Let X1 Xn be a random sample from the uniform distribution o

Let X1,. .,Xn be a random sample from the uniform distribution over the interval (Theta1,Theta2). Is the estimator X(n)-X(1) a consistent estimator for the range of the distribution?

Solution

x1 is uniform with parameter theta1

and x2 with theta 2

x1 has pdf as 1 and x2 pdf as 2

Consider

xn-x1

= Difference in x values

probability for this is the reciprocal of the difference.

Hence xn-x1 is not a range.

Actual 1/xn - 1/x1 can be taken as reciprocal of the range.