(a) (4 points) Find the method of moments estimator for θ. (b) (4 points) Find the maximum likeli...
Suppose X1, X2, ..., Xn is an iid sample from fx(r ja-θ(1-z)0-11(0 1), where x θ>0. (a) Find the method of moments (MOM) estimator of θ. (b) Find the maximum likelihood estimator (MLE) of θ (c) Find the MLE of Po(X 1/2) d) Is there a function of θ, say T 0), for which there exists an unbiased estimator whose variance attains the Cramér-Rao Lower Bound? If so, find it and identify the corresponding estimator. If not, show why not.
Bernoulli distribution with parameter θ . a) Use the method of moments to obtain an estimator of θ b) Obtain the maximum likelihood estimator (MLE) of θ.
Let X1,X2,...,Xn be iid exponential random variables with unknown mean β. (b) Find the maximum likelihood estimator of β. (c) Determine whether the maximum likelihood estimator is unbiased for β. (d) Find the mean squared error of the maximum likelihood estimator of β. (e) Find the Cramer-Rao lower bound for the variances of unbiased estimators of β. (f) What is the UMVUE (uniformly minimum variance unbiased estimator) of β? What is your reason? (g) Determine the asymptotic distribution of the...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
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
Advanced Statistics, I need help with (c) and (d) 2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
, , Yn is a random sample from a distribution with pdf f,0% θ)-22, 3. (20 points) If Y., Y2, 0 Syse, a. find cÝ, where c is a constant, that is an unbiased estimator of θ; and b. show that the variance of is less than the Cramér-Rao lower bound for fr (y; 0) c. Why isn't this a violation of the Cramér-Rao inequality? , , Yn is a random sample from a distribution with pdf f,0% θ)-22, 3....
Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1 where I(.) denotes the indicator function. (a) Find a 2-dimensional sufficient statistic for the mode (b) Suppose θ is a known constant. Find the MLE for v. (d) Suppose v-1. Find the MLE for and determine its asymptotic distribution. Carefully justify your answer and state any theorems that you use. (e) Suppose1. Find the asymptotic distribution of the MLE estimator of exp[- Let Xi,... ,Xn be i.i.d with pdf θνθ θ+1...
be a random sample from the density 16 1. Let Xi, . f(x; β) otherwise 8(1-/4). You may suppose that E(X)(/ (a) Find a sufficient statistic Y for B and Var(X) C21 C2] 031 (b) Find the maximum likelihood estimator B of B and show that it is a function (c) Determine the Rao-Cramér lower bound (RCLB) for the variance of unbiased (d) Use the following data and maximum likelihood estimator to give an approxi- 2.66, 2.02, 2.02, 0.76, 1.70,...