U-statistic

About: U-statistic is a research topic. Over the lifetime, 1209 publications have been published within this topic receiving 32898 citations.

Maximal Inequalities for Degenerate $U$-Processes with Applications to Optimization Estimators

Abstract: Maximal inequalities for degenerate $U$-processes of order $k, k \geq 1$, are established. The results rest on a moment inequality (due to Bonami) for $k$th-order forms and on extensions of chaining and symmetrization inequalities from the theory of empirical processes. Rates of uniform convergence are obtained. The maximal inequalities can be used to determine the limiting distribution of estimators that optimize criterion functions having $U$-process structure. As an application, a semiparametric regression estimator that maximizes a $U$-process of order 3 is shown to be $\sqrt n$-consistent and asymptotically normally distributed.

Mathematical Statistics with Mathematica

Abstract: Imagine computer software that can find expectations of arbitrary random variables, calculate variances, invert characteristic functions, solve transformations of random variables, calculate probabilities, derive order statistics, find Fisher's Information and Cramer-Rao Lower Bounds, derive symbolic (exact) maximum likelihood estimators, perform automated moment conversions, and so on. Imagine that this software was wonderfully easy to use, and yet so powerful that it can find corrections to mainstream reference texts and solve new problems in seconds. Then, imagine a book that uses that software to bring mathematical statistics to life...

Moment inequalities for functions of independent random variables

Abstract: A general method for obtaining moment inequalities for functions of independent random variables is presented It is a generalization of the entropy method which has been used to derive concentration inequalities for such functions [Boucheron, Lugosi and Massart Ann Probab 31 (2003) 1583-1614], and is based on a generalized tensorization inequality due to Latala and Oleszkiewicz [Lecture Notes in Math, 1745 (2000) 147-168] The new inequalities prove to be a versatile tool in a wide range of applications We illustrate the power of the method by showing how it can be used to effortlessly re-derive classical inequalities including Rosenthal and Kahane-Khinchine-type inequalities for sums of independent random variables, moment inequalities for suprema of empirical processes and moment inequalities for Rademacher chaos and U-statistics Some of these corollaries are apparently new In particular, we generalize Talagrand's exponential inequality for Rademacher chaos of order 2 to any order We also discuss applications for other complex functions of independent random variables, such as suprema of Boolean polynomials which include, as special cases, subgraph counting problems in random graphs

Applications of Multivariate Polykays to the Theory of Unbiased Ratio-Type Estimation

Abstract: The multivariate polykays, or multipart k-statistics, are obtained as a slight extension of results given by Tukey [4] for the univariate polykays. The relationship between this system and the system of multivariate symmetric means is indicated and multiplication formulas are given. An application of these results to the construction of unbiased ratio-type estimators and variance estimates for finite populations is given as a further illustration of the usefulness of polykays.

The Efficient Construction of an Unbiased Random Sequence

Abstract: We consider procedures for converting input sequences of symbols generated by a stationary random process into sequences of independent, equiprobable output symbols, measuring the efficiency of such a procedure when the input sequence is finite by the expected value of the ratio of output symbols to input symbols. For a large class of processes and a large class of procedures we give an obvious information-theoretic upper bound to efficiency. We also construct procedures which attain this bound in the limit of long input sequences without making use of the process parameters, for two classes of processes. In the independent case we generalize a 1951 result of von Neumann and 1970 results of Hoeffding and Simons for independent but biased binary input, gaining a factor of 3 or 4 in efficiency. In the finite-state case we generalize a 1968 result of Samuelson for two-state binary Markov input, gaining a larger factor in efficiency.