L¼ (3:61)

aþ1

j¼1 xj

This yields the MLE of a

" #À1

X

n

Xj

^

a ¼n , (3:62)

log

^

x0

j¼1

80 PARETO DISTRIBUTIONS

whereas for the threshold x0 the estimator is given by

x0 ¼ X1:n : (3:63)

^^

It should be noted that, since x0 x and x0, x0 overestimates x0 . If this estimator is

^

used to solve for a in (3.62), it is seen that a is also overestimated.

Some direct calculations show that

na

^

E(a ) ¼ n . 2,

,

nÀ2

n 2 a2

^

var(a ) ¼ n . 3,

,

(n À 2)2 (n À 3)

yielding

a2 (n2 þ 4n À 12)

^

MSE(a ) ¼ , n . 3:

(n À 2)2 (n À 3)

^

The corresponding results for x0 are

nx0 a 1

n.

E(^ 0 ) ¼

x , ,

a

na À 1

nx0 a2 2

n.

var(^ 0 ) ¼

x , ,

(na À 1)2 (na À 2) a

yielding

2x2 2

0

:

n.

MSE(^ 0 ) ¼

x ,

a

(na À 1)(na À 2)

^

Both estimators are consistent (Quandt, 1966b). Since 1=a is asymptotically

^

ef¬cient in the exponential case, the same is true of a . On the other hand, n(X1:n À

x0 ) follows asymptotically an Exp(0, x0 =a) distribution and hence is biased. Saksena

and Johnson (1984) showed that the maximum likelihood estimators are jointly

complete.

If the parameter a is known, then T ¼ X1:n will be the complete suf¬cient statistic

with p.d.f.

na na

f (t) ¼ x, x0 t, (3:64)

naþ1 0

t

a Par(na, x0 ) distribution [cf. (3.34)].

81

3.6 ESTIMATION

The MLE for the shape parameter a, for known scale x0 , equals

n

a ¼ Pn

^ :

i¼1 log(Xj =x0 )

This estimator is also complete and suf¬cient. Incidentally, its distribution has

the p.d.f.

(an)n À(nþ1) À(an)=x

, x . 0,

f (x) ¼ x e (3:65)

G(n)

which can be recognized as the density of a Vinci (1921) distribution that is

discussed in greater detail in Chapter 5. It should be noted that the situation where x0

is known is not uncommon in actuarial applications, where, for example, in

reinsurance the reinsurer is only concerned with losses above a certain

predetermined level, the retention level.

If the parameters a, x0 are both unknown, then the complete suf¬cient statistic is

(T , S), where

X

n

Xi

T ¼ X1:n , S¼ , (3:66)

log

X1:n

i¼1

which has the density function

nan xna snÀ2 eÀas

0

f (s, t) ¼ , x0 t, 0 s, (3:67)

Á

G(n À 1) t naþ1

^

^

showing that a and x0 are mutually independent. [This is also a direct consequence

of the independence of exponential spacings, the property underlying characteri-

^

zation (3.44).] Hence, the marginal density of a is now given by

°anÞnÀ1 Àn Àan=x

x . 0,

f (x) ¼ xe , (3:68)

G(n À 1)

again an inverse gamma (Vinci) distribution.

In view of the importance of grouped data in connection with size distributions,

we brie¬‚y mention relevant work pertaining to the classical Pareto distribution.

Maximum likelihood estimation from grouped data when x0 is known was studied