J. Wolfowitz

Bio: J. Wolfowitz is an academic researcher. The author has contributed to research in topics: Normal distribution & Noncentral chi-squared distribution. The author has an hindex of 1, co-authored 1 publications receiving 110 citations.

On the deviations of the empiric distribution function of vector chance variables

Abstract: BY J. KIEFER(') AND J. WOLFOWITZp)

Differential privacy and robust statistics

Abstract: We show by means of several examples that robust statistical estimators present an excellent starting point for differentially private estimators. Our algorithms use a new paradigm for differentially private mechanisms, which we call Propose-Test-Release (PTR), and for which we give a formal definition and general composition theorems.

Information-theoretic asymptotics of Bayes methods

Abstract: In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. The authors examine the relative entropy distance D/sub n/ between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D/sub n//n converges to zero at rate (log n)/n. The constant c, which the authors explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stock-market portfolio selection. >

Distribution Free Tests of Independence Based on the Sample Distribution Function

Abstract: Certain tests of independence based on the sample distribution function (d.f.) possess power properties superior to those of other tests of independence previously discussed in the literature. The characteristic functions of the limiting d.f.'s of a class of such test criteria are obtained, and the corresponding d.f. is tabled in the bivariate case, where the test is equivalent to one originally proposed by Hoeffding [4]. A discussion is included of the computational problems which arise in the inversion of characteristic functions of this type. Techniques for computing the statistics and for approximating the tail probabilities are considered.

K-Sample Analogues of the Kolmogorov-Smirnov and Cramer-V. Mises Tests

Abstract: The main purpose of this paper is to obtain the limiting distribution of certain statistics described in the title. It was suggested by the author in [1] that these statistics might be useful for testing the homogeneity hypothesis $H_1$ that $k$ random samples of real random variables have the same continuous probability law, or the goodness-of-fit hypothesis $H_2$ that all of them have some specified continuous probability law. Most tests of $H_1$ discussed in the existing literature, or at least all such tests known to the author before [1] in the case $k > 2$, have only been shown to have desirable consistency or power properties against limited classes of alternatives (see e.g., [2], [3], [4] for lists of references on these tests), while those suggested here are shown to be consistent against all alternatives and to have good power properties. Some test statistics whose distributions can be computed from known results are also listed.

On Weak Convergence of Stochastic Processes with Multidimensional Time Parameter

Abstract: The well-known space $D\lbrack 0, 1\rbrack$ is generalized to $k$ time dimensions and some properties of this space $D_k$ are derived. Then, following the "classical" lines as presented in Billingsley [1], a Skorohod-metric, tightness criteria and some other results concerning weak convergence are given. The theory is applied to prove weak convergence of two generalizations of the one-dimensional empirical process and of the Kolmogorov-Smirnov test statistic of independence. Stochastic processes with multidimensional time parameter and their weak convergence have been investigated by several authors. Dudley [4] established a theory of convergence of stochastic processes with sample functions in nonseparable metric spaces. Later on, Wichura [11] (see also Wichura [12]) modified the concepts of Dudley and developed them systematically. He applied his theory to a space which is with minor changes our space $D_k$. Weak convergence in the sense of Wichura [12] and ours differ usually, but both concepts coincide if the limit process has--with probability one--continuous sample functions only. From here it follows that the results of Dudley and Wichura concerning weak convergence of multivariate empirical processes are equivalent to ours. At least two further authors proved the convergence of multivariate empirical processes, namely LeCam [8] and Bickel [1]. Our proof follows the classical approach of Parthasarathy [9] using an argument of Kuelbs [7] to carry over the proof from 1 to $k$ dimensions. Kuelbs however deals properly with the "interpolated sum" process for two-dimensional time parameter. The space $D_k$ seems to be defined for the first time in connection with multivariate processes by Winkler [13], yet his investigations are not concerned with weak convergence. Another generalization of the space $D\lbrack 0, 1\rbrack$ and the Skorohod metric to functions on more general spaces than $E_k$ is given in the paper [10] of Straf, in which there are applications to genuinely discontinuous limit processes.