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1. Suppose that X Unif(0, 30) and we draw a random sample X1,..., Xn Find the...
2. In class, I showed the below picture. Here, I have changed the vertical axis from...
2. In class, I showed the below picture. Here, I have changed the vertical axis from variance to SD. In this new picture, how can we visualize the MSE? How does this way of seeing the MSE help us decide which of two (possibly biased) estimators is more efficient? SD 02 θ5 Bias (B) 0 (unbiased)
1. Suppose that X ~Unif (0, 30) and we draw a random sample Xi,..., Xn Find...
1. Suppose that X ~Unif (0, 30) and we draw a random sample Xi,..., Xn Find the MME and compute its relative efficiency to θ2-2X1-9X2.
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....
Additional Question Fix θ > 0 and let X1, . . . , Xn i.i.d. ∼...
Additional Question Fix θ > 0 and let X1, . . . , Xn i.i.d. ∼ Unif[0, θ]. We saw in class that the MLE of θ, ˆθMLE = max(X1, . . . , Xn), is biased. I give two other estimators of θ, which can be made unbiased by appropriate choice of constants C1, C2: ˆθ1 = C1 max(X1, . . . , Xn) and ˆθ2 = C2Σxi We have two questions: (1) Find values of C1, C2 for...
Please give detailed steps. Thank you. 5. Let {X1, X2,..., Xn) denote a random sample of...
Please give detailed steps. Thank you.
5. Let {X1, X2,..., Xn) denote a random sample of size N from a population d escribed by a random variable X. Let's denote the population mean of X by E(X) - u and its variance by Consider the following four estimators of the population mean μ : 3 (this is an example of an average using only part of the sample the last 3 observations) (this is an example of a weighted average)...
Fix θ > 0 and let X1, . . . , Xn tid. Unif0.θ]. We saw...
Fix θ > 0 and let X1, . . . , Xn tid. Unif0.θ]. We saw in class that the MLE of θ, θΜLE- max(Xi,... , Xn), is biased. I give two other estimators of θ, which can be made unbiased by appropriate choice of constants C1, C2 1C max(Xi,... ,Xn) and We have two questions (1) Find values of C1, C2 for which these estimators are unbiased. Note that C1, C2 may depend on n (2) Which of these...
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...
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...
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows...
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
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...
2. In class, I showed the below picture. Here, I have changed the vertical axis from variance to SD. In this new picture, how can we visualize the MSE? How does this way of seeing the MSE help us decide which of two (possibly biased) estimators is more efficient? SD 02 θ5 Bias (B) 0 (unbiased)
1. Suppose that X ~Unif (0, 30) and we draw a random sample Xi,..., Xn Find the MME and compute its relative efficiency to θ2-2X1-9X2.
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....
Please give detailed steps. Thank you.
5. Let {X1, X2,..., Xn) denote a random sample of size N from a population d escribed by a random variable X. Let's denote the population mean of X by E(X) - u and its variance by Consider the following four estimators of the population mean μ : 3 (this is an example of an average using only part of the sample the last 3 observations) (this is an example of a weighted average)...
Fix θ > 0 and let X1, . . . , Xn tid. Unif0.θ]. We saw in class that the MLE of θ, θΜLE- max(Xi,... , Xn), is biased. I give two other estimators of θ, which can be made unbiased by appropriate choice of constants C1, C2 1C max(Xi,... ,Xn) and We have two questions (1) Find values of C1, C2 for which these estimators are unbiased. Note that C1, C2 may depend on n (2) Which of these...
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...
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...
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
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...
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