Mathematical Statistics: An Introduction
Wiebe R. Pestman · 1998
Probability theory: probability spaces stochastic variables product measures and statistical independence functions of stochastic vectors expectation, variance and covariance of stochastic variables distribution functions and probability distributions moments, moment generating functions and characteristic function the central limit theorem exercises. Statistics and their probability distributions, estimation theory: introduction the Gamma distribution and the Chi-2-distribution the t-distribution statistics to measure differences in mean the F-distribution the Beta distribution populations which are not normally distributed Bayesian estimation estimation theory in a more general framework maximum likelihood estimation, sufficiency exercises. Hypothesis testing: the Neyman-Pearson theory hypothesis tests concerning normally distributed populations the Chi-2 test on goodness of fit the Chi-2 test on statistical independence exercises. Simple regression analysis: the method of least squares construction of an unbiased estimator of Sigma-2 normal regression analysis Pearson's product-moment correlation coefficient the sum of squares of errors as a measure of the amount of linear structure exercises. Normal analysis of variance: one-way analysis of variance two-way analysis of variance exercises. Non-parametric methods: the sign test Wilcoxon's signed-rank test Wilcoxon's rank-sum test the runs test rank correlation tests the Kruskal-Wallis test Friedman's test exercises. Stochastic analysis and its applications in statistics: the empirical distribution function associated with a sample convergence of stochastic variables the Glivenko-Cantelli theorem the Kolmogorov-Smirnov test statistic metrics on the set of distribution functions smoothing techniques robustness of statistics trimmed means, the median, and their robustness statistical functionals the von Mises derivative influence functions Bootstrap methods estimation of densities by mean of kernel densities estimation of densities by means of histograms exercises. Vectorial statistics: linear algebra the expectation vector and the covariance operator of stochastic vectors vectorial samples the vectorial normal distribution conditional probability distributions that emanate from Gaussian ones vectorial samples from Gaussian distributed populations normal correlation analysis multiple regression analysis the multiple correlation coefficient exercise. Appendices: Lebesgue's convergence theorems product measures conditional probabilities the characteristic function of the Cauchy distribution metric spaces, equicontinuity the Fourier transform and the existence of stoutly tailed distribution functions. List of elementary probability densities frequently used symbols statistical tables references.