4. Let X,x, X, be a random sample from a uniform distribution on the interval (0,0)...
Let X1,X2Xn be a random sample from a uniform distribution on the interval (0,0) (a) Show that the density function of Xcp-minXXXn) is given by n-1 72 0 otherwise (b) Use (a) to calculate E[Xcu]. Calculate the bias, B(6). Find a function of Xo) that is an unbiased estimator of 0
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
2. Let X1, X2,. ., Xn be a random sample from a uniform distribution on the interval (0-1,0+1). . Find the method of moment estimator of θ. Is your estimator an unbiased estimator of θ? . Given the following n 5 observations of X, give a point estimate of θ: 6.61 7.70 6.98 8.36 7.26
5. Suppose that X1, X2, , Xn s a random sample from a uniform distribution on the interval (9,8 + 1). (a) Determine the bias of the estimator X, the sample mean. (b) Determine the mean-square error of X as an estimator of θ. (c) Find a function, a, of that is an unbiased estimator of θ. Determine the mean-square error of θ.
3. Let X, X,. X, be a random sample from a distribution with density f(x)- ) if 0 < x < θ 0 otherwise (a) (b) Determine the density of =max(X1,X2, ,Xn} Use the result of (a) to show that θ is a biased estimator of θ. Determine the bias, B(6) Calculate the mean-square error, MSE(6). (c)
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...
20. Let Xi, X2, function Xn be a random sample from a population X with density C")pr(1-0)rn-r for x = 0, 1.2, , m f(x:0) = 0 otherwise, , where 0 〈 θく1 is parameter. Show that unbiased estimator of θ for a fixed m. is a uniform minimum variance 20. Let Xi, X2, function Xn be a random sample from a population X with density C")pr(1-0)rn-r for x = 0, 1.2, , m f(x:0) = 0 otherwise, , where...
.A Uniform random variable is defined on the interval (8,0+1). Let YY, be a random sample taken from this distribution and Y is an estimator of θ. (a) Compute the bias of Y (b) Find a function of that is an unbiased estimator of θ. (c) Find MSE(Y
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...
1. Let X1, X2,...,x. be a random sample from the unif(0,0) distribution (a) Find an unbiased estimatior of O based on the sample mean X (b) Find an unbiased estimator of based on the sample maximum X (c) Which estimator is better in terms of variance?