7. (12 points) Let Yı,Y2, ..., Yn be a random sample from Gamma(a,b), where a =...
Let Yı, Y2, ...,Yn be an iid sample from a population distribution described by the pdf fy(y|0) = (@+ 1) yº, o<y<1 for 0> - -1. (a) Find the MOM estimator of 0. (b) Find the maximum likelihood estimator (MLE) of 0. (c) Find the MLE of the population mean E(Y) = 0 +1 0 + 2 You do not need to prove that the above is true. Just find its MLE.
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...
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...
(1 point) Let Yı, Y2, ..., Yn be a random sample from the probability density function f(yla) = |aya-2/5° f(y ) 0 <y< 5 otherwise 0 for > -1. Find an estimator for a using the method of moments.
Let Y1,Y2, …… Yn be a random sample from the distribution f(y) = θxθ-1 where 0 < x < 1 and 0 < θ < ∞. Show that the maximum likelihood estimator (MLE) for θ is
Let Yı, Y2, ..., Yn iid N (4,02), where the population mean and population variance o2 are both unknown. Show that the Method of Moments (MOM) estimators of u and o2 are given by n û =Ý ΣΥ, n i=1 п n - - 1 ô2 - = s2 Ü(Y; – 7) n п i=1 Note: In this case, (Y, S2) is a sufficient statistic for (u, 02). The MOM estimators of u and o2 are therefore functions of a...
iid Let Yı, Y2, ..., Yn N(u,), where the population mean y and population variance o are both unknown. Show that the Method of Moments (MOM) estimators of u and o? are given by n i =Y, Y n =1 72 = n-1 S2 (Y; -Y) n n i=1 Note: In this case, (Y, S?) is a sufficient statistic for (u, o?). The MOM estimators of u and o2 are therefore functions of a sufficient statistic.
QUESTION 7 Let Y, Y2, ....Yn denote a random sample of size n from a population whose density is given by (a) Find an estimator for θ by the maximum likelihood method. (b) Find the maximum likelihood estimator for E( Y4).
QUESTION 7 Let Y,, Y2,..., Yn denote a random sample of size n from a population whose density is given by (a) Find an estimator for 0 by the maximum likelihood method. (b) Find the maximum likelihood estimator for E(Y4).
QUESTION 5 Suppose that Yı, Y2,.., Yn independent variables such that where β is an unknown parameter, X1, x2-.., xn are known real numbers, and el,e2 independent random errors each with a normal distribution with mean 0 and variance ơ2 ,en are (a) Show that is an unbiased estimator of β. What is the variance of the estimator? (b) Given that the probability density function of Y is elsewhere, show that the maximum likelihood estimator of β is not the...