Show transcribed image text b ~ Pois(A). In Example 2.21 we found the UMVUE Exercise 5.3 Let Yi, (1-l)V ''" for p-P(Y1). Is it consistent? 1n Example 2.21 This is a less intuitive example. Suppose we have records Yi,…,Y, on the annual numbers of earthquakes in a certain seismic region over the last n years. It is reasonable to assume that the Y, are i.i.d. Pois(X). We want to estimate the probability that the following year will not be "quieÅ¥" (seismically active) and there will be at least one earthquake in the region. Let p-P(Y > 1) = 1-e-a. The sufficient statistic W = Yi +â€¦+Kn is the parameter is again the probability of a certain event (seismically active year), the corresponding indicator variable T = 1 if Yi > 1 and zero otherwise, is an unbiased estimator for p. Then, total number of earthquakes over n years. Since the unknown Unlike the previous example, it would be quite hard to obtain the above unbiased estimator by "intuitive" arguments Compare T1 with the-MLE p of p. The MLE of Î» is Î»-7 (see Exercise 2.13) and therefore p 1-e-r. The two estimators are different but note that for large n Does "Rao-Blackwellization" necessarily yield an UMVUE? Generally not. To guarantee UMVUE an additional requirement of completeness on a sufficient statis- tic W is needed. For interested and advanced readers we discuss it in more detail in Section 2.6.5. However, under mild conditions, it can be shown that if the distribution of the data belongs to the p-parametric exponential family (see Definition 1.6), where p- dim(0), and an unbiased estimator is a function of the corresponding sufficient statistic (ri (y), , Î£ 1Tp0%)), it is an UMVUE (see Section 2.6.5 for more detail). Thus, in particular, the unbiased estimator (1-1) , K for p = P(Y > 1) for

b ~ Pois(A). In Example 2.21 we found the UMVUE Exercise 5.3 Let Yi, (1-l)V ''" for p-P(Y1). Is it consistent? 1n Example 2.21 This is a less intuitive example. Suppose we have records Yi,…,Y, on the annual numbers of earthquakes in a certain seismic region over the last n years. It is reasonable to assume that the Y, are i.i.d. Pois(X). We want to estimate the probability that the following year will not be "quieÅ¥" (seismically active) and there will be at least one earthquake in the region. Let p-P(Y > 1) = 1-e-a. The sufficient statistic W = Yi +â€¦+Kn is the parameter is again the probability of a certain event (seismically active year), the corresponding indicator variable T = 1 if Yi > 1 and zero otherwise, is an unbiased estimator for p. Then, total number of earthquakes over n years. Since the unknown Unlike the previous example, it would be quite hard to obtain the above unbiased estimator by "intuitive" arguments Compare T1 with the-MLE p of p. The MLE of Î» is Î»-7 (see Exercise 2.13) and therefore p 1-e-r. The two estimators are different but note that for large n Does "Rao-Blackwellization" necessarily yield an UMVUE? Generally not. To guarantee UMVUE an additional requirement of completeness on a sufficient statis- tic W is needed. For interested and advanced readers we discuss it in more detail in Section 2.6.5. However, under mild conditions, it can be shown that if the distribution of the data belongs to the p-parametric exponential family (see Definition 1.6), where p- dim(0), and an unbiased estimator is a function of the corresponding sufficient statistic (ri (y), , Î£ 1Tp0%)), it is an UMVUE (see Section 2.6.5 for more detail). Thus, in particular, the unbiased estimator (1-1) , K for p = P(Y > 1) for