1. Let X1, X2,...,x. be a random sample from the unif(0,0) distribution (a) Find an unbiased...

Question

Question

Answer 1

Answer #1

The following images have the complete solution...So check them out!!

XI, --Xn wuurf (0,0). if (a)= 1 ocazo. is the common pof for all xis. Now E(X)= Q ,H! EC ) = -(E(X,) + . TEX)) 7EX 3 n E(2x)

Now, xn) will have the pdf :- f (x ) = n! [- F(x)]**(f(n))} (+1) af! , x6(0,0) where F (2) 2 CDf of x. F(n) 0, 0) at [to, o,

pdf will be (x) = 2 x 1 2.0 ) st n- put of Xim, will be, f(x) = a (-2) + ((0,0) f.com) (- )44(0,0). Now, E(Xus) - 5 . 0 10-20

:: E(X) = a. nt) - (n+1) Xen is unbiased for o. Now, van (2.7) = 4 von (7) x,) t-avar (x) - 4 a var (x1) = 4 0-0) OLIC E C 10

:. van(xmi)? E( X 2 ) (n+ 1) (nt 2) (n+1) 2 Clotis come inten] 2 2ut x -n-y. 2) (uti) (nt) .varl Xina Von (2x). 3 n (n+1) ² (

cheers :)

Answer 2

Similar Homework Help Questions

Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution....

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).
1. (20 points) Let X1....X be a random sample from a uniform distribution over [0,0]. (a)...

1. (20 points) Let X1....X be a random sample from a uniform distribution over [0,0]. (a) (4 points) Find the maximum likelihood estimator (MLE) 0 MLE for 0. (b) (3 points) Is the MLE ONLE unbiased for 0? If yes, prove it: If not, construct an unbiased estimator 0, based on the MLE. (c) (4 points) Find the method of moment estimator (MME) OM ME for 8. (d) (3 points) Is the MME OMME tnbiased for 6? If yes, prove...
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the...

Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0 unbiased estimator of e. estimator eCYis an (c) Get the lower bound for the variance of the unbiased estimator found in (b) Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0...

6. Let X1...., X, be a random sample from Unif(0,0). 1 a. Find E(4X1X3). b. Find...

6. Let X1...., X, be a random sample from Unif(0,0). 1 a. Find E(4X1X3). b. Find a minimal sufficient statistic for . c. Find a (1 - a)100% confidence set for using the sufficient statistic in (b).
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over...

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...
1. Suppose that {X1, ... , Xn} is a random sample from a normal distribution with...

1. Suppose that {X1, ... , Xn} is a random sample from a normal distribution with mean p and and variance o2. Let sa be the sample variance. We showed in lectures that S2 is an unbiased estimator of o2. (a) Show that S is not an unbiased estimator of o. (b) Find the constant k such that kS is an unbiased estimator of o. Hint. Use a result from Student's Theorem that (n − 1)52 ~ x?(n − 1)...

7. Section 6.4, Exercise 1 Let X. X be a random sample from the U(0,0) distribution, and let , 2X...

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 Bernoulli(6) distribution with prob- ability fun...

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...
Let X1,X2,,X be a random sample from a distribution function f(x,8) = θ"(1-8)1-r for x =...

Let X1,X2,,X be a random sample from a distribution function f(x,8) = θ"(1-8)1-r for x = 0,1 (a) Show that Y = Σ.1X, is a sufficient statistic for θ. (i) Find a function of Y that is an unbiased estimate for θ (ii) Hence, explain why this function is the minimum variance unbiased estimator(MVUE) for θ (c) Is1-the MVUE for Please explain.

, X,.), the minimum of the sample. mean θ. Let X(1)-min( X1.x2, (a) show that the...

, X,.), the minimum of the sample. mean θ. Let X(1)-min( X1.x2, (a) show that the estimator θ nx(1) is an unbiased estimator of θ. (hint: you were asked to derive the distribution of Xa for a random sample from an exponential distribution on assignment 2 -you may use the result). (b) X, the sample mean, is also an unbiased estimator of o. Which of the unbiased Suppose you observe the following random sample from the population: Calculate point estimates...