2 1.4793 21.1011 2 0.6833 0.6833 . . .

0.05 13 22.8602 2 2.2974 21.8646

0.025 17 23.0834 2 2.5636 22.1764 2 1.8460 21.5424 2 1.2466 20.9398

2 0.5833 0.5833 . . .

0.01 29 23.5038 2 3.0486 22.7227 2 2.4569 22.2258 2 2.0166 21.8216

2 1.0898 20.8971 2 0.6861 20.4341 0.4341. . .

21.6355 2 1.4541 21.2736

Source: Schader and Schmid (1986).

may be expressed in terms of the decomposition

IX (s) ¼ IN (s) þ IX jN (s):

The relative loss of information is now given by

[@pj (s)=@s]2 =pj (s) s2

IX (s) À IN (s)

L¼ ¼1À 1 À In °sÞ:

Eu [@ log f (X js)=@s]2

IX (s) n

Thus, one requires, for given g [ (0, 1) and s; the smallest integer k Ã such that

L(s; k Ã ; aÃ , . . . , aÃ ) g:

1 kÀ1

Schader and Schmid (1986) found that only for k odd there is a unique global

optimum of this optimization problem and provided optimal class boundaries zÃ j

based on the least odd number of classes k Ã for which the loss of information is

less than or equal to a given value of g; for g ¼ 0:1; 0.05, 0.025, and 0:01: See

Table 4.2. From the table, optimal class boundaries aÃ for a lognormal distribution

j

with parameters m and s can be obtained upon setting aÃ :¼ exp°szÃ þ mÞ:

j j

It is interesting that compared to the Pareto distribution (cf. Section 3.6),

a considerably larger number of classes is required for a given loss of information.

4.7 THREE- AND FOUR-PARAMETER LOGNORMAL

DISTRIBUTIONS

If there exists a l [ IR such that Z ¼ log (X À l) follows a normal distribution, then

X is said to follow a three-parameter lognormal distribution. For this to be the case, it

is clearly necessary that X take any value exceeding l but have zero probability of

taking any value below l: Thus, the p.d.f. of X is

& '

1 1 2

p¬¬¬¬¬¬ exp À 2 [ log (x À l) À m] , x . l:

f (x) ¼ (4:30)

2s

(x À l) 2ps

122 LOGNORMAL DISTRIBUTIONS

As a size distribution, this distribution was already considered by Gibrat (1931). The

distribution is obtained through one of the three transformations in Johnsons™s

(1949) translation system; see (2.76) in Chapter 2.

The location characteristics of the three-parameter form are greater by l than

those of the LN(m, s 2 ) distribution. The mean is at l þ exp(m þ s 2 =2); the median

at l þ exp(m); and the mode at l þ exp(m À s 2 ): The quantiles are displaced from

F À1 (u) to l þ F À1 (u); 0 , u , 1: The kth moment about l is

k2s 2

k

E[(X À l) ] ¼ exp km þ (4:31)

2

so that the moments about the mean and hence the shape factors remain unchanged.

If the threshold parameter l is known, as is, for example, the case in actuarial

applications when it represents a deductible for claim amounts, estimation can, of

course, proceed as described in Section 4.6 after adjusting the data by l: If this

parameter is unknown, there are considerable additional dif¬culties. Since the books

by Cohen and Whitten (1988) and Crow and Shimizu (1988) contain several

chapters studying estimation problems associated with the three-parameter

lognormal distribution, we shall be rather brief here. The main dif¬culty appears

to be that likelihood methods lead to an estimation problem with an unbounded

likelihood. Speci¬cally, Hill (1963) demonstrated that there exists a path along

which the likelihood function

Y

n

L(x1 , . . . , xn ; m, s 2 , l) ¼ f (xi ; m, s 2 , l) (4:32)

i¼1

tends to þ1 as (m, s 2 , l) approaches (À1, 1, x1:n ):

One way of circumventing this dif¬culty consists of considering the observations

as measured with error (being recorded only to the nearest unit of measurement),

leading to a multinomial model as suggested by Giesbrecht and Kempthorne (1976).

With individual data considered error-free, Hill (1963) and Grif¬ths (1980)

justi¬ed using estimates corresponding to the largest ¬nite local maximum of the

likelihood function. Smith (1985, p. 88) noted the existence of a local maximum that

de¬nes an asymptotically normal and ef¬cient estimator. On differentiating the

logarithm of the likelihood (4.32), we obtain the estimating equations

1Xn

@log L

[ log(xi À l) À m] ¼ 0, (4:33)

¼2

s i¼1

@m

1Xn

@log L n

[ log(xi À l) À m]2 ¼ 0, (4:34)

¼À þ 3

s s i¼1

@s

and

1 X log(xi À l) À m X 1

n n

@log L

¼ 0: (4:35)

¼2 þ

s i¼1 xi À l xi À l

@l i¼1

123

4.7 THREE- AND FOUR-PARAMETER LOGNORMAL DISTRIBUTIONS

Eliminating m and s from these equations, we get (Cohen, 1951)

"

X X X

n n n

1

^ ^ ^

log2 (xi À l)

g(l ) :¼ log(xi À l) À

^

xi À l

i¼1 i¼1 i¼1

( )2 #

1X X log(xi À l)

n n ^

^

log(xi À l) Àn ¼ 0, (4:36)

þ

^

n i¼1 xi À l

i¼1

a highly nonlinear equation in one variable. The probably most commonly used

approach to ML estimation seems to be solving (4.36) by, for example, Newton “

Raphson and subsequently obtaining estimates of m and s 2 from this solution

^

(namely, the mean and variance of the logarithms of the data adjusted by l).

However, a practical dif¬culty appears to be that the search for the local MLEs