Selected Topics in Characteristic Functions
This work examines inequalities for characteristic functions and the theory of empirical characteristic function. The first chapter contains basic general results concerning characteristic functions. The second chapter deals with various inequalities for characteristic functions, and in the third chapter empirical characteristic functions and their applications to statistics are studied. A collection of various examples, counterexamples and assertions demonstrating interesting properties of characteristic functions is also included.
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absolutely continuous distribution acteristic bounded variation ch.f char charac characteristic func characteristic function f(t conditions of Theorem consider converges COROLLARY corresponding characteristic functions corresponding distribution cosx covariance matrix Csorgo Denote density estimation density function p(x dF(x differentiable discrete unimodal distribution F distribution function F(x empirical characteristic function empirical distribution function estimator pn(x example exists a characteristic Feuerverger finite fn(t function with characteristic hence implies inequality integer integrable interval J-oo kernel estimator Lebesgue measure Lemma Let F(x Let the conditions Let us prove lim sup m-dimensional multi-dimensional multivariate non-negative normal distribution obtain oo oo parameter positive integer positive numbers probability density function proof of Theorem random variable random vector right-hand side satisfying conditions satisfying the conditions sequence sinx teristic function test statistic tic functions tion un(t underlying distribution uniform distribution unimodal distribution univariate Ushakov values virtue of Theorem
Page 340 - Heathcote, CR ( 1972). A test of goodness of fit for symmetric random variables. Austral. J. Statist. 14, 172-181.