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class03_18

# class03_18 - 1.017/1.010 Class 18 Small Sample Statistics...

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1.017/1.010 Class 18 Small Sample Statistics Small Samples When sample size N is small the estimator of a distributional property a (mean, variance, 90 percentile, etc.) is generally not normal. In this case, the CDF’s of the estimate a and standardized statistic z (used to derive confidence intervals and hypothesis tests) can be approximated with stochastic simulation . ˆ In order to generate random replicates in the stochastic simulation we need to specify the property a (or parameters that are related to it): For estimating confidence intervals we assume a a ˆ = (the estimate computed from the actual data). For testing hypotheses we assume 0 a a = (the hypothesized parameter value). The stochastic simulation uses many N rep random sample replicates, each of length N , to generate N rep estimates. The desired estimate and standardized statistic CDFs are derived from this ensemble of estimates. Example – Small-sample two-sided confidence Intervals for the mean of an exponential distribution Consider a small sample that is thought to be drawn from an exponential distribution with unknown parameter a : [ x 1 , x 2 , x 3 , x 4 , x 5 ] = [0.05 1.46 0.50 0.72 0.11 ] The sample mean is an unbiased estimator of a : 57 . 0

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class03_18 - 1.017/1.010 Class 18 Small Sample Statistics...

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