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.
Homework Answers
TOPIC:Maximum likelihood estimator,Unbiased estimator and method of moment estimator.
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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...
Let X1, X2, ..., Xn be a random sample with probability density function a) Is ˜θ unbiased for θ? Explain. b) Is ˜θ consistent for θ? Explain. c) Find the limiting distribution of √ n( ˜θ − θ). need...
Let X1, X2, ..., Xn be a random sample with probability density
function
a) Is ˜θ unbiased for θ? Explain.
b) Is ˜θ consistent for θ? Explain.
c) Find the limiting distribution of √ n( ˜θ − θ).
need only C,D, and E
Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
Let Xi,... , Xn be a random sample from a normal random variable X with E(X)...
Let Xi,... , Xn be a random sample from a normal random variable X with E(X) 0 and var(X)-0, i.e., X ~N(0,0) (a) What is the pdf of X? (b) Find the likelihood function, L(0), and the log-likelihood function, e(0) (c) Find the maximun likelihood estimator of θ, θ (d) Is θ unbiased?
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability fun...
Advanced Statistics, I need help with (c) and (d)
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random var...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ...
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ E Θ : (0.00) (a) Calculate the distribution function Fo of Xi (b) Let x1, .., xn be a realization of X1, Xn. What is the log-likelihood- function for the parameter θ? (c) Calculate the maximum-likelihood-estimator θ(x1, , xn) for the unknown parameter θ
6-7. Let θ > 1 and let X1,X2, ,Xn be a random sample from the distri- bution with probability den...
Will thumbs up if done neatly and
correctly!
6-7. Let θ > 1 and let X1,X2, ,Xn be a random sample from the distri- bution with probability density function f(x; θ-zind, 1 < x < θ. 6. a) Obtain the maximum likelihood estimator of θ, θ b) Is a consistent estimator of θ? Justify your answer
6-7. Let θ > 1 and let X1,X2, ,Xn be a random sample from the distri- bution with probability density function f(x; θ-zind, 1
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x;...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3e-dız?, x > 0. a. Find E(X), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for \, Gamma for the function, and pi for the mathematical constant 11. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/ I. Hint 1: Consider u = 1x2 or u = x2....
4. The Uniform (0,20) distribution has probability density function if 0 x 20 f (x) 20 0, otherwi...
4. The Uniform (0,20) distribution has probability density function if 0 x 20 f (x) 20 0, otherwise, , where 0 > 0. Let X,i,.., X, be a random sample from this distribution. Not cavered 2011 (a) [6 marks] Find-4MM, the nethod of -moment estimator for θ for θ? If not, construct-an unbiased'estimator forg based on b) 8 marks Let X(n) n unbia estimator MM. CMM inbiase ( = max(X,, , Xn). Let 0- be another estimator of θ. 18θ...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x;...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3 e-tz, x > 0. a. Find E(XK), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for 1, Gamma for the function, and pi for the mathematical constant i. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/n. Hint 1: Consider u = 1x2 or u = x2....
Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x <...
Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x < oo. (a) Find the method of moments estimator of θ. (b) Find the maxinum likelihood estimator of θ
Let X1, X2, ..., Xn be a random sample with probability density
function
a) Is ˜θ unbiased for θ? Explain.
b) Is ˜θ consistent for θ? Explain.
c) Find the limiting distribution of √ n( ˜θ − θ).
need only C,D, and E
Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
Let Xi,... , Xn be a random sample from a normal random variable X with E(X) 0 and var(X)-0, i.e., X ~N(0,0) (a) What is the pdf of X? (b) Find the likelihood function, L(0), and the log-likelihood function, e(0) (c) Find the maximun likelihood estimator of θ, θ (d) Is θ unbiased?
Advanced Statistics, I need help with (c) and (d)
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ E Θ : (0.00) (a) Calculate the distribution function Fo of Xi (b) Let x1, .., xn be a realization of X1, Xn. What is the log-likelihood- function for the parameter θ? (c) Calculate the maximum-likelihood-estimator θ(x1, , xn) for the unknown parameter θ
Will thumbs up if done neatly and
correctly!
6-7. Let θ > 1 and let X1,X2, ,Xn be a random sample from the distri- bution with probability density function f(x; θ-zind, 1 < x < θ. 6. a) Obtain the maximum likelihood estimator of θ, θ b) Is a consistent estimator of θ? Justify your answer
6-7. Let θ > 1 and let X1,X2, ,Xn be a random sample from the distri- bution with probability density function f(x; θ-zind, 1
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3e-dız?, x > 0. a. Find E(X), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for \, Gamma for the function, and pi for the mathematical constant 11. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/ I. Hint 1: Consider u = 1x2 or u = x2....
4. The Uniform (0,20) distribution has probability density function if 0 x 20 f (x) 20 0, otherwise, , where 0 > 0. Let X,i,.., X, be a random sample from this distribution. Not cavered 2011 (a) [6 marks] Find-4MM, the nethod of -moment estimator for θ for θ? If not, construct-an unbiased'estimator forg based on b) 8 marks Let X(n) n unbia estimator MM. CMM inbiase ( = max(X,, , Xn). Let 0- be another estimator of θ. 18θ...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3 e-tz, x > 0. a. Find E(XK), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for 1, Gamma for the function, and pi for the mathematical constant i. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/n. Hint 1: Consider u = 1x2 or u = x2....
Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x < oo. (a) Find the method of moments estimator of θ. (b) Find the maxinum likelihood estimator of θ
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