Let X,..., Xn be an independent and identically distributed sequence of random variables from a population...
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Let X₁,..., Xn be an independent and identically distributed sequence of random variables from a population in (g(x | 0)| 0>0}, where g(x | 0) = 0x⁰-1, 0 < x < 1. (a) [5 points] Show that the expectation of X₁ is equal to 1 (b) [10 points] Show that the method of moment estimator for 00 is given by ÔMM X 1-X Is this an unbiased estimator of 00? Hint: The function h(x) = ₁ is convex for all 0 < x < 1. (c) [10 points] Show that the maximum likelihood for 80 is given by ÔML n Σi=1 In(X₁) (d) [10 points] Show that {ge | 0> 0} is an exponential family and use the exponential family representation to calculate a sufficient statistic for 00. (e) [10 points] Using (d), argue that the estimator ODB==¹0ML is the UMVUE of 00- Hint: You may use that Eo (OML) = 10. (f) (10 points] Calculate the Cramér-Rao lower bound for any unbiased estimator of Does Windows ODB attain this bound? Does this contradict the claim in (e)? Go to Settings to activate Win Hint: You may use that Varo (OML) = (n-1)(n-20. Let X₁,..., Xn be an independent and identically distributed sequence of random variables from a population in (g(x | 0)| 0>0}, where g(x | 0) = 0x⁰-1, 0 < x < 1. (a) [5 points] Show that the expectation of X₁ is equal to 1 (b) [10 points] Show that the method of moment estimator for 00 is given by ÔMM X 1-X Is this an unbiased estimator of 00? Hint: The function h(x) = ₁ is convex for all 0 < x < 1. (c) [10 points] Show that the maximum likelihood for 80 is given by ÔML n Σi=1 In(X₁) (d) [10 points] Show that {ge | 0> 0} is an exponential family and use the exponential family representation to calculate a sufficient statistic for 00. (e) [10 points] Using (d), argue that the estimator ODB==¹0ML is the UMVUE of 00- Hint: You may use that Eo (OML) = 10. (f) (10 points] Calculate the Cramér-Rao lower bound for any unbiased estimator of Does Windows ODB attain this bound? Does this contradict the claim in (e)? Go to Settings to activate Win Hint: You may use that Varo (OML) = (n-1)(n-20.
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Mathematical Statistics With Applications In R
ISBN: 9780124171138
2nd Edition
Authors: Chris P. Tsokos, K.M. Ramachandran
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