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1. (40) Suppose that X1, X2, .. , Xn, forms an normal distribution with mean /u and variance o2, both unknown: independ...
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2,...
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...
4. Let X1, X2, ...,Xn be a random sample from a normal distribution with mean 0...
4. Let X1, X2, ...,Xn be a random sample from a normal distribution with mean 0 and unknown variance o2. (a) Show that U = <!-, X} is a sufficient statistic for o?. [4] (c) Show that the MLE of o2 is Ô = 2-1 X?. [4] (c) Calculate the mean and variance of Ô from (b). Explain why ő is also the MVUE of o2. [6]
1.Suppose X1, X2, .., Xn is a random sample from N(", 02) 10 pts] If o2...
1.Suppose X1, X2, .., Xn is a random sample from N(", 02) 10 pts] If o2 1, u is unknown. Find the MLE of a. b. [10 pts If o2 = 1, p is unknown. f = X is an estimator of u. What is the MSE of this estimator? Now assume o2 is unknown. The following data is a set of observations of X1,..., Xn. Use the dataset to answer (c), (d) and (e) 11 8 9 7 6...
Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a...
Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)
Let X1,, Xn be independent and identically distributed random variables with unknown mean μ and unknown...
Let X1,, Xn be independent and identically distributed random variables with unknown mean μ and unknown variance σ2. It is given that the sample variance is an unbiased estimator of ơ2 Suggest why the estimator Xf -S2 might be proposed for estimating 2, justify your answer
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
1. Suppose that {X1, ... , Xn} is a random sample from a normal distribution with...
1. Suppose that {X1, ... , Xn} is a random sample from a normal distribution with mean p and and variance o2. Let sa be the sample variance. We showed in lectures that S2 is an unbiased estimator of o2. (a) Show that S is not an unbiased estimator of o. (b) Find the constant k such that kS is an unbiased estimator of o. Hint. Use a result from Student's Theorem that (n − 1)52 ~ x?(n − 1)...
Consider a random sample X1, ..., Xn from a normal distribution with known mean 0 and...
Consider a random sample X1, ..., Xn from a normal distribution with known mean 0 and unknown variance 0 = 02 (a) Write the likelihood and log-likelihood function (b) Derive the maximum likelihood estimator for 6 (c) Show that the Fisher information matrix is I(O) = 2014 (d) What is the variance of the maximum likelihood estimator for @? Does it attain the Cramer-Rao lower bound? (e) Suppose that you are testing 0 = 1 versus the alternative 0 #...
2. Let X1,..., Xn be i.i.d. according to a normal distribution N(u,02). (a) Get a sufficient...
2. Let X1,..., Xn be i.i.d. according to a normal distribution N(u,02). (a) Get a sufficient statistic for u. Show your work. (b) Find the maximum likelihood estimator for u. (c) Show that the MLE in part (b) is an unbiased estimator for u. (d) Using Basu's theorem, prove that your MLE from before and sº, the sample variance, are independent. (Hint: use W; = X1-0 and (n-1)32)
1. (50 points) Suppose X1, ..., Xn form a random sample from a N(u,02) distribution with...
1. (50 points) Suppose X1, ..., Xn form a random sample from a N(u,02) distribution with p.d.f. Fe 202, for – V2110 <x< . Assume that o = 2 is known. a) (10 points) Derive the 90% confidence interval for u that has the shortest length. You must show all details including the pivot you use. b) (8 points) Show that the sample mean is an efficient estimator for u. Assume in (c)- (f) that the prior distribution of u...
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...
4. Let X1, X2, ...,Xn be a random sample from a normal distribution with mean 0 and unknown variance o2. (a) Show that U = <!-, X} is a sufficient statistic for o?. [4] (c) Show that the MLE of o2 is Ô = 2-1 X?. [4] (c) Calculate the mean and variance of Ô from (b). Explain why ő is also the MVUE of o2. [6]
1.Suppose X1, X2, .., Xn is a random sample from N(", 02) 10 pts] If o2 1, u is unknown. Find the MLE of a. b. [10 pts If o2 = 1, p is unknown. f = X is an estimator of u. What is the MSE of this estimator? Now assume o2 is unknown. The following data is a set of observations of X1,..., Xn. Use the dataset to answer (c), (d) and (e) 11 8 9 7 6...
Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)
Let X1,, Xn be independent and identically distributed random variables with unknown mean μ and unknown variance σ2. It is given that the sample variance is an unbiased estimator of ơ2 Suggest why the estimator Xf -S2 might be proposed for estimating 2, justify your answer
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
1. Suppose that {X1, ... , Xn} is a random sample from a normal distribution with mean p and and variance o2. Let sa be the sample variance. We showed in lectures that S2 is an unbiased estimator of o2. (a) Show that S is not an unbiased estimator of o. (b) Find the constant k such that kS is an unbiased estimator of o. Hint. Use a result from Student's Theorem that (n − 1)52 ~ x?(n − 1)...
Consider a random sample X1, ..., Xn from a normal distribution with known mean 0 and unknown variance 0 = 02 (a) Write the likelihood and log-likelihood function (b) Derive the maximum likelihood estimator for 6 (c) Show that the Fisher information matrix is I(O) = 2014 (d) What is the variance of the maximum likelihood estimator for @? Does it attain the Cramer-Rao lower bound? (e) Suppose that you are testing 0 = 1 versus the alternative 0 #...
2. Let X1,..., Xn be i.i.d. according to a normal distribution N(u,02). (a) Get a sufficient statistic for u. Show your work. (b) Find the maximum likelihood estimator for u. (c) Show that the MLE in part (b) is an unbiased estimator for u. (d) Using Basu's theorem, prove that your MLE from before and sº, the sample variance, are independent. (Hint: use W; = X1-0 and (n-1)32)
1. (50 points) Suppose X1, ..., Xn form a random sample from a N(u,02) distribution with p.d.f. Fe 202, for – V2110 <x< . Assume that o = 2 is known. a) (10 points) Derive the 90% confidence interval for u that has the shortest length. You must show all details including the pivot you use. b) (8 points) Show that the sample mean is an efficient estimator for u. Assume in (c)- (f) that the prior distribution of u...
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