1. Let X1, X2,...,X, be a random sample from each of the distributions having the to...
1.2. Let X 1,X2,. , Xn represent a random sample from each of the distributions having the following pdfs: (a) f(x:0-829-1, 0 < x < 1, 0 < θ < oo, zero elsewhere. (b) f(x:0) = e-u-0), θ S1 < oo,-oo < θ < oo, zero elsewhere. Note this is a nonregular case. In each case find the mile θ of θ.
Let X1, X2, ..., Xn represent a random sample from each of the distributions having the following pdf. Please find the maximum likelihood estimator for each case: (c) f(x; θ)--e-x/e,0 < x < 00, 0 < θ < oo, zero elsewhere (d) f(x; θ) e- , θ x < 00,-00 < θ < 00, zero elsewhere In each case, find the mie of a (x-6)
Find the maximum likelihood estimator θ(hat) of θ. Let X1,X2,...Xn represent a random sample from each of the distributions having the following pdfs or pmfs: (a) f(x; θ)-m', (b) f(x; θ)-8x9-1,0 < x < 1,0 < θ < 00, zero elsewhere ere-e x! θ < 00, zero elsewhere, where f(0:0) x-0, 1,2, ,0 -1
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
3. Let X1 , X2, . . . , Xn be a randon sample from the distribution with pdf f(r;0) = (1/2)e-z-8,-X < < oo,-oc < θ < oo. Find the maximum likelihood estimator of θ.
6. Let X1, X2,.. , Xn denote a random sample of size n> 1 from a distribution with pdf f(x; 6) = 6e-8, 0<x< 20, zero elsewhere, and 0 > 0. Le Y = x. (a) Show that Y is a sufficient and complete statistics for . (b) Prove that (n-1)/Y is an unbiased estimator of 0.
7. Let X, X, be a random sample with common pár 1 2 f(x) θ e-A, x > 0,0 > 0, 0 elsewhere. (a) Find the maximum likelihood estimator of θ, denoted by (b) Determine the sampling distribution of θ (c) Find Eô) and Var(). (d) What is the maximum value of the likelihood function? θ .
Let X1, X2, ...... Xn be a random sample of size n from EXP() distribution , , zero , elsewhere. Given, mean of distribution and variances and mgf a) Show that the mle for is . Is a consistent estimator for ? b)Show that Fisher information . Is mle of an efficiency estimator for ? why or why not? Justify your answer. c) what is the mle estimator of ? Is the mle of a consistent estimator for ? d) Is...
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...
Let X1,X2,...,Xn denote a random sample from the Rayleigh distribution given by f(x) = (2x θ)e−x2 θ x > 0; 0, elsewhere with unknown parameter θ > 0. (A) Find the maximum likelihood estimator ˆ θ of θ. (B) If we observer the values x1 = 0.5, x2 = 1.3, and x3 = 1.7, find the maximum likelihood estimate of θ.