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,-
. 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-
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))
Exactly6.4-2. X1,X2,...,Xn ∼ N(μ,σ). Assumeμisknown; show that
θ=nμ)2 is the MLE for σ2 and show that it is unbiased.
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...
4. Let X1,X2, ,Xn be a randonn sample from N(μ, σ2) distribution, and let s* Ση! (Xi-X)2 and S2-n-T Ση#1 (Xi-X)2 be the estimators of σ2 (i) Show that the MSE of s is smaller than the MSE of S2 (ii) Find E [VS2] and suggest an unbiased estimator of σ.
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
4. Xi ,i = 1, , n are iid N(μ, σ2). (a) Find the MLE of μ, σ2. Are these unbiased estimators of μ and of σ2 respectively? Aside: You can use your result in (b) to justify your answer for the bias part of the MLE estimator of σ2 (b) In this part you will show, despite that the sample variance is an unbiased estimator of σ2, that the sample standard deviation is is a biased estimator of σ....
, X,' up N(μ, σ2), with σ2 known. Let μη-Xn + 5. Let Xi, of u be an estimator (a) Is ,hi an unbiased estimator for μ? (b) For a particular fixed n, find the distribution of (c) Find the mean squared error (MSE) of . (d) Prove that μη is consistent for μ
. If X1, X2,..., Xn are independent random variables with common mean μ and variances σ1, σ2, . . ., σα , prove that Σί (Xi-T)2/[n(n-1)] is an ว. 102n unbiased estimate of var[X] 3. Suppose that in Exercise 2 the variances are known. LeTw Σί uiXi