5.13. Suppose X1, X2, , xn are iid N(μ, σ2), where-oo < μ < 00 and...
Suppose that X1,X2, ,Xn are iid N(μ, σ2), where both parameters are unknown. Derive the likelihood ratio test (LRT) of Ho : σ2 < σ1 versus Ho : σ2 > σ.. (a) Argue that a LRT will reject Ho when w(x)S2 2 0 is large and find the critical value to confer a size α test. (b) Derive the power function of the LRT
5. Let X1, X2, . .. , Xn be independently distributed as N(μ, σ2). Define 7t n-1 ー1 Users/rumi3/Downloads/Linear-Regression-Analysis-Seber.pdf MOMENT GENERATING FUNCTIONS AND INDEPENDENCE 13 and n-1 2(n-1 i=1 (a) Prove that var[S2-2c4/(n-1). b) Show that Q is an unbiased estimate ofa (c) Find the variance of Q and hence show that as n → oo, the effi- ciency of Q relative to S2 is
. Xi , X2, . . . , xn iid N(μ, σ2). Assume μ is known; show that θ Exactly 6.4-2 A)2 is the MLE for σ2 and show that it is unbiased. -ni(x-
Let X = (X1, . . . , Xn) be a random sample of size n with mean μ and variance σ2. Consider Tm i=1 (a) Find the bias of μη(X) for μ. Also find the bias of S2 and ỡXX) for σ2. (b) Show that Hm(X) is consistent. (c) Suppose EIXI < oo. Show that S2 and ỡXX) are consistent. Let X = (X1, . . . , Xn) be a random sample of size n with mean μ...
4. Suppose Yi, Yn are iid randonn variables with E(X) = μ, Var(y)-σ2 < oo. For large n, find the approximate distribution of p = n Σηι Yi, Be sure to name any theorems you used.
Suppose Xi, X2, ,Xn is an iid N(μ, c2μ2 sample, where c2 is known. Let μ and μ denote the method of moments and maximum likelihood estimators of μ, respectively. (a) Show that ~ X and μ where ma = n-1 Σηι X? is the second sample (uncentered) moment. (b) Prove that both estimators μ and μ are consistent estimators. (c) Show that v n(μ-μ)-> N(0, σ ) and yM(^-μ)-+ N(0, σ ). Calculate σ and σ . Which estimator...
please answer with full soultion. with explantion. (4 points) Let Xi, , Xn denote a randon sample from a Normal N(μ, 1) distribution, with 11 as the unknown parameter. Let X denote the sample mean. (Note that the mean and the variance of a normal N(μ, σ2) distribution is μ and σ2, respectively.) Is X2 an unbiased estimator for 112? Explain your answer. (Hint: Recall the fornula E(X2) (E(X)Var(X) and apply this formula for X - be careful on the...
1. Let X1, X2, , Xn be independent Normal μ, σ2) random variables. Let y,-n Σ_lx, denote a sequence of random variables (a) Find E(y,) and Var(y,) for all n in terms of μ and σ2. (b) Find the PDF for Yn for alln. (c) Find the MGF for Yn for all n.
3. Let X1, X2, ,Xn be a random sample from N(μ, σ2), and k be a positive integer. Find E(S2). In particular, find E(S2) and var(s2).
Exercice 5. Let Xi, ,Xn be iid normal randon variables : Xi ~ N(μ, σ2). We denote 4 Tl Show that (İ) ils2 (i.e., that x is independent of 82). (ii) x ~ N(μ, σ2/n). (iii) !뷰 ~ เลี้-1