Point Estimation October 25, 20051POINT ESTIMATIONStatistics, Realizations, Parameters, and EstimatorsmeansampleltheoreticaourisXsample onefor mean theofn realizatioourisxmean population trueisµmean population theofestimator an isˆµGeneric Parameter• θ as a generic parameter we want to estimate• Moore & McCabe use “V” as I use θPoint Estimation October 25, 20052A Good Point EstimateProblems with EstimatorsBias VariabilityBiasUnbiasedBiasedPoint Estimation October 25, 20053Defining and Measuring BiasUnbiased Estimator:()θθ=ˆE()θθ−=ˆEbiasStatistical Bias Defined:Distribution of Sample Means0 5 10 15Percent40 50 60 70Mean of Feeling Thermometer: Bushµ = 54.94mean of means = 54.93mean of medians = 54.95Estimating the Variance()2σ<MSDE()2limσ=∞→MSDEn()221σ=⎟⎟⎠⎞⎜⎜⎝⎛−−∑nXXEPoint Estimation October 25, 20054EfficiencyEfficiency: Two DistributionsMeasuring Relative Efficiency1221ˆvarˆvarˆtocomparedˆθθθθ=Example:57.157.1MediantoMean22==nnXXσσPresumes an underlying normal distributionPoint Estimation October 25, 20055Mean Squared Error: Combining Efficiency and BiasDefining MSE()22ˆbiasvarianceEMSE +=−=θθGeneralized Efficiency12ˆ ˆ θθMSEMSE=Other Concepts• Robust Estimation• Consistency• SufficiencynXXσσ=trimmed meanbiweighted meanPoint Estimation October 25, 20056Evaluating Bias and EfficiencyThree estimators of µ:321332123211414141ˆ412141ˆ313131ˆXXXXXXXXX++=++=++=θθθCheck Biasµµµµθ=++=++=313131)(31)(31)(31)ˆ(3211XEXEXEEµµµµθ=++=++=412141)(41)(21)(41)ˆ(3212XEXEXEEµµµµθ43414141)(41)(41)(41)ˆ(3213=++=++= XEXEXEECheck Efficiency22222222ˆ313131311XXXXσσσσσθ=⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛=222222222ˆ831664121412XXXXXσσσσσσθ==⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛=22222222ˆ1634141413XXXXσσσσσθ=⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛=Point Estimation October 25, 20057Check MSEц=10, σ2=252biasefficiencyMSE +=33.802531ˆMSE1=+=θ375.902583ˆMSE2=+=θ94.1026.669.4)104310(25163ˆMSE23=+=−+=θCheck MSEц=6,