09 01 2024 E TS LAB

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BAYESIAN STATISTICS A. Guglielmi & M. Gianella 09.01.2024 Properly justify all your answers. Use the indicator function to denote the support of a distr. . Exercise 1 The strain resistance of a new material, Titanium and Nickel (TiNi), is tested in a laboratory. Different experimental units of TiNi are strained until they break. Let Ti be the failure time of unit i, expressed in hours, i.e., the time until the unit breaks, for i = 1, . . . , n. The lab staff asks for your assistance in modeling and analyzing the observed failure times as an expert in Bayesian statistics. The lab staff believes the new material improves resistance as time increases. Hence, you come up with the following distribution for Ti: Ti | α = (Wi)1/α = α p Wi, with α > 0, where Wi | θ iid ∼ E 1 θ  , i = 1, . . . , n, with θ > 0. Here E 1 θ  denotes the exponential distribution with mean θ. 1. Verify that the conditional density of Ti, given (α, θ), is fTi(t | α, θ) = α θ tα−1 exp  − tα θ  I(0,+∞)(t), (1) i.e., Ti | (α, θ) is distributed according to the Weibull density with parameters ( α, θ); here we write Ti | (α, θ) iid ∼ W eib(α, θ), i = 1, . . . , n. 2. Compute, for any Ti ∼ W eib(α, θ), its k-th moment E[T k i | α, θ] for any positive integer k. Then derive the conditional mean E[Ti | α, θ] and variance Var[Ti | α, θ]. (Hint: the change-of-variable y = tα might be useful.) 3. Compute the associated conditional survival function STi(t | α, θ) = P(Ti > t | α, θ). You fix α = 0.5, consistent with prior information, though you assume θ random. 4. Compute the likelihood L(θ; t1, . . . , tn). Then find a conjugate prior distribution π(θ), depending on two hyperparameters ( a, b) such that Eπ[θ] = b/(a − 1). Remember to write down the range of values of prior hyperparameters (a, b) yielding…