5102-Solution-Set-01

5102-Solution-Set-01 - SOLUTIONS FOR PROBLEM SET 1 STAT...

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Unformatted text preview: SOLUTIONS FOR PROBLEM SET 1 STAT 5102-004 Given a parametric family of distributions, for parameter value θ and observable value y , let lik( θ | y ) = f ( y | θ ) be the likelihood at θ given y . Varying the parameter describes the likelihood function. (1) For any positive θ , a continuous uniform random variable Y over (0 ,θ ) has density f ( y | θ ) = 1 θ I (0 ,θ ) ( y ) in the real line. The indicator maps to one if the inequalities 0 < y < θ < ∞ are all satisfied and zero otherwise. Thus, for any positive y and θ , I (0 ,θ ) ( y ) = I ( y, ∞ ) ( θ ) and lik( θ | y ) = 1 θ I ( y, ∞ ) ( θ ) . The following figure graphs the function. θ lik ( θ |y 29 ● ● y 2y 3y 4y 1 y (2) Densities of distributions in an exponential family have the form h ( y )exp ( yθ- κ ( θ ) ) . Parametric Free Canonical Cumulant Family Parameter Parameter Function Binomial( n,p ) p ∈ (0 , 1) θ = logit p κ ( θ ) = n log(1 + e θ ) Poisson( λ ) λ ∈ (0 , ∞ ) θ = log λ κ (...
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5102-Solution-Set-01 - SOLUTIONS FOR PROBLEM SET 1 STAT...

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