# STA 7334 Fall 2010: Class Diary

## 1 2010-08-24 (1 period)

Syllabus. Several simple, motivational examples: (1) t-test with nonnormal data; and (2) symmetric location estimation (ARE(mean, median)).

## 2 2010-08-26 (2 periods)

Score test ("solve the quadratic") interval for a binomial proportion. The asymptotic distribution of a sample quantile (central order statistic), emphasizing the need for a central limit theorem for triangular arrays. The asymptotic distirbution of the least squares estimator in ordinary regression with nonnormal errors, emphasizing again the need for a multivariate central limit theorem for triangular arrays.

## 3 2010-08-31 (1 period)

Random elements of metric spaces and weak convergence of probability measures.

## 4 2010-09-02 (2 periods)

Portmanteau theorem. Three modes of stochastic convergence for random elements of a metric space: almost sure convergence, convergence in probability, and convergence in distribution. Relationships among the modes of convergence. Continuous mapping theorem.

## 5 2010-09-07 (1 period)

More on continuous mapping theorem. Convergence in distribution to a constant implies convergence in probability. Abstract version of Slutsky's theorem and some corollaries.

## 6 2010-09-09 (2 periods)

The characteristic function for distributions on Euclidean space. Review univariate results: uniquely identifies distribution, L\'evy's continuity theorem. Generalize uniqueness result to multivariate case using convolution with N(0, σ I). Use the same convolution method to prove that convergence of characteristic functions is equivalent to weak convergence (almost L\'evy's theorem. Cram\'er-Wold device and applications.

## 7 2010-09-14 (1 period)

Uniform tightness. Prohorov's theorem. Define "little-oh-p" and "big-oh-p" and prove basic results.

## 8 2010-09-16 (2 periods)

Finish up with big-oh-p and little-oh-p. Prove delta method under too strong and then weaker assumptions. Simple examples. Cases where first derivative is zero.

## 9 2010-09-21 (1 period)

Sample moments. Derive limiting distribution of sample variance and prove that normal theory test of variance is not robust to departures from normality.

## 10 2010-09-23 (2 periods)

Derive limiting distributin of sample correlation coefficient. Consider special case of bivariate normal for tests and confidence intervals. Variance stabilizing transformations. Fisher's-z and the arcsin-square-root transformations.

## 11 2010-09-28 (1 periods)

Uniform delta method.

## 12 2010-09-30 (2 periods)

Non-differentiable transformations (example). Methods of moments.

## 13 2010-10-05 (1 period)

Bahadur representation for sample quantiles. Joint limiting distribution of several sample quantiles.

## 14 2010-10-07 (2 periods)

Example: limiting distribution of the sample inter-quartile range. Go over problems from the first two assignments.

## 16 2010-10-14 (2 periods)

Date: 2010-10-11 19:41:32 EDT

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