Problem 2: Let (X1,... Xn) denote a random variable from X having density fx(x) = 1/...
Let (X1, ..., Xn) denote a random variable from X having a Log-normal density fX(x) = φ(Ln(x) − m)/x, x > 0 where m is an unknown parameter. Show (1/n) sigma(i=1 to n) of Ln(Xi) is a MVU estimator for m.
Problem 1: Let (Xi,..., Xn) denote a random variable from X having a Log-normal density fx (x) = d(L m)/ x, x 〉 0 n(x) - where m is an unknown parameter. Show n-1 Σ'al Ln(X) is a MVU estimator for m. Problem 1: Let (Xi,..., Xn) denote a random variable from X having a Log-normal density fx (x) = d(L m)/ x, x 〉 0 n(x) - where m is an unknown parameter. Show n-1 Σ'al Ln(X) is a...
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...
Let X1, . . . , Xn be a random sample from a population X with p.d.f fθ(x) = θ xθ−1 , for 0 < x < 1 0, otherwise, where θ > 1 is parameter. Find the MLE of 1/θ. If it is an unbiased estimator of 1/θ, compare its variance with the Cramer-Rao lower bound.
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...
3. Let X1, X2, . . . , Xn be independent samples of a random variable with the probability density function (PDF): fX(x) = θ(x − 1/ 2 ) + 1, 0 ≤ x ≤ 1 ,0 otherwise where θ ∈ [−2, 2] is an unknown parameter. We define the estimator ˆθn = 12X − 6 to estimate θ. (a) Is ˆθn an unbiased estimator of θ? (b) Is ˆθn a consistent estimator of θ? (c) Find the mean squared...
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...
Suppose X1, X2, , xn is an iid sample from fx(x10)-θe_&z1 (a) For n 2 2, show that (x > 0), where θ > 0 . n- is the uniformly minimum variance unbiased estimator (UMVUE) of θ (b) Calculate varo(0). Comment, in particular, on the n 2 case. (c) Show that vars(0) does not attain the Cramer-Rao Lower Bound (CRLB) on the variance of all unbiased estimators of T(9-0 (d) For this part only, suppose that n 1, 11T(X) is...
Let X1, X2, ..., Xn denote a random sample of size n from a population whose density fucntion is given by 383x-4 f S x f(x) = 0 elsewhere where ß > 0 is unknown. Consider the estimator ß = min(X1, X2, ...,Xn). Derive the bias of the estimator ß.
Let X1, X2, ..., Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) max(X1,X2, ...,Xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for e.