k kY

E(X ) ¼ E(e ) ¼ exp km þ k s : (4:5)

2

Hence, the lognormal distribution is one of the few well-known distributions that

possesses moments of all orders, positive, negative, and fractional. However, the

sequence of integer moments does not characterize the lognormal distribution;

see Heyde (1963) for a continuous distribution having the same moments as the

lognormal and Leipnik (1991) for a discrete counterexample. This somewhat

unexpected property is of interest in probability theory and has attracted the

attention of numerous researchers.

The low-order moments and other basic characteristics of the lognormal

distribution are as follows. [For compactness the notation v ¼ exp(s 2 ) is used in

several formulas.] From (4.5), the mean of the lognormal distribution is

m þ s2

E(X ) ¼ exp (4:6)

2

and the variance is given by

2 2

var(X ) ¼ e2mþs (es À 1) ¼ e2m v(v À 1): (4:7)

Hence, the coef¬cient of variation equals

p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ p¬¬¬¬¬¬¬¬¬¬¬¬

exp s 2 À 1 ¼ v À 1:

CV(X ) ¼ (4:8)

The coef¬cient of skewness is

p¬¬¬¬¬¬¬¬¬¬¬¬

a3 ¼ (v þ 2) v À 1

and the coef¬cient of kurtosis equals

a4 ¼ v4 þ 2v3 þ 3v2 À 3:

Note that a3 . 0 and a4 . 3; that is, the distribution is positively skewed and

leptokurtic.

The geometric mean is given by

È É

xgeo ¼ exp E(log X ) ¼ em , (4:9)

which coincides with the median (see below).

113

4.3 MOMENTS AND OTHER BASIC PROPERTIES

The distribution is unimodal, the mode being at

xmode ¼ exp(m À s 2 ): (4:10)

A comparison of (4.6) and (4.10) illustrates the effect of the shape parameter s: For a

¬xed scale exp m; an increase in s moves the mode toward zero while at the same

time the mean increases, and both movements are exponentially fast. This means

that the p.d.f. becomes fairly skewed for moderate increases in s (compare with

Figure 4.1).

By construction, the lognormal quantiles are given by

F À1 (u) ¼ exp{m þ s FÀ1 (u)}, 0 , u , 1, (4:11)

where F is the c.d.f. of the standard normal distribution. In particular, the median is

xmed ¼ F À1 (0:5) ¼ exp(m), (4:12)

which is a direct consequence of the symmetry of the normal distribution.

Thus, the mean “ median“ mode inequality

E(X ) . xmed . xmode

is satis¬ed by the lognormal distribution.

The lognormal distribution is closed under power transformations, in the sense

that

X $ LN(m, s 2 ) ¼ X r $ LN(rm, r2 s 2 ), for all r [ IR: (4:13)

)

Note that power transformation is a popular device in applications of statistical

distribution theory.

The entropy is

1 1

E{Àlog f (X )} ¼ þ m À log p¬¬¬¬¬¬ : (4:14)

2 s 2p

The form of the characteristic function has been a long-standing challenging

problem; fairly recently Leipnik (1991) provided a series expansion in terms of

Hermite functions in a logarithmic variable. It is of special interest because the

formula for the characteristic function of its generator”the normal distribution”is

a basic fact in probability theory discovered possibly 150 years ago. It follows from

more general results of Bondesson (1979) that the distribution is in¬nitely divisible.

114 LOGNORMAL DISTRIBUTIONS

The mean excess (or mean residual life) function has an asymptotic

representation of the form (e.g., Embrechts, Kluppelberg, and Mikosch 1997)

¨

s 2x

e(x) ¼ [1 þ o(1)]: (4:15)

log x À m

Its asymptotically linear increase re¬‚ects the heavy-tailed nature of the lognormal

distribution.

Sweet (1990) has studied the hazard rate of lognormal distributions. Figure 4.2

exhibits the typical shape, which is unimodal with r(0) ¼ 0 [in fact, all derivatives of

r(x) are zero at x ¼ 0], and a slow decrease to zero as x ! 1:

The value of x that maximizes r(x) is

xM ¼ exp(m þ zM s),

where zM is given by (zM þ s) ¼ w(zM )=[1 À F(zM )]: Thus, Às , zM , Às þ

sÀ1 ; and therefore

exp(m À s 2 ) , xM , exp(m À s 2 þ 1):

Lognormal hazard rates: m ¼ 0 and s ¼ 0:5, 0:75, 1:0 (from top).

Figure 2

115

4.5 LORENZ CURVE AND INEQUALITY MEASURES

As s ! 1; xM ! exp(m À s 2 ); and so for large s;

exp(m À s 2 =2)

p¬¬¬¬¬¬ :

max r(x) %

s 2p

x

Similarly, as s ! 0; xM ! exp(m À s 2 þ 1); and so for small s;

max r(x) % {s 2 exp(m À s 2 þ 1)}À1 :

x

The properties of the order statistics from lognormal parent distributions can

usually only be derived numerically. Recently, Balakrishnan and Chen (1999)

provided comprehensive tables on moments, variances, and covariances of order

statistics for all sample sizes up to 25 and for several choices of the shape parameter s:

4.4 CHARACTERIZATIONS

The lognormal distribution can conveniently be characterized by the maximum