Let X1, X2,..., Xn be a sample from the inverse Gaussian pdf,

a. Show that the MLEs of μ and λ are

b. Tweedie (1957) showed that n and n are independent, n having an inverse Gaussian distribution with parameters μ and nλ, and nλ/n having a X2n-1 distribution. Schwarz and Samanta (1991) give a proof of these facts using an induction argument.
(i) Show that 2 has an inverse Gaussian distribution with parameters μ and 2λ, 2λ/2 has a X21distribution, and they are independent.
(ii) Assume the result is true for n = k and that we get a new, independent observation x. Establish the induction step used by Schwarz and Samanta (1991), and transform the pdf f(x, k k) to f(x, k+1, k+1). Show that this density factors in the appropriate way and that the result of Tweedie follows.

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