

IULIE 2Fcontent%2F725024%2FviewConter 1. (12 points) Let Y1, ... ,Y, be random sample from normal with mean...
10. Let Y1,..., Y, be a random sample from a distribution with pdf 0<y< elsewhere f(x) = { $(0 –» a) Find E(Y). b) Find the method of moments estimator for 8. c) Let X be an estimator of 8. Is it an unbiased estimator? Find the mean square error of X. Show work
Suppose
that Y1 , Y2 ,..., Yn denote a random sample of size n from a
normal population with mean μ and variance 2 .
Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....
1. Let Y1, . . . ,Y,, be a random sample from a population with density function 0, otherwise (a) Find the method of moments estimator of θ (b) Show that Yan.-max(Yi, . . . ,%) is sufficient for 02] (Hint: Recall the indicator function given by I(A)1 if A is true and 0 otherwise.) (c) Determine the density function of Yn) and hence find a function of Ym) that is an unbiased estimator of θ (d) Find c so...
Let Y1,K,Y n denote a random sample from a Poisson distribution with parameter λ . a. Find a sufficient statistics for λ. b. Find the minimum variance unbiased estimator(MVUE) of λ2 .
Ouestion 7 (10 points)Suppose Y..... y denote a random sample of
size n from an exponential distribu-| tion with mean 9.a) (5
points)Find the bias and MSE of the estimator B1 = nY().b) (3
points)Consider another estimator B, =Y. Find the efficiency of 6,
relative to 62.e) (7 points)Prove that 2 is a pivotal quantity and
find a 95% confidence interval for 8.
Question 7 (10 points) Suppose Y1, ..., Yn denote a random sample of size n from an...
Let X1, ..., X10 be a random sample from a population with mean y and variance o?. Consider the following estimators for ji: X1 +...+ X10 3X1 - 2X3 + 3X10 Ô1 = @2 10 2 Are these estimators unbiased (i.e. is their expectation equal to u)? A. Both estimators are unbiased. C. Only the second is unbiased. E. Insufficient information. B. Both estimators are biased. D. Only the first is unbiased.
Suppose Y1, ..., Yn denote a random sample of size n from an exponential distribu- tion with mean 0. a) (5 points) Find the bias and MSE of the estimator ôz = nY1). b) (3 points) Consider another estimator ôz = Y. Find the efficiency of ôı relative to 62. c) (7 points) Prove that 297 Yi is a pivotal quantity and find a 95% confidence interval for 0.
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a
population with Rayleigh distribution (Weibull distribution with
parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ
> 0, y > 0
Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn},
and ˆθ2 = 1 n Xn i=1 Y 2 i .
ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased
estimators, and in...
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
Q- If Y1,…,Yn is an i.i.d. random sample from a population with mean μY and variance σ2Y, which of the following is not true? 1) Y1,…,Yn are identically distributed random variables 2) Y1,…,Yn are mutually independent random variables 3) Var(Y¯)=σ2Y 4) E(Y¯)=μY