Homework Answers
, xn is an iid sample from fx(x10)-θe-8z1(x > 0), where θ > 0. Suppose X1,...
, xn is an iid sample from fx(x10)-θe-8z1(x > 0), where θ > 0. Suppose X1, X2, For n 2 2, n- is the uniformly minimum variance unbiased estimator (UMVUE) of 0 (d) For this part only, suppose that n-1. If T(Xi) is an unbiased estimator of e, show that Pe(T(X) 0)>0
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...
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 X1, X2, , xn is an iid sample from fx(x10)-θe_&z1 (a) For n 2 2,...
Suppose X1, X2, , xn is an iid sample from fx(x10)-θe_&z1 (a) For n 2 2, show that (x > 0), where θ > 0 . n- is the uniformly minimum variance unbiased estimator (UMVUE) of θ (b) Calculate varo(0). Comment, in particular, on the n 2 case. (c) Show that vars(0) does not attain the Cramer-Rao Lower Bound (CRLB) on the variance of all unbiased estimators of T(9-0 (d) For this part only, suppose that n 1, 11T(X) is...
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 Xi, , xn be a sample from fx (x10) = e-(z-8),T 〉 θ, θ e(-00,00)...
Let Xi, , xn be a sample from fx (x10) = e-(z-8),T 〉 θ, θ e(-00,00) 1-Identify a sufficient statistic for θ if it exists. 2.Identify a Minimal sufficient statistic for θ it it exists. O, 0O
5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show th...
1.(c)
2.(a),(b)
5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
Suppose X1, X2, ..., Xn is an iid sample from fx(r ja-θ(1-z)0-11(0 1), where x θ>0....
Suppose X1, X2, ..., Xn is an iid sample from fx(r ja-θ(1-z)0-11(0 1), where x θ>0. (a) Find the method of moments (MOM) estimator of θ. (b) Find the maximum likelihood estimator (MLE) of θ (c) Find the MLE of Po(X 1/2) d) Is there a function of θ, say T 0), for which there exists an unbiased estimator whose variance attains the Cramér-Rao Lower Bound? If so, find it and identify the corresponding estimator. If not, show why not.
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ ...
difficult……
2and4 thanks
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ ...
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ. (b) Find an...
, xn is an iid sample from fx(x10)-θe-8z1(x > 0), where θ > 0. Suppose X1, X2, For n 2 2, n- is the uniformly minimum variance unbiased estimator (UMVUE) of 0 (d) For this part only, suppose that n-1. If T(Xi) is an unbiased estimator of e, show that Pe(T(X) 0)>0
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...
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 X1, X2, , xn is an iid sample from fx(x10)-θe_&z1 (a) For n 2 2, show that (x > 0), where θ > 0 . n- is the uniformly minimum variance unbiased estimator (UMVUE) of θ (b) Calculate varo(0). Comment, in particular, on the n 2 case. (c) Show that vars(0) does not attain the Cramer-Rao Lower Bound (CRLB) on the variance of all unbiased estimators of T(9-0 (d) For this part only, suppose that n 1, 11T(X) is...
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 Xi, , xn be a sample from fx (x10) = e-(z-8),T 〉 θ, θ e(-00,00) 1-Identify a sufficient statistic for θ if it exists. 2.Identify a Minimal sufficient statistic for θ it it exists. O, 0O
1.(c)
2.(a),(b)
5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
Suppose X1, X2, ..., Xn is an iid sample from fx(r ja-θ(1-z)0-11(0 1), where x θ>0. (a) Find the method of moments (MOM) estimator of θ. (b) Find the maximum likelihood estimator (MLE) of θ (c) Find the MLE of Po(X 1/2) d) Is there a function of θ, say T 0), for which there exists an unbiased estimator whose variance attains the Cramér-Rao Lower Bound? If so, find it and identify the corresponding estimator. If not, show why not.
difficult……
2and4 thanks
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ. (b) Find an...
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