TOPIC: Bias and MSE based on a sample size of size 'n' from an Uniform distribution.
.A Uniform random variable is defined on the interval (8,0+1). Let YY, be a random sample...
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 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
4. Let X,x, X, be a random sample from a uniform distribution on the interval (0,0) (a) Show that the density function of XnX,X2 Xn is given by 0 otherwise (b) Use (a) to calculate E[X)). Caleulate the bias, B). Find a function of X) that is an unbiased estimator of .
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. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the method of ,y,). We moment estimator θ-2Y and the maximum likelihood estimator θ-maxx,Yo, derived the Bias and MSE for both estimators. With the intent to correct the bias of the mle θ we proposed the following new estimator -Imax where the subscript u stands for "unbiased." (a) Find the MSE of (b) Compare the MSE of θυ to the MSE of θ, the original...
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...
7. Section 6.4, Exercise 1 Let X. X be a random sample from the U(0,0) distribution, and let , 2X and mx X, be estimators for 0. It is given that the mean and variance of oz are (a) Give an expression for the bias of cach of the two estimators. Are they unbiased? (b) Give an expression for the MSE of cach of the two estimators. (c) Com pute the MSE of each of the two ctrnators for n...
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
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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...