Question Five Consider n independent identically distribution (iid) observations from Gamma(k, )) distribution with density function f (x K, X) = 1 and Moment Generating Function: Mx(t) = () a [1 mark] Determine the cumulant generating function for a single observation from a Gamma(k, X). Kx (t) = (1 - kt)-* O Kx(t) = - log(k + At) O Kx (t) = (1 - At) O Kx(t) = -Alog(1 - kt) O Kx(t) = -klog(1 - At)b) [2 marks] The first derivative of the cumulant generating function is K'x(t) = KA (1 - At) Compute the second derivative of the cumulant generating function. K K'x (t) = 7 (1 - At ) 2 O K* (t) = (1 - At)2 O Kx (t) = (1 - At)2 O KX2 Kx (t ) = 1 - At O Kx (t) = (1 - At)2\f[3 marks] It is known that Kx (f ) = -klog( ), K* (t ) = and nts = K -nK Find the first order saddlepoint approximation for the density of X. Hint: The first order saddlepoint approximation is f( ) =. n enkx (i)-nix 2TK" (t) O f ( ) =1 27 ne (ned) ment (na)-le nin. O f( ) = 1 2T " (nk ) men ( na)ni-le nin . O f ( * ) = 1/27 - (nkid) ken (na)"-le ni O f (z) = 1 27 (nkid) mend (na)-le nin. Oe [3 marks] Suppose X = (X1, X2, ..., Xn) are a random sample from a Bernoulli(0). We are interested in estimating T(0) = 0(1 - 0) and we use the MLE T(X) = X(1 - X). Find the theoretical bias when using ' to estimate T. O O - (1 - 0 ) O n O -10(1 - 0) n O There is no bias