The saddlepoint approximation of the distribution of the sample mean in Example 9.9 can be used as a theoretical alternative of the bootstrap computation. Define the empirical cumulant generating...

The saddlepoint approximation of the distribution of the sample mean in Example 9.9 can be used as a theoretical alternative of the bootstrap computation. Define the empirical cumulant generating function For the following observations compare the bootstrap distribution of the sample mean with the saddlepoint approximation (9.8) based on the empirical cumulant function. Suppose y 1 , . . . , y n are an iid sample from the inverse Gaussian distribution with density (a) Assuming ? is known, find the saddlepoint approximation of the density of the MLE of µ. (b) Assuming µ is known find the saddlepoint approximation of the density of the MLE of ?. (c) For µ = 1, ? = 1, and n = 10, show how good is the approximation in (b) by performing a Monte Carlo simulation similar to the one in Example 9.8.