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.
Exactly6.4-2. X1,X2,...,Xn ∼ N(μ,σ). Assumeμisknown; show that θ=nμ)2 is the MLE for σ2 and show that...
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))
Let X1, X2, · · · , Xn ∼ N(µ, σ2 ). Show that the MLE of µ is efficient.
Exactly 6.4-2. Χι , X2, . . . , Xn i d N(μ, σ2). A p)2 is the MLE for σ2 and show that it is unbiased. t-ri 1 (Xi ssume μ is known: show tha
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).
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...
2. Suppose that X1, X2, . . . , Xn are iid. N(0, σ) with density function f (xlo) Find the Fisher information I(o) a. b. Now, call: σ2 your parameter, with this new parametrization, f(x19)-E-e-28 Find the Fisher information 1(8) 1(ог). Is 1(σ*)-1 (σ)? c. Find o2MOM d. Find σ2MLE e. Find Elo-MLE]. Show that σ2MLEls unbiased f. Find Var[σ 2MLEİ. Does σ2MLE attain the CRLB?
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown. Let ρ-r/of and g m/n, and consider the problem of unbiased estimation of u In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown....