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Question 3 [Sans R Say that you observe five random variables from the continuous uniform distribution...
R] Question 3 [Sans Say that you observe a two random variables, x1 and x2, from...
R] Question 3 [Sans Say that you observe a two random variables, x1 and x2, from the continuous uniform distribution of [0,0].The continuous uniform distribution has density 0 otherwise. We would like to find an estimator for θ based on the data x1 and x2 What is the mcan squared error of the estimator i+ 2 (b). Let i - max(1, a2). The density of the estimator is 0 otherwise. hat is the mean squared error of the estimator r?...
Problem 2 [Sans R] Say you observe 2 independent random variables, labeled ri and r2, that...
Problem 2 [Sans R] Say you observe 2 independent random variables, labeled ri and r2, that come from a uniform distribution on the numbers 1, ···.0. This distribution is described by . . . ,0 0 otherwise. (a). What is the mean of in terms of θ? (b). What is the variance of xi in terms of θ? (c). What is the mean of T-21222 în terms of θ? (d). What is the variance of x-n22 in terms of θ?
Question 2 [Sans R You have four friends. Each of them will flip a fair coin...
Question 2 [Sans R You have four friends. Each of them will flip a fair coin θ times and record the number of heads. You do not know 6. But at the end your friends report 18 12 (a). What is a good distribution to use for this data? (b). What is the method of moments estimate for 0? (c). What is the ınaximilin likelihood estimate for θ? Keep in mind that θ must be an 7 integer, i.c. 1,2,3,....
EtxXn be an i.l.d. sample from a uniform( -0.5,0+ 0.5) distribution. (a) Find a method of moments...
etxXn be an i.l.d. sample from a uniform( -0.5,0+ 0.5) distribution. (a) Find a method of moments estimate of θ (b) Suppose n- 2 and the data are 0.6,0.9 Find a formula for the likelihood function, and also sketch the likelihood function. (c) Note that when there are n observations, the maximum likelihood function does imum. Show that one possible maximum is the midrange 2 (d) Find the mean squared errors for the method of moments estimator and midrange. (e)...
4. Let X be continuous random variable whose PDF is given by for r>1 otherwise, where...
4. Let X be continuous random variable whose PDF is given by for r>1 otherwise, where 0 is an unknown scalar parmeter. (a) Find the maximum likelihood estimator of . (b) Find a method-of-moments estimator of θ for the case when θ > 1. (c) Why can we not find a method-of-moments estimator when θ < 1? 151 151 151
Suppose we observe independent pairs (Xi,Yi) where each (Xi,Yi) has a uniform distribution in the circle...
Suppose we observe independent pairs (Xi,Yi) where each (Xi,Yi) has a uniform distribution in the circle of unknown radius θ and centered at (0,0) in the plane. Show that for the method of moments estimator and the maximum likelihood estimator, it is the case that the distribution of ˆθ/θ does not depend on θ. Explain why this means we can write MSEθ(ˆ θ) = θ2*MSE1(ˆ θ) where here the subscript θ means “under the assumption that the true value is...
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the...
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the method of ,y,). We moment estimator θ-2Y and the maximum likelihood estimator θ-maxx,Yo, derived the Bias and MSE for both estimators. With the intent to correct the bias of the mle θ we proposed the following new estimator -Imax where the subscript u stands for "unbiased." (a) Find the MSE of (b) Compare the MSE of θυ to the MSE of θ, the original...
2 Let X1, X2, ..., X, be independent continuous random variables from the following distribution: (*)...
2 Let X1, X2, ..., X, be independent continuous random variables from the following distribution: (*) - ar-(-) where I 21 and a > 1 You may use the fact: E[X] = -1 2.1 Show that the maximum likelihood estimator of a is â MLE - Srlos Xi 2.2 Show that the method moment estimator for a is: &mom = 1 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency?
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0....
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
2 Let X1, X2, ..., X., be independent continuous random variables from the following distribution: f(x)=ox-(-V...
2 Let X1, X2, ..., X., be independent continuous random variables from the following distribution: f(x)=ox-(-V where = 1 and a > 1 You may use the fact: E(X)= -1 2.1 Show that the maximum likelihood estimator of a isante = sok X. 2.2 Show that the method moment estimator for a is: & mom = 1 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency?
R] Question 3 [Sans Say that you observe a two random variables, x1 and x2, from the continuous uniform distribution of [0,0].The continuous uniform distribution has density 0 otherwise. We would like to find an estimator for θ based on the data x1 and x2 What is the mcan squared error of the estimator i+ 2 (b). Let i - max(1, a2). The density of the estimator is 0 otherwise. hat is the mean squared error of the estimator r?...
Problem 2 [Sans R] Say you observe 2 independent random variables, labeled ri and r2, that come from a uniform distribution on the numbers 1, ···.0. This distribution is described by . . . ,0 0 otherwise. (a). What is the mean of in terms of θ? (b). What is the variance of xi in terms of θ? (c). What is the mean of T-21222 în terms of θ? (d). What is the variance of x-n22 in terms of θ?
Question 2 [Sans R You have four friends. Each of them will flip a fair coin θ times and record the number of heads. You do not know 6. But at the end your friends report 18 12 (a). What is a good distribution to use for this data? (b). What is the method of moments estimate for 0? (c). What is the ınaximilin likelihood estimate for θ? Keep in mind that θ must be an 7 integer, i.c. 1,2,3,....
etxXn be an i.l.d. sample from a uniform( -0.5,0+ 0.5) distribution. (a) Find a method of moments estimate of θ (b) Suppose n- 2 and the data are 0.6,0.9 Find a formula for the likelihood function, and also sketch the likelihood function. (c) Note that when there are n observations, the maximum likelihood function does imum. Show that one possible maximum is the midrange 2 (d) Find the mean squared errors for the method of moments estimator and midrange. (e)...
4. Let X be continuous random variable whose PDF is given by for r>1 otherwise, where 0 is an unknown scalar parmeter. (a) Find the maximum likelihood estimator of . (b) Find a method-of-moments estimator of θ for the case when θ > 1. (c) Why can we not find a method-of-moments estimator when θ < 1? 151 151 151
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the method of ,y,). We moment estimator θ-2Y and the maximum likelihood estimator θ-maxx,Yo, derived the Bias and MSE for both estimators. With the intent to correct the bias of the mle θ we proposed the following new estimator -Imax where the subscript u stands for "unbiased." (a) Find the MSE of (b) Compare the MSE of θυ to the MSE of θ, the original...
2 Let X1, X2, ..., X, be independent continuous random variables from the following distribution: (*) - ar-(-) where I 21 and a > 1 You may use the fact: E[X] = -1 2.1 Show that the maximum likelihood estimator of a is â MLE - Srlos Xi 2.2 Show that the method moment estimator for a is: &mom = 1 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency?
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
2 Let X1, X2, ..., X., be independent continuous random variables from the following distribution: f(x)=ox-(-V where = 1 and a > 1 You may use the fact: E(X)= -1 2.1 Show that the maximum likelihood estimator of a isante = sok X. 2.2 Show that the method moment estimator for a is: & mom = 1 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency?
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