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.
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Ouestion 7 (10 points)Suppose Y..... y denote a random sample of
size n from an exponential...
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.
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x...
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...
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...
X denote the mean of a random sample of size 25 from a gamma type distribu-...
X denote the mean of a random sample of size 25 from a gamma type distribu- tion with a = 4 and β > 0. Use the Central Limit theorem to find an approximate 0.954 confidence interval for μ, the mean of the gallina distribution. Hint: Use the random variable (X-43)/?7,/432/25. 6. Let Yi < ½ < < }, denote the order statistics of a randon sample of size n from a distribution that has pdf f(z) = 4r3/04, O...
. 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 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....
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).
Consider X1,X2, , Xn be an iid random sample fron Unif(0.0). Let θ = (끄+1) Y where Y = max(X1, x...
Consider X1,X2, , Xn be an iid random sample fron Unif(0.0). Let θ = (끄+1) Y where Y = max(X1, x. . . . , X.). It can be easily shown that the cdf of Y is h(y) = Prp.SH-()" 1. Prove that Y is a biased estimator of θ and write down the expression of the bias 2. Prove that θ is an unbiased estimator of θ. 3. Determine and write down the cdf of 0 4. Discuss why...
QUESTION 6 Let Y., Y. , Yn denote a random sample of size n from a...
QUESTION 6 Let Y., Y. , Yn denote a random sample of size n from a population whose density is given by , Yn) and θ2 = Ỹ. Two estimators for θ are θ,-nY(1) where Y(1)-min (h, ½, (a) Show that θ1 and θ2 are both unbiased estimators of θ (b) Find the efficiency ofa relative to θ2.
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.
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...
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...
X denote the mean of a random sample of size 25 from a gamma type distribu- tion with a = 4 and β > 0. Use the Central Limit theorem to find an approximate 0.954 confidence interval for μ, the mean of the gallina distribution. Hint: Use the random variable (X-43)/?7,/432/25. 6. Let Yi < ½ < < }, denote the order statistics of a randon sample of size n from a distribution that has pdf f(z) = 4r3/04, O...
. 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
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....
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).
Consider X1,X2, , Xn be an iid random sample fron Unif(0.0). Let θ = (끄+1) Y where Y = max(X1, x. . . . , X.). It can be easily shown that the cdf of Y is h(y) = Prp.SH-()" 1. Prove that Y is a biased estimator of θ and write down the expression of the bias 2. Prove that θ is an unbiased estimator of θ. 3. Determine and write down the cdf of 0 4. Discuss why...
QUESTION 6 Let Y., Y. , Yn denote a random sample of size n from a population whose density is given by , Yn) and θ2 = Ỹ. Two estimators for θ are θ,-nY(1) where Y(1)-min (h, ½, (a) Show that θ1 and θ2 are both unbiased estimators of θ (b) Find the efficiency ofa relative to θ2.
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