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a) Let X₁, X₂, ..., Xn denote a random sample from a probability density function given by 1 f(x; a,0) = (T(a)@ª)xª-¹e ,x>0 Where a > 0 is known. x == 0, elsewhere i) Find the maximum likelihood estimator (MLE), say ê of 0. ii) Is the MLE ê a consistent estimator of 0. iii) Use factorisation theorem to find the sufficient statistic for 0?.

a) Let X₁, X₂, ..., Xn denote a random sample from a probability density function given by 1 f(x; a,0) = (T(a)@ª)xª-¹e ,x>0 Where a > 0 is known. x == 0, elsewhere i) Find the maximum likelihood estimator (MLE), say ê of 0. ii) Is the MLE ê a consistent estimator of 0. iii) Use factorisation theorem to find the sufficient statistic for 0?.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 23E
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a) Let X₁, X₂, ..., Xn denote a random sample from a probability density function given by f(x; α,0) = Where a > 0 is known. 1 (T(a)ṇa)xª ,x > 0 0, elsewhere + e i) Find the maximum likelihood estimator (MLE), say ô of 0. ii) Is the MLE ê a consistent estimator of 0. iii) Use factorisation theorem to find the sufficient statistic for 0?. iv) Show whether or not U = ₁X₁ is a pivotal quantity. If yes, use it to =1 calculate a 90% confidence interval for 0 with n = 5 and a = 2.
Transcribed Image Text:a) Let X₁, X₂, ..., Xn denote a random sample from a probability density function given by f(x; α,0) = Where a > 0 is known. 1 (T(a)ṇa)xª ,x > 0 0, elsewhere + e i) Find the maximum likelihood estimator (MLE), say ô of 0. ii) Is the MLE ê a consistent estimator of 0. iii) Use factorisation theorem to find the sufficient statistic for 0?. iv) Show whether or not U = ₁X₁ is a pivotal quantity. If yes, use it to =1 calculate a 90% confidence interval for 0 with n = 5 and a = 2.
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