4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distr...
Consider a random sample (X1, Y1),(X2, Y2), . . . ,(Xn, Yn) where Y | X = x is modeled by a N(β0 + βx, σ2 ) distribution, where β0, β1 and σ 2 are unknown. (a) Prove that the mle of β1 is an unbiased estimator of β1. (b) Prove that the mle of β0 is an unbiased estimator of β0.
Question 3 15 marks] Let X1,..,X be independent identically distributed random variables with pdf common ) = { (#)%2-1/64 0 fx (a;e) 0 where 0 >0 is an unknown parameter X-1. Show that Y ~ T (}, ); (a) Let Y (b) Show that 1 T n =1 is an unbiased estimator of 0-1 ewhere / (0; X) is the log- likeliho od function; (c) Compute U - (d) What functions T (0) have unbiased estimators that attain the relevant...
Suppose that the random variables Y Y, satisfy where xi, ,Xn are fixed constants and Ei, ,En are îid N(0, σ2), where σ2 is a fixed constant. (a) What distribution do Yi,.., Yn follow? What is your reasoning? (b) Find the MLEs for α and β.
2 Let X1, X2, ...,X, be independent continuous random variables from the following distribution: f(3) = ox-(0-1) where : > 1 and a > 1 You may use the fact: E[X]- .- 2.1 Show that the maximum likelihood estimator of a is ômle = Ei log Xi 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency? 2.4 Show that the fisher information in the whole sample is: 1(a)= 2.5 What Cramer Rao lower bound...
8.7-11. Let Y1,Y2, ...,Yn be n independent random variables with normal distributions N(Bx;,02), where X],x2,...,xn are known and not all equal and B and 2 are unknown parameters (a) Find the likelihood ratio test for Ho: B = 0 against H: B+0. (b) Can this test be based on a statistic with a well-known distribution?
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...