20 06 2023 E TS

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BAYESIAN STATISTICS A. Guglielmi & M. Beraha 20.06.2023 Properly justify all your answers. Use the indicator function to denote the support of a distr. . Exercise 1 The values below are an observed sample from lifetimes (expressed in years) of the same type of coffee machine, called City, in the Department of Mathematics: 2.40 3 .10 4 .03 1 .41 3 .39 5 .19 1 .16 2 .01 6 .02 3 .15 The failure rate of the City coffee machine is investigated by a group of PhD students, who consider the following conditional distribution X1, . . . , Xn | θ i.i.d ∼ Exp(θ), θ > 0, (1) where Xi is the lifetime (in years) of the i-th City coffee machine in the Department. 1. Verify that, in this case, the failure rate of Xi under (1) is constant and it is a function of θ, which we denote here by τ(θ). 2. Derive the Jeffreys’ prior for the failure rate τ(θ), and the corresponding posterior distribution, with the observed data. Compute a point estimate of τ(θ) under this posterior. 3. Find a conjugate prior density πC for θ under model (1), explicitly deriving the posterior hyperparameters. Specifically, verify that the conjugate prior can be characterized by two hyperparameters, the shape and the rate, such that E πC(θ) = shape/rate. Then compute the posterior mean of θ and interpret both hyperparameters under the equivalent sample approach. 4. Derive the prior marginal density of X1 for the model (1) under the conjugate prior πC found at point 3. Moreover, compute the associated quantile function, that is the function QX1(p) = F −1 X1 (p) per p ∈ (0, 1), where FX1 is the prior marginal distribution function of X1 and F −1 X1 is its inverse function. 5. To fix hyperparameters in πC assume an empirical Bayes procedure as follows: fix the shape hyperparam- eters equal to 5, and set the rate such…