﻿ 厦门大学数学科学学院

短课程：统计决策理论

1. Statistical Decision Theory

In modern statistics, all statistical inferences are viewed decision based on data. This talk introduces the decision theoretic framework of statistics, including action space, decision rule, loss function and risk functional. We illustrate the decision framework through detailing parameter estimation and hypothesis testing. Also, we point out the potential challenges to be confronted with under this framework.

Key words  Decision rule; Loss function; Probability model; Risk functional.

2. Prediction --- from viewpoint of decision theory

To predict a random variable is the eternal topic in statistics. In this talk, we derive fundamental theorem of prediction: the conditional expectation as the best prediction in the context of covariates. Specifically, we introduce the best linear prediction and the decomposition of the mean squared prediction error.

Key words  Mean squared prediction error; Conditional expectation; Best linear prediction

3. Statistical Regression --- from viewpoint of decision theory

Under the decision framework, this talk develops the best prediction in the context of simple linear regression and multiple linear regression. As an illustration, we show how to get the least square estimate directly based on the analytical form of the best linear prediction.

Key words  Bivariate normal distribution; Least square estimate; Multivariate normal distribution; Multiple correlation coefficient

4. Decision criteria under the statistic decision framework

This talk introduces the two global criteria to build the optimal decision rule. The Bayes rule pays attention to the expected risk based on the posterior distribution, and the Minimax rule focus on the minimizing the maximum risk of the decision rules. As an illustration, we details the two optimal rules in the application of oil drilling.

Key words  Prior distribution, Risk averse; Bayes risk; Randomized rule; Maximum risk

5. Bayes decision rule under the statistic decision framework

This talk has a discussion on the fundamental theory the Bayes rules as well as other important aspects. We mainly illustrate the Bayes rule in the context of parameter estimation, hypothesis testing and classification. As an illustration, we details the two optimal rules in the application of oil drilling.

Key words  Classification, Prior distribution, Bayes risk; Randomized decision rule

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