Stat 5101 Lecture Notes - School of Statistics

208 Stat 5101 (Geyer) Course Notes(b)Show thatn ( θ − X (n))D−→ Exp(1/θ),as n →∞.Hints This is a rare problem (the only one of the kind we will meet in thiscourse) when we can’t use the LLN or the CLT to get convergence in probabilityand convergence in distribution results (obvious because the problem is notabout X n and the asymptotic distribution we seek isn’t normal). Thus we needto derive convergence in distribution directly from the definition (Definition 6.1.1in these notes or the definition on p. 135 in Lindgren).Hint for Part (a): Show that the c. d. f. of X (n) converges to the c. d. f. ofthe constant random variable θ. (Why does this do the job?)Hint for Part (b): DefineY n = n ( θ − X (n))(the random variable we’re trying to get an asymptotic distribution for). Deriveits c. d. f. F Yn (y). What you need to show is thatF Yn (y) → F (y),for all ywhere F is the c. d. f. of the Exp(1/θ) distribution. The fact from calculus(lim 1+ x ) n=exn→∞ nis useful in this.You can derive the c. d. f. of Y n from the c. d. f. of X (n) , which is given inthe first displayed equation (unnumbered) of Section 7.6 in Lindgren.7-8. Suppose X 1 , ..., X n are i. i. d. N (µ, σ 2 ). What is the probability that|X n − µ| > 2S n / √ n if n = 10?7-9. Suppose X 1 , ..., X n are i. i. d. N (µ, σ 2 ). What is the probability thatS 2 n > 2σ 2 if n = 10?7-10. R and Mathematica and many textbooks use a different parameterizationof the gamma distribution. They writef(x | α, β) =1β α Γ(α) xα−1 e −x/β (7.39)rather thanf(x | α, λ) =λαΓ(α) xα−1 e −λx (7.40)Clearly the two parameterizations have the same first parameter α, as the notationsuggests, and second parameters related by λ =1/β.(a) Show that β is the usual kind of scale parameter, that if X has p. d. f.(7.39), then σX has p. d. f. f(x | α, σβ), where again the p. d. f. is definedby (7.39).