Populations, Parameters, Statistics, and Sampling
Populations, Parameters, Statistics, and Sampling
Populations, Parameters, Statistics, and Sampling
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Other Properties<br />
• Some statistics are designed with other properties in mind<br />
beyond probability (MLE) – in these examples, G is<br />
sample statistic that estimates θ<br />
– unbiased<br />
• expected value of G over all samples is θ (E[G] = θ)<br />
– x is unbiased estimate of μ<br />
– for binomial, P is unbiased estimate of p<br />
– however, S 2 is a biased estimate of σ 2<br />
– consistency<br />
• as N increases, should approach population parameters<br />
– relative efficiency<br />
• the variance of the estimator as compared to other estimators<br />
– mean is more efficient than median for normal distributions<br />
– sufficient<br />
• G contains all the information in the sample that can be used to find θ<br />
– often a set of sufficient statistics is required (e.g., mean <strong>and</strong> corrected<br />
variance)
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