Department of Statistics

Email: wichura@galton

Distribution functions and friends: inverse distribution functions, representing functions, quantile functions, Q/Q plots, probability and inverse probability transformations.

Change-of-variables for probability densities: univariate and multivariate, standard examples.

Computer algebra: Maple

Measures of location: mean, median, mode: $g$-means, medians, and modes; Jensen, Holder, Cauchy-Schwarz inequalities.

Measure of spread: SD, mean absolute deviation, Gini's mean difference; interrelations, examples.

Integration to the limit: MCT, DCT, the sandwich theorem.

Switching the order of summation/integration: Fubini's theorem

Moment generating functions (real argument): examples, characterization of the region of finiteness, power series expansion.

Integration of complex valued functions.

Moment generating functions (complex argument).

Characteristic functions: examples, inversion formulas, uniqueness theorems,

Convergence in distribution: criteria for convergence.

Limit theorems: expansion of the characteristic function about $t=0$, weak law of large numbers, global CLT, local CLT for densities.

Cumulants and cumulant generating functions: examples, properties.

Edgeworth expansion for densities and friends: expansions for the cdf, the normalizing transformation, the Cornish-Fisher transformation.

The Euler-Maclaurin summation formula: Sheppard's correction for cumulants, The Kolassa/McCullagh Edgeworth expansion for lattice distributions.

Saddlepoint expansion: exponential tilting, saddle point expansion for densities.

Last update: 1/04/01