Populations, Parameters, Statistics, and Sampling

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Other Properties • Some statistics are designed with other properties in mind beyond probability (MLE) – in these examples, G is sample statistic that estimates θ – unbiased • expected value of G over all samples is θ (E[G] = θ) – x is unbiased estimate of μ – for binomial, P is unbiased estimate of p – however, S 2 is a biased estimate of σ 2 – consistency • as N increases, should approach population parameters – relative efficiency • the variance of the estimator as compared to other estimators – mean is more efficient than median for normal distributions – sufficient • G contains all the information in the sample that can be used to find θ – often a set of sufficient statistics is required (e.g., mean and corrected variance)
Sampling Distribution of the Mean • Mean is unbiased, consistent, efficient, and combined with σ 2 is sufficient in many circumstances – the mean of the sampling distribution of means is the same as the population mean • However, the sampling distribution of the mean is not identical to population distribution • for example, variance of means σ M 2 = σ 2 / N – σ M is the standard error of the mean – a standard score of the sample mean is therefore Z M x − μ x − μ = = σ σ / N M – in order to compare the observed mean to the expected mean given some hypothesis, we need the standard error of the mean – suppose x = 15, μ = 10, σ = 4 and N = 16 » what’s the probability of this deviant, or greater?
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Page 3: Maximum Likelihood (MLE) • In cal
Page 7: Other Issues with Estimators • Po

Populations, Parameters, Statistics, and Sampling

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