Homework Answers
TOPIC: Maximum likelihood estimator(MLE),Unbiased estimator and MSE of an estimator.
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7. Let X1 , Xn be i.i.d. with the density p(r,0) = a*(1 - 0)1-k1{x =...
7. Let X1, · · · , Xn be i.i.d. with the density p(x, θ) = θ k (1 − θ) 1−k I{x = 0, 1} (a) Find the ML estimator of θ. (...
7. Let X1, · · · , Xn be i.i.d. with the density p(x, θ) = θ k
(1 − θ) 1−k I{x = 0, 1}
(a) Find the ML estimator of θ.
(b) Is it unbiased ?
(c) Compute its MSE
7. Let Xi, . . . , Xn be i.id, with the density p(z,0)-gk(1-0)1-k1(z-0, 1) (b) Is it unbiased? (c) Compute its MSE
7. Let Xi, . . . , Xn be i.id, with the density p(z,0)-gk(1-0)1-k1(z-0, 1)...
pr(1-0)1-k1(a 0, 1} Let Xi, . . . ,X, be i.id. with the density p(x, θ) (a) Find the ML estimator of . b) Is it unbiase...
pr(1-0)1-k1(a 0, 1} Let Xi, . . . ,X, be i.id. with the density p(x, θ) (a) Find the ML estimator of . b) Is it unbiased? (c) Compute its MSE
pr(1-0)1-k1(a 0, 1} Let Xi, . . . ,X, be i.id. with the density p(x, θ) (a) Find the ML estimator of . b) Is it unbiased? (c) Compute its MSE
Let X1,...X be i.i.d with density f()(1/0)exp(-/0) for r >0 and 0> 0. a. Find the...
Let X1,...X be i.i.d with density f()(1/0)exp(-/0) for r >0 and 0> 0. a. Find the pitman estimator of 0 b. Show that the pitman estimator has smaller risk than the UMVUE of when the loss function is (t-0)2 02 Suppose C. f(x)= 0exp(-0x) and that 0 has a gamma prior with parameters a and p, find the Bayes estimator of 0 d. Find the minimum Bayes risk e. Find the minimax estimator of 0 if one exists. 1
Let...
Hint of HW2: 4. Let X = (X1, ..., Xn) be an i.i.d. sample from the...
Hint of HW2:
4. Let X = (X1, ..., Xn) be an i.i.d. sample from the shifted exponential distri- bution with density fa,1(x) = de-1(z–a) 1(x > a), 0 := (a, 1) E O = R (0,00). Use Neyman-Fisher's theorem to: (a) show that S = X(1) is an SS for the family {fa,1}QER; (b) find an SS for the family {f1,1}x>0; (c) find an SS for the family {fa,x}o=(0,1)€0. (d) In part (a), use the procedure from Rao- Blackwell's...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function...
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p....
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known ΣΧ; is an unbiased estimator for p. that = n 2. Suggest an unbiased estimator for pa. (Hint: use the fact that the sample variance is unbiased for variance.) 3. Show that p= ΣΧ,+2 n+4 is a biased estimator for p. 4. For what values of p, MSE) is smaller than MSE)?
6. Let X1, , Xn be i.i.d. N(u,a2) (a) Find the sample analogue estimator of 0....
6. Let X1, , Xn be i.i.d. N(u,a2) (a) Find the sample analogue estimator of 0. (b) Find the ML estimator of 0
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 < x < 1, > 0 a) Find a sufficient statistic for . Is the statistic minimal sufficient? b) Find the MLE for and verify that it is a function of the statistic in a) c) Find IX() and hence give the CRLB for an unbiased estimator of . pdf means probability distribution function We were unable to transcribe this...
Suppose X1, X2, . . . , Xn (n ≥ 5) are i.i.d. Exp(µ) with the...
Suppose X1, X2, . . . , Xn (n ≥ 5) are i.i.d. Exp(µ) with the density f(x) = 1 µ e −x/µ for x > 0. (a) Let ˆµ1 = X. Show ˆµ1 is a minimum variance unbiased estimator. (b) Let ˆµ2 = (X1 +X2)/2. Show ˆµ2 is unbiased. Calculate V ar(ˆµ2). Confirm V ar(ˆµ1) < V ar(ˆµ2). Calculate the efficiency of ˆµ2 relative to ˆµ1. (c) Show X is consistent and sufficient. (d) Show ˆµ2 is not consistent...
7. Let X1, · · · , Xn be i.i.d. with the density p(x, θ) = θ k
(1 − θ) 1−k I{x = 0, 1}
(a) Find the ML estimator of θ.
(b) Is it unbiased ?
(c) Compute its MSE
7. Let Xi, . . . , Xn be i.id, with the density p(z,0)-gk(1-0)1-k1(z-0, 1) (b) Is it unbiased? (c) Compute its MSE
7. Let Xi, . . . , Xn be i.id, with the density p(z,0)-gk(1-0)1-k1(z-0, 1)...
pr(1-0)1-k1(a 0, 1} Let Xi, . . . ,X, be i.id. with the density p(x, θ) (a) Find the ML estimator of . b) Is it unbiased? (c) Compute its MSE
pr(1-0)1-k1(a 0, 1} Let Xi, . . . ,X, be i.id. with the density p(x, θ) (a) Find the ML estimator of . b) Is it unbiased? (c) Compute its MSE
Let X1,...X be i.i.d with density f()(1/0)exp(-/0) for r >0 and 0> 0. a. Find the pitman estimator of 0 b. Show that the pitman estimator has smaller risk than the UMVUE of when the loss function is (t-0)2 02 Suppose C. f(x)= 0exp(-0x) and that 0 has a gamma prior with parameters a and p, find the Bayes estimator of 0 d. Find the minimum Bayes risk e. Find the minimax estimator of 0 if one exists. 1
Let...
Hint of HW2:
4. Let X = (X1, ..., Xn) be an i.i.d. sample from the shifted exponential distri- bution with density fa,1(x) = de-1(z–a) 1(x > a), 0 := (a, 1) E O = R (0,00). Use Neyman-Fisher's theorem to: (a) show that S = X(1) is an SS for the family {fa,1}QER; (b) find an SS for the family {f1,1}x>0; (c) find an SS for the family {fa,x}o=(0,1)€0. (d) In part (a), use the procedure from Rao- Blackwell's...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known ΣΧ; is an unbiased estimator for p. that = n 2. Suggest an unbiased estimator for pa. (Hint: use the fact that the sample variance is unbiased for variance.) 3. Show that p= ΣΧ,+2 n+4 is a biased estimator for p. 4. For what values of p, MSE) is smaller than MSE)?
6. Let X1, , Xn be i.i.d. N(u,a2) (a) Find the sample analogue estimator of 0. (b) Find the ML estimator of 0
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