E. L. Lehmann

Bio: E. L. Lehmann is an academic researcher from Stanford University. The author has contributed to research in topics: Minimax & Sample space. The author has an hindex of 5, co-authored 5 publications receiving 1297 citations. Previous affiliations of E. L. Lehmann include University of Chicago & University of California.

The Use of Maximum Likelihood Estimates in {\chi^2} Tests for Goodness of Fit

Abstract: The usual test that a sample comes from a distribution of given form is performed by counting the number of observations falling into specified cells and applying the χ2 test to these frequencies. In estimating the parameters for this test, one may use the maximum likelihood (or equivalent) estimate based (1) on the cell frequencies, or (2) on the original observations. This paper shows that in (2), unlike the well known result for (1), the test statistic does not have a limiting χ2-distribution, but that it is stochastically larger than would be expected under the χ2 theory. The limiting distribution is obtained and some examples are computed. These indicate that the error is not serious in the case of fitting a Poisson distribution, but may be so for the fitting of a normal.

The Power of Rank Tests

Abstract: Simple nonparametric classes of alternatives are defined for various nonparametric hypotheses. The power of a number of such tests against these alternatives is obtained and illustrated with some numerical results. Optimum rank tests against certain types of alternatives are derived, and optimum properties of Wilcoxon’s one- and two-sample tests and of the rank correlation test for independence are proved.

The use of Previous Experience in Reaching Statistical Decisions

Abstract: Instead of minimizing the maximum risk it is proposed to re-strict attention to decision procedures whose maximum risk does not exceed the minimax risk by more than a given amount. Subject to this restriction one may wish to minimize the average risk with respect to some guessed a priori distribution suggested by previous experience. It is shown how Wald’s minimax theory can be modified to yield analogous results concerning such restricted Bayes solutions. A number of examples are discussed, and some extensions of the above criterion are briefly considered.

Testing Multiparameter Hypotheses

Abstract: Let the distribution of some random variables depend on real parameters θ1, • • •, θs and consider the hypothesis H: θi ≦ θi*, i = 1, • • •,s. It is shown under certain regularity assumptions that unbiased tests of H do not exist. Tests of minimum bias and other types of minimax tests are derived under suitable monotonicity conditions. Certain related multidecision problems are discussed and two-sided hypotheses are considered very briefly.

On the Existence of Least Favorable Distributions

Abstract: Sufficient conditions for the existence of a least favorable dis-tribution were given by Wald in his work on general decision theory. It is shown here that for problems of hypothesis testing and more generally for multiple decision problems involving a finite number of decisions, the result holds under a much weaker restriction than Wald’s assumption of a compact parameter space.

Regression Models and Life-Tables

Abstract: The analysis of censored failure times is considered. It is assumed that on each individual arc available values of one or more explanatory variables. The hazard function (age-specific failure rate) is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time. A conditional likelihood is obtained, leading to inferences about the unknown regression coefficients. Some generalizations are outlined.

On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown

Abstract: The standard tables used for the Kolmogorov-Smirnov test are valid when testing whether a set of observations are from a completely-specified continuous distribution. If one or more parameters must be estimated from the sample then the tables are no longer valid. A table is given in this note for use with the Kolmogorov-Smirnov statistic for testing whether a set of observations is from a normal population when the mean and variance are not specified but must be estimated from the sample. The table is obtained from a Monte Carlo calculation. A brief Monte Carlo investigation is made of the power of the test.

Econometric Models for Count Data with an Application to the Patents-R&D Relationship

Abstract: This paper focuses on developing and adapting statistical models of counts (nonnegative integers) in the context of panel data and using them to analyze the relationship between patents and R & D expenditures. Since a variety of other economic data come in the form of repeated counts of some individual actions or events, the methodology should have wide applications. The statistical models we develop are applications and generalizations of the Poisson distribution. Two important issues are (i) Given the panel nature of our data, how can we allow for separate persistent individual (fixed or random) effects? (ii) How does one introduce the equivalent of disturbances-in-the-equation into the analysis of Poisson and other discrete probability functions? The first problem is solved by conditioning on the total sum of outcomes over the observed years, while the second problem is solved by introducing an additional source of randomness, allowing the Poisson parameter to be itself randomly distributed, and compounding the two distributions. Lastly, we develop a test statistic for the presence of serial correlation when fixed effects estimators are used in nonlinear conditional models.

An Approximately Unbiased Test of Phylogenetic Tree Selection

Abstract: An approximately unbiased (AU) test that uses a newly devised multiscale bootstrap technique was developed for general hypothesis testing of regions in an attempt to reduce test bias. It was applied to maximum-likelihood tree selection for obtaining the confidence set of trees. The AU test is based on the theory of Efron et al. (Proc. Natl. Acad. Sci. USA 93:13429-13434; 1996), but the new method provides higher-order accuracy yet simpler implementation. The AU test, like the Shimodaira-Hasegawa (SH) test, adjusts the selection bias overlooked in the standard use of the bootstrap probability and Kishino-Hasegawa tests. The selection bias comes from comparing many trees at the same time and often leads to overconfidence in the wrong trees. The SH test, though safe to use, may exhibit another type of bias such that it appears conservative. Here I show that the AU test is less biased than other methods in typical cases of tree selection. These points are illustrated in a simulation study as well as in the analysis of mammalian mitochondrial protein sequences. The theoretical argument provides a simple formula that covers the bootstrap probability test, the Kishino-Hasegawa test, the AU test, and the Zharkikh-Li test. A practical suggestion is provided as to which test should be used under particular circumstances.

Psychology and Economics

Abstract: UNTVERSITY OF CALIFORNIA AT BERKELEY Department of Economics Berkeley, CaHfornia 94720-3880 Working Paper No. 97-251 Psychology and Economics Matthew Rabin Department of Economics University of California, Berkeley January 1997 Key words: bounded rationality, decision making, fairness, framing effects, heuristics and biases, preferences, psychology, reciprocity, reference-dependent utility JEL Classification: A12, B49, D i l , D60, D81, D83, D91 This is a longer version of an essay under preparation for possible publication in the Journal of Economic Literature. I thank John Pencavel and anonymous referees for earlier comments on its structure and content. For comments on this draft, I thank Steven Blatt, Colin Camerer, Peter Diamond, Erik Eyster, Ernst Fehr, Danny Kahneman, George Loewenstein, Ted O'Donoghue, and John Pencavel. For helpful conversations over the past several years on topics covered in this essay, I thank George Akerlof, Gary Chamess, Eddie Dekel, Peter Diamond, David Laibson, David I. Levine, George Loewenstein, Rob MacCoun, James Montgomery, Vai-Lam Mui, Drazen Prelec, and especially Colin Camerer, Danny Kahneman, and Richard Thaler. Co-authors on research related to the topics of this essay include David Bowman, Deborah Minehart, Ted O'Donoghue, and Joel Schrag. Helpful research assistance was provided by Gadi Barlevy, Nikki Blasberg, Gail Brennan, Paul Ellickson, April Franco, Marcus Heng, Bruce Hsu, Jin Woo Jung, and especially Steven Blatt, Jimmy Chan, Erik Eyster, and Clara Wang. I am extremely grateful for financial support from the Russell Sage and Alfred P. Sloan Foundations.