05 09 2022 E TS

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BAYESIAN STATISTICS A. Guglielmi & M. Beraha 05.09.2022 Properly justify all your answers . Use the indicator function to denote the support of a distr.. Exercise 1 Let X1, X2, . . . , Xn be (conditionally) independent and identically distributed random variables following the inverse-Gaussian density with parameters µ, τ > 0: f(x; µ, τ) = r τ 2πx3 exp  − τ µ2 2x (x − 1 µ)2  I(0,+∞)(x). (1) Note that, for any i = 1, . . . , n, E(Xi|µ, τ) = 1 µ , Var(Xi|µ, τ) = 1 τ µ3 . 1. Compute the likelihood L(µ, τ; x1, . . . , xn) for non-negative datapoints ( x1, . . . , xn). Assume henceforth that µ is known and equal to 1 . 2. Find a conjugate prior density π (w.r.t. the Lebesgue measure on (0 , +∞)) for τ and derive the hyperparameters of the posterior distribution. (Hint: denote by (α, β) the hyperparameters of the prior density π.) 3. Assume now α = ν 2, β = ν 2 + (ν − 1)2 2 . Then fix ν in the prior density π(τ), and henceforth fix the prior hyperparameters α and β, such that a priori (marginally) Var( X1) = 7. If there are many such ν’s, choose the largest value. For failure times X1, . . . , X100 in the context of reliability of certain electronic devices under high stress condition, we assume the conditional distribution (1) (with µ = 1); summary statistics of the data are the following: 100X 1 1 xi = 857.5 100X 1 xi = 97 . 4. Under the available data, compute the hyperparameters of the posterior density π(τ |data) de- rived at point 2. with ν as determined at point 3. 5. Use a suitable limit distribution to derive a 95% highest posterior density (HPD) interval for τ. 6. By using the posterior odds, test the hypotheses H0 : τ ≤ 0.1 vs H1 : τ > 0.1. Which decision do you support? Solution of EX. 1 1. For x1, . . . , xn > 0, the likelihood is L(µ, τ; x1, . . . , xn) = nY…