3. Application of Delta Method on Gamma Variables

(d) 0/4 points (graded) Using the previous part, find confidence intervals for a with asymptotic level 90% using both the "solving" and the "plug-in" methods. Use n = 25, and Xn = 4.5. (Enter your answers accurate to 2 decimal places. Use the Gaussian estimate q0.05 ~ 1.6448 for best results.) I solve = 4.172 4.828 I plug-in = 4.3275 X 4.3402 X STANDARD NOTATIONThe Gamma distribution Gamma (a, /) with paramters a > 0, and / > 0 is defined by the density fap (x) = I (a) x-lex, for allx > 0. The I function is defined by 00 T(s) = x-le *dx. As usual, the constant - is a normalization constant that gives for fap (x) dx = 1. In this problem, let X1. ... . X'n bei.i.d. Gamma variables with B = -for some a > 0. CX That is, X1, .... Xn ~ Gamma (a, - ) random variables for some a > 0. The pdf for X, is therefore fa (x) = Tojoux exa for all x 2 0