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(Q1) Let X1, Xn be random sample drawn from a random variable X with pdf f(x; 0) = S0x0-1, 0

(Q1) Let X1, Xn be random sample drawn from a random variable X with pdf f(x; 0) = S0x0-1, 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
Section13.3: Special Probability Density Functions
Problem 30E
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(Q1) Let X1, Xn be random sample drawn from a random variable X with pdf f(x; 0) = S0x0-1, 0<x<1 [, otherwise We can show that the maximum likelihood estimate (MLE) of 0 is given by n On ΣΕ, log (X;) Let Ylog(X). By Law of Large Numbers, Y→ Plog(x) = - when n→ ∞. (i) Show that the Fisher information is given by I(0) = n/0². (ii) Based on large-sample property of MLE, compute 95% confidence interval for 0. (iii) Use the following data to get the 95% confidence interval for 0. data<-c(0.0009, 0.0768, 0.0006, 0.2371, 0.0027, 0.0246, 0.9751, 0.7831, 0.3975, 0.0002, 0.1190, 0.2481, 0.0001, 0.4908, 0.0087, 0.5189, 0.4380, 0.7013, 0.0662, 0.1851, 0.2441, 0.7435, 0.1209, 0.0523, 0.0171) Hint: Use R to compute 95% CI (using the formula in (ii)). (1) (2)
Transcribed Image Text:(Q1) Let X1, Xn be random sample drawn from a random variable X with pdf f(x; 0) = S0x0-1, 0<x<1 [, otherwise We can show that the maximum likelihood estimate (MLE) of 0 is given by n On ΣΕ, log (X;) Let Ylog(X). By Law of Large Numbers, Y→ Plog(x) = - when n→ ∞. (i) Show that the Fisher information is given by I(0) = n/0². (ii) Based on large-sample property of MLE, compute 95% confidence interval for 0. (iii) Use the following data to get the 95% confidence interval for 0. data<-c(0.0009, 0.0768, 0.0006, 0.2371, 0.0027, 0.0246, 0.9751, 0.7831, 0.3975, 0.0002, 0.1190, 0.2481, 0.0001, 0.4908, 0.0087, 0.5189, 0.4380, 0.7013, 0.0662, 0.1851, 0.2441, 0.7435, 0.1209, 0.0523, 0.0171) Hint: Use R to compute 95% CI (using the formula in (ii)). (1) (2)
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