Suppose that X1, X2,....Xn ~ iid Poisson ().
Define two estamtors for
.
a) Show
b) Show the variances of the estimators. Provide the relative
efficiency (the fraction of two MSEs) of
and draw your conculsion)
Suppose that X1, X2,....Xn ~ iid Poisson (). Define two estamtors for . a) Show b)...
Suppose X1,. , Xn are iid Poisson(A) random variables. Show by direct calculation without using any theoremm in mathematical statistics, that (a) Ση! Xi/n is an unbiased estimator for λ. (b) X is optimal in MSE among all unbiased estimators. This is to say, let T be another unbiased estimator, then EA(X) EA(T2
Suppose that X1, X2, .., Xn are iid Poisson observations, each having common pdf 0 e-8 0, otherwise. Find the UMVUE of τ(0)-g2.
Suppose Y-X1-X2 where X1, x2 are iid Poisson(11) (a) Show that Y has moment generating function My (t) = e11(ette-t-2) (b) Even though you can do it from other results, use the mgf in (a) to find Var(Y).
Suppose X1 and X2 are iid Poisson(θ) random variables and let T = X1 + 2X2. (a) Find the conditional distribution of (X1,X2) given T = 7. (b) For θ = 1 and θ = 2, respectively, calculate all probabilities in the above conditional distribution and present the two conditional distributions numerically.
Suppose X1, X2, ..., Xn are independent and identically distributed (iid) with a Uniform -0,0 distri- bution for some unknown e > 0, i.e., the Xi's have pdf Suppose X1, X2,..., Xn are independent and identically distributed (iid f(3) = S 20, if –0 < x < 0; 20 0, otherwise. (a) (4 pts) Briefly explain why or why not this is an exponential family (b) (5 pts) Find one meaningful sufficient statistic for 0. (By "meaningful”, I mean it...
Solve the following two parts: (Hint: Use Complete Sufficient Statistic) Suppose X1 , X2, of λ2 and the UMVUE of (-1)(-1) Suppose X1 , X2, and UMVUE of 1/g? a. , Xn are iid Poisson distribution with mean λ. Find the UMVUE b. , Xn are iid Uniform[0, θ]. Assumen 3. Find UMVUE of θ3
Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It is well-known that X is an unbiased estimator for l because I = E(X). 1. Show that X1+Xn is also an unbiased estimator for \. 2 2. Show that S2 (Xi-X) = is also an unbaised esimator for \. n-1 3. Find MSE(S2). (We will need two facts) E com/questions/2476527/variance-of-sample-variance) 2. Fact 2: For Poisson distribution, E[(X – u)4] 312 + 1. (See for...
Let X1,X2,...,Xn be iid exponential random variables with unknown mean β. (b) Find the maximum likelihood estimator of β. (c) Determine whether the maximum likelihood estimator is unbiased for β. (d) Find the mean squared error of the maximum likelihood estimator of β. (e) Find the Cramer-Rao lower bound for the variances of unbiased estimators of β. (f) What is the UMVUE (uniformly minimum variance unbiased estimator) of β? What is your reason? (g) Determine the asymptotic distribution of the...
= e B and cumu- Let X1, X2, ..., Xn be a random sample where X; has a probability density function f(x) lative density function F(x) = 1 - e B. Consider the following two estimators of B: х Bi = X1 B2 = (a) [2 points] Compute the relative efficiency of @1 to ộ2. (b) [1 point] Interpret the relative efficiency of ß1 to ß2 for a sample of size n = - 30.
Suppose X1, X2, . . . , Xn are a random sample from a Uniform(0, θ) distribution, where θ > 0. Consider two different estimators of θ: R1 = 2X¯ R2 =(n + 1)/n max(X1, . . . , Xn) (a) For each of the estimators R1 and R2, assess whether it is an unbiased estimator of θ. (b) Compute the variances of R1 and R2. Under what conditions will R2 have a smaller variance than R1?