Fundamentals of Probability and Statistics for Engineers

284 Fundamentals of Probability and Statistics for EngineersXX 2rX 1Figure 9. 3 Measurement X, for Example 9.13moments of X can be directly generated from Equation (9.77). We remark that,although an estimator for 2 is not required, it is nevertheless an unknownparameter and must be considered together with . In the applied literature, anunknown parameter for which the value is of no interest is sometimes referredto as a nuisance parameter.Two moment equations are needed in this case. However, we see fromEquation (9.77) that the odd-order moments of X are quite complicated. Forsimplicity, the second-order and fourth-order moment equations will be used.We easily obtain from Equation (9.77)) 2 ˆ ‡ 2 2 ;…9:78† 4 ˆ 2 ‡ 8 2 ‡ 8 4 :The two moment equations are^ ‡ 2 c )2 ˆ M 2 ;^ 2 ‡ 8 ^ c 2 ‡ 8 c 22 ˆ M 4 :…9:79†Solving for^ ,wehave^ ˆ…2M2 2 M 4 † 1=2 : …9:80†Incidentally, a moment estimatorc 2 for 2 , if needed, is obtained from Equations(9.79) to bec 2 ˆ 12 …M 2^†:…9:81†Combined Moment Estimators. Let us take another look at Example 9.11 forthe purpose of motivating the following development. In this example, anestimator for has been obtained by using the first-order moment equation.Based on the same sample, one can obtain additional moment estimators for by using higher-order moment equations. For example, since 2 ˆ 2/ 2 ,thesecond-order moment equation, 2 ˆ M 2 ;TLFeBOOK