Let X1,…, Xn be a sample of iid Bin(1, ?) random variables, and let T = X(1 − X) be an estimator of Var(Xi ) = ?(1 − ?). Determine
E(T).
Bias(T; ?(1 − ?)).
Let X1,…, Xn be a sample of iid Bin(1, ?) random variables, and let T =...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) = P(X1 = -1) = 1/2. Let Sn = X1 + X2 + ... + xn. (a) Find ES, and var(Sn); (b) Show that Sn is a martingale.
1. Let X1,... , Xn be IID random points from Exp(1/B). The PDF of Exp(1/B) is for x 〉 0. Let X,-1 Σー X, be the sample average. Let 3 be the parameter of interest that we want to estimate. Xi be the sample average. Let B be the parameter of (a) (1 pt) What is the bias and variance of using the sample average Xn as the estimator of 3? (b) (0.5 pt) What is the mean square error...
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
Let Xi,..., Xn be iid random variables with distribution Bern(p) (a) Is the statistic 름 Σ. ? (b) Is the statistic (Σ¡X 2? Xi an unbiased estimator of p i) an unbiased estimator of p Let Xi,..., Xn be iid random variables with distribution Bern(p) (a) Is the statistic 름 Σ. ? (b) Is the statistic (Σ¡X 2? Xi an unbiased estimator of p i) an unbiased estimator of p
2. Let X1, X2,. . , Xn denote independent and identically distributed random variables with variance σ2, which of the following is sufficient to conclude that the estimator T f(Xi, , Xn) of a parameter 6 is consistent (fully justify your answer): (a) Var(T) (b) E(T) (n-1) and Var(T) (c) E(T) 6. (d) E(T) θ and Var(T)-g2. 72 121
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
Let X1,…, Xn be a sample of iid random variable with pdf f (x; ?) = 1/(2x−?+1) on S = {?, ? + 1, ? + 2,…} with Θ = ℕ. Determine a) a sufficient statistic for ?. b) F(1)(x). c) f(1)(x). d) E[X(1)].
Let Xi, .Χίο be a sample of iid Bin( 1, θ) random variables, and let e-{i : Σοί HA : θ 0.8. Determine a) the size of this critical region. b) the power of this critical region for 0 0.8. x,2 9} be a critical region for testing Ho:0 0.6 versus Let Xi, .Χίο be a sample of iid Bin( 1, θ) random variables, and let e-{i : Σοί HA : θ 0.8. Determine a) the size of this critical...
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
Let {Xn} be a sequence of iid random variables 1. (20 points) Let {Xn} be a sequence of iid random variables with common pdf f(x) = - =e-x2/2,x ER. Then find the limit in probability of the sequence of random variables {Y} where Yo: 31x11. i=1