![Let X, , x, be a random sample from some density which has mean μ and variance σ2. Show that Σ a, X, is an unbiased estimator of/e for any set of known constants a, , . . . , a, satisfying Σ a,-1. If Σ a.-1, show that var [ Σ a, xl] is minimized for ai = 1/n, i = 1, [HINT: Prove that Σ a-Σ (al-IMF + 1/n when Σ al = 1 .] (a) (b) , n. rt](http://img.homeworklib.com/questions/9ff46350-9a70-11eb-bab1-29f11a45735d.png?x-oss-process=image/resize,w_560)
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Let X, , x, be a random sample from some density which has mean μ and...
Let X,, X,,...X be a random sample of size n from a normal distribution with parameters a. Derive the Cramer-Rao lower...
Let X,, X,,...X be a random sample of size n from a normal distribution with parameters a. Derive the Cramer-Rao lower bound matrix for an unbiased estimator of the vector of parameters (μ, σ2). b. Using the Cramer-Rao lower bound prove that the sample mean X is the minimum variance unbiased estimator of u Is the maximum likelihood estimator of σ--σ-->|··( X,-X ) unbiased? c.
Let X,, X,,...X be a random sample of size n from a normal distribution with...
x, and S1 are the sample mean and sample variance from a population with mean μ|...
x, and S1 are the sample mean and sample variance from a population with mean μ| and variance ơf. Similarly, X2 and S1 are the sample mean and sample variance from a second population with mean μ and variance σ2. Assume that these two populations are independent, and the sample sizes from each population are n,and n2, respectively. (a) Show that X1-X2 is an unbiased estimator of μ1-μ2. (b) Find the standard error of X, -X. How could you estimate...
(4 points) Let Xi, , Xn denote a randon sample from a Normal N(μ, 1) distribution, with 11 as the...
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 Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ...
1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n We use V, to estimate λ. (a) Show that is an unbiased estimator for λ. (b) Let ơin be the variance of V,, . Show that lin ơi,-
1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n...
please answer the questions easily Suppose X1, X2, X3 is a random sample from a normal...
please answer the questions easily
Suppose X1, X2, X3 is a random sample from a normal population with mean μ and variance (a) I,'ind i.he variallex, of Y , x..:.: Xy/X.t as an ( tinai." r of μ (b) Find the variance of Z-A+x2+x3 as an estimator of μ. (c) Which estimator is more efficient (i.e. has the smallest variance)? Consider a random sample of size n from a normal population with known mean μ and unknown variance σ2. Let...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution,...
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...
Let X,,X.X be a random sample of size n from a random variable with mean and variance given by (μ, σ2) a Show that the...
Let X,,X.X be a random sample of size n from a random variable with mean and variance given by (μ, σ2) a Show that the sample meanX is a consistent estimator of mean 1(X-X)2 converges in probability Show that the sample variance of ơ2-02- b. 1n to Ơ2 . Clearly state any theorems or results you may have used in this proof.
Let X,,X.X be a random sample of size n from a random variable with mean and variance given...
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and...
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and variance σ , with known μ and unknown σ' . You are given that Σ(X-μ)2 is sufficient for σ a) Find El Σ(X-μ). |. Show all steps. Use the fact that: Var(Y)-E(P)-(BY)' i-1 b) Find the MVUE of σ.
4(25 points) Let X be a random variable with mean μ = E(X) and σ2 V(X). Let X = n Σ_1Xī be X2 + X...
4(25 points) Let X be a random variable with mean μ = E(X) and σ2 V(X). Let X = n Σ_1Xī be X2 + Xs) be the average of the the sample mean from a random sample (X X. Let X (X first three observations. (a) Prove that X is an unbiased estimator for μ. Prove that X is also an unbiased estimator for μ. (b) Explain that X is a consistent estimator for μ. Explain why X is not...
4. Let ,, , xn be independent and suppose that E(X.) k,0 + bi, for known...
4. Let ,, , xn be independent and suppose that E(X.) k,0 + bi, for known constants ki and bi, and Var(X) = σ2, i 1, , n. (a) Find the least squares estimator θ of θ. (b) Show that θ is unbiased. c) Show that the variance of θ is Var(8)-: T (e) Show that the variance of is Var() (d) Show that Tn Σ(x,-ke-W2 = Σ(x,-k9-b)2 + Σ ka@ー0)2 i-1 -1 ー1 (e) Hence show that Ti 121
Let X,, X,,...X be a random sample of size n from a normal distribution with parameters a. Derive the Cramer-Rao lower bound matrix for an unbiased estimator of the vector of parameters (μ, σ2). b. Using the Cramer-Rao lower bound prove that the sample mean X is the minimum variance unbiased estimator of u Is the maximum likelihood estimator of σ--σ-->|··( X,-X ) unbiased? c.
Let X,, X,,...X be a random sample of size n from a normal distribution with...
x, and S1 are the sample mean and sample variance from a population with mean μ| and variance ơf. Similarly, X2 and S1 are the sample mean and sample variance from a second population with mean μ and variance σ2. Assume that these two populations are independent, and the sample sizes from each population are n,and n2, respectively. (a) Show that X1-X2 is an unbiased estimator of μ1-μ2. (b) Find the standard error of X, -X. How could you estimate...
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 Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n We use V, to estimate λ. (a) Show that is an unbiased estimator for λ. (b) Let ơin be the variance of V,, . Show that lin ơi,-
1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n...
please answer the questions easily
Suppose X1, X2, X3 is a random sample from a normal population with mean μ and variance (a) I,'ind i.he variallex, of Y , x..:.: Xy/X.t as an ( tinai." r of μ (b) Find the variance of Z-A+x2+x3 as an estimator of μ. (c) Which estimator is more efficient (i.e. has the smallest variance)? Consider a random sample of size n from a normal population with known mean μ and unknown variance σ2. Let...
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...
Let X,,X.X be a random sample of size n from a random variable with mean and variance given by (μ, σ2) a Show that the sample meanX is a consistent estimator of mean 1(X-X)2 converges in probability Show that the sample variance of ơ2-02- b. 1n to Ơ2 . Clearly state any theorems or results you may have used in this proof.
Let X,,X.X be a random sample of size n from a random variable with mean and variance given...
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and variance σ , with known μ and unknown σ' . You are given that Σ(X-μ)2 is sufficient for σ a) Find El Σ(X-μ). |. Show all steps. Use the fact that: Var(Y)-E(P)-(BY)' i-1 b) Find the MVUE of σ.
4(25 points) Let X be a random variable with mean μ = E(X) and σ2 V(X). Let X = n Σ_1Xī be X2 + Xs) be the average of the the sample mean from a random sample (X X. Let X (X first three observations. (a) Prove that X is an unbiased estimator for μ. Prove that X is also an unbiased estimator for μ. (b) Explain that X is a consistent estimator for μ. Explain why X is not...
4. Let ,, , xn be independent and suppose that E(X.) k,0 + bi, for known constants ki and bi, and Var(X) = σ2, i 1, , n. (a) Find the least squares estimator θ of θ. (b) Show that θ is unbiased. c) Show that the variance of θ is Var(8)-: T (e) Show that the variance of is Var() (d) Show that Tn Σ(x,-ke-W2 = Σ(x,-k9-b)2 + Σ ka@ー0)2 i-1 -1 ー1 (e) Hence show that Ti 121
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