2. Let X 1, , Xn be iid from the distribution modeled by 8-2 fx (1:0)-(9....
Suppose X1, X2, ..., Xn is an iid sample from fx(r ja-θ(1-z)0-11(0 1), where x θ>0. (a) Find the method of moments (MOM) estimator of θ. (b) Find the maximum likelihood estimator (MLE) of θ (c) Find the MLE of Po(X 1/2) d) Is there a function of θ, say T 0), for which there exists an unbiased estimator whose variance attains the Cramér-Rao Lower Bound? If so, find it and identify the corresponding estimator. If not, show why not.
5. Find a method-of-moments estimator (MME) of θ based on a randorn sample Xi, ,Xn from each of the following distributions 040<1 (b) f(r:0)-(0 + 1)re-2,T > 1, θ > 0
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1 +0)-r(0-1) (1-2), 0 < x < 1, θ > 0. the estimator T-4 is a method of moments estimator for θ. It can be shown that the asymptotic distribution of T is Normal with ETT θ and Var(T) 0042)2 Apply the integral transform method (provide an equation that should be solved to obtain random observations from the distribution) to generate a sam ple of...
Let X be a random variable with cdf FX (x:0), expected value EIX-μ and variance VlX- σ2. Let X1,X2, , Xn be an id sample drawn according to FX(x,8) where Fx (x,8) =万 for all x E (0,0). Let max(X1, X2, , X.) be an estimator of θ, suggested from pure common sense. Remember that if Y = max(X1, X2, , Xn). Then it can be shown that the cdf Fy () of Y is given by Fr(u) (Fx()" where...
Suppose that Xi, X2,., Xn is an iid sample from (1- 0) In 0 0, X(T 0, herwise, where the parameter θ satisfies 0 θ 1. (a) Estimate θ using the method of moments (MOM) and using the method of maximum likelihood. Note: I am not sure if you can get closed form expressions for either estimator, but that is OK. Just write out the equation(s) that would need to be solved (numerically) to
, xn is an iid sample from fx(x10)-θe-8z1(x > 0), where θ > 0. Suppose X1, X2, For n 2 2, n- is the uniformly minimum variance unbiased estimator (UMVUE) of 0 (d) For this part only, suppose that n-1. If T(Xi) is an unbiased estimator of e, show that Pe(T(X) 0)>0
Suppose that X1, X2, ,Xn is an iid sample from Íx (x10), where θ Ε Θ. In each case below, find (i) the method of moments estimator of θ, (ii) the maximum likelihood estimator of θ, and (iii) the uniformly minimum variance unbiased estimator (UMVUE) of T(9) 0. exp fx (x10) 1(0 < x < 20), Θ-10 : θ 0}, τ(0) arbitrary, differentiable 20 (d) n-1 (sample size of n-1 only) ー29 In part (d), comment on whether the UMVUE...
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ, ie. I. What is the Maximum Likelihood estimate θ of θ? 2. Show that the maximum likelihood estimator of θ is unbiased. 3. We're looking to cstimate the variance θ (1-9) of Xi . x being the empirical average 2(1-2). Check that T is not unli ator propose an unbiased estimator of θ(1-0).
2. Let Xi,... ,Xn be a random sample from a distribution with p.d.f for 0 < x < θ f(x; 0) - 0 elsewhere . (a) Find an estimator for θ using the method of moments. (b) Find the variance of your estimator in (a).
Suppose X1, X2, , Xn is an iid sample from a uniform distribution over (θ, θΗθ!), where (a) Find the method of moments estimator of θ (b) Find the maximum likelihood estimator (MLE) of θ. (c) Is the MLE of θ a consistent estimator of θ? Explain.