# Department of Mathematics and Statistics

Shortly before spring classes begin in January, students will take a comprehensive exam on the theoretical foundations of statistics. There will be a two hour exam on the material of 201A and a two hour exam on the material of 201B. All students taking the exam will receive copies of previous examinations.

## the second part, we discuss the design of an interactive statistical

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### You saved the day! I have statistical data!

The adds a curve to the plot representing a moving average over the data. It is **not** a trend line! also adds some type of margin of error around this smoothing line (I admit that I have not looked deeply into the internals of ggplot2). If we interpret the margin of error loosely as a confidence interval, we can make a statistical conclusion of this graph. Recall that a basic one-sample confidence interval with population standard deviation known is

### 3. Calculate the test statistic.

Descriptive statistics also help the researcher speak to the limits of who the results can be generalized to. For example, knowing that 100% of the participants were male would imply that they can not necessarily generalize to females.

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ABSTRACT. This thesis presents certain recent methodologies and some new results for the statistical analysis of probability distributions on non-Euclidean manifolds. The notions of Frechet mean and variation as measures of center and spread are introduced and their properties are discussed. The sample estimates from a random sample are shown to be consistent under fairly broad conditions. Depending on the choice of distance on the manifold, intrinsic and extrinsic statistical analyses are carried out. In both cases, sufficient conditions are derived for the uniqueness of the population means and for the asymptotic normality of the sample estimates. Analytic expressions for the parameters in the asymptotic distributions are derived. The manifolds of particular interest in this thesis are the shape spaces of k-ads. The statistical analysis tools developed on general manifolds are applied to the spaces of direct similarity shapes, planar shapes, reflection similarity shapes, affine shapes and projective shapes. Two-sample nonparametric tests are constructed to compare the mean shapes and variation in shapes for two random samples. The samples in consideration can be either independent of each other or be the outcome of a matched pair experiment. The testing procedures are based on the asymptotic distribution of the test statistics, or on nonparametric bootstrap methods suitably constructed. Real life examples are included to illustrate the theory.

**Xiang Zhu Wins ASA Award**

Xiang Zhu's work on enrichment analysis of genome-wide association summary statistics has won an ASA student paper award. Xiang is advised by and will present this paper at the 2017 Joint Statistical Meeting in Baltimore. This work is supported by grants from the Gordon and Betty Moore Foundation and the National Institutes of Health. The manuscript is available at .