5. (11 marks] Consider n independent identically distributed (iid) observations from a x distribution 2 a) [1 mark] Dete

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5 11 Marks Consider N Independent Identically Distributed Iid Observations From A X Distribution 2 A 1 Mark Dete 1 (114.54 KiB) Viewed 26 times

5. (11 marks] Consider n independent identically distributed (iid) observations from a x distribution 2 a) [1 mark] Determine the cumulant generating function for a single observation from a x;, distribution. b) [1 mark] Compute the first and second derivative of the cumulant generating function. c) [1 mark] Find the saddlepoint Ê. d) [2 marks] Evaluate Ky(f) and K” () and simplify as much as possible. e) [3 marks] Show that the first order saddlepoint approximation for the density of X is npp-np/2emp/2qnp/2-1e-nc/2 f(T) VAP Hint: The first order saddlepoint approximation is n f(ā) a -e{nKx(t)-ntī} V 27K”(6) = f) [2 marks] Hence, determine the first order saddlepoint approximation for the density of the sum of n iid observations S = 2-1 X; from this distribution. Hint: Consider using the density transformation formula dx fs(s) = fx(x(s)) ds = ) g) (1 mark] What is the approximate distribution of this saddlepoint approxima- tion? Keep in mind the famous Stirling roximation of the Gamma function that states 41 (np np/2 пр r(2) ༄། e-np/2