10.3. Suppose in Problem 10.2, imputations are randomly drawn with replacement from the r respondents’ values.

(a) Show that y* is unbiased for the population mean Y .

(b) Show that conditional on the observed data, the variance of y* is ms2 Rð1 r 1Þ=n2, and that the expectation of s2 * is s2 Rð1 r 1Þ½1 þ rn 1ðn 1Þ 1 .

(c) Show that conditional on the sample sizes n and r (and the population Y values), the variance of y* is the variance of yR times ½1 þ ðr 1Þn 1ð1 r=nÞð1 r=NÞ 1 , and show that this is greater than the expectation of U* ¼ s2 *ðn 1 N 1Þ.

(d) Assume r and N=r are large, and show that interval estimates of Y based on U* as the estimated variance of y* are too short by a factor ð1 þ nr 1 rn 1Þ 1=2 . Note that there are two reasons: n > r, and y* is not as efficient as yR. Tabulate true coverages and true significance levels as functions of r=n and nominal level.

10.4. Suppose multiple imputations are created using the method of Problem 10.3 D times, and let y ðdÞ * and UðdÞ * be the values of y* and U* for the dth imputed data set. Let  y* ¼ PD d¼1 y ðdÞ * =D, and T* be the multiple imputation estimate of variance of  y*. That is, T* ¼ U * þ ð1 þ D 1 ÞB*; where U * ¼ P D d¼1 UðdÞ * =D; B* ¼ P D d¼1 ð y ðdÞ *  y*Þ 2 :

(a) Show that, conditional on the data, the expected value of B* equals the variance of y*.

(b) Show that the variance of  y* (conditional on n, r, and the population Y values) is D 1VarðY *Þþð1 D 1ÞVarðyRÞ, and conclude that  y* is more efficient than the single-imputation estimate y*.

(c) Tabulate values of the relative efficiency of  y* to yR for different values of D, assuming large r and N=r.

(d) Show that the variance of  y* (conditional on n, r, and the population Y values) is greater than the expectation of T* by approximately s2 Rð1 r=nÞ 2 =r.

(e) Assume r and N=r are large, and tabulate true coverages and significance levels of the multiple imputation inference. Compare with the results in Problem 10.3, part (d).