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Two continuous random variables have the joint density function 23,4 , where T > 0 and U > 0 are parameters. Consider estimating =TU using Y. Is Y a consistent
estimator for ? Prove your claim.
Hint: Derive the mean squared error of the estimator.
Question 2
The coskewness of three random variables *, 1, Z is defined as ?where \3 and H3 are the mean and standard deviations of *
respectively, and \4,H4,\],H] are similarly defined. Suppose we have a random sample of size N
from <, the joint CDF of .
a) Derive the plug-in estimator for the coskewness of *, 1, Z.
b) Write a Matlab code to construct bootstrap percentile and pivotal intervals for coskewness.
Assume that
1 0.3 0.3
0.3 1 0.3
0.3 0.3 1
_e, draw a random sample of size 100 from this
distribution, then construct 95% bootstrap and percentile intervals for coskewness based on the
sample.
Question 3
Write Matlab codes to generate random samples from the gamma distribution using a) the
inverse transform method and b) the rejection sampling method with samples from an
exponential distribution with rate parameter 6 = 1/fT as presented in the lecture notes.
Compare the time taken to generate 10g
random values from the gamma distribution with
parameters f, T = h1,1 , 10,1 , 1,10 , 10,10 i.
Hint: use tic and toc to record computation time.
Question 4
Suppose that *, … , * is a random sample from <. Derive the maximum likelihood estimator of
the parameter T and prove its consistency or otherwise when
a) < is a normal distribution with mean \ and variance H
b) < is a Poisson distribution with mean 6, and T = 6.
Two continuous random variables have the joint density function 23,4 , where T > 0 and U > 0 are parameters. Consider estimating =TU using Y. Is Y a consistent
estimator for ? Prove your claim.
Hint: Derive the mean squared error of the estimator.
Question 2
The coskewness of three random variables *, 1, Z is defined as ?where \3 and H3 are the mean and standard deviations of *
respectively, and \4,H4,\],H] are similarly defined. Suppose we have a random sample of size N
from <, the joint CDF of .
a) Derive the plug-in estimator for the coskewness of *, 1, Z.
b) Write a Matlab code to construct bootstrap percentile and pivotal intervals for coskewness.
Assume that
1 0.3 0.3
0.3 1 0.3
0.3 0.3 1
_e, draw a random sample of size 100 from this
distribution, then construct 95% bootstrap and percentile intervals for coskewness based on the
sample.
Question 3
Write Matlab codes to generate random samples from the gamma distribution using a) the
inverse transform method and b) the rejection sampling method with samples from an
exponential distribution with rate parameter 6 = 1/fT as presented in the lecture notes.
Compare the time taken to generate 10g
random values from the gamma distribution with
parameters f, T = h1,1 , 10,1 , 1,10 , 10,10 i.
Hint: use tic and toc to record computation time.
Question 4
Suppose that *, … , * is a random sample from <. Derive the maximum likelihood estimator of
the parameter T and prove its consistency or otherwise when
a) < is a normal distribution with mean \ and variance H
b) < is a Poisson distribution with mean 6, and T = 6.