Show that S = ∑ Xi is sufficient for θ a. by using equation(10.2.1), b. by the factorization criterion of equation (10.2.3). 14. Consider a random sample of size n from a uniform distribution, Xi ~ UNIF(θ, 2θ); θ > 0.
Can you find a single sufficient statistic for θ?
Can you find a pair of jointly sufficient statistics for θ?
2. Show that the following families of distributions belong to the regular exponential class, and for each case use this information to find complete sufficient statistics based on a random sample X1,…,Xn. b. POI(µ); µ > 0. d. N(µ, σ^2); – infinity < µ < infinity, σ^2 > 0. f. GAM(θ, κ); θ > 0, κ > 0. g. BETA( θ 1, θ 2); θ 1 > 0, θ 2 > 0. Bonus: Estimation in a simple linear regression with no intercept. Suppose that the random variables Y1,…,Yn satisfy: Yi = βXi + ϵi, i=1,…n. Where X1,…Xn are fixed constants, and ϵ1,…,ϵn are iid N(0, σ^2 ), where β and σ^2 are unknown. a. What is the distribution of the Yi’s? Use this distribution to write down the joint pdf and find a two-dimensional sufficient statistic for ( β , σ^2 ) b. Find the MLE of β .
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