avoids full estimation of the covariance matrix by assuming that the p components admit a logical ordering such that the dependence between components is related to their displacement. The test is shown to be competitive with other recently developed methods under ARMA and long-range dependence structures and to achieve superior power for heavy-tailed data. The test does not assume equality of covariance matrices between the two populations, is robust to heteroscedasticity in the component variances, and requires very little computation time, which allows its use in settings with very large p. An analysis of mitochondrial calcium concentration in mouse cardiac muscles over time and of copy number variations in a glioblastoma multiforme dataset from The Cancer Genome Atlas are carried out to illustrate the test. Supplementary materials for this article are available online.

A Progressive Block Empirical Likelihood Method for Time Series

This article develops a new blockwise empirical likelihood (BEL) method for stationary, weakly dependent time processes, called the progressive block empirical likelihood (PBEL). In contrast to the standard version of BEL, which uses data blocks of constant length for a given sample size and whose performance can depend crucially on the block length selection, this new approach involves a data-blocking scheme where blocks increase in length by an arithmetic progression. Consequently, no block length selections are required for the PBEL method, which implies a certain type of robustness for this version of BEL. For inference of smooth functions of the process mean, theoretical results establish the chi-squared limit of the log-likelihood ratio based on PBEL, which can be used to calibrate confidence regions. Using the same progressive block scheme, distributional extensions are also provided for other nonparametric likelihoods with time series in the family of Cressie–Read discrepancies. Simulation evidence indicates that the PBEL method can perform comparably to the standard BEL in coverage accuracy (when the latter uses a “good” block choice) and can exhibit more stability, without the need to select a usual block length. Supplementary materials for this article are available online.

A Smooth Block Bootstrap for Statistical Functionals and Time Series

Unlike with independent data, smoothed bootstraps have received little consideration for time series, although data smoothing within resampling can improve bootstrap approximations, especially when target distributions depend on smooth population quantities (e.g., marginal densities). For approximating a broad class statistics formulated through statistical functionals (e.g., LL‐estimators, and sample quantiles), we propose a smooth bootstrap by modifying a state‐of‐the‐art (extended) tapered block bootstrap (TBB). Our treatment shows that the smooth TBB applies to time series inference cases not formally established with other TBB versions. Simulations also indicate that smoothing enhances the block bootstrap.

Non-Parametric Spectral Density Estimation Under Long-Range Dependence: SPECTRAL DENSITY ESTIMATION UNDER LRD

One major aim of time series analysis, particularly in the physical and geo†sciences, is the estimation of the spectral density function. With weakly dependent time processes, non†parametric, kernel†based methods are available for spectral density estimation, which involves smoothing the periodogram by a kernel function. However, a similar non†parametric approach is presently unavailable for strongly, or long†range, dependent processes. In particular, as the spectral density function under long†range dependence commonly has a pole at the origin, kernel†based methods developed for weakly dependent processes (i.e., with bounded spectral densities) do not apply readily for long†range dependence without suitable modification. To address this, we propose a non†parametric kernel†based method for spectral density estimation, which is valid under both weak and strong dependence. Based on the initial or pilot estimator of the long†memory parameter, the method involves a frequency domain transformation to dampen the dependence in periodogram ordinates and mimic kernel†based estimation under weak dependence. Under mild assumptions, the proposed non†parametric spectral density estimator is shown to be uniformly consistent, and general expressions are provided for rates of estimation error and optimal kernel bandwidths. The method is investigated through simulation and illustrated through data examples, which also consider bandwidth selection.

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