A random variable Y is said to have a Singh-Maddala distribution with parameter θ > 0 if the density function of Y is f(y|θ) = 2θy (1 + y2)θ+1 for y > 0. Suppose that Y1,Y2,...,Yn is a random sample from a Singh-Maddala(θ) population where θ > 0 is a parameter. (a) Verify that f(y|θ) belongs to an exponential family in traditional form. (b) Compute the likelihood function
A random variable Y is said to have a Singh-Maddala distribution with parameter θ > 0 if the density
f(y|θ) = 2θy
(1 + y2)θ+1
for y > 0. Suppose that Y1,Y2,...,Yn is a random sample from a Singh-Maddala(θ) population where θ > 0 is a parameter.
(a) Verify that f(y|θ) belongs to an exponential family in traditional form.
(b) Compute the likelihood function L(θ) for this random sample.
(c) Determine θˆMLE, the maximum likelihood estimator of θ. You may use any method that we discussed in class for finding the MLE.
(d) Find the Fisher information I(θ) in a single observation from this density.
(e) Using the standard normal approximation for the distribution of a maximum likelihood es- timator based on the Fisher information, construct an approximate 95% confidence interval for θ. Note that z0.025 = 1.96.
Note that the Singh-Maddala distribution is also known as the Burr Type XII distribution and is used in economics and actuarial science to model income sizes.
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