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QUESTION 3 Let Y1, Y2, ..., Yn denote a random sample of size n from a...
QUESTION 5 Let Y , Y2, , Yn denote a random sample of size n from...
QUESTION 5 Let Y , Y2, , Yn denote a random sample of size n from a population whose density is given by (a) Find the method of moments estimator for β given that α is known. Find the mean and variance of p (b) (c) show that β is a consistent estimator for β.
Suppose that Y1 , Y2 ,..., Yn denote a random sample of size n from a...
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....
Let Y,, Y2, .., Yn denote a random sample of size n from a population whose...
Let Y,, Y2, .., Yn denote a random sample of size n from a population whose density is given by Find the method of moments estimator for α.
QUESTION 7 Let Y, Y2, ....Yn denote a random sample of size n from a population...
QUESTION 7 Let Y, Y2, ....Yn denote a random sample of size n from a population whose density is given by (a) Find an estimator for θ by the maximum likelihood method. (b) Find the maximum likelihood estimator for E( Y4).
QUESTION 7 Let Y,, Y2,..., Yn denote a random sample of size n from a population...
QUESTION 7 Let Y,, Y2,..., Yn denote a random sample of size n from a population whose density is given by (a) Find an estimator for 0 by the maximum likelihood method. (b) Find the maximum likelihood estimator for E(Y4).
Question 3 [25] , Yn denote a random sample of size n from a Let Y,...
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
Let X1, X2, ..., Xn denote a random sample of size n from a population whose...
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.
QUESTION8 Let Y,,Y2, ..., Yn denote a random sample of size n from a population whose...
QUESTION8 Let Y,,Y2, ..., Yn denote a random sample of size n from a population whose density is given by (a) Find the maximum likelihood estimator of θ given α is known. (b) Is the maximum likelihood estimator unbiased? (c) is a consistent estimator of θ? (d) Compute the Cramer-Rao lower bound for V(). Interpret the result. (e) Find the maximum likelihood estimator of α given θ is known.
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a...
. 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...
Suppose Y1, ..., Yn denote a random sample of size n from an exponential distribu- tion...
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.
QUESTION 5 Let Y , Y2, , Yn denote a random sample of size n from a population whose density is given by (a) Find the method of moments estimator for β given that α is known. Find the mean and variance of p (b) (c) show that β is a consistent estimator for β.
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....
Let Y,, Y2, .., Yn denote a random sample of size n from a population whose density is given by Find the method of moments estimator for α.
QUESTION 7 Let Y, Y2, ....Yn denote a random sample of size n from a population whose density is given by (a) Find an estimator for θ by the maximum likelihood method. (b) Find the maximum likelihood estimator for E( Y4).
QUESTION 7 Let Y,, Y2,..., Yn denote a random sample of size n from a population whose density is given by (a) Find an estimator for 0 by the maximum likelihood method. (b) Find the maximum likelihood estimator for E(Y4).
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
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.
QUESTION8 Let Y,,Y2, ..., Yn denote a random sample of size n from a population whose density is given by (a) Find the maximum likelihood estimator of θ given α is known. (b) Is the maximum likelihood estimator unbiased? (c) is a consistent estimator of θ? (d) Compute the Cramer-Rao lower bound for V(). Interpret the result. (e) Find the maximum likelihood estimator of α given θ is known.
. 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...
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.
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