7-52. Let XX.. . . . . X be uniformly distributed on the interval 1 2 0 to a. Re

Question

7-52. Let XX.. . . . . X be uniformly distributed on the interval 1 2 0 to a. Recall that the maximum likelihood estimator of a is (a) Argue intuitively why à cannot be an unbiased estimator (b) Suppose that E(a) na/ (n1). Is it reasonable that â con- max(Xi). for a sistently underestimates a? Show that the bias in the esti- mator approaches zero as n gets large. (c) Propose an unbiased estimator for a. (d) Let Y=max(Xi). Use the fact that Y y if and only if each X; S y to derive the cumulative distribution function of Y. Then show that the probability density function of Y is ny 0SySa 0, otherwise Use this result to show that the maximum likelihood esti- mator for a is biased (e) We have two unbiased estimators for a: the moment where estimator al=2x and à,-[(n +1) / n] max(Xi), (Xi is the largest observation in a random sample of size n. It can be shown that V(al)=a2/(3n) and that V(d»)=a2/[n(n +2)]. Show that if n> 1, a, is a better estimator than a. In what sense is it a better estimator of a?

Explanation / Answer

a) this is because a^ is always less than a .
hence it can't be unbiased estimator

b)
bias = E(a^) -a
= an/(n+1) -a
= - a /(n+1)
clearly as n tend to infinity
1/(n+1) tend to 0
hence
bias tend to 0

c)
unbiased estimator for a is (n+1)/n a^

d)

Y = max(Xi)

P(Y < y)

P(max(X_i) < y)

= P(X1 < y, X2 < y ,..Xn < y)

= P(X1 < y) P(X2 < y) ...P(Xn < y)

= (y/a)^n

pdf = d/dy F(y)

= n (y/a)^(n-1) * 1/a

= n y^(n-1) /a^n

e)
better estimator is the one which is unbiased and has less variance
whence n > 1
3n <= n(n+2)
hence
Var(a1^) >= Var(a2^2)    {note that 3n and n(n+2) was in denominator}
hence a2^ is better
it is better in sense of consistency

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