Annals Of Statistics Research Articles

Two-stage rank estimation of quantile index models

This paper estimates a class of models which satisfy a monotonicity condition on the conditional quantile function of the response variable. This class includes as a special case the monotonic transformation model with the error term satisfying a conditional quantile restriction, thus allowing for very general forms of conditional heteroscedasticity. A two-stage approach is adopted to estimate the relevant parameters. In the first stage the conditional quantile function is estimated nonparametrically by the local polynomial estimator discussed in Chaudhuri (Journal of Multivariate Analysis 39 (1991a) 246–269; Annals of Statistics 19 (1991b) 760–777) and Cavanagh (1996, Preprint). In the second stage, the monotonicity of the quantile function is exploited to estimate the parameters of interest by maximizing a rank-based objective function. The proposed estimator is shown to have desirable asymptotic properties and can then also be used for dimensionality reduction or to estimate the unknown structural function in the context of a transformation model.

CONSISTENT MODEL SPECIFICATION TESTS

We point out the close relationship between the integrated conditional moment tests in Bierens (1982, Journal of Econometrics 20, 105–134) and Bierens and Ploberger (1997, Econometrica 65, 1129–1152) with the complex-valued exponential weight function and the kernel-based tests in Härdle and Mammen (1993, Annals of Statistics 21, 1926–1947), Li and Wang (1998, Journal of Econometrics 87, 145–165), and Zheng (1996, Journal of Econometrics 75, 263–289). It is well established that the integrated conditional moment tests of Bierens (1982) and Bierens and Ploberger (1997) are more powerful than kernel-based nonparametric tests against Pitman local alternatives. In this paper we analyze the power properties of the kernel-based tests and the integrated conditional moment tests for a sequence of “singular” local alternatives, and show that the kernel-based tests can be more powerful than the integrated conditional moment tests for these “singular” local alternatives. These results suggest that integrated conditional moment tests and kernel-based tests should be viewed as complements to each other. Results from a simulation study are in agreement with the theoretical results.

A conversation with Johannes H. B. Kemperman

Johannes H. B. Kemperman, born in 1924, received his Bachelor of Science in 1945 and Ph.D. in 1950, each in mathematics and physics, from the University of Amsterdam. From 1948 to 1951 he was a Research Associate at the Mathematical Centre in Amsterdam. During 1951–1953 he was a Visiting Professor on a Fulbright grant at Purdue University, and subsequently he joined the faculty and stayed at Purdue for 10 years, becoming a full Professor in 1959. In 1961 he went to the University of Rochester, becoming the Fayerweather Professor of Mathematics in 1970. He stayed at Rochester for 25 years. In 1985 he joined the faculty at Rutgers University as a Professor in the Statistics Department and also a voting member of the Mathematics Department. He retired to emeritus status at Rutgers in 1995. He served in editorial posts at the Annals of Mathematical Statistics, Annals of Probability, Annals of Statistics, Aequationes Mathematicae and Stochastic Processes and Applications. He is a Fellow of the Institute of Mathematical Statistics and the American Association for the Advancement of Sciences and is a Correspondent of the Royal Dutch Academy of Sciences (Amsterdam) . He has produced 23 Ph.D. students, 3 books and over 100 publications in analysis, number theory, group theory, probability, statistics, functional equations, mathematical biology and other areas.

Bias reduction in autoregressive models

It is well known that standard estimators of an AR(p) model are biased in finite samples, yet little is done in practice to remove the bias. Apparently small biases have important implications for the estimation of the impulse response function, which is a nonlinear function of the original coefficients. This note shows how to obtain estimators adjusted for first order bias based on Stine and Shaman’s fixed point characterisation of the bias (Stine, R.A., Shaman, P., 1989. A fixed point characterisation for bias of autoregressive estimators. Annals of Statistics 17, 1275-1284).

Time-dependent ROC curves for censored survival data and a diagnostic marker.

ROC curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker, X, for a binary disease variable, D. However, many disease outcomes are time dependent, D(t), and ROC curves that vary as a function of time may be more appropriate. A common example of a time-dependent variable is vital status, where D(t) = 1 if a patient has died prior to time t and zero otherwise. We propose summarizing the discrimination potential of a marker X, measured at baseline (t = 0), by calculating ROC curves for cumulative disease or death incidence by time t, which we denote as ROC(t). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodate censored data. A simple estimator is based on using the Kaplan-Meier estimator for each possible subset X > c. However, this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in X. An alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of (X, T), where T represents survival time (Akritas, M. J., 1994, Annals of Statistics 22, 1299-1327). We present an example where ROC(t) is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the ROC(t) curve displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials.

BLOCKING MULTIPLE SOURCES OF ERROR IN SMALL ANALYTIC STUDIES

Click to increase image sizeClick to decrease image sizeKey Words: Experimental designPractical significanceRobust designAnalytic studiesBlocking. Additional informationNotes on contributorsRamón V. LeónRamon Le6n is currently associate professor, Department of Statistics, University of Tennessee, Knoxville (UT). He earned a Ph.D. in statistics from Florida State University. Before coming to UT (from 1981 to 1991), he worked at Bell Laboratories, then part of AT&T and now part of Lucent Technologies. In his last management posi tion at Bell Laboratories, he was responsible for technology planning and deployment (including research, software development, consulting, and training) in robust product and process design (Taguchi methods), statistical process control, and statistical experimental design. Before that, he headed a group with similar responsibilities but in the area of reliability engineering. Before joining Bell Laboratories, he taught in the departments of statistics of Rutgers University and The Florida State University. His articles have appeared in Technometrics, The Annals of Statistics, The Annals of Probability, Mathematics of Operations Research, Statistic Sinica, and The Journal of Mathematical Analysis and Applications.Robert W. MeeRobert Mee is currently professor and head, Department of Statistics, University of Tennessee, Knoxville (UT). He earned a Ph.D. in statistics from Iowa State University, and he served on the faculty of Southern Methodist University and the University of South Alabama before coming to UT in 1989. He also worked for 1 year as a Fulbright Professor with the Universitatea Politechnica Timisoara, Romania. Dr. Mee coordinates the “Basics of Multifactor Experimentation” Institute, a short course for engineers offered by UT's Center for Executive Education. He previously served on the editorial board for the Journal of Quality Technology and Technometrics and is now a member of the Technometrics Management Committee.

A hybrid method for improved critical points for multiple comparisons

We present new methods for computing conservative critical points for simultaneous confidence intervals and bounds in the normal-theoretic fixed effects general linear model. The methods use a representation of the joint error probability attributed to Uusipaikka (unpublished manuscript, 1984), and popularized by Naiman (Annals of Statistics, 14, 1986, 896–906). This representation expresses the joint error probability of the confidence bounds as an expectation of surface areas of unions of disks with random radii on the sphere. The representation can be used to compute critical points by most well-known conservative methods; these can be regarded as using different upper bounds for the surface area of the union of disks. Our new hybrid methods compute the critical points using the minimum of several of these existing upper bounds, as well as some new bounds, and so improve upon all such methods included in the minimum expression. The new methods are a substantial improvement over the Hunter-Worsley and Šidák methods. Over 216 test cases the sample size savings of the new methods to the Hunter-Worsley method ranged from a worst-case loss of 3% to savings as large as 65%. Compared to the Šidák method, sample size savings ranged from a worst-case loss of 2% to savings as large as 95%. Savings of 5–10% were typical against both existing methods. One of the new methods, the “capped tubes” critical point, can be computed relatively cheaply without a large sacrifice in relative efficiency.

Relative coarsening at random

Many types of data are often incompletely observed. How incompletely is typically randomly determined. Heitjan and Rubin (Annals of Statistics, 1991) proposed a condition, “coarsened at random” or CAR, ensuring ignorability of this randomness in discrete sample spaces. In general sample spaces CAR comes in two flavors according to whether it is defined in terms of probabilities or densities. In this paper, CAR defined in terms of densities, called relative CAR, is discussed as a condition for ignorability in a statistical model allowing for partial observation of random elements determining the degree of incompleteness in the observation of the data.

A Bayesian model for local smoothing in kernel density estimation

A new procedure is proposed for deriving variable bandwidths in univariate kernel density estimation, based upon likelihood cross-validation and an analysis of a Bayesian graphical model. The procedure admits bandwidth selection which is flexible in terms of the amount of smoothing required. In addition, the basic model can be extended to incorporate local smoothing of the density estimate. The method is shown to perform well in both theoretical and practical situations, and we compare our method with those of Abramson (The Annals of Statistics 10: 1217–1223) and Sain and Scott (Journal of the American Statistical Association 91: 1525–1534). In particular, we note that in certain cases, the Sain and Scott method performs poorly even with relatively large sample sizes. We compare various bandwidth selection methods using standard mean integrated square error criteria to assess the quality of the density estimates. We study situations where the underlying density is assumed both known and unknown, and note that in practice, our method performs well when sample sizes are small. In addition, we also apply the methods to real data, and again we believe our methods perform at least as well as existing methods.

Consistency of the BIC order estimator

We announce two results on the problem of estimating the order of a Markov chain from observation of a sample path. First is that the Bayesian Information Criterion (BIC) leads to an almost surely consistent estimator. Second is that the Bayesian minimum description length estimator, of which the BIC estimator is an approximation, fails to be consistent for the uniformly distributed i.i.d. process. A key tool is a strong ratio-typicality result for empirical k k -block distributions. Complete proofs are given in the authors’ article to appear in The Annals of Statistics.

A better way to bootstrap pairs

In this paper we are interested in heteroskedastic regression models, for which an appropriate bootstrap method is bootstrapping pairs, proposed by Freedman (Annals of Statistics, 9 (1981) 1218–1228). We propose an ameliorate version of it, with better numerical performance.

LONG AND SHORT MEMORY CONDITIONAL HETEROSKEDASTICITY IN ESTIMATING THE MEMORY PARAMETER OF LEVELS

Semiparametric estimates of long memory seem useful in the analysis of long financial time series because they are consistent under much broader conditions than parametric estimates. However, recent large sample theory for semiparametric estimates forbids conditional heteroskedasticity. We show that a leading semiparametric estimate, the Gaussian or local Whittle one, can be consistent and have the same limiting distribution under conditional heteroskedasticity as under the conditional homoskedasticity assumed by Robinson (1995, Annals of Statistics 23, 1630–61). Indeed, noting that long memory has been observed in the squares of financial time series, we allow, under regularity conditions, for conditional heteroskedasticity of the general form introduced by Robinson (1991, Journal of Econometrics 47, 67–84), which may include long memory behavior for the squares, such as the fractional noise and autoregressive fractionally integrated moving average form, and also standard short memory ARCH and GARCH specifications.

Adaptive Regression Splines in the Cox Model

We develop a method for constructing adaptive regression spline models for the exploration of survival data. The method combines Cox's (1972, Journal of the Royal Statistical Society, Series B 34, 187-200) regression model with a weighted least-squares version of the multivariate adaptive regressi on spline (MARS) technique of Friedman (1991, Annals of Statistics 19, 1-141) to adaptively select the knots and covariates. The new technique can automatically fit models with terms that represent nonlinear effects and interactions among covariates. Applications based on simulated data and data from a clinical trial for myeloma are presented. Results from the myeloma application identified several important prognostic variables, including a possible nonmonotone relationship with survival in one laboratory variable. Results are compared to those from the adaptive hazard regression (HARE) method of Kooperberg, Stone, and Truong (1995, Journal of the American Statistical Association 90, 78-94).

CONSTRAINED SMOOTHING SPLINES

We use smoothing splines to introduce prior information in nonparametric models. The type of information we consider is based on the belief that the regression curve is similar in shape to a parametric model. The resulting estimator is a convex sum of a fit to data and the parametric model, and it can be seen as shrinkage of the smoothing spline toward the parametric model. We analyze its rates of convergence and we provide some asymptotic distribution theory. Because the asymptotic distribution is intractable, we propose to carry out inference with the estimator by using the method proposed by Politis and Romano (1994, Annals of Statistics 22, 2031-2050). We also propose a data-driven technique to compute the smoothing parameters that provides asymptotically optimal estimates. Finally, we apply our results to the estimation of a model of investment behavior of the U.S. telephone industry and we present some Monte Carlo results.

A conversation with I. Richard Savage (with the assistance of Bruce Spencer)

I. Richard Savage was born in Detroit on October 26, 1925. He attended Northern High School in Detroit and received a bachelor's degree in mathematics at the University of Chicago when he was nineteen. Subsequently, he received a master's degree in mathematics from the University of Michigan (1945) and a Ph.D. in mathematical statistics from Columbia in 1953. While at Columbia, he met JoAnn Osherow and they were married in 1950. They have two daughters, Martha, born in 1951, and Donna, born in 1953. His career began with a three­year (1951–1954) stint as a mathematical statistician in the National Bureau of Standards, followed by a three­ year visiting appointment in the Department of Statistics at Stanford University. From 1957 to 1963, Savage was a professor of Statistics, Biostatistics and Economics at the University of Minnesota. He then spent the next eleven years on the faculty of Florida State University in the Department of Statistics. From 1974 until his retirement in 1990, he was a professor in the Department of Statistics at Yale University. His research interests include rank order statistics, statistics and pub­ lic policy and Bayesian statistics. He served as Coeditor of the Journal of the American Statistical Association (1968, as Editor of the Annals of Statistics (1974–1977), and as President of the American Statistical Association (1984). Because of his commitment to advance proper use of statistics to shape public policy, he has been heavily involved with the Committee on National Statistics of the National Research Council. The book Statistics and Public Policy, edited by Bruce Spencer, was published in 1997 in honor of Savage's contributions to statistics.

Semiparametric Regression Analysis for Recurrent Event Interval Counts

This paper deals with analysis of data from longitudinal studies where the rate of a recurrent event characterizing morbidity is the primary criterion for treatment evaluation. We consider clinical trials which require patients to visit their clinical center at successive scheduled times as part of follow-up. At each visit, the patient reports the number of events that occurred since the previous visit, or an examination reveals the number of accumulated events, such as skin cancers. The exact occurrence times of the events are unavailable and the actual patient visit times typically vary randomly about the scheduled follow-up times. Each patient's record thus consists of a sequence of clinic visit dates, event counts corresponding to the successive time intervals between clinic visits, and baseline covariates. We propose a semiparametric regression model, extending the fully parametric model of Thall (1988, Biometrics 44, 197-209), to estimate and test for covariate effects on the rate of events over time while also accounting for the possibly time-varying nature of the underlying event rate. Covariate effects enter the model parametrically, while the underlying time-varying event rate is modelled nonparametrically. The method of Severini and Wong (1992, Annals of Statistics 20, 1768-1802) is used to construct asymptotically efficient estimators of the parametric component and to specify their asymptotic distribution. A simulation study and application to a data set are provided.

Sequential Analysis of Censored Survival Data From Three Treatment Groups

In this paper, we propose a simple means of designing and analyzing a sequential procedure for comparing survival data from three treatments with the goal of eventually identifying the best treatment. Our procedure consists of the concatenation of two sequential tests, as is suggested by Siegmund (1993, Annals of Statistics 21, 464-483) for instantaneous normal responses. The first sequential test is a global test that attempts to detect an overall treatment effect. If one is found, the least promising treatment is eliminated and a second sequential test attempts to identify the better of the two remaining treatments. Although there are three different information time scales to consider corresponding to each pairwise comparison, we show that under certain conditions they may be approximated by a single time scale. This enables us to gain insight into the problem of censored survival data from the more easily understood case of instantaneous normal data. Also, it eliminates the need for intensive computations and simulations for the design and analysis of the procedure.

Applications of Improved Variance Estimators in a Multivariate Normal Mean Vector Estimation

Consider the problem of estimating a normal mean vector when i.i.d observations are available from a p-dimensional normal distribution with an unknown mean vector and an unknown diagonal dispersion matrix proportional to the identity matrix. By using the improved variance estimation techniques we propose wide classes of shrinkage mean estimators which are uniformly better than the James-Stein estimator. Some of our improved mean estimators are completely new and are not covered by Kubokawa's (1994; A Unified Approach to Improving Equivariant Estimators. Annals of Statistics) result. Numerical results are provided to study the risk performance of some of these improved mean estimators.

Fuzzy common knowledge

In Agreeing to Disagree ( The Annals of Statistics 4, No. 6), R.J. Aumann introduced the notion of common knowledge and proved that “if two people have the same prior probabilities and their posterior probabilities for an event A are common knowledge, then these posterior probabilities are equal.” In this paper we introduce the concept of “fuzzy common knowledge” and generalize the previous theorem. The extension seems to give a game model which is more adequate for those situations in which subjective beliefs are predominant.

Jackknife Estimators of Variance for Parameter Estimates from Estimating Equations with Applications to Clustered Survival Data

An estimate of a parameter vector beta is often obtained by setting a "score" vector equal to zero and solving for beta. Estimating equations of this type include maximum likelihood, quasi-likelihood (McCullagh, 1983, Annals of Statistics 11, 59-67), and generalized estimating equations (Liang and Zeger, 1986, Biometrika 73, 13-22). White (1982, Econometrica 50, 1-26) proposed a variance estimator for beta that is robust to model misspecification. We show that a "one-step" jackknife estimator of variance is asymptotically equivalent to the variance estimator proposed by White. The one-step variance estimator may be preferred when the appropriate computer packages are not available to compute White's estimator directly. This jackknife estimator is very useful in our example with clustered survival data.