6. Suppose we have i.id. Xi, , Xn ~ N(μ, σ2). In the class, we learned that Σί i m(Xi-X) X2-1. Use this fact and answer the following questions. (a) Consider an estimator σ-c Σηι (Xi-X)2. Find its mean and variance.
5.13. Suppose X1, X2, , xn are iid N(μ, σ2), where-oo < μ < 00 and σ2 > 0. (a) Consider the statistic cS2, where c is a constant and S2 is the usual sample variance (denominator -n-1). Find the value of c that minimizes 2112 var(cS2 (b) Consider the normal subfamily where σ2-112, where μ > 0. Let S denote the sample standard deviation. Find a linear combination cl O2 , whose expectation is equal to μ. Find the...
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...
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
is taken from N(μ, σ2), where the mean 2. A randorn sample X1, X2, , xn of size μ is a known real num ber. Show that the m axim urn likelihood estimator for σ2 is ớmle n Σ.i(Xi μ)2 and that this estimator is an unbiased estinator of σ2. (I lint: Σ.JX _ μ)-g. Σ.i My L and Σ. (Xcpl, follows X2(n))
X1, X2, . . . , Xn i.i.d. ∼ N (µ, σ2 ). Assume µ is known; show that ˆθ = 1 n Pn i=1(Xi− µ) 2 is the MLE for σ 2 and show that it is unbiased. Exactly 6.4-2. Xi, X2, . . . , xn i d. N(μ, μ)2 is the MLE for σ2 and show that it is unbiased. r'). Assume μ is known; show that θ- n Ση! (X,-
Suppose that X1, ..., Xn is a random sample from a normal distribution with mean μ and variance σ2. Two unbiased estimators of σ2 are 1?n 1 i=1 σˆ12 =S2 = n−1 Find the efficiency of σˆ12 relative to σˆ2. (Xi −X̄)2, and σˆ2= 2(X1 −X2)2
σ2). 6. Suppose X1, Yİ, X2, Y2, , Xn, Y, are independent rv's with Xi and Y both N(μ, All parameters μί, 1-1, ,n, and σ2 are unknown. For example, Xi and Yi muay be repeated measurements on a laboratory specimen from the ith individual, with μί representing the amount of some antigen in the specimen; the measuring instrument is inaccurate, with normally distributed errors with constant variability. Let Z, X/V2. (a) Consider the estimate σ2- (b) Show that 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).