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Central Dogma of Statistical Inference

Population

Sample

Draw samples from the population

How samples can tell us about the population?

Estimate parameters of population from samples

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Sample vs Population Statistics

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Properties of Expectation and Variance

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Useful Properties of Normal Random Variable

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Examples of Statistical Inference: �Height measurement

Population

Sample

Randomly recruit 100 individuals from Purdue and measure their heights

How do these 100 individuals (sample) tell us about the heights of everyone at Purdue (population)?

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Height measurement

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What if we change the size of samples?

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What about the shape of the sampling distribution?

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Central Limit Theorem (CLT)

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Theoretical Calculation

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Confidence Interval

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Interpretation

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Theoretical Calculation

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Monte Carlo for 100 Sampling

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Principle of Parameter Estimation

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Method of Moments

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Maximal Likelihood Estimator (MLE)

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Notes

Aspects	Method of Moments (MoM)	Maximum Likelihood Estimation (MLE)
Basic Idea	Match sample moments to theoretical moments	Choose parameter values that maximize the likelihood of the observed data
Computation	Often simpler, involves solving algebraic equations	Usually more complex, involves optimization (may require calculus or numerical methods)
Efficiency	Generally less efficient (higher variance)	Statistically efficient under regular conditions
Robustness	Can be more robust when model assumptions are slightly off	Can be sensitive to model misspecification

Covered in more details in a mathematical statistics class

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