Let X be a random variable with density f (x) Cav-1 (0 < x < 1) ne constant C e R and 0 > 0. From the second subquestion on, you 1 an i.i.d. sample X1,..., X, from this distribution. 72 etermine the constant C as a function of 0.

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Let X be a random variable with density f (x) = Crv8–1 (0 < x < 1)
for some constant Ce R and 0 > 0. From the second subquestion on, you
rely on an i.i.d. sample X1,..., X, from this distribution.
a. Determine the constant C as a function of 0.
b. Determine the MLE for the parameter 0.
c. Determine the asymptotic distribution of the MLE for large sample size
n.
d. Construct an asymptotic confidence interval for 0 of confidence level
0.95.
e. Provide a plugin estimator of the tail probability P(X > 0.99) (hence
plugging in the estimator of 0 in the theoretical formula for P(X >
0.99)).
f. Compute the asymptotic bias and variance of your estimator of P(X >
0.99) using the delta method. [Recall here that a" = (log a)a*]
Transcribed Image Text:1. Let X be a random variable with density f (x) = Crv8–1 (0 < x < 1) for some constant Ce R and 0 > 0. From the second subquestion on, you rely on an i.i.d. sample X1,..., X, from this distribution. a. Determine the constant C as a function of 0. b. Determine the MLE for the parameter 0. c. Determine the asymptotic distribution of the MLE for large sample size n. d. Construct an asymptotic confidence interval for 0 of confidence level 0.95. e. Provide a plugin estimator of the tail probability P(X > 0.99) (hence plugging in the estimator of 0 in the theoretical formula for P(X > 0.99)). f. Compute the asymptotic bias and variance of your estimator of P(X > 0.99) using the delta method. [Recall here that a" = (log a)a*]
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