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Unformatted text preview: 7-1 CHAPTER 7 Section 7-27-1. E( )()==&amp;&amp;=&amp;&amp;&amp;&amp;===nnXEnnXEXniinii22121221211E( )( )==&amp;&amp;=&amp;&amp;&amp;&amp;===nnXEnnXEXniinii11112, 1Xand 2Xare unbiased estimators of . The variances are V( )2n21=Xand V( )n22=X; compare the MSE (variance in this case), 212/2/)()(2221===nnnnMSEMSESince both estimators are unbiased, examination of the variances would conclude that X1is the better estimator with the smaller variance. 7-2. E( )===+++=)7(71))(7(71)()()(717211XEXEXEXE&amp;E( )=+-=++=]2[21)()()2(217612XEXEXEa) Both 1and 2are unbiased estimates of since the expected values of these statistics are equivalent to the true mean, . b) V( )()227212721171)7(491)()()(717...==+++=+++=XVXVXVXXXV&amp;7)(21=VV( )()))()()(4(41)()()2(212246146124612XVXVXVXVXVXVXXXV++=++=+-== ()144222++= 1462()23)(22=VSince both estimators are unbiased, the variances can be compared to decide which is the better estimator. The variance of 1is smaller than that of 2,1is the better estimator. 7-3. Since both 1and 2are unbiased, the variances of the estimators can be examined to determine which is the better estimator. The variance of 2is smaller than that of 1thus 2may be the better estimator. Relative Efficiency = 5.2410)()()()(2121===VVMSEMSE7-4. Since both estimators are unbiased: 7-2 Relative Efficiency = 2122/37/)()()()(222121===VVMSEMSE7-5. 5.2410)()()()(2121===VVMSEMSE7-6. =)(1E2/)(2=E-=)(2EBias= 2-= -2V)(1= 10 V)(2= 4 For unbiasedness, use 1since it is the only unbiased estimator. As for minimum variance and efficiency we have: Relative Efficiency =222121))(())((BiasVBiasV++where, Bias for 1is 0. Thus, Relative Efficiency =()()1042401622++-&amp;&amp;=+If the relative efficiency is less than or equal to 1, 1is the better estimator. Use 1, when 401612()+40162+()242 -4 899.or 4 899.If -&lt;&lt;4 8994 899..then use 2. For unbiasedness, use 1. For efficiency, use 1when -4 899.or 4 899.and use 2when -&lt;&lt;4 8994 899... 7-7. =)(1ENo bias )(12)(11MSEV===)(2ENo bias )(10)(22MSEV==)(3EBias 6)(3=MSE[note that this includes (bias2)] To compare the three estimators, calculate the relative efficiencies: 2.11012)()(21==MSEMSE, since rel. eff. &gt; 1 use 2as the estimator for 2612)()(31==MSEMSE, since rel. eff. &gt; 1 use 3as the estimator for 8.1610)()(32==MSEMSE, since rel. eff. &gt; 1 use 3as the estimator for Conclusion: 3is the most efficient estimator with bias, but it is biased....
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