Suppose X and Y are independent, Pareto-distributed, with cumulative distributions given by FX(x;
)=1x1,FY(y;)=1y1, with x,y1 and ,>0. Let Z=min{X,Y} and define the (non)censoring indicator ={10if
X<Yotherwise (This type of censoring is often known as "type I censoring.") (a) (10 marks) Obtain
the density function of Z(fZ) and the frequency function of (f). What are the distributions of Z and ?
(b) (5 marks) Let Z1,,Zn be a random sample from fZ(z;), with =+, and let 1,,n be a random sample
from f(d;p), with p=/(+). Derive the maximum liklihood estimators of and p. (c) (8 marks) Appealing
to the asymptotic normality of the maximum likelihood estimator, provide a 95% confidence
interval for and for p..
Suppose X and Y are independent Paretodistributed with cu.pdf
1. Suppose X and Y are independent, Pareto-distributed, with cumulative distributions given by FX(x;
)=1x1,FY(y;)=1y1, with x,y1 and ,>0. Let Z=min{X,Y} and define the (non)censoring indicator ={10if
X<Yotherwise (This type of censoring is often known as "type I censoring.") (a) (10 marks) Obtain
the density function of Z(fZ) and the frequency function of (f). What are the distributions of Z and ?
(b) (5 marks) Let Z1,,Zn be a random sample from fZ(z;), with =+, and let 1,,n be a random sample
from f(d;p), with p=/(+). Derive the maximum liklihood estimators of and p. (c) (8 marks) Appealing
to the asymptotic normality of the maximum likelihood estimator, provide a 95% confidence
interval for and for p.
