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 which these estimators are unbiased. Note that C1, C2 may depend on n.
(2) Which of these estimators is “better”? Provide some justification. Note that this question has many possible answers, as there are many ways that one estimator can be “better” than another. You should supply a reasonable criterion, then figure out which one is better according to this criterion.
Homework Answers
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....
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...
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 X1. . . . Xn be i.i.d Uniform over the interval (θ, θ + 1].Show...
Let X1. . . . Xn be i.i.d Uniform over the interval (θ, θ + 1].Show that X(1)+X(n) )/2- 1/2 is also an unbiased estimator of θ, whereX(1) is the minimum order statistic and X(n) is the maximum order statistic. If X - 1/2 is also an unbiased estimator of θ which of the two estimators would you prefer to use.
Suppose X1, X2, . . . , Xn are a random sample from a Uniform(0, θ) distribution, where θ > 0. Consider two different...
Suppose X1, X2, . . . , Xn are a random sample from a Uniform(0, θ) distribution, where θ > 0. Consider two different estimators of θ: R1 = 2X¯ R2 =(n + 1)/n max(X1, . . . , Xn) (a) For each of the estimators R1 and R2, assess whether it is an unbiased estimator of θ. (b) Compute the variances of R1 and R2. Under what conditions will R2 have a smaller variance than R1?
7. Let X1, · · · , Xn be i.i.d. with the density p(x, θ) = θ k (1 − θ) 1−k I{x = 0, 1} (a) Find the ML estimator of θ. (...
7. Let X1, · · · , Xn be i.i.d. with the density p(x, θ) = θ k
(1 − θ) 1−k I{x = 0, 1}
(a) Find the ML estimator of θ.
(b) Is it unbiased ?
(c) Compute its MSE
7. Let Xi, . . . , Xn be i.id, with the density p(z,0)-gk(1-0)1-k1(z-0, 1) (b) Is it unbiased? (c) Compute its MSE
7. Let Xi, . . . , Xn be i.id, with the density p(z,0)-gk(1-0)1-k1(z-0, 1)...
Let X1, ..., Xn be IID observations from Uniform(0, θ). T(X) = max(X1, . . ....
Let X1, ..., Xn be IID observations from Uniform(0, θ). T(X) = max(X1, . . . Xn) is a sufficient statistic (additionally, T is the MLE for θ). Find a (1 − α)-level confidence interval for θ. [Note: The support of this distribution changes depending on the value of θ, so we cannot use Fisher’s approximation for the MLE because not all of the regularity assumptions hold.]
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)...
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ...
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ E Θ : (0.00) (a) Calculate the distribution function Fo of Xi (b) Let x1, .., xn be a realization of X1, Xn. What is the log-likelihood- function for the parameter θ? (c) Calculate the maximum-likelihood-estimator θ(x1, , xn) for the unknown parameter θ
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function...
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
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....
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...
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...
7. Let X1, · · · , Xn be i.i.d. with the density p(x, θ) = θ k
(1 − θ) 1−k I{x = 0, 1}
(a) Find the ML estimator of θ.
(b) Is it unbiased ?
(c) Compute its MSE
7. Let Xi, . . . , Xn be i.id, with the density p(z,0)-gk(1-0)1-k1(z-0, 1) (b) Is it unbiased? (c) Compute its MSE
7. Let Xi, . . . , Xn be i.id, with the density p(z,0)-gk(1-0)1-k1(z-0, 1)...
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)...
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ E Θ : (0.00) (a) Calculate the distribution function Fo of Xi (b) Let x1, .., xn be a realization of X1, Xn. What is the log-likelihood- function for the parameter θ? (c) Calculate the maximum-likelihood-estimator θ(x1, , xn) for the unknown parameter θ
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
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