Chapter 7 Introduction to statistical inference v4 PartI...Probability and Statistics Chapter 7: Introduction to Statistical Inference. K Van Steen 6. 7 Analysis of two-way tables

Probability and Statistics Chapter 7: Introduction to Statistical Inference

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CHAPTER 7: INTRODUCTION TO STATISTICAL INFERENCE

1 Introduction

2 Parametric point estimation

2.1 Methods of finding estimates

Method of moments

Maximum likelihood

Other


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2.2 Properties of point estimates

Mean Squared Error

Consistency and BAN

Sufficient statistics

2.3 Unbiased estimation

2.4 Highlight: Bayes estimators (postponed)

Posterior distribution - Loss-function invariance -

Minimax estimator


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3 Parametric interval estimation

3.1 Estimating with confidence

3.2 Methods of finding confidence intervals

3.3 Highlight: Bayesian interval estimation (postponed)


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4 Hypothesis testing

4.1 Tests of significance

4.2 Hypothesis types

Simple hypothesis

Composite hypothesis

4.3 Assumptions about hypothesis tests

4.4 Examples

4.5 Use and abuse of tests

4.6 Multiple testing


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5 Inference for distributions

5.1 Inference for the mean of a population

5.2 Comparing two means

5.3 Optional topics in comparing distributions

6 Inference for proportions

6.1 inference for a single proportion

6.2 Comparing two proportions


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7 Analysis of two-way tables

7.1 Inference for two-way tables

7.2 Formulas and models for two-way tables and Goodness-of-fit

8 How to select the appropriate test?


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1 Introduction

• Statistical inference means drawing conclusions based on data. There are a

many contexts in which inference is desirable, and there are many

approaches to performing inference.

• One important inferential context is parametric models. For example, if you

have noisy (x; y) data that you think follow the pattern y = β0 + β1x + error ,

then you might want to estimate β0, β1, and the magnitude of the error.

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• In earlier chapters, we indicated that a sample from the distribution of a

population is useful in making inferences about the population itself.

• Two important problems in statistical inference are estimation and testing

(of hypotheses).

• In point estimation, the value of some statistic (i.e. a function of random

variables – see later) represents or estimates the unknown parameter of

interest.

• In interval estimation, two statistics are defined so that they span an

interval for which the probability can be determined that it contains the

unknown parameter

Probability and Statistics

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Definition of an estimator


Definition of an estimator

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2.1 Methods of finding estimates

Method of moments



estimates


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Example 1 of the method of moments




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Method-of-moment estimators are not unique


moment estimators are not unique


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Maximum probability


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Maximum likelihood


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Intuition behind Maximum Likelihood (ML)


Intuition behind Maximum Likelihood (ML) estimators


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Definition of ML estimator


Definition of ML estimator


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Example of ML estimators


Example of ML estimators


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The invariance property of ML estimators


The invariance property of ML estimators


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Bayesian approach


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Other criteria to estimate parameters


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Mean Squared Error




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Consistency and BAN



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Loss and Risk function



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Interpretation



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before


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before


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Sufficient statistics



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Hence …



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How to find UMVUEs?

Lower bound for the variance


Lower bound for the variance


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(proof for your information)


(proof for your information)


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Uses of the theorem



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2.4 Highlight: Bayes estimators (postponed)


Highlight: Bayes estimators (postponed)


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3 Parametric interval estimation

3.1 Estimating with confidence

Overview of inference


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Statistical confidence


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Confidence intervals


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Implications


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Standardizing the normal curve using z


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Varying confidence levels


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How do we find specific z* values?


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Link between confidence level and margin of error


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Impact of sample size


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Sample size and experimental design


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3.2 Methods of finding confidence intervals

Statistical method


Methods of finding confidence intervals


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(Figure A)


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(Figure A)


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(Figure B)


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(Figure B)


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4 Hypothesis testing

4.1 Tests of significance

Reasoning of significance tests


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Stating hypotheses


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One-sided and two-sided tests


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How to choose?


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The p-value


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Interpreting a p-value


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Tests for a population mean


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P-value in one-sided and two-sided tests


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The significance level α


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Confidence intervals to test hypotheses


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Logic of confidence interval test


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More about the relationship to confidence intervals


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Permutation / randomization tests


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Bootstrap tests


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4.2 Hypothesis types

Simple hypothesis and Composite hypothesis

• A simple hypothesis is a hypothesis which specifies the population

distribution completely.

• Examples

1. H0: X ~ Bi(100,1/2), i.e. p is specified

2. H0: X ~ N(5,20), i.e. µ and are specified

• A composite hypothesis is a hypothesis which does not specify the

population distribution completely.

• Examples

1. X ~ Bi(100,p) and H1: p > 0.5

2. X ~ N(0, ) and H1: unspecified

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Chapter 7 Introduction to statistical inference v4 PartI...Probability and Statistics Chapter 7: Introduction to Statistical Inference. K Van Steen 6. 7 Analysis of two-way tables