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531f10SIM - STAT 531 Simulation Study H Kim Department of...

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STAT 531: Simulation Study H. Kim Department of Mathematics and Statistics University of Calgary Fall 2010 H.Kim 1/30
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Simulation A numerical technique for conducting experiments on the computer Monte Carlo (MC) simulation : computer experiment involving random sampling from probability distributions invaluable in statistics. . . usually, when statisticians talk about “simulations,” they mean “Monte Carlo simulations” Fall 2010 H.Kim 2/30
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Rationale In statistics properties of statistical methods must be established so that the methods may be used with confidence exact analytical derivations of properties are rarely possible large sample approximations to properties are often possible, however. . . evaluation of the relevance of the approximation to (finite) sample sizes likely to be encountered in practice is needed analytical results may require assumptions (e.g., normality), but what happens when these assumptions are violated ? analytical results, even large sample ones, may not be possible Fall 2010 H.Kim 3/30
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Usual Issues Under various conditions is an estimator biased in finite samples? is it still consistent under departures from assumptions? what is its sampling variance ? how does it compare to competing estimators on the basis of bias, precision, etc.? does a procedure for constructing a confidence interval for a parameter achieve the advertised nominal level of coverage ? does a hypothesis testing procedure attain the advertised level or size? if it does, what power is possible against different alternatives to the null hypothesis? do different test procedures deliver different power? How to answer these questions in the absence of analytical results? Fall 2010 H.Kim 4/30
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Monte Carlo Simulation an estimator or test statistic has a true sampling distribution under a particular set of conditions (finite sample size, true distribution of the data, etc.) ideally, we would want to know this true sampling distribution in order to address the issues on the previous slide but derivation of the true sampling distribution is not tractable approximate the sampling distribution of an estimator or test statistic under a particular set of conditions Fall 2010 H.Kim 5/30
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How to Approximate A typical Monte Carlo simulation involves the following generate S independent data sets under the conditions of interest compute the numerical value of the estimator/test statistic T (data) for each data set T 1 ,..., T S if S is large enough, summary statistics across T 1 ,..., T S should be good approximations to the true sampling properties of the estimator/test statistic under the conditions of interest e.g., for an estimator for a parameter θ : T s is the value of T from the s th data set, s = 1 ,..., S , the sample mean over S data sets is an estimate of the true mean of the sampling distribution of the estimator Fall 2010 H.Kim 6/30
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Simulations for Properties of Estimators Simple example: Compare three estimators for the mean μ of a distribution based on i.i.d.
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