STAT 420 (Introduction to Statistics)
| DESCRIPTION |
This course introduces some of the
key ideas of mathematical statistics
related to the good performance and optimality of statistical
procedures
(parameter estimates and hypothesis tests) and covers many
examples.
The main objective is to learn how data arising from probability
distributions
with some unknown parameters can be used to narrow down or draw
inferences
about
those unknown parameters. Students will learn how to construct optimal
tests and estimators in many settings, particularly those involving
normally
distributed or large-sample data. |
| PREREQUISITES |
Stat 410. Note: STAT 420 is cross-listed with SURV
420. Credit will be granted for only one of the following:
STAT 420 or SURV 420.
|
| TOPICS |
Probability Review:
Densities, change-of-variable, expectation,
moment generating functions, conditional expectation and variance, best
mean-squared-error predictors. (Optional: multivariate normal
distribution)
(1.5-2 weeks)
Limit Theorems:
Central Limit Theorem (plus optional
supplementary
discussion of multivariate CLT). 'Delta method'. (1.5 weeks)
Sampling Distributions Related to the Normal:
Distributions of sample mean and
variance;
x2 , t, F distributions. (Supplementary material on
limiting
distribution of Pearson chi-squared goodness-of-fit distribution) (2
weeks)
Estimation:
Problem of point estimation.
Likelihood.
Method of moments and maximum likelihood estimators (MLE's). Cramer-Rao
bound, Fisher information. Asymptotic normal distributions of
moments
estimators and MLE's. Large-sample Confidence Intervals based
upon
moments and ML estimators. Relative efficiency. confidence
intervals for mean, variance and two-sample parameters for normal
distributions.
(3.5 weeks)
Exponential families and sufficient statistics:
Definition of exponential family and
sufficient
statistics. Factorization theorem. Completeness,
Rao-Blackwell
Theorem. (1.5-2 weeks)
Hypothesis testing:
Definitions and formulation of Neyman-Pearson
theory. Duality between tests and confidence intervals.
Optimal
(Neyman-Pearson) simple vs. simple rejection regions. Power
functions.
UMP tests. P-value. Generalized likelihood ratio tests and
examples.
(3 weeks)
Miscellaneous Topics:
Material chosen from among: order and rank
statistics; linear regression; data analysis; simulation and bootstrap
methods; Bayesian procedures. (<2 weeks)
|
| TEXT |
Text(s)
typically used in this course. |
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