362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0), x =...
362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0), x = 0, 1, 2, . .. , zero elsewhere, where 0 0s I (a) Find the m. l.e., 6, of 0. X, is a complete sufficient statistic for 0. (b) Show that (c) Determine the unbiased minimum variance estimator of 0.
362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0),...
7.77. If X1, X2,.., X, is a random sample from a distribution with p.d.f. f(x;0)=0*xe-, 0 <x< 00, zero elsewhere, where 0 e< ao: (a) Find the m.l.e., 6. of 0. Is 6 unbiased? X and then compute E(0). Hint: First find the p.d.f. of Y = (b) Argue that Y is a complete sufficient statistic for 8. (c) Find the unbiased minimum variance estimator of 0. (d) Show that X/Y and Y are (e) What is the distribution of...
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
7.41. Let X. X. ..., X, denote a random sample from a distribution that is N(0, 0). Then Y- X is a complete sufficient statistic for 0. Find the unbiased minimum variance estimator of .
2.a. Let X1, X2, ..., X., be a random sample from a distribution with p.d.f. (39) f( 0) = (1 - 1) if 0 < x <1 elsewhere ( 1 2.) = where 8 > 0. Find a sufficient statistic for 0. Justify your answer! Hint: (2(1-)). b. Let X1, X2,..., X, be a random sample from a distribution with p.d.f. (1:0) = 22/ if 0 < I< elsewhere where 8 >0. Find a sufficient statistic for 8. Justify your...
2. Let Xi,... ,Xn be a random sample from a distribution with p.d.f for 0 < x < θ f(x; 0) - 0 elsewhere . (a) Find an estimator for θ using the method of moments. (b) Find the variance of your estimator in (a).
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Let X,,. X, be a random sample from a Poisson (a) (a). 2. distribution. Find the sufficient statistic for A. (25 marks) Let X,X,X, be a random sample from a gamma (k, B) (b). P.1 distribution with k is fixed. DefineX X, n피 based upon unbiased ness, consistency Evaluate (0). and efficiency is a minimum variance unbiased estimator for B Show that (ii). (75 marks) (2)3
Let X,,. X, be a random sample from...
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0 < x < 1), where 0 > 0. Find the maximum likelihood estimator of u = 8/1 b) Find a sufficient statistic and check completeness. (c) Find the UMVUE(uniformly minimum variance unbiased estimator of each of the following : 0,1/0,4 = 0/(1+0).
1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0 < x < 1), where 0 > 0. Find the maximum likelihood estimator of u = 8/1 b) Find a sufficient statistic and check completeness. (c) Find the UMVUE(uniformly minimum variance unbiased estimator of each of the following : 0,1/0,4 = 0/(1+0).
Let X1,X2,,X be a random sample from a distribution function f(x,8) = θ"(1-8)1-r for x = 0,1 (a) Show that Y = Σ.1X, is a sufficient statistic for θ. (i) Find a function of Y that is an unbiased estimate for θ (ii) Hence, explain why this function is the minimum variance unbiased estimator(MVUE) for θ (c) Is1-the MVUE for Please explain.