![[20 marks] Let xi, . . . , Xn be a random sample drawn independently from a one-parameter curved normal distribution which ha](http://img.homeworklib.com/images/2502db47-ee5b-410b-a185-3cd02a464a3a.png?x-oss-process=image/resize,w_560)
![(d) [3 marks] Find the maximum likelihood estimator θ2 of. (You do not need to perform the second derivative test.) (e) 3 mar](http://img.homeworklib.com/images/f47544bb-c820-4add-86f0-e5af60434602.png?x-oss-process=image/resize,w_560)
(d) [3 marks] Find the maximum likelihood estimator θ2 of. (You do not need to perform the second derivative test.) (e) 3 marks Find the Fisher information T( (f) [3 marks] Is θ2 an MVUE of θ? Justify your answer.
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[20 marks] Let xi, . . . , Xn be a random sample drawn independently from a one-parameter curved normal distribution which has density -oo 〈 x 〈 oo, θ > 0, 2πθ nx, and r2 - enote T-1 Tn (d) [3...
2. 20 marks] Let z1,., xn be a random sample drawn independently from a one-parameter curved normal distribution which...
2. 20 marks] Let z1,., xn be a random sample drawn independently from a one-parameter curved normal distribution which has density -oo < x < 00, θ>0, , riid i.e., X r, and 2,2-1 Γη (e) 3 marks Find the Fisher information Z(0) (f) [3 marks] Is θ2 an MVUE of θ? Justify your answer (g) 3 marks] Assume that T = 1.32 and x-3.76 for a random sample of size n = 100. Find the Wald 95% confidence interval...
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ,...
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ, ie. I. What is the Maximum Likelihood estimate θ of θ? 2. Show that the maximum likelihood estimator of θ is unbiased. 3. We're looking to cstimate the variance θ (1-9) of Xi . x being the empirical average 2(1-2). Check that T is not unli ator propose an unbiased estimator of θ(1-0).
3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have th...
3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have the joint probability function where x = (zi, 2 2:23), θ = (θί, θ2), n = x1 + 2 2 + x3, θι, θ2 > 0 and 1-0,-26, > 0. (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks] Find that the Fisher information matrix I(0) (c) [4 marks] Show that θ is an MVUE. (d)...
3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have the joint probability function where x = (zi, 2 2:23), θ = (θί, θ2), n = x1 + 2 2...
3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have the joint probability function where x = (zi, 2 2:23), θ = (θί, θ2), n = x1 + 2 2 + x3, θι, θ2 > 0 and 1-0,-26, > 0. (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks] Find that the Fisher information matrix I(0) (c) [4 marks] Show that θ is an MVUE. (d)...
A (3 pt) Let Xi, ,X, are drawn from the distribution ftheta(z) = F 404 (r+0) , for 0 < x < oo and...
A (3 pt) Let Xi, ,X, are drawn from the distribution ftheta(z) = F 404 (r+0) , for 0 < x < oo and 0 < θ < oo. We define Y = 3X an estimator for θ. Verify whether this estimator is unbiased? Find the MSE of Y. Hint: E(x)E(X B (3 pt) Let X,.., X, are drawn from the distribution fo) for O < x < 00 and 0 < θ < oo. We define Y = 2X...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
Problem 4 Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ...
Problem 4 Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ (0,1), Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b. Find he Maximum likelihood estimator of θ? Is it asymptotically normal in this sample? Find the asymptotic variance Consider the following hypotheses: H0: X is...
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood fun...
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood function, L(θ), and the log-likelihood function, `(θ). (b) Find the maximum likelihood estimator of θ, ˆθ. (c) Is ˆθ unbiased? (d) What is the distribution of X? Find the moment estimator of θ, ˜θ.
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0...
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
20. Let Xi, X2, function Xn be a random sample from a population X with density C")pr(1-0)rn-r fo...
20. Let Xi, X2, function Xn be a random sample from a population X with density C")pr(1-0)rn-r for x = 0, 1.2, , m f(x:0) = 0 otherwise, , where 0 〈 θく1 is parameter. Show that unbiased estimator of θ for a fixed m. is a uniform minimum variance
20. Let Xi, X2, function Xn be a random sample from a population X with density C")pr(1-0)rn-r for x = 0, 1.2, , m f(x:0) = 0 otherwise, , where...
2. 20 marks] Let z1,., xn be a random sample drawn independently from a one-parameter curved normal distribution which has density -oo < x < 00, θ>0, , riid i.e., X r, and 2,2-1 Γη (e) 3 marks Find the Fisher information Z(0) (f) [3 marks] Is θ2 an MVUE of θ? Justify your answer (g) 3 marks] Assume that T = 1.32 and x-3.76 for a random sample of size n = 100. Find the Wald 95% confidence interval...
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ, ie. I. What is the Maximum Likelihood estimate θ of θ? 2. Show that the maximum likelihood estimator of θ is unbiased. 3. We're looking to cstimate the variance θ (1-9) of Xi . x being the empirical average 2(1-2). Check that T is not unli ator propose an unbiased estimator of θ(1-0).
3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have the joint probability function where x = (zi, 2 2:23), θ = (θί, θ2), n = x1 + 2 2 + x3, θι, θ2 > 0 and 1-0,-26, > 0. (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks] Find that the Fisher information matrix I(0) (c) [4 marks] Show that θ is an MVUE. (d)...
3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have the joint probability function where x = (zi, 2 2:23), θ = (θί, θ2), n = x1 + 2 2 + x3, θι, θ2 > 0 and 1-0,-26, > 0. (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks] Find that the Fisher information matrix I(0) (c) [4 marks] Show that θ is an MVUE. (d)...
A (3 pt) Let Xi, ,X, are drawn from the distribution ftheta(z) = F 404 (r+0) , for 0 < x < oo and 0 < θ < oo. We define Y = 3X an estimator for θ. Verify whether this estimator is unbiased? Find the MSE of Y. Hint: E(x)E(X B (3 pt) Let X,.., X, are drawn from the distribution fo) for O < x < 00 and 0 < θ < oo. We define Y = 2X...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
20. Let Xi, X2, function Xn be a random sample from a population X with density C")pr(1-0)rn-r for x = 0, 1.2, , m f(x:0) = 0 otherwise, , where 0 〈 θく1 is parameter. Show that unbiased estimator of θ for a fixed m. is a uniform minimum variance
20. Let Xi, X2, function Xn be a random sample from a population X with density C")pr(1-0)rn-r for x = 0, 1.2, , m f(x:0) = 0 otherwise, , where...
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