aÃ ¼ 1 À a: ^ (3:72)

n

These are, in fact, the UMVU estimators of the parameters. This is a direct

consequence of the fact that (a, x0 ) is a suf¬cient statistic and (aÃ , xÃ ) is a function

^^ 0

^^

of (a, x0 ). These estimators were obtained by Like (1969), with simpli¬ed

s

derivations later given by Baxter (1980).

The variances of the UMVUEs are found to be

a2

Ã

, n . 3,

var(a ) ¼ (3:73)

nÀ3

and

x2 2

0

var(xÃ ) :

n.

, (3:74)

¼

0

a(n À 1)(an À 2) a

Equation 3.73 shows that the UMVUE almost attains the Cramer“ Rao bound

´

2

for a, which is a =n.

Being a rescaled ML estimator, the UMVUE of a evidently also follows an

inverse gamma distribution. In addition, note that, although x0 is necessarily

positive, xÃ is negative if a , 1=(n À 1), that is, if 2na=a . 2n(n À 1)a. Since

^ ^

0

^

2na=a follows an inverse gamma distribution, this occurs with nonzero probability.

However, this probability appears to be quite small in practice.

If x0 is known, the UMVUE of a is

1

Ã

^

a ¼ 1 À a,

n

with variance

a2

var(aÃ ) ¼ , n . 2:

nÀ2

Similarly, for known a the UMVUE of x0 is

& '

1

xÃ ¼ X1:n 1À ,

0

na

with variance

x2 2

0

var(xÃ ) ¼ :

n.

,

0

an(an À 2) a

85

3.6 ESTIMATION

Moothathu (1986) has studied the UMVU estimation of the quantiles, the mean,

as well as the geometric and harmonic means, for both known and unknown x0 . [For

known x0 , the estimators of the moments, the geometric mean, and the median were

derived by Kern (1983) somewhat earlier.] The estimators are conveniently

expressed in terms of various special functions.

The estimators of the mean are given by

x0 1 F1 (1; n; nS1 ) if x0 is known (3:75)

S2

X1:n 1 F1 (1; n À 1; nS2 ) À 1 F1 (1; n À 1; nS2 ) if x0 is unknown, (3:76)

nÀ1

where 1 F1 is Kummer™s con¬‚uent hypergeometric function.

Here

1X n

Xi

S1 ¼ , (3:77)

log

n i¼1 x0

1X n

Xi

:

S2 ¼ (3:78)

log

n i¼1 X1:n

The estimators for the uth quantile are

x0 0 F1 [À; n; ÀnS1 log(1 À u)] if x0 is known, (3:79)

&

X1:n 0 F1 [À; n À 1; ÀnS2 log(1 À u)]

'

S2

0 F 1 [À; n; ÀnS2 log(1 À u)] if x0 is unknown, (3:80)

À

nÀ1

where 0 F1 is a Bessel function. (See p. 288 for a de¬nition of p Fq :)

Being equal to the (1 À 1=e)th quantile, the geometric mean is therefore esti-

mated by

x0 0 F1 (À; n; nS1 ) if x0 is known, (3:81)

& '

S2

X1:n 0 F1 (À; n À 1; nS2 ) À 0 F1 (À; n; nS2 ) if x0 is unknown: (3:82)

nÀ1

In the case of the harmonic mean, the corresponding estimators are

x0 (1 þ S1 ) if x0 is known, (3:83)

S2

X1:n 1 F1 (À1; n À 1; ÀnS2 ) À 1 F1 (À1; n; ÀnS2 ) if x0 is unknown: (3:84)

nÀ1

86 PARETO DISTRIBUTIONS

All of the previously given estimators are moreover strongly consistent.

Further results are available for the coef¬cient of variation, for which the

UMVUEs are (Moothathu, 1988)

1

S1 1 F 1 ; n þ 1; 2nS1 if x0 is known, (3:85)

2

1

S2 1 F1 ; n; 2nS2 if x0 is unknown, (3:86)

2

the mode (Moothathu, 1986), the skewness and kurtosis coef¬cients (Moothathu, 1988),

the p.d.f. and c.d.f. (Asrabadi, 1990), and the mean excess function (Rytgaard, 1990).

Independently, Woo and Kang (1990) obtained the UMVUEs for a whole class of

p