1.(c)
2.(a),(b)
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5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show th...
Problem 5: 8 points] Let Xi, , xn be IID from N(0,g2), for θ > 0. For this model, both X and cS are unbiased estimators of 0, where c- V-1)/2) (a) Prove that for any a, the estimator aX (1- a) (cS) is an unbiased estimator of (b) Find the value of a that produces the estimator with the minimum variance. (c) Show that (X, S2) is sufficient for θ but it is not a complete sufficient statistic. V2T(n/2)...
Let Xi,..., Xn be iid random variables with distribution Bern(p) (a) Is the statistic 름 Σ. ? (b) Is the statistic (Σ¡X 2? Xi an unbiased estimator of p i) an unbiased estimator of p Let Xi,..., Xn be iid random variables with distribution Bern(p) (a) Is the statistic 름 Σ. ? (b) Is the statistic (Σ¡X 2? Xi an unbiased estimator of p i) an unbiased estimator of p
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...
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.
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find a complete sufficient statistic for 0. d. Find the UMVUE for θ. Make sure you indicate how you know it is the UMVUE. Let Xi , X2,. … X, denote a random sample of size n...
Solve the following two parts: (Hint: Use Complete Sufficient Statistic) Suppose X1 , X2, of λ2 and the UMVUE of (-1)(-1) Suppose X1 , X2, and UMVUE of 1/g? a. , Xn are iid Poisson distribution with mean λ. Find the UMVUE b. , Xn are iid Uniform[0, θ]. Assumen 3. Find UMVUE of θ3
Letter f and g only. 44 Let X,..., X. be a random sample from (a) Find a sufficient statistic. (b) Find a maximum-likelihood estimator of θ. (c) Find a method-of-moments estimator of θ. (d) Is there a complete sufficient statistic? If so, find it. (e) Find the UMVUE of 0 if one exists. (f) Find the Pitman estimator for the location parameter θ. (g) Using the prior density g(0)--e-n,๑)(8), find the posterior Bayes estimator Of θ. 44 Let X,..., X....
, Xn iid. N 5. Let Xi, (μ, σ2), μ E R and σ2 > 0 are both unknown. Find an asymp- totically likelihood ratio test (LRT) of approximate size α for testing μ-σ 2 H1:ťtơ2 Ho : versus , Xn iid. N 5. Let Xi, (μ, σ2), μ E R and σ2 > 0 are both unknown. Find an asymp- totically likelihood ratio test (LRT) of approximate size α for testing μ-σ 2 H1:ťtơ2 Ho : versus
, 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
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