2. Suppose you decide to randomly generate numbers from X ~ Unif(0, ). Your friend will...
2. Suppose you decide to randomly generate numbers from X ~ Unif (0,0). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use θMLE-max Xi-X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead he'll say the second largest number: θ-Xn-1. Determine the bias of this estimator by carefully finding the density function...
Fix θ > 0 and let Xi, , x, i d. Unif[0.0]. We saw in class that the MLE of θ, oMLE- I give two other estimators of θ, which can be made unbiased by appropriate choice of -C1 max(Xs , . . . , X,) max(X., Xn), is biased. constants C1,C2 We have two questions: (1) Find values of C1, C2 for which these estimators are unbiased. Note that Ci,C2 may depend on n (2) Which of these estimators...
3. Suppose you have X Binom(n, p) where n is known and p is unknown. Typically, people use p = _ to estimate p, where X = Xi is simply a sample of size l. This might represent simultaneously flipping n coins (just once!) and counting the number of heads you see, where each coin has Pheads - p. Now, if both n and p are known, we know the variance V of X is just np(1-p) If p is...