Statistics 242b

## Statistical Inference (Statistics 242/542) Spring 1996

Instructor: David Pollard (pollard@stat.yale.edu; http://www.stat.yale.edu/~pollard)
Office: Statistics Department, 24 Hillhouse Avenue. Phone: 432-0666
Office hours: Monday 12:00--2:00

Teaching Assistant: Alex Thiry (thiry@stat.yale.edu)
Office hours: Please email to arrange office hours

Prerequisites: Statistics 241a and linear algebra, at the level of Math 220.

Problem set about every two weeks, counting for 40% of final grade
Midterm test, counting for a token 10% of final grade
Final exam, counting for 50% of final grade

Homework policy:
Late homework accepted only with a Dean's note. No work accepted after distribution of solution sheet.

### Aim of the course

"Principles of statistical analysis: maximum likelihood, sampling distributions, estimation, confidence intervals, tests of significance, regression, analysis of variance, and the method of least squares."

Text: Rice "Mathematical Statistics and Data Analysis"

I hope to cover most of the material in Chapters 7 to 12, and 14, of Rice, with some material from Chapters 13 and 15 if time permits. I would expect to devote about the same amount of time to each topic as I did when I last used the Rice text, in 1990, but with some modifications to allow for more discussion of computing issues. I strongly recommend that students who take 242 also enrol in Statistics 200b.

• Bickel and Doksum "Mathematical Statistics: Basic Ideas and Selected Topics" (Holden-Day, 1977) [ A text at the next level up from 242. A good source for more detailed information.]

• Chambers, Cleveland, Kleiner, and Tukey "Graphical Methods for Data Analysis" (Wadsworth, 1983) [ Good discussion of topics such as Q-Q plots.]

• Tanur, Mosteller, Kruskal, Link, Pieters, and Rising "Statistics: A Guide to the Unknown" (Holden-Day, 1972). [A collection of short articles that illustrate many aspects of statistics. I might steal from it.]

• Stuart "Basic Ideas of Scientific Sampling" (Griffin, 1968). [Good discussion of sampling, with lots of simple numerical examples.]

### Material covered in 1990

Here is the breakdown of the course for Spring 1990. I intend to follow the same order of topics this year, but with some reapportionment of time spent on each chapter.

Survey sampling (Chapter 7; Lectures 1-6)

• Simple random sampling (with and without replacement); normal approximations; variance corrections; confidence intervals for mean; estimation of variance.
• Ratio estimators
• Stratified sampling; optimal allocation.
Estimation (Chapter 8; Lectures 7-11)
• Method of moments (for fitting negative binomial); general method of moments (M-estimators) with normal approximation; confidence intervals; MLE as optimal M-estimator.
• Information inequality; MLE in cases like uniform(0,theta); MLE for normal distribution; negative binomial MLE; sufficiency.
• Special supplement on generalized method of moments

Hypothesis testing and goodness-of-fit (Chapter 9; Lectures 12-17)
• Type 1 and type 2 errors; power functions; Neyman-Pearson lemma
• Chi-square test; likelihood ratio test
• Q-Q plots etc
There was some overlap between the last topic for Chapter 9 and the topics for Chapter 10.

Summarizing data (Chapter 10; Lectures 18-20)

• Histograms and density estimators; kernel estimation.
• Estimates of location; median; trimmed mean; more on M-estimators.
Comparing two samples (Chapter 11; Lectures 21-24)
• Facts concerning multivariate normal; t-tests; nonparametric tests
Analysis of variance (Chapter 12; Lectures 25-30)
• One-way ANOVA; geometric interpretation; generalized ANOVA table; least squares
• Two-way tables (Box-Cox survival data); transformations and plots
Regression (Chapter 14; Lectures 31-34)
• Fit a+bx; estimation of linear functions of parameters; variances; t-tests
• Multiple regression; more least squares.
• Material that is now covered in Statistics 312a.
Decision theory and Bayesian inference (Chapter 15; Lectures 35-36)
• Posterior distributions.
• Betting against confidence intervals, and other puzzles.