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3. Suppose X1, X2, , Xn are iid based on the random variable modeled by 2,0-1...
Suppose X1, X2, . . . , Xn are iid based on the random variable modeled...
Suppose X1, X2, . . . , Xn are iid based on the random variable modeled by where 0 ≤ x ≤ 1 and α > 0. a. Find an equation that the MLE for α must satisfy. Note: You will not be able to explicitly solve for the MLE as in other problems. b. If you are told E(X) = 1/2 and Var(X) = 1/(8α + 4), find the MME for α. This problem is a nice example where...
Let X1, X2, ..., Xn be iid random variables from a Uniform(-0,0) distribution, where 8 >...
Let X1, X2, ..., Xn be iid random variables from a Uniform(-0,0) distribution, where 8 > 0. Find the MLE of 0.4
Suppose that X1, X2,....Xn is an iid sample of size n from a Pareto pdf of...
Suppose that X1, X2,....Xn is an iid sample of size n from a Pareto pdf of the form 0-1) otherwise, where θ > 0. (a) Find θ the method of moments (MOM) estimator for θ For what values of θ does θ exist? Why? (b) Find θ, the maximum likelihood estimator (MLE) for θ. (c) Show explicitly that the MLE depends on the sufficient statistic for this Pareto family but that the MOM estimator does not
2. Suppose that X1, X2, . . . , Xn are iid. N(0, σ) with density...
2. Suppose that X1, X2, . . . , Xn are iid. N(0, σ) with density function f (xlo) Find the Fisher information I(o) a. b. Now, call: σ2 your parameter, with this new parametrization, f(x19)-E-e-28 Find the Fisher information 1(8) 1(ог). Is 1(σ*)-1 (σ)? c. Find o2MOM d. Find σ2MLE e. Find Elo-MLE]. Show that σ2MLEls unbiased f. Find Var[σ 2MLEİ. Does σ2MLE attain the CRLB?
4.(120) Let X1,,,Xn be iid r(, 1) and g(u) given. Let 6n be the MLE of g(4) (1)(60) Find the asymptotic distributio...
4.(120) Let X1,,,Xn be iid r(, 1) and g(u) given. Let 6n be the MLE of g(4) (1)(60) Find the asymptotic distribution of 6, (2)(60) Find the ARE of T Icc(X) w.r.t. on P(X1> c), c > 0 is i n i1 5.(80) Let X1, ,,Xn be iid with E(X1) = u and Var(X1) limiting distribution of nlog (1 +). o2. Find the where T n(X - 4)/s. - 1 -
4.(120) Let X1,,,Xn be iid r(, 1) and g(u)...
Suppose X1, X2, , Xn is an iid sample from a uniform distribution over (θ, θΗθ!),...
Suppose X1, X2, , Xn is an iid sample from a uniform distribution over (θ, θΗθ!), where (a) Find the method of moments estimator of θ (b) Find the maximum likelihood estimator (MLE) of θ. (c) Is the MLE of θ a consistent estimator of θ? Explain.
2. Let X 1, , Xn be iid from the distribution modeled by 8-2 fx (1:0)-(9....
2. Let X 1, , Xn be iid from the distribution modeled by 8-2 fx (1:0)-(9. θ):r"-"(1-2) dr where 0 < x < 1 and θ > 1 Find the MME (method of moments estimate/estimator) for 0
4. Let X1, X2, ..., Xn be iid from the Bernoulli distribution with common probability mass...
4. Let X1, X2, ..., Xn be iid from the Bernoulli distribution with common probability mass function Px(x) = p*(1 – p)1-x for x = 0,1, and 0 < p < 1 14 a. (4) Find the MLE Ôule of p.
Suppose that X1,X2,. X are iid random variables with pdf ,220 (a) Find the maximum likelihood...
Suppose that X1,X2,. X are iid random variables with pdf ,220 (a) Find the maximum likelihood estimate of the parameter a (b) Find the Fisher Information of X1,X2,.. ., Xn and use it to estimate a 95% confidence interval on the MLE of a (c) Explain how the central limit theorem relates to (b).
Suppose that X1, X2, ..., Xn is an iid sample, each with probability p of being...
Suppose that X1, X2, ..., Xn is an iid sample, each with probability p of being distributed as uniform over (-1/2,1/2) and with probability 1 - p of being distributed as uniform over (a) Find the cumulative distribution function (cdf) and the probability density function (pdf) of X1 (b) Find the maximum likelihood estimator (MLE) of p. c) Find another estimator of p using the method of moments (MOM)
Let X1, X2, ..., Xn be iid random variables from a Uniform(-0,0) distribution, where 8 > 0. Find the MLE of 0.4
Suppose that X1, X2,....Xn is an iid sample of size n from a Pareto pdf of the form 0-1) otherwise, where θ > 0. (a) Find θ the method of moments (MOM) estimator for θ For what values of θ does θ exist? Why? (b) Find θ, the maximum likelihood estimator (MLE) for θ. (c) Show explicitly that the MLE depends on the sufficient statistic for this Pareto family but that the MOM estimator does not
2. Suppose that X1, X2, . . . , Xn are iid. N(0, σ) with density function f (xlo) Find the Fisher information I(o) a. b. Now, call: σ2 your parameter, with this new parametrization, f(x19)-E-e-28 Find the Fisher information 1(8) 1(ог). Is 1(σ*)-1 (σ)? c. Find o2MOM d. Find σ2MLE e. Find Elo-MLE]. Show that σ2MLEls unbiased f. Find Var[σ 2MLEİ. Does σ2MLE attain the CRLB?
4.(120) Let X1,,,Xn be iid r(, 1) and g(u) given. Let 6n be the MLE of g(4) (1)(60) Find the asymptotic distribution of 6, (2)(60) Find the ARE of T Icc(X) w.r.t. on P(X1> c), c > 0 is i n i1 5.(80) Let X1, ,,Xn be iid with E(X1) = u and Var(X1) limiting distribution of nlog (1 +). o2. Find the where T n(X - 4)/s. - 1 -
4.(120) Let X1,,,Xn be iid r(, 1) and g(u)...
Suppose X1, X2, , Xn is an iid sample from a uniform distribution over (θ, θΗθ!), where (a) Find the method of moments estimator of θ (b) Find the maximum likelihood estimator (MLE) of θ. (c) Is the MLE of θ a consistent estimator of θ? Explain.
2. Let X 1, , Xn be iid from the distribution modeled by 8-2 fx (1:0)-(9. θ):r"-"(1-2) dr where 0 < x < 1 and θ > 1 Find the MME (method of moments estimate/estimator) for 0
4. Let X1, X2, ..., Xn be iid from the Bernoulli distribution with common probability mass function Px(x) = p*(1 – p)1-x for x = 0,1, and 0 < p < 1 14 a. (4) Find the MLE Ôule of p.
Suppose that X1,X2,. X are iid random variables with pdf ,220 (a) Find the maximum likelihood estimate of the parameter a (b) Find the Fisher Information of X1,X2,.. ., Xn and use it to estimate a 95% confidence interval on the MLE of a (c) Explain how the central limit theorem relates to (b).
Suppose that X1, X2, ..., Xn is an iid sample, each with probability p of being distributed as uniform over (-1/2,1/2) and with probability 1 - p of being distributed as uniform over (a) Find the cumulative distribution function (cdf) and the probability density function (pdf) of X1 (b) Find the maximum likelihood estimator (MLE) of p. c) Find another estimator of p using the method of moments (MOM)
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