4. Let X1,X2, ,Xn be a randonn sample from N(μ, σ2) distribution, and let s* Ση!...
4. Let X1,X2, x 2) distribution, and let sr_ Ση:1 (Xi-X)2 and S2 n-l Σηι (Xi-X)2 be the estimators of σ2. (i) Show that the MSE of S" is smaller than the MSE of S2 (ii) Find ElvS2] and suggest an unbiased estimator of σ. n be a random sample from N (μ, σ
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
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
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...
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).
, 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
1. Let X1, . . . , Xn be a sample of size n from a distribution with expectation μ (2X1 + X2 + . . . + Xn-1 + 2Xn)/(n+1)l be an estimator and variance σ . and let μ- for μ. Is it unbiased? asymptotically unbiased? consistent?
8. Let X, X2, , xn all be be distributed Normal(μ, σ2). Let X1, X2, , xn be mu- tually independent. a) Find the distribution of U-Σǐ! Xi for positive integer m < n b) Find the distribution of Z2 where Z = M Hint: Can the solution from problem #2 be applied here for specific values of a and b?