(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
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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ad2bb3e-a31d-4eda-9e39-bcf1d8a1082c%2F8222571d-2b2c-4256-9cd2-d42f724e7523%2F33j26kg_processed.png&w=3840&q=75)
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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