2007-01-01, Brown, Lawrence D

Outside the field of statistics, the literature on observational studies offers advice about research designs or strategies for judging whether or not an association is causal, such as multiple operationalism or a dose-response relationship. These useful suggestions are typically informal and qualitative. A quantitative measure, design sensitivity, is proposed for measuring the contribution such strategies are then evaluated in terms of their contribution to design sensitivity. A related method for computing the power of a sensitivity analysis is also developed.

2017-03-01, Ernst, Philip A, Brown, Lawrence D, Shepp, Larry, Wolpert, Robert L

We consider the class, ℂp, of all zero mean stationary Gaussian processes, {Yt : t ∈ (—∞, ∞)} with p derivatives, for which the vector valued process {(Yt(0) ,...,Yt(p)) : t ≥ 0} is a p + 1-vector Markov process, where Yt(0) = Y(t). We provide a rigorous description and treatment of these stationary Gaussian processes as limits of stationary AR(p) time series.

2016-11-01, Brown, Lawrence D, Levine, Michael, Wang, Lie

A multivariate semiparametric partial linear model for both fixed and random design cases is considered. In either case, the model is analyzed using a difference sequence approach. The linear component is estimated based on the differences of observations and the functional component is estimated using a multivariate Nadaraya–Watson kernel smoother of the residuals of the linear fit. We show that both components can be asymptotically estimated as well as if the other component were known. The estimator of the linear component is shown to be asymptotically normal and efficient in the fixed design case if the length of the difference sequence used goes to infinity at a certain rate. The functional component estimator is shown to be rate optimal if the Lipschitz smoothness index exceeds half the dimensionality of the functional component argument. We also develop a test for linear combinations of regression coefficients whose asymptotic power does not depend on the functional component. All of the proposed procedures are easy to implement. Finally, numerical performance of all the procedures is studied using simulated data.

1997, Brown, Lawrence D, Low, Mark G, Zhao, Linda H

Fixed parameter asymptotic statements are often used in the context of nonparametric curve estimation problems (e.g., nonparametric density or regression estimation). In this context several forms of superefficiency can occur. In contrast to what can happen in regular parametric problems, here every parameter point (e.g., unknown density or regression function) can be a point of superefficiency. We begin with an example which shows how fixed parameter asymptotic statements have often appeared in the study of adaptive kernel estimators, and how superefficiency can occur in this context. We then carry out a more systematic study of such fixed parameter statements. It is shown in four general settings how the degree of superefficiency attainable depends on the structural assumptions in each case.

2008-01-01, Brown, Lawrence D

Batting average is one of the principle performance measures for an individual baseball player. It is natural to statistically model this as a binomial-variable proportion, with a given (observed) number of qualifying attempts (called “at-bats”), an observed number of successes (“hits”) distributed according to the binomial distribution, and with a true (but unknown) value of pi that represents the player’s latent ability. This is a common data structure in many statistical applications; and so the methodological study here has implications for such a range of applications. We look at batting records for each Major League player over the course of a single season (2005). The primary focus is on using only the batting records from an earlier part of the season (e.g., the first 3 months) in order to estimate the batter’s latent ability, pi, and consequently, also to predict their batting-average performance for the remainder of the season. Since we are using a season that has already concluded, we can then validate our estimation performance by comparing the estimated values to the actual values for the remainder of the season. The prediction methods to be investigated are motivated from empirical Bayes and hierarchical Bayes interpretations. A newly proposed nonparametric empirical Bayes procedure performs particularly well in the basic analysis of the full data set, though less well with analyses involving more homogeneous subsets of the data. In those more homogeneous situations better performance is obtained from appropriate versions of more familiar methods. In all situations the poorest performing choice is the naïve predictor which directly uses the current average to predict the future average. One feature of all the statistical methodologies here is the preliminary use of a new form of variance stabilizing transformation in order to transform the binomial data problem into a somewhat more familiar structure involving (approximately) Normal random variables with known variances. This transformation technique is also used in the construction of a new empirical validation test of the binomial model assumption that is the conceptual basis for all our analyses.

1990, Brown, Lawrence D

It is a pleasure and an embarrassment to read a historical story in which one plays an integral role. From my perspective the story has been accurately related, but I do have some miscellaneous comments to make which are related to the general topic.

1974, Brown, Lawrence D, Cohen, Arthur

Consider the problem of estimating a common mean of two independent normal distributions, each with unknown variances. Note that the problem of recovery of interblock information in balanced incomplete blocks designs is such a problem. Suppose a random sample of size m is drawn from the first population and a random sample of size n is drawn from the second population. We first show that the sample mean of the first population can be improved on (with an unbiased estimator having smaller variance), provided m ≧ 2 and n ≧ 3. The method of proof is applicable to the recovery of information problem. For that problem, it is shown that interblock information could be used provided b ≧ 4. Furthermore for the case b = t = 3, or in the common mean problem, where n = 2, it is shown that the prescribed estimator does not offer improvement. Some of the results for the common mean problem are extended to the case of K means. Results similar to some of those obtained for point estimation, are also obtained for confidence estimation.

1985, Brown, Lawrence D, Farrell, R. H

Basic decision theory for discrete random variables of the multivariate geometric (power series) type is developed. Some properties of Bayes estimators that carry over in the limit to admissible estimators are obtained. A stepwise generalized Bayes representation of admissible estimators is developed with estimation of the mean of a multivariate Poisson random variable in mind. The development carries over to estimation of the mean of a multivariate negative Binomial random variable. Due to the natural boundary of the parameter space there is an interesting pathology illustrated to some extent by the examples given. Examples include one to show that admissible estimators with somewhere infinite risk do exist in two or more dimensions.

1993, Brown, Lawrence D

A bound is given for the Bayes risk of an estimator under truncated squared error loss. The bound derives from an information inequality for the risk under this loss. It is then used to provide new proofs for some classical results of asymptotic theory.

1992, Brown, Lawrence D, Marden, John I

In this paper we give necessary conditions and sufficient conditions for a test to be locally unbiased, we define local admissibility and we characterize local admissibility in hypothesis testing problems with simple null hypotheses. Applications are presented involving same-sign alternatives, ordered alternatives and independence testing of several variables.