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Recall that ?-n/ ??-1 log Xi is the mle of ? for a beta(8.1) distribution.Also -_...
6.2.1 2. Recall that θ--r/ Σ (θ, 1 ) distribution. Also, W - i-1 log Xi...
6.2.1 2. Recall that θ--r/ Σ (θ, 1 ) distribution. Also, W - i-1 log Xi has the gamma distribution Г(n, 1/ ) -1 log X, is the mle of θ for a beta (a) Show that 2θW has a X2(2n) distribution. (b) Using part (a), find ci and c2 so that (6.2.35) for 0 < α < 1 . Next, obtain a (1-a) 100% confidence interval for θ.
3. Suppose that ai . ,,an are a random sample from a N( ,02) distribution. Recall...
3. Suppose that ai . ,,an are a random sample from a N( ,02) distribution. Recall that the MLE in this case is [a, σ]T = [x, V (n-1)s2/n]T and the information matrix is Consider the data s2-4.84 with n 16 (a) Use the delta-method to obtain an approximate 95% confidence interval for log(o) (b) Obtain an approximate 95% confidence interval for σ2 using the confidence interval from (a). Compare to the exact interval, [2.21,15.77], and approximate interval [0.43, 10.50...
5. Let X1,X2,. Xn be a random sample from a Beta(0, 1) distribution. Recall that W...
5. Let X1,X2,. Xn be a random sample from a Beta(0, 1) distribution. Recall that W -Σ-1 logXi has the gamma distribution Γ(n,1/8) a) Show that 2θW has a χ"(2n) distribution b) Using part a), find c1 and c2 so that P (cı < 쯩 < c2)-1-α, for 0 < α obtain a (1-a) 100% CI for 20n 1, and then
For each of the following simulation studies, please try two different sample sizes (n 30 and n 3...
Using Rstudio to this question. Begin with
set.seed(38257890)
For each of the following simulation studies, please try two different sample sizes (n 30 and n 300). When comparing the estimators, you need to consider both of the bias and variance of the estimates across 100 simulated samples with the same sizes. Please choose your own parameter(s) for the distributions 1. Conduct a simulation study to compare the method of moment estimator and MLE for the parameters of a Beta distribution...
For each of the following simulation studies, please try two different sample sizes (n = 30 and n...
For each of the following simulation studies, please try two different sample sizes (n = 30 and n = 300). When comparing the estimators, you need to consider both of the bias and variance of the estimates across 100 simulated samples with the same sizes. Please choose your own parameter(s) for the distributions. 1. Conduct a simulation study to compare the method of moment estimator and MLE for the parameters of a Beta distribution or Gamma distribution. (You only need...
For each of the following simulation studies, please try two different sample sizes (n = 30 and n...
For each of the following simulation studies, please try two different sample sizes (n = 30 and n = 300). When comparing the estimators, you need to consider both of the bias and variance of the estimates across 100 simulated samples with the same sizes. Please choose your own parameter(s) for the distributions. 1. Conduct a simulation study to compare the method of moment estimator and MLE for the parameters of a Beta distribution or Gamma distribution. (You only need...
Let Xi iid∼ N(0, θ) for i = 1, ..., n. a) Find the MLE for...
Let Xi iid∼ N(0, θ) for i = 1, ..., n.
a) Find the MLE for θ. Call it
b) Is biased?
c) Is
consistent?
d) Find the variance of
(e) What is the asymptotic distribution of ?
4. Let Xi" pois(1) for i = 1, ..., n. (a) Find the MLE for d....
4. Let Xi" pois(1) for i = 1, ..., n. (a) Find the MLE for d. Call it û (b) What is the asymptotic distribution of î? (c) Find an estimator for P(X1 < 1) (hint: write down P(X1 < 1) using the distribution. Can you use a property of MLEs to get this? )
4. Xi ,i = 1, , n are iid N(μ, σ2). (a) Find the MLE of...
4. Xi ,i = 1, , n are iid N(μ, σ2). (a) Find the MLE of μ, σ2. Are these unbiased estimators of μ and of σ2 respectively? Aside: You can use your result in (b) to justify your answer for the bias part of the MLE estimator of σ2 (b) In this part you will show, despite that the sample variance is an unbiased estimator of σ2, that the sample standard deviation is is a biased estimator of σ....
6. Let Xi 1,... ,Xn be a random sample from a normal distribution with mean u and variance ơ2 whi...
6. Let Xi 1,... ,Xn be a random sample from a normal distribution with mean u and variance ơ2 which are both unknown. (a) Given observations xi, ,Xn, one would like to obtain a (1-a) x 100% one-sided confidence interval for u as a form of L E (-00, u) the expression of u for any a and n. (b) Based on part (a), use the duality between confidence interval and hypothesis testing problem, find a critical region of size...
6.2.1 2. Recall that θ--r/ Σ (θ, 1 ) distribution. Also, W - i-1 log Xi has the gamma distribution Г(n, 1/ ) -1 log X, is the mle of θ for a beta (a) Show that 2θW has a X2(2n) distribution. (b) Using part (a), find ci and c2 so that (6.2.35) for 0 < α < 1 . Next, obtain a (1-a) 100% confidence interval for θ.
3. Suppose that ai . ,,an are a random sample from a N( ,02) distribution. Recall that the MLE in this case is [a, σ]T = [x, V (n-1)s2/n]T and the information matrix is Consider the data s2-4.84 with n 16 (a) Use the delta-method to obtain an approximate 95% confidence interval for log(o) (b) Obtain an approximate 95% confidence interval for σ2 using the confidence interval from (a). Compare to the exact interval, [2.21,15.77], and approximate interval [0.43, 10.50...
5. Let X1,X2,. Xn be a random sample from a Beta(0, 1) distribution. Recall that W -Σ-1 logXi has the gamma distribution Γ(n,1/8) a) Show that 2θW has a χ"(2n) distribution b) Using part a), find c1 and c2 so that P (cı < 쯩 < c2)-1-α, for 0 < α obtain a (1-a) 100% CI for 20n 1, and then
Using Rstudio to this question. Begin with
set.seed(38257890)
For each of the following simulation studies, please try two different sample sizes (n 30 and n 300). When comparing the estimators, you need to consider both of the bias and variance of the estimates across 100 simulated samples with the same sizes. Please choose your own parameter(s) for the distributions 1. Conduct a simulation study to compare the method of moment estimator and MLE for the parameters of a Beta distribution...
For each of the following simulation studies, please try two different sample sizes (n = 30 and n = 300). When comparing the estimators, you need to consider both of the bias and variance of the estimates across 100 simulated samples with the same sizes. Please choose your own parameter(s) for the distributions. 1. Conduct a simulation study to compare the method of moment estimator and MLE for the parameters of a Beta distribution or Gamma distribution. (You only need...
For each of the following simulation studies, please try two different sample sizes (n = 30 and n = 300). When comparing the estimators, you need to consider both of the bias and variance of the estimates across 100 simulated samples with the same sizes. Please choose your own parameter(s) for the distributions. 1. Conduct a simulation study to compare the method of moment estimator and MLE for the parameters of a Beta distribution or Gamma distribution. (You only need...
Let Xi iid∼ N(0, θ) for i = 1, ..., n.
a) Find the MLE for θ. Call it
b) Is biased?
c) Is
consistent?
d) Find the variance of
(e) What is the asymptotic distribution of ?
4. Let Xi" pois(1) for i = 1, ..., n. (a) Find the MLE for d. Call it û (b) What is the asymptotic distribution of î? (c) Find an estimator for P(X1 < 1) (hint: write down P(X1 < 1) using the distribution. Can you use a property of MLEs to get this? )
4. Xi ,i = 1, , n are iid N(μ, σ2). (a) Find the MLE of μ, σ2. Are these unbiased estimators of μ and of σ2 respectively? Aside: You can use your result in (b) to justify your answer for the bias part of the MLE estimator of σ2 (b) In this part you will show, despite that the sample variance is an unbiased estimator of σ2, that the sample standard deviation is is a biased estimator of σ....
6. Let Xi 1,... ,Xn be a random sample from a normal distribution with mean u and variance ơ2 which are both unknown. (a) Given observations xi, ,Xn, one would like to obtain a (1-a) x 100% one-sided confidence interval for u as a form of L E (-00, u) the expression of u for any a and n. (b) Based on part (a), use the duality between confidence interval and hypothesis testing problem, find a critical region of size...
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