Specifically Part (3)Homework Answers
3)
The probability density function of X is given by:
The likelihood function is given by:
which cannot be used to calculate MLE.
Now, 
where 
are the order statistics.
So, MLE is given by: 
Thus, we can see that there are several possible choices for MLE for this problem.
Add Answer to:
Specifically Part (3)
Based on 6.4-13. Let Xi, . . . , Χη 1.1 a. Unif(9-1.0...
Fix θ > 0 and let Xi, , x, i d. Unif[0.0]. We saw in class...
Fix θ > 0 and let Xi, , x, i d. Unif[0.0]. We saw in class that the MLE of θ, oMLE- I give two other estimators of θ, which can be made unbiased by appropriate choice of -C1 max(Xs , . . . , X,) max(X., Xn), is biased. constants C1,C2 We have two questions: (1) Find values of C1, C2 for which these estimators are unbiased. Note that Ci,C2 may depend on n (2) Which of these estimators...
Let Xi iid∼ Unif(0, θ) for i = 1, 2, 3 and = 4Y(1) = 2Y(2)...
Let Xi iid∼ Unif(0, θ) for i = 1, 2, 3 and
=
4Y(1)
=
2Y(2)
θ* = (4/3)Y(3)
(a) For each estimator compute the bias?
(b) For each estimator compute the variance?
(c) Which estimator is best and why?
3. Let Xi,... , X,n be a random sample from a population with pdf 0, otherwise,...
3. Let Xi,... , X,n be a random sample from a population with pdf 0, otherwise, where θ > 0. a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ (c) Find the pdf of θ in (b).
Additional Question i.i.d. ˆ Fix θ > 0 and let X1,...,Xn ∼ Unif[0,θ]. We saw in...
Additional Question i.i.d. ˆ Fix θ > 0 and let X1,...,Xn ∼
Unif[0,θ]. We saw in class that the MLE of θ, θMLE = max(X1, . . .
, Xn), is biased. I give two other estimators of θ, which can be
made unbiased by appropriate choice of constants C1, C2:
ADDITIONAL QUESTION Fix θ 0 and let Xi, . . . , Xn iid. Unifl0.0]. We saw in class that the MLE of θ, θΜ1E- max(Xi,..., Xn), is biased....
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible (graded, results hidden) Assume we do not actually get to observe X, . . . , Xn. to estimate based on this new data. Instead let Yİ , . . . , Y, be our observations where Yi-l (X·S 0.5) . our goals What distribution does Yi follow? First, choose the type of the distribution: Bernoulli Poisson Norma Exponential Second, enter the parameter of...
Please give detailed steps. Thank you. 5. Let {X, : i-1..n^ denote a random sample of...
Please give detailed steps. Thank you.
5. Let {X, : i-1..n^ denote a random sample of size n from a population described by a random varaible X following a Poisson(θ) distribution with PDF given by θ and var(X) θ (i.e. you do not You may take it as given that E(X) need to show these) a. Recall that an estimator is efficient, if it satisfies 2 conditions: 2) it achieves the Cramer-Rao Lower Bound (CLRB) for unbiased estimators: Show that...
Number 2 only PLEASE 1. [40] 6.4-5. Let Xi, X2..,Xn be a random sample from dis-...
Number 2 only PLEASE
1. [40] 6.4-5. Let Xi, X2..,Xn be a random sample from dis- tributions with the given probability density functions. In each case, find the maximum likelihood estimator . 6.4-10. Let X1, X2,... ,Xn be a random sample of size n from a geometric distribution for which p is the probabil- ity of success. (a) Use the method of moments to find a point estimate 2. [20] for p. 100] 6.5-3. The midterm and final exam scores...
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...
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assum...
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1...
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1 +0)-r(0-1) (1-2), 0 < x < 1, θ > 0. the estimator T-4 is a method of moments estimator for θ. It can be shown that the asymptotic distribution of T is Normal with ETT θ and Var(T) 0042)2 Apply the integral transform method (provide an equation that should be solved to obtain random observations from the distribution) to generate a sam ple of...
Fix θ > 0 and let Xi, , x, i d. Unif[0.0]. We saw in class that the MLE of θ, oMLE- I give two other estimators of θ, which can be made unbiased by appropriate choice of -C1 max(Xs , . . . , X,) max(X., Xn), is biased. constants C1,C2 We have two questions: (1) Find values of C1, C2 for which these estimators are unbiased. Note that Ci,C2 may depend on n (2) Which of these estimators...
Let Xi iid∼ Unif(0, θ) for i = 1, 2, 3 and
=
4Y(1)
=
2Y(2)
θ* = (4/3)Y(3)
(a) For each estimator compute the bias?
(b) For each estimator compute the variance?
(c) Which estimator is best and why?
3. Let Xi,... , X,n be a random sample from a population with pdf 0, otherwise, where θ > 0. a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ (c) Find the pdf of θ in (b).
Additional Question i.i.d. ˆ Fix θ > 0 and let X1,...,Xn ∼
Unif[0,θ]. We saw in class that the MLE of θ, θMLE = max(X1, . . .
, Xn), is biased. I give two other estimators of θ, which can be
made unbiased by appropriate choice of constants C1, C2:
ADDITIONAL QUESTION Fix θ 0 and let Xi, . . . , Xn iid. Unifl0.0]. We saw in class that the MLE of θ, θΜ1E- max(Xi,..., Xn), is biased....
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible (graded, results hidden) Assume we do not actually get to observe X, . . . , Xn. to estimate based on this new data. Instead let Yİ , . . . , Y, be our observations where Yi-l (X·S 0.5) . our goals What distribution does Yi follow? First, choose the type of the distribution: Bernoulli Poisson Norma Exponential Second, enter the parameter of...
Please give detailed steps. Thank you.
5. Let {X, : i-1..n^ denote a random sample of size n from a population described by a random varaible X following a Poisson(θ) distribution with PDF given by θ and var(X) θ (i.e. you do not You may take it as given that E(X) need to show these) a. Recall that an estimator is efficient, if it satisfies 2 conditions: 2) it achieves the Cramer-Rao Lower Bound (CLRB) for unbiased estimators: Show that...
Number 2 only PLEASE
1. [40] 6.4-5. Let Xi, X2..,Xn be a random sample from dis- tributions with the given probability density functions. In each case, find the maximum likelihood estimator . 6.4-10. Let X1, X2,... ,Xn be a random sample of size n from a geometric distribution for which p is the probabil- ity of success. (a) Use the method of moments to find a point estimate 2. [20] for p. 100] 6.5-3. The midterm and final exam scores...
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...
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1 +0)-r(0-1) (1-2), 0 < x < 1, θ > 0. the estimator T-4 is a method of moments estimator for θ. It can be shown that the asymptotic distribution of T is Normal with ETT θ and Var(T) 0042)2 Apply the integral transform method (provide an equation that should be solved to obtain random observations from the distribution) to generate a sam ple of...
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