1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where...
I need the answer for (ii) 1. A certain continuous distribution has cumulative distribution function (CDF) given by F(a)-0, <0 where θ is an unknown parameter, θ > 0. Let X, be the sample miean and X(n) = max {Xu X2, ,..} 0) Show that n +, is an unbiased stimator of o Find its mean squnare error and check whether θι, is consistent for θ. (ii) Show that 2n- Xn) is a consistent estimator of fe (iii) Assume n...
explan the answer 10: A certain continuous distribution has cumulative distribution function (CDF) given by F(r) 0, <0 where θ is an unknown parameter, θ > 0. (i) Find (a) the p.d.f., (b) the mean and (e) the variance of this distribution. (ii) Suppose that X (Xi, X2, Xn) is a random sample from this distribu- tion and let Y max(Xi, XXn). Find the CDF and p.d.f. of Y. Hence find the value of a for which EloY)
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...
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...
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...
Let X be a random variable with cdf FX (x:0), expected value EIX-μ and variance VlX- σ2. Let X1,X2, , Xn be an id sample drawn according to FX(x,8) where Fx (x,8) =万 for all x E (0,0). Let max(X1, X2, , X.) be an estimator of θ, suggested from pure common sense. Remember that if Y = max(X1, X2, , Xn). Then it can be shown that the cdf Fy () of Y is given by Fr(u) (Fx()" where...
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
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...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)? 8. Let X1,...,Xn denote a random...