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(3) Suppose that E (0,) θ, Ε(92) θ,V(4) of, and V(92)-σ . Assume that 0, and...
(3) Suppose that E (0,) θ, Ε(92) θ,V(91) of, and V(02) σ . Assume that 0,...
(3) Suppose that E (0,) θ, Ε(92) θ,V(91) of, and V(02) σ . Assume that 0, and θ2 are independent. Consider the following estimator: 6, - a+(1 -@ (a) Show that @g is unbiased for θ (b) Find the value of a that minimizes the variance of 03 (c) Which estimator would you use? θί,02, or th when using the value of a found in part (b)
(3) Suppose that E(4) θ, E(4) θ, V(4) σ. and V(0) σ3. Assume that θί and...
(3) Suppose that E(4) θ, E(4) θ, V(4) σ. and V(0) σ3. Assume that θί and 02 are independent. Consider the following estimator: (a) Show that θ3 is unbiased for θ (b) Find the value of a that minimizes the variance of θ3 (c) which estimator would you use? θ, θ2, or 6, when using the value of a found in part (b)
. Suppose that 6, and o2 are both unbiased estimators of e. a) b) e) Show...
. Suppose that 6, and o2 are both unbiased estimators of e. a) b) e) Show that theestimator θ t914(1-t)a, is also an unbiased estimator of θ for any value of the constant t. Suppose V[6]:ơİ and v[62] of. Ifa, anda,are independent, find an expression for V[d] in terms of t, σ' and σ Find the value of t that produces an estimator of the form 6 ะเอิ,+(1 that has the smallest possible variance. (Your final answer will be in...
Suppose that X1, X2, ,Xn is an iid sample from Íx (x10), where θ Ε Θ....
Suppose that X1, X2, ,Xn is an iid sample from Íx (x10), where θ Ε Θ. In each case below, find (i) the method of moments estimator of θ, (ii) the maximum likelihood estimator of θ, and (iii) the uniformly minimum variance unbiased estimator (UMVUE) of T(9) 0. exp fx (x10) 1(0 < x < 20), Θ-10 : θ 0}, τ(0) arbitrary, differentiable 20 (d) n-1 (sample size of n-1 only) ー29 In part (d), comment on whether the UMVUE...
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
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...
5. Suppose X and Y are random variables such that E(X)=E(Y) = θ, Var(X) = σ...
5. Suppose X and Y are random variables such that E(X)=E(Y) = θ, Var(X) = σ and Var(Y)-吆 . Consider a new random variable W = aX + (1-a)Y (a) Show that W is unbiased for θ. (b) If X and Y are independent, how should the constant a be chosen in order to minimize the variance of W?
S -E θ2 θ, Var | θί σ. Var 021-σ3. and Cov | θι.02 ov 61,02...
S -E θ2 θ, Var | θί σ. Var 021-σ3. and Cov | θι.02 ov 61,02 σ12. Consider the uppose that E [1 unbiased estimator What value should be chosen for the constant a in order to minimize the variance and thus mean squared error of 03 as an estimator of θ? Note: The second derivative of the variance function is positive, which you can figure out by knowing that the correlation coefficient ρ- 012 is between-1 and 1; however,...
Suppose that Xi, X2, ..., Xn is an iid sample from where θ > 0. (a)...
Suppose that Xi, X2, ..., Xn is an iid sample from where θ > 0. (a) Show that is a complete and sufficient statistic for σ (b) Prove that Y1-X11 follows an exponential distribution with mean σ (c) Find the uniformly minimum variance unbiased estimator (UMVUE) of T(o-o", where r is a fixed constant larger than 0.
Suppose thatX1,...Xn are IID with pdf f(x;θ) = 1 /2θ if -θ<x<θ otherwise =0 (a) Find...
Suppose thatX1,...Xn are IID with pdf f(x;θ) = 1 /2θ if -θ<x<θ otherwise =0 (a) Find an unbiased estimator of θ. You must prove that your estimator is unbiased. (b) Find the variance of the estimator in (a).
(3) Suppose that E (0,) θ, Ε(92) θ,V(91) of, and V(02) σ . Assume that 0, and θ2 are independent. Consider the following estimator: 6, - a+(1 -@ (a) Show that @g is unbiased for θ (b) Find the value of a that minimizes the variance of 03 (c) Which estimator would you use? θί,02, or th when using the value of a found in part (b)
(3) Suppose that E(4) θ, E(4) θ, V(4) σ. and V(0) σ3. Assume that θί and 02 are independent. Consider the following estimator: (a) Show that θ3 is unbiased for θ (b) Find the value of a that minimizes the variance of θ3 (c) which estimator would you use? θ, θ2, or 6, when using the value of a found in part (b)
. Suppose that 6, and o2 are both unbiased estimators of e. a) b) e) Show that theestimator θ t914(1-t)a, is also an unbiased estimator of θ for any value of the constant t. Suppose V[6]:ơİ and v[62] of. Ifa, anda,are independent, find an expression for V[d] in terms of t, σ' and σ Find the value of t that produces an estimator of the form 6 ะเอิ,+(1 that has the smallest possible variance. (Your final answer will be in...
Suppose that X1, X2, ,Xn is an iid sample from Íx (x10), where θ Ε Θ. In each case below, find (i) the method of moments estimator of θ, (ii) the maximum likelihood estimator of θ, and (iii) the uniformly minimum variance unbiased estimator (UMVUE) of T(9) 0. exp fx (x10) 1(0 < x < 20), Θ-10 : θ 0}, τ(0) arbitrary, differentiable 20 (d) n-1 (sample size of n-1 only) ー29 In part (d), comment on whether the UMVUE...
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
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...
5. Suppose X and Y are random variables such that E(X)=E(Y) = θ, Var(X) = σ and Var(Y)-吆 . Consider a new random variable W = aX + (1-a)Y (a) Show that W is unbiased for θ. (b) If X and Y are independent, how should the constant a be chosen in order to minimize the variance of W?
S -E θ2 θ, Var | θί σ. Var 021-σ3. and Cov | θι.02 ov 61,02 σ12. Consider the uppose that E [1 unbiased estimator What value should be chosen for the constant a in order to minimize the variance and thus mean squared error of 03 as an estimator of θ? Note: The second derivative of the variance function is positive, which you can figure out by knowing that the correlation coefficient ρ- 012 is between-1 and 1; however,...
Suppose that Xi, X2, ..., Xn is an iid sample from where θ > 0. (a) Show that is a complete and sufficient statistic for σ (b) Prove that Y1-X11 follows an exponential distribution with mean σ (c) Find the uniformly minimum variance unbiased estimator (UMVUE) of T(o-o", where r is a fixed constant larger than 0.
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