4. (11 marks) Let X1, X2, . . . , Xn be a random sample from an exponential popu

Question

4. (11 marks) Let X1, X2, . . . , Xn be a random sample from an exponential population whose
density function is given by

f(x) = (1/?) e^(?x/?) , x > 0 where ? > 0.

(d) Find the MLE of the first quartile of this distribution.
(e) Find the likelihood ratio test for testing

H0 : ? = ?0 versus H1 : ? not= ?0

for given ?0 > 0. Show that this test is defined by
Reject H0 if ?(hat) < c1 or ?(hat) > c2,

where c1 and c2 are some constants.

(f) Find the most powerful test of size ? for testing

H0 : theat = theta0 versus H1 : theta = theta1

where 0 < ?0 < ?1 are given parameter values.

(g) Show that this family has the monotone likelihood ratio property. Then, show that
the test in part

(f) is the uniformly most powerful test of size ? for testing

H0 : theata=theta0 versus H1 : theta= theta1

for given 0 < ?0 < ?1

(h) Using the exact distribution of summation(i=1 to n ) of Xi, construct a two-sided 100(1??)% confidence interval for ?.

Explanation / Answer

n. Let X1, X2,..., Xn be a random sample from a distribution that depends on one or more unknown parameters ?1, ?2,..., ?m with probability density (or mass) function f(xi; ?1, ?2,..., ?m). Suppose that (?1, ?2,..., ?m) is restricted to a given parameter space ?. Then:

(1) When regarded as a function of ?1, ?2,..., ?m, the joint probability density (or mass) function of X1, X2,..., Xn:

L(?1,?2,…,?m)=n?i=1f(xi;?1,?2,…,?m)L(?1,?2,…,?m)=?i=1nf(xi;?1,?2,…,?m)

((?1, ?2,..., ?m) in ?) is called the likelihood function.

(2) If:

[u1(x1,x2,…,xn),u2(x1,x2,…,xn),…,um(x1,x2,…,xn)][u1(x1,x2,…,xn),u2(x1,x2,…,xn),…,um(x1,x2,…,xn)]

is the m-tuple that maximizes the likelihood function, then:

^?i=ui(X1,X2,…,Xn)?^i=ui(X1,X2,…,Xn)

is the maximum likelihood estimator of ?i, for i = 1, 2, ..., m.

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