- Answer From E Onwards Please 1 (116.22 KiB) Viewed 33 times
Answer from e onwards please
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Answer from e onwards please
Answer from e onwards please
Question 1. Consider a sample of >2 independent and identically distributed random variables Y... Y, from an exp(A) distribution with probability density function f(y) exp(-xy) A> 0. We will use the notation 8. (a) Let Y₁ min(Y₁...Y) be the smallest observation in the sample, and this sample is collected from a distribution where y(y) and Fy(y) are the probability density function and distribution function respectively. Show that Fy(9) 1-1-Fy(9)]" Hint: Evaluate P(Y() >y). (b) Show that has probability density function (pdf): fro(@)=n]1 - Fy(y)]"−¹ƒv (y)- (c) Hence show that Yu follows an exp[rà) distribution if the sample is from an exp(A) distribution. (d) Show that the following two estimators are both unbiased estimators of 6: and 0₂-Y where Y is the sample mean. Hint: You may need to use common probability distributions PDF on iLearn. (c) Derive the relative efficiency of 8, compared to . Which estimator do you prefer? (f) Determine whether , and are consistent estimators of 6. log-likelihood function, as functions. (g) Write down the likelihood function and of A. (h) Find a sufficient statistic for A. (i) Derive the maximum likelihood estimator (MLE) of X. Comment on your result in relation to the previous part.
Question 1. Consider a sample of >2 independent and identically distributed random variables Y... Y, from an exp(A) distribution with probability density function f(y) exp(-xy) A> 0. We will use the notation 8. (a) Let Y₁ min(Y₁...Y) be the smallest observation in the sample, and this sample is collected from a distribution where y(y) and Fy(y) are the probability density function and distribution function respectively. Show that Fy(9) 1-1-Fy(9)]" Hint: Evaluate P(Y() >y). (b) Show that has probability density function (pdf): fro(@)=n]1 - Fy(y)]"−¹ƒv (y)- (c) Hence show that Yu follows an exp[rà) distribution if the sample is from an exp(A) distribution. (d) Show that the following two estimators are both unbiased estimators of 6: and 0₂-Y where Y is the sample mean. Hint: You may need to use common probability distributions PDF on iLearn. (c) Derive the relative efficiency of 8, compared to . Which estimator do you prefer? (f) Determine whether , and are consistent estimators of 6. log-likelihood function, as functions. (g) Write down the likelihood function and of A. (h) Find a sufficient statistic for A. (i) Derive the maximum likelihood estimator (MLE) of X. Comment on your result in relation to the previous part.
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