distribucion log normal

C H A P T E R 1 4

Lognormal Distributions

1 INTRODUCTION

The idea of a transformation such that the transformed variable is normally distributed was encountered in Chapter 12. In Section 4.3 of that chapter some specific transformations were introduced. Of these the most commonly used, and the only one of sufficient importance to merit a separate chapter, isthe simple logarithmic transformation.

If there is a number such that Z - is normally distributed,the distribution of X is said to be lognormal. For this to be the case, it is clearly necessary that X take any value exceeding but have zero probabilityof taking any value less than The term "lognormal"can also be applied tothe distribution of X if - is normally distributed, X having zeroprobability of exceeding 8. However, since replacement of X by (and 8by - reduces this situation to the first, we will consider only the first case.

The distribution of X can be defined by the equation

where is a unit normal variable and y, 6 and are parameters. From

(14.1) it follows that the probability density function of X is

(We take 6 without loss of generality, since - has the same distribu-tion as An alternative notation replaces y and 6 by the expected value and standard deviation a of Z = - 8). The two sets of parameters are

, related by the equations

208

so that (14.1) becomes

LOGNORMAL DISTRIBUTIONS

and (14.2) becomes

The lognormal distribution is sometimes called the antilognormal distribu-tion. This name has some logical basis in that it is not the distribution of thelogarithm of a normal variable (this is not even always real) but of an

exponential-that is, antilogarithmic-functionof such a variable. However,"lognormal" is most commonly used, and we will follow this practice. Theminor variants logarithmic- or logarithmico-normal have been used, as havethe names of pioneers in its development, notably (1879) and

and Gibrat (1930) (seeSection 2). When applied to economic data, particularly production func-tions, it is sometimes called the Cobb-Douglas distribution

It can be seen that a change in the value of the parameter affects onlythe location of the distribution. It does not affect the variance or the shape

(or any property depending only on differences between values of thevariable and its expected value). It is convenient to assign a particular valuefor ease of algebra, with the understanding that many of the results soobtained can be transferred to the more general distribution. In many

applications 8 is "known" to be zero (so that = or X is a"positive random variable"). This important case has been given the nametwo-parameter distribution (parameters y, S or a). For this distribution (14.1) becomes

and becomes

log X -=

The general case (with 8 not necessarily zero) can be called theparameter lognormal distribution (parameters y, 6 , 8 or a ,

HISTORICAL REMARKS 209

A so-called four parameter lognormal distribution has been defined by

Since (14.5) can be rewritten

with = - log A, it is really only a three-parameter lognormal that isdefined by (14.1).

2 HISTORICAL REMARKS

(1879) pointed out that, if .. . , are independent positiverandom variables and

then

n

log1

and if the independent random variables log are such that a central limittype of result applies, then the standardized distribution of log would tendto a unit normal distribution as n tends to infinity. The limiting distribution of would then be (two-parameter) lognormal. In an accompanying paper

(1879) obtained expressions for the mean, median, mode, variance, and certain percentiles of the distribution.

Subsequent to this, little material was published relating to thedistribution until 1903 when Kapteyn again considered its genesis on the linesdescribed above. [Fechner (1897) mentioned the use of the distribution in thedescription of psychophysical phenomena but gave little emphasis to this

topic.] Kapteyn and van Uven (1916) gave a graphical method (based on theuse of sample quantiles) for estimating parameters, and in the following yearsthere was a considerable increase in published information on the lognormaland related distributions. (1917) obtained formulas for the highermoments (in a study of the distribution of ages at first marriage), while van Uven considered transformations to normality from a more generalpoint of view; (1919) obtained approximate formulas for the standarddeviations of estimators obtained by the method of moments. Estimation from percentile points was described by Davies (1925, 1929) and tables to


facilitate estimation from sample moments were published by Yuan (1933).Unbiased estimators based on sample moments were constructed by Finney

(1941).From 1930 onward fields of application of the lognormal distribution have

increased steadily. Gibrat (1930, 1931) found the distribution usefully repre-

senting the distribution of size for varied kinds of 'natural' economic units. Gibrat's of proportional effect" can be expressed as

where . . . , is a sequence of random variables and is a set of mutually independent random variables, statistically independent of

Formula leads to

For small compared to 1, we have

log = log +1

Using a multiplicative analogue to the additive central limit theorem, wearrive at the conclusion that is asymptotically lognormally distributed with two parameters. Variations and of Gibrat's argument haveappeared in the literature. Kalecki assumed that the regression of

+ on log is linear, with

+ Z,) = log + with = 0,

leading to

Soon after Gibrat's work, Gaddum (1933) and Bliss found thatdistributions of critical dose (dose just causing reaction) for a number ofdrugs could be represented with adequate accuracy by a (two-parameter)lognormal distribution. On the basis of these observations, a highly developedmethod of statistical analysis of "quantal" (all-or-none) response data hasbeen elaborated. (The term analysis" has been given to such analy-ses, although is often understood to apply to a special transformed value, + 5, where is an observed proportion.)

Lognormal distributions have also been found to be applicable to distribu-tions of particle size in naturally occurring aggregates [Hatch Hatch

MOMENTS AND OTHER PROPERTIES

and Choute and with Bol'shev, Prohorov, andRudinov Kalinske Kolmogorov Kottler

and Wise among many others]. It is possible to give a generaltheoretical basis for this application on lines similar to those of Gibrat'sarguments. Consider a quantity A, subjected to a large number of successiveindependent subdivisions, the jth resulting in a proportion of the quantitysurviving, so that after divisions the surviving quantity is . . . X,. If

. . , are each uniformly distributed (see Chapter 26) over the intervalto 1, then the distribution of this quantity will be approximately lognormal.

[See Halmos and for more complete accounts of thisargument.]

Further applications, in agricultural, entomological, and even literary research were described by Cochran Williams (1937, Grundy

(1958, and Pearce Koch (1966, 1969) discussed

mechanisms that might generate lognormal distributions in a variety ofbiological and pharmacological situations.

This broadening of areas of practical usefulness of the distribution wasfollowed by (and presumably associated with) renewed interest in estimation of its parameters. (Section 4 of this chapter contains technical accounts ofestimation procedures.) Even in 1957 there were so many doubtful points in this regard that Aitchison and Brown (1957) devoted a substantial part of

their book to discussion of problems of estimation, on account of "unre-solved difficulties." Many of these difficulties, especially those concerningmaximum likelihood estimation, have now been solved, or at least clarified. A recent paper [Nakamura contains a description of construction of anadequate theoretical framework. Wu (1966) has shown that lognormal distri-butions can arise as limiting distributions of order statistics when order andsample size increase in certain relationships.

The book by Aitchison and Brown was very useful to us when

organizing material for this chapter in the first edition. It can still be

recommended for supplementary reading together with a more recent com-pendium on lognormal distributions, edited by Crow and Shimizucontaining contributions from several experts. In revising the present chapter, we have tended to include topics which are less thoroughly covered in thiscompendium.

3 MOMENTS AND OTHER PROPERTIES

Most of the following discussion will be in terms of the two-parameterdistribution, using (14.4) rather than (14.3). The rth moment of X about zero is


The expected value of X is

+ =

where = and the variance is

The rth central moment of X [Wartmann is

In particular

= - +

= - + + 3w2

The shape factors are

and

Neither of these depends on Note that a, and a, > 3-that is, thedistributions are positively skewed and are leptokurtic.

Equations and may be regarded as parametric equations ofa curve in the plane. This curve is called the lognormal line and is

MOMENTS AND OTHER PROPERTIES 213

shown in Figure 12.1 of Chapter 12, where it separates the regions forJohnson and distributions. The coefficient of variation is - Italso does not depend on The distribution of X is unimodal; the mode is at

Mode ( X ) = (14.10)

From (14.4) it is clear that the value such that = a , isrelated to the corresponding percentile of the unit normal distribution bythe formula:

= +In particular

Median (X) = = (since = 0).

Comparison of and shows that

Median ( X ) Mode ( X )

and

Mode (X) Median ( X )

The Theil coefficient is

Since the coefficient of variation is - the asymmetry index is(from

= w + 2 = (Coefficient of + 3. (14.14)Coefficient of variation

In hydrological literature special attention has been devoted to relationsbetween Median and [Burges, Lettenmeier, and Bates

Charbeneau Burges and HoshiThe standardized deuiate is, for example,

[compare and Some values of are shown inTable 14.1.

Slifker and Shapiro (1980) have noted that for lognormal distributions

= 1 for all a , (14.16)

Tab

le14.1

Sta

nd

ard

ized

Poin

tsof

Logn

orm

al

Dis

trib

uti

on

s=

(Up

per

Tai

l)99

.95

99.9

99.7

599

.59

997

.595

90

75

50

(Low

erT

ail)

0.05

0.1

0.25

0.5

12.5

51

0.

25

(Med

ian)


and they have suggested using this as a basis for deciding whether alognormal distribution is appropriate.

From Table 14.1 we see that as a tends to zero (or to infinity), thestandardized lognormal distribution tends to a unit normal distribution, and

For a small,

It is to be expected that

1. will change relatively slowly with (for small) if = 1a = 0.84 or

2. the standardized distance,

will change relatively slowly if = 9/2 a = 0.983 or 0.017). Thesefeatures also are indicated in Table 14.1. It is also clear from this table thatas increases the lognormal distribution rapidly becomes markedly mal. Table 14.2, which gives values of a, and a,, also indicates how rapidlythe skewness and kurtosis increase with a. It is for this reason that onlyrelatively small values of are used in Table 14.1.

It can be seen in Table 14.1 that for larger there is a high probabilitydensity below the expected value, leading to small numerical standardizeddeviations. Conversely there is a long positive tail with large standardizeddeviations for upper percentiles. Figure 14.1 shows some typical probabilitydensity functions (standardized in each case so that the expected value is

zero and the standard deviation is 1).

Table 14.2 Values of and a, for Distribution (14.2)'

0.1 0.30 3.16

0.2 0.61 3.680.3 0.95 4.640.4 1.32 6.260.5 1.75 8.90

LOGNORMAL

Figure 14.1 Standardized Dis-

tribution o 2

Wise (1952) has shown that the probability density function of the

parameter distribution has two points of inflection at

Just as the sum of two independent normal variables is also normally

distributed, so the product of two independent (two-parameter) lognormallydistributed variables is also lognormally distributed. If

are independent unit normal variables, then

and so

is a unit normal variable. In particular, if . . ., is a random sample from the distribution (14.2)' (with = so that log X, is distributed nor-mally with expected value and variance then log X, =

where is the sample geometric mean of X,, ..., isdistributed normally with expected value and variance The sample

geometric mean therefore has a lognormal distribution with parameters5, [in or y = - =

We introduce the notation to denote a distribution defined by

(14.4) so that has the distribution + + if hasthe distribution (for any and are mutually independent.Then the distribution of where is any positive constant, is


+ log Clearly, if . . . , are a mutually independent set, k 2then is distributed as + log

Distributions of sums of independent random variables, each havinglognormal distribution, have received special attention in the literature.These distributions are of importance in many areas of telecommunication,statistical detection problems, multihop scatter systems, and so on.

(1960) approximates the distribution of sum by a lognormal distri-bution with the same first two moments. Barakat (1976) considers the sum

= of N distributed as He expresses thecharacteristic function of each X, in the form

mi tye

He then uses the expansion

to obtain the characteristic function in the'form

Values of the coefficients are given in Barakat's Table 1; the are"quasi-Hermite" polynomials. The pdf is obtained by inverting (14.19) using

quadrature.The somewhat less ambitious target of obtaining the distribution of the

sum of two independent lognormal variables was considered by Naus (1969).He obtained the moment-generating function of the distribution of the sumof two independent variables. and Yeh consider thesame problem when the two lognormal variables are not necessarily identi-

cally distributed.The following (condensed) analysis of the problem indicates the difficulties

involved: Suppose that is distributed = and that andare mutually independent. The joint pdf of and is

218 LOGNORMAL DISTRIBUTIONS

Changing to variables = + we have the Jacobian

and the joint pdf of and is

Hence

Setting = we obtain

The integral in (14.20) may be evaluated by quadrature.The preceding discussion has been limited to variables with two-parameter

lognormal distributions. The extra parameter, 8 in or moves

("translates") the whole distribution by an amount in the direction of increasing X. Many properties-particularly variance and shape-remainunchanged, but the reproductive property does not hold if 8 for either

or

MOMENTS AND OTHER PROPERTIES

5.0

0 1 2 3 4 5 6 7 8 9 1 0

14.2 Hazard Rate of the Lognormal Distribution, with = 0; = =

The information-generating function corresponding to is

The entropy is

A recent reference for the characteristic function of is(1991). The distribution is not determined by its moments. A result ofBondesson (1979) is that any distribution with pdf of form

with completely monotone on satisfying some mild regularityconditions is infinitely divisible (see Chapter 1). This result enables us todeduce that lognormal distributions are infinitely divisible.

Sweet (1990) has studied the hazard rate of lognormal distributions. Figure 14.2, taken from that paper, exhibits some of his results for From this figure it can be seen that = and that runs to amaximum and thereafter decreases slowly to zero. The value of t thatmaximizes is


where + = - = which is Mills's ratio(see Chapter

From + and hence

As - and so for large

- + and so for small

= (u2

I

4 ESTIMATION

4.1 Known

If the value of the parameter is known (it is often possible to take equalto zero), then estimation of the parameters and (or and presentsfew difficulties beyond those already discussed in connection with the normaldistribution. In fact, by using values of = - the problem isreduced to that of estimation of parameters of a normal distribution. Manyspecialized problems, such as those relating to truncated distributions orcensored samples, or the use of order statistics, may be reduced to corre-sponding problems (already discussed in Chapter for the normal distribu-tion. Maximum likelihood estimation is exactly equivalent to maximum likeli-hood estimation for normal distributions, so that the maximum likelihood estimators for and respectively, are

where = n (assuming of course that . . . , are indepen-dent random variables each having the same lognormal distribution withknown).

There are a few situations where techniques appropriate to normal distri-butions cannot be applied directly. One is the construction of best linearunbiased estimators, using order statistics of the original variables

ESTIMATION

other is that where the original values are not available, but the data arein groups of equal width. While the results can immediately be written downin the form of grouped observations from a normal distribution, the groupsappropriate to the transformed variables will not be of equal width. In suchcircumstances methods of the kind described in Chapter 12, Section 2, mightbe applied.

Sometimes it may be desired to estimate not but rather theexpected value + and the variance - of the vari-able X. Since and = - - are jointly complete andsufficient statistics for and if the UMVU estimator of

= + is

If a function has expected value it is the minimum varianceunbiased estimator of Finney (1941) obtained such estimators of theexpected value and variance in the form of infinite series

respectively, where

Unfortunately, the series in (14.29) converges slowly (except for very smallvalues of Finney recommends using the approximation

which (he states) should be safe for n > 50 in and n 100 in provided that the coefficient of variation - is less than 1

(corresponding to a 0.83).

Finney showed that [with =


Value of

Efficiency

Efficiencyof

Figure 14.3 The Efficiencyof and of in Large Samples

By comparison, for the unbiased estimators of the expected value andvariance, and = - - respectively,

Figure 14.3 [taken from Finney shows approximate values of the"efficiency"ratios 100 and 100 as afunction of It will be noted that while is reasonably efficient comparedwith M, considerable reduction in variance is achieved by using V in place of

[See also Peters (1963) has constructed best quadraticestimators of log = + r2u2/2, using order statistics based on log

Confidence limits for and/or a can of course be constructed from the I

transformed values Z,, Z,, ..., using techniques in Chapter 13. In partic-ular (when 0 = since the coefficient of variation of X is -

confidence limits (with confidence coefficient for this quantity are

- - and -where denotes the point of the distribution with v degrees offreedom, to be described in Chapter 18. [See Koopmans, Owen, andRosenblatt Similar arguments could be used to construct confidence limits for any other monotonic function of such as and

4.2 Unknown

Estimation problems present considerable difficulty when 0 is not known. As

might be expected, estimation of 0 is particularly inaccurate. This parameter is a "threshold value," below which the cumulative distribution function is

ESTIMATION 223

Table 14.3 Percentile Points of Standardized Lognormal Distributions

Lower

10% 25% Median

zero and above which it is positive. Such values are often-one might say"usually" to estimate.

However, estimation of parameters is often not as important as estimation

of probabilities-in particular, of cumulative distribution functions.Table 14.3 shows the median and upper and lower 10% and 25% points oflognormal distributions with widely differing values of 0, but with and a sochosen that each distribution is standardized has zero expected value

and unit standard deviation; see also Table 14.1).There can be considerable variation in 0 with little effect on the per-

centiles and little effect on values of the cumulative distribution functions for fixed values of X. Insensitivity to variation in is most marked for largenegative values of These observations and Table 14.3 should correct anyfeeling of depression caused by the inaccuracy of estimation of 0. Of coursethere can be situations where the accuracy of estimation of 0 itself is a

primary consideration. In these cases special techniques, both of experimen-tation and analysis, may be needed. These will be briefly described at the endof this section.

Maximum likelihood estimation of 0, and a might be expected, to beattained by the following tedious, but straightforward, procedure. Since for given the likelihood function is maximized by taking

= -

1

[cf.


one might take a sequence of values of 8, calculate the maximized likelihoodcorresponding to each, and then try to estimate numerically the value 8 of 8by maximizing the maximized likelihood. However, Hill (1963) has shown thatas tends to ..., the maximized likelihood tends to

infinity. Formally this would seem to indicate that we should accept the "estimates": 6 = .. . , = = Hill resolved this

difficulty by introducing a prior joint distribution for 8, and and then using theorem. This leads to the conclusion that of theformal maximum likelihood equations should be used, with 8 satisfying

where = - - and satisfy with 8

replaced byFormal calculation of limiting variances gives

Formulas equivalent to these have been given by Cohen and Hill

but is unlikely to be valid [see Hill and below].We may note that Harter and Moore (1966) carried out a sampling

experiment, with = 4, = 2, and = 10. Their results can be representedapproximately, for sample size n in the range 50 to 200 by

(note the factor n2 in the last formula). If is known, their results give= 4.1-not much less than if 8 is not known-but n is now

only about 2.1.Tiku (1968) has suggested an approximate linearization of the maximum

likelihood equations based on the approximate formula

- = + (14.36)

*It is clear that we must take . . . ,

ESTIMATION 225

for Mills's ratio. Appropriate values of a and depend on the range ofvalues of x. (See discussion of a similar technique in Chapter 13, Section 8.4.)

Over the last 20 years there has been further consideration of maximumlikelihood estimation for the three-parameter lognormal and its difficulties.We are inclined to agree with the opinion expressed by Griffiths that"the method of maximum likelihood has been wrongly discredited because ofsupposed computational difficulties and theoretical uncertainties."

Calitz (1973) suggests using Cohen's method to solve the maximumlikelihood equations. He solves (14.34) numerically using theRaphson method, which will converge provided the initial value of 8 ischosen so that log Using the value 8 =

- where is the least of the observed values of X, providedsatisfactory results. Lambert's (1964) failure to solve this problem-whichtriggered many denunciations of the maximum likelihood method-was dueto his choice of method of iteration. He tried to iterate on all three maximum

likelihood equations simultaneously.(1976) proposed using penalty functions, maximizing the function

with given by subject to constraints of form

. . . ,

The constant c is chosen to be a large positive number (although its value can be problem dependent). (1991) points out that a necessary condition for the maximum likelihood equations to "degenerate" so that theycannot be solved is that

where = + = be a decreasing function of a.

Noting that all data are in fact subject to grouping (and so are, in

discrete), Giesbrecht and Kempthorne (1976) have shown that the problem ofan unbounded likelihood function can be removed by taking a suitable

discretized model. Griffiths utilizes the discretization procedure toobtain a confidence interval for the threshold parameter 8. Provided that thegroup width is sufficiently small, the likelihood will be practically of the sameform for both the (approximate) continuous and (correct) discrete models,except in the neighborhood of the singularity in the continuous model. Griffiths (1980) also discusses construction of approximate confidence inter-vals for 8.

Dahiya and Guttman (1982) construct a "shortest prediction interval" oflevel 1 - a for a two-parameter lognormal variable. [This is not an estimated


interval, it is simply the interval (a, such that b - a is minimized subject toX < b] = 1- a.] Clearly we must have a = + and b =

+ with

(to provide level 1 - a). Subject to (14.38) we have to minimize b - a, orequivalently

From - = whence

To minimize (14.391, we equate

to zero.Inserting the value of from we obtain

whence -A2/2 = - that is,

The values of A and B are easily determined from (14.38) and butDahiya and Guttman provide tables to facilitate the process. These authors also consider construction of shortest intervals for themedian [Of course, if it is known that = confidence rvals for

are easily constructed using normal theory results (see Chapter

Since

log x= logi = l

and

ESTIMATION 227

are sufficient for the parameters of a minimum variance unbiased estimator of

= log = +

is

= log X +

a minimum variance unbiased estimator of the variance of

(since log X and are independent) is [Land

Among other proposed modifications of likelihood estimation we note the suggestions of Cohen and (1980) based on introduction of order statistics: They propose replacing the likelihood equationa = o

where is the rth order statistic among .., and is the rth orderstatistic among n mutually independent unit normal variables, or by

Other modifications include: the equation is replaced by

-= S

2 with =n -

The actual estimating equations for are

and


Nevertheless, uncertainties and difficulties in the use of (14.33) andlead us to consider methods other than maximum likelihood.

If the first three sample moments are equated to the correspondingpopulation values, we obtain the formulas

whence

From G and then can be determined. From

and finally can be determined fromYuan (1933) provided a table of values of the right-hand side of (14.46) to

aid in the solution of that equation. and Brown (1957) give a table for directly to four decimal places for = Without

using such tables, an iterative method using equation (14.46) in the form

is quite easy to apply, or the explicit solution

can be used.This method is easy to apply but liable to be inaccurate because of

sampling variation in [However, (1981) found it to be quite reliable.] Even when is known, estimation of population variance by samplevariance is relatively inaccurate (see Figure 14.3). Approximate formulas for

ESTIMATION 229

the variances of the estimators were obtained by They are

where = - 1 = - 1 = (square of coefficient of variation).A modification of the method of moments is to estimate by this method

and then use estimators M, V [see and of ex-pected value and variance, applied to the variables -

Alternatively, if one is prepared to accept some loss in efficiency, it might be possible to use a relatively simple method of calculating estimates. Certainspecial forms of the method of percentile points give very simple formulas forthe estimators. Using the relationship [see (14.1) and

the following formula can be obtained:

If estimated values of the are inserted in the solution of theresulting equation for must be effected by trial and error. However, if wechoose = (corresponding to the median) and = 1 - a,, then (notingthat = 0, and = the equations are (asterisk denoting "estimated value"):

= + exp

whence

and


so that no solution by trial and error is needed. Aitchison and Brownsuggest taking a, = 0.95, but it is likely that any value between 0.90 and 0.95will give about the same accuracy. Lambert suggested using values ofthe parameters obtained by this method as a starting point for an iterative formal solution of the maximum likelihood equations. He used the value

= 61/64 = 0.953.It is possible for (14.50) to give a negative value of but this happens

only if

which is very unlikely to be the case if the distribution has substantial positiveskewness.

If the sample median is replaced by the sample arithmetic mean and equation by

then from and

From this equation S* can be found numerically. Aitchison and Brown(1957) give graphs of the function on the right-hand side of for

= 0.95, 0.90, and 0.80, which helps to give an initial value for the solution.This method of estimation is called Kemsley method (1952). It can also givea negative value for though this is unlikely to occur.

A method suggested by Boswell, Ord, and Patil (1979) is easy to apply:the lower, 50th and upper percentiles of the normal

variable Z = - These are - and + respectively,where a = = 1 - The corresponding (unknown) percentiles ofX will be = - + 8, = + 8, and = ++ 8. Solving for 8, from these three equations we obtain

Estimating and from the corresponding percentiles of lognor-mal data we obtain an estimator of 8.

Boswell, Ord, and Patil (1979) recommended choosing in the range of1.5 to 2. Aitchison and Brown (1957) made an experimental comparison ofresults of using the method of moments (14.461, the method of quantiles

ESTIMATION 231

and Kemsley's method to estimate 8. They came to the conclusion that (14.50) with a,= 0.95 is slightly better than (14.51) and thatboth are considerably better than the method of moments.

Other methods of estimating 8 have been suggested. method(1951) is based on the idea that in a large sample at least one observed value of X should be not much greater than the threshold value 8. The leastsample value is equated to the formula for the + point ofthe distribution, giving the relationship

= + exp ---I

This is then combined with the first two maximum likelihood equations(14.33).

There are natural modifications of this method, to allow for cases when there are several equal (or indistinguishable, as in grouped data) minimumvalues. Using a variant of this an initial value of (for use in some iterative process) may be chosen as minus some arbitrary (usually rathersmall) value.

Cohen, and Ding's moment estimation methoduses the equations (in an obvious notation)

Equation is obtained from (14.53) by replacing -+ by Table 14.4 gives values of and

- [More detailed tables of are available in Harter (1961);see also Tippett (1925) and Chapter 13, Section

Table 14.4 Valuesof and


The explicit form of the estimating equations is

The above three equations give

From o* (and so a*= can be determined numerically.Cohen, and Ding (1985) and Cohen and (1988) provide

graphs of as a function of for n = 10, 15, 20, 30, 50, 100, and 400.A pioneering paper by Cheng and (1982) introduced a new method

of estimation-maximum product-of-spacings This overcomes diffi-culties involved in maximum likelihood estimation for the three-parameterlognormal distribution. The authors claim that Giesbrecht and Kempthorne's(1976) grouping and discretization method does not appear to identify the

essential difficulty. "Whether it is possible in practice or not, it is stilllegitimate to hypothesize a situation where the observations are truly contin-uously distributed, when discretization is merely an approximation to theactual situation." The authors contend that in maximum likelihood estima-

tion for a continuous density one maximizes, approximately

replacing the probabilities on the right-hand side by the first-order approxi-mations

The quantities

- = i = . . , n +

where = < . . . +,= constitute an orderedrandom sample of size n, are called spacings of The MPS method

chooses that maximizes

Under certain assumptions on the MPS estimators are consistent.For the three-parameter lognormal distribution, if the true parameter =

lies in any set in - < < < then the estimatorsare consistent as which is a stronger result than is available formaximum likelihood estimators.

For the lognormal model we have to maximize

where = - - = - = Instead of directmaximization Cheng and Arnin (1982) obtained the restricted maximum of

with respect to and for a sequence of values of 8, and thenmaximized with respect to 8. They used = = and= - with = - - in an iterativeprocedure, and reported fast convergence, usually in three or four iterations, using the stopping rule

Evans and Shaban discuss estimation of parameters of the form

where a and b are arbitrary constants. Other papers on this subject byNeyman and Scott Mehran Bradu and Mundlak

Shimizu and Iwase and Shimizu (1983) are summarized in Crow and Shimizu

Zellner and Rukhin (1986) studied Bayesian estima-tion of parameters of the two-parameter lognormal distribution. The first two authors utilized improper priors a l/uc, c Rukhin utilizes generalized prior density which is also "uniform" in Wang, Ma, and Shi(1992) discuss, among other things, Bayesian estimation of the mean =

+ of lognormal distribution Using a)a as ajoint noninformative prior on a), they derive the marginal posterior for 8


in the form

where = log and = - - [which is

Zellner's result].For censored samples I . . . < with known and using

prior normal distribution on 5, they obtain

- log

where

The integral I has no closed form solution and must be numerically evaluated. These authors use two-parameter lognormal distributions forindependent variables = with independent "noninformative prior"distributions for and in analysis of environmental data.

The reliability function for two parameter lognormal distributions is given

The Bayesian estimator based on a sample of observations X,, . . ., is

given by [cf. Padgett and Wei

(log - m')-

8' + +

where is a random variable with t-distribution with degrees offreedom, provided that has the "normal-gamma prior" prior

is and prior is with density =

ESTIMATION 235

The joint prior is

a-

Here

where = log and a' = a + As a 0, andcorresponding to Jeffreys's "vague" prior, we have

.t -

+ + I

(1989) obtains a very similar result directly assuming the joint prior:

aComparison with the maximum likelihood estimator

log t -

and with the minimum variance unbiased estimator

where is the incomplete beta function ratio and w = +t - t with = and S 2

= -

- shows, as might be expected, that the estimator has thesmallest variance when the assumed priors are actually the true priors. Theestimator is, however, slightly biased.


4.3 Graphical Estimation

Wise (1952) has put forward the following interesting suggestion for a graphical method of estimating the parameters. It is particularly useful whenthe "rate of increase" of probability probability density) is observed rather than actual frequencies, though it can also be used in the latter case,with sufficiently extensive data. He starts by observing that the tangents atthe two points of inflexion of the probability density function remain close tothe curve for a considerable length, and so should be estimable graphicallywith fair accuracy. The modulus of the ratio of the slopes is

exp 2 + a 1 + .IWise provides tables of the logarithm of this quantity to four decimal

places for a = to aid in estimating a from the observed slopes ofthe inflection tangents. The initial point is estimated from the values

of at the points where these tangents cut the horizontal axis (see Figure 14.4). If then is estimated from the formula

1 + - - (14.62)

where

= + +

Time

Figure 14.4 Geometrical Method of Estimating Lognormal Parameters

TABLES AND GRAPHS

with

Wise also provides values of L to four decimal places for a

Finally is estimated from the formula

+ - - .

Sequential estimation and testing have been discussed by Zacks (1966) and

TomlinsonIssues of testing normality versus lognormality has received attention in

the literature for the last 20 years. See Kotz Rademaker, and and more recently and Hwang (1991).

5 TABLES AND GRAPHS

Aitchison and Brown (1957) give values of the coefficient of variation, a,,(a,- 3), the ratios [from of mean to median and mean tomode and the probability that X does not exceed for

(When = values appropriate to the unit normal distribution are shown.)All values [except for coefficient of variation and (a, - 3) when = 0.051are given to four significant figures.

Tables of percentile points of (two-parameter) lognormal distributions have been published by Moshman Upper and lower 5%,1%, and 0.5% values to 4 decimal places, for a, = and byBroadbent ratios of upper and lower 5% and 1% values to theexpected value, to 4 decimal places, for coefficient of variation -equal to (Note that this corresponds to 0.15 approxi-mately.)

Tables of random lognormal deviates have been published by Hyreniusand Gustafsson (1962). These were derived from tables of random normaldeviates (see Chapter 13) and are given to 2 decimal places for distributionswith skewness 0.2, 0.5, 1.0, and 2.0 (corresponding approximately toa = 0.006, 0.16, 0.31, and 0.55).

Graph paper with the horizontal (abscissa) scale logarithmic and thevertical (ordinate) scale normal [marked with proportion P at a distance from the origin equal to where P = is called probabilitypaper. If X has a two-parameter lognormal distribution, plotting P =

x] as ordinate against as abscissa will give a straight-line plot. The


slope of the line is 6 it meets the horizontal axis at -

From sample plots these parameters may be estimated by observing the slopeof a fitted line and its intersection with the horizontal axis. If a third parameter is needed, the plotted points tend to curve up, away from a straight line, as decreases. It is possible, with some patience, to obtain a graphical estimate of by trial and error [as suggested by Gibratchanging the value of until a plot using - in place of can be mostclosely fitted by a straight line. For a subjective method of this kind, it is notpossible to obtain even an approximate formula for the standard deviation ofthe estimator of 8. However, the method seems to give quite useful resultsand can certainly be used to obtain initial values for use in iterative pro-cesses.

Graph paper of the kind described above is useful in analysis ofresponse data. Moments, product moments, and percentage points of

various order statistics for standard lognormal distribution have beencomputed and tabulated for all samples of size 20 or less by Gupta,McDonald, and Galarneau (1974).

6 APPLICATIONS

Section 2 of this chapter indicates several fields in which lognormal distribu-tions have been found applicable, and the references in the present sectioncontain further examples. Application of the distribution is not only based on empirical observation but can, in some cases, be supported by theoreticalargument-for example, in the distribution of particle sizes in natural aggre-gates (see Section and in the closely related distribution of dust concen-tration in industrial atmospheres [Kolmogorov (1941); Tomlinson

Geological applications have been described byChayes Miller and and Prohorov

(1961) gives a number of examples of such applications and in-cludes some further references.

The three-parameter lognormal distribution was introduced to geology byKrige (1960) for modeling gold and uranium grades, and it is now widelyregarded as the "natural" parametric model for low-concentration deposits. The lognormal model for minerals present in low concentrations has been experimentally verified for many minerals [Krige Harbaugh and Ducastaing Its use for gold deposits was pioneered by SichelFurther applications, mentioned by include duration ofsickness absence and physicians' consultation time. Wise (1966) has described application to dye-dilution curves representing concentration of indicator as a function of time. Hermanson and Johnson (1967) found that it gives a goodrepresentation of flood flows, although extreme value distributions (see Chapter 22) are more generally associated with this field. Medical applica-tions are summarized by Royston who mentions, in particular,

APPLICATIONS

ing the weights of children [Rona and Altman and construction ofage-specific reference ranges for clinical variables [Royston Royston(1992) fits lognormal distributions to observations of antibody concentration in of type group B streptococcus

Leipnik (1991) mentions applications of sums of independent lognormalvariables in telecommunication and studies of effects of the atmosphere on radar signals. Molecular particles are modeled in radar noise theory ashaving normal velocity distribution, while dust particles of a given type areassigned lognormal distributions.

The lognormal distribution has also been found to be a serious competitor to the Weibull distribution (Chapter 21) in representing lifetime distributions (for manufactured products). Among our references, Adams Epstein(1947, Feinlieb Goldthwaite Gupta andand Berry (1961) refer to this topic. Other applications in quality control aredescribed by Morrison and Rohn Aitchison and

, Brown (1957) cite various applications such as the number of persons in aI

census occupation class and the distribution of incomes in econometrics, thedistribution of stars in the universe, and the distribution of the radicalcomponent of Chinese characters.

The 1974 distribution of wealth in the United Kingdom was studied byChester (1979) who fitted a two-parameter lognormal distribution to datareported in the Inland Revenue Statistics Theoretical grounds for this application of lognormal distributions are developed by Sargan (1957) andPestieau and Possen (1979) [see Gibrat (1930,

The two-parameter is, in at least one important respect, a more realistic representation of distributions of characters like weight, height,and density than is the normal distribution. These quantities cannot takenegative values, but a normal distribution ascribes positive probability to suchevents, while the two-parameter lognormal distribution does not. Further-more, by taking small enough, it is possible to construct a lognormal distribution closely resembling any normal distribution. Hence, even if anormal distribution is felt to be really appropriate, it might be replaced by asuitable lognormal distribution. Such a replacement is convenient when obtaining confidence limits for the coefficient of variation. Koopmans, Owen,and Rosenblatt (1964) pointed out that if the normal distribution is replacedby a lognormal distribution, then confidence limits for the coefficient ofvariation are easily constructed (as described in Section 4 of this chapter). Wise (1966) has pointed out marked similarities in shape between appropri-ately chosen lognormal distributions and inverse Gaussian (Chapter 15) andgamma (Chapter 17) distributions.

The lognormal distribution is also applied, in effect, when certain approxi-mations to the distribution of Fisher's z = log F are used (see Chapter 27).It is well known that the distribution of is much closer to normality than isthat of F Logarithmic transformations are also often used in attempts to "equalize variances" [Curtiss Pearce If


the standard deviation of a character is expected to vary from locality tolocality) in such a that the coefficient of variation remains roughlyconstant with standard deviation roughly proportional to expectedvalue), then application of the method of statistical differentials (as described in Chapter 1) indicates that by use of logarithms of observed values, thedependence of standard deviation on expected value should be substantiallyreduced. Very often the transformed variables also have a distribution more nearly normal than that of the original variables.

Ratnaparkhi and Park (1986)-starting from Yang's (1978) deterministic

model for the rate of change in the residual strength with respect to the number of fatigue cycles

where is the residual strength at cycle n, is a nonnegative function

of n, and c is a physical parameter-proposed using a lognormal distribution for the initial (or "ultimate") strength. They deduced from this that the random variable N (the fatigue failure cycle) has a three-parameter lognor-mal distribution.

The relationship between leaving a company and employees tenure hasbeen described by two-parameter lognormal distributions with great success[Young It was found that the lognormal modelaccurately defines the leaving behavior for any "entry cohort" (defined as a group of people of about the same quality, performing roughly the same workand joining the company at about the same time). Agrafiotis estimated

the early leavers for each entry cohort of a company based on the lognormalhypothesis by fitting an appropriate unweighted least-squares regression.

and Wells (1972) point out that recent work in analyzing automo-bile insurance losses has shown that the lognormal distribution can beeffectively used to fit the distribution for individual insurance claim pay-ments. Many applications in biochemistry, including mechanisms generating the lognormal distribution can be found in Masuyama (1984) and its refer-ences.

Recently, for the U.S. population of men and women aged 17-84 years,Brainard and Burmaster (1992) showed that the lognormal distribution fitsthe marginal histograms of weight (in cells representing 10-lb intervals) for both genders.

7 CENSORING, TRUNCATED LOGNORMAL AND RELATED DISTRIBUTIONS

As pointed out in Section 4, estimation for the two-parameter lognormal distribution, and the three-parameter distribution with known value for

CENSORING, TRUNCATED LOGNORMAL AND RELATED DISTRIBUTIONS 241

presents no special difficulties beyond those already encountered with normaldistributions in Chapter 13. The same is true for censored samples from truncated forms of these distributions [see, and Khatri

If log X has a normal distribution with expected value and standarddeviation truncated from below at log then the rth moment of Xabout zero is

1 - r u )( r th moment if not truncated) , (14.65)

-

with = (log - [Quensel For the three-parameter distri-bution, with not known, when censoring or truncation has been applied,estimation presents considerable difficulty.

Tiku (1968) has used his approximate linearization formula (described in Section 4) to simplify the maximum likelihood equations for the truncatedlognormal; and Young have considered estimation from groupeddata.

Thompson (1951) gives details of fitting a "truncated lognormal distribu-tion" that is a mixed distribution defined by

Tables are provided to assist estimation by moments.Harter and Moore (1966) gave the results of some sampling experiments in

which values of parameters were estimated for lognormal distributions cen-sored by omission of proportions at the lower and at the upper limits of

the range of variation, with q, = or 0.01 and = or 0.5. (The case= = corresponds to a complete, untruncated lognormal distribution.

These results have already been noted in Section 4.)Taking = 4, u = 2, and = 10, Table 14.5 shows the values that were

obtained for the variances of estimators determined by solving the maximumlikelihood equations. the results are applicable to a censored

sample from lognormal distribution they should give a useful indication ofthe accuracy to be expected with truncated lognormal distributions. Thesubstantial increase in with even a small amount of censoring =

0.01) at the lower limits is notable. It would seem reasonable to suppose thatvariances and covariances are approximately inversely proportional to samplesize, if estimates are needed for sample sizes larger than 100. The arithmeticmeans of the estimators were also calculated. They indicated a positive bias


Table 14.5 Variances and Covariances of Maximum Likelihood Estimators

for Censored Three-ParameterLognormal Distribution

0.00 0.5 0.0600 0.0961 0.2109 0.0628 0.0146 0.0296

0.01 0.0 0.0428 0.0351 0.2861 0.0064 0.0126 -0.0047

0.01 0.5 0.0628 0.1244 0.3011 0.0199 0.0239 0.03750.0416 0.0312 0.1733 0.0232 -0.0015 -0.0032

Note: In the sampling experiment with sample size of 100, = 4, = 2, and = 10 (the value of

was not used in the analysis).

"No censoring.

in of about 0.8-0.9 when = 0.01 (whether or not = and a positivebias of about 0.3 when = 0.5 with = There was also a positive bias ofabout 0.5 in when = 0.01 and = 0.5.

It is of interest to compare the figures in Table 14.5 with corresponding values in Table 14.6, where the value 10 of was supposed known and usedin the analysis. The variance of is considerably reduced (by comparisonwith Table but that of is not greatly changed. The effect of varyingis much smaller.

Progressively censored sampling of three-parameter lognormal distribu-tions has been studied by Gajjar and (1969) and Cohen Let Ndesignate the total sample size of items subject to a life or fatigue testing and n the number that fail (resulting in completely determined life spans).Suppose that censoring removal) occurs in k stages at timesj = . . ,k, and that surviving items are removed (censored) fromfurther observation at the jth stage. Thus N = n +

Cohen (1976) discusses Type I censoring, with and the numberof survivors at these times are represented by random variables. For k-stageType I progressively censored sampling the likelihood function, in an obviousnotation, is

Table 14.6 Maximum Likelihood Estimators for Censored Three-Parameter

Lognormal Distribution Where Value of Is Supposed Known

"No censoring.


For the lognormal distribution the local maximum likelihood estimat-ing equations are

where = = - with

Cohen suggests selecting trial values for 6, solving the first two equa-tions with = for and using standard Newton-Raphson procedure,and then substituting these values into the third equation. If the thirdequation is not satisfied for any value of in the permissible interval

he recommends application of a modified maximum likelihoodmethod (which has proved satisfactory in many applications) in which thethird equation is replaced by

= - +

where = + and is the rth order statistic among the X'sfor some r 1.

The discretized form of the (truncated) lognormal distribution has been found to offer a competitive alternative to logarithmic series distributions in some practical situations (see Chapter 7). Quensel has described thelogarithmic Gram-Charlier distribution in which log X has a Gram-Charlierdistribution.

Kumazawa and Numakunai (1981) introduced a hybrid distribu-tion defined by the cdf,

For = 0, the equation (14.68) represents a normal distribution withparametrs and for = it gives the lognormal distribution


Figure 14.5 Frequency Curves of the Normal, Lognormal, and Hybrid Lognormal Distribution,

and Respectively

[or Thus (14.68) can be viewed asthe cdf of a hybrid normal-lognormal distribution. Figure 14.5 providescomparison of the three curves for standard values of these parameters.

For small 1 the curve of approaches that of and for

> 1.5 it decreases faster than both and Data of occupa-tional radiation exposures, as reported in the late 1970s by the U.S. NuclearRegulatory Commission, are shown to be far better fitted by the hybridlognormal distribution than by lognormal or normal.

Based on the relation

where = - = l /a 2, and = Kumazawa andNumakunai (1981) designed hybrid lognormal probability paper that provides

a quick method of judging whether given data can be fitted by a hybridlognormal distribution.

The and systems of distributions [Johnson discussed in

Chapter 12, are related to the lognormal distribution in that the lognormaldistribution is a limiting form of either system and in that the "lognormalline" is the border between the regions in the plane corresponding tothe two systems.


The distribution corresponding to the transformation, defined by

which has four parameters and A, was considered by van Uven asearly as 1917. It has been termed the four-parameter lognormal distributionby and Brown (1957). This name is not so well recognized as thetwo-parameter or three-parameter lognormal nomenclature. In view of theexistence of other four-parameter transformations the useful-

ness of this name is doubtful.Lambert uses the form of pdf,

which is a reparametrization of distribution (Chapter 12,Section 4.3). The random variable - - has thedistribution. Johnson (1949) carried out estimation of 8 and using quantilesand a method based on quantiles and points of the range. (See also Chapter

Estimation becomes a difficult problem when all four parameters must be estimated from observations. Lambert (1970) tackles maximum likelihoodestimation of the parameters. The difficulties associated with the behavior ofthe likelihood function at = min or = are overcome byassuming that observations are multiples of some unit 6 a recordedobservation may differ from its true value by up to The likelihoodequation is

if either 7 or 8.

(The parameter space is defined by < <


< < At any maximum of the following equations are satisfied:

=-c - - - - p},P

Solving the first two equations for and yields

Then

If the values and that maximize are found, then the setwill be the solution of equations (14.721, and as long as

- 6/2 and + these may be taken as the maximum likelihoodestimates.

To locate the greatest value of the likelihood, Lambert (1970) recommendsthat "the function log be plotted over a range of values of andExamination of such a plot enables one either to locate a region in which thegreatest value of log lies or to observe in which direction to shift to find this value; in this case the function was recalculated over a suitable newregion. Once a region containing the greatest value is found, changing theintervals in and at which log is calculated enables one to locate thegreatest value with any desired accuracy and so to obtain and Lambertconcludes that "although techniques have been developed for estimating, from the likelihood, the parameters of the four-parameter lognormal

CONVOLUTION OF NORMAL AND LOGNORMAL DISTRIBUTIONS 247

and these appear to provide satisfactory estimates with artificialsamples, the method has not so far yielded useful estimates."

We are unaware of any further development along these lines in estimat-ing the Johnson four-parameter lognormal distribution using maximum likeli-hood method. Wasilewski studied mixtures of several two parameterlognormal distributions = 1,. .., with pdf

1 - (log -, 0, (14.73)

The rth moment about zero of (14.73) is

Wasilewski (1988) discusses estimation of the parameters by themethod of moments, equating for r = -k, -k + 1,.. . ,k - 1, k the valuesof E[X r] to the observed values

8 CONVOLUTION OF NORMAL AND LOGNORMAL DISTRIBUTIONS

(1991) investigated, in detail, convolutions of normal and lognormaldistributions. His work was motivated by application of lognormaltions-namely, in measuring lognormally distributed quantities with instru-ments that give quick and inexpensive readings but have substantial randommeasurement errors. If + a ) is distributed as and is a measurement of with an unbiased normally distributed error Y

independent of so that X = Z + Y and then

log - 5


(using the transformation t = log The parameters a and are the location and scale parameters of the convolution.

The canonical form = 0, = 1) is

The moments of are easily calculated from the relation

where denotes a unit normal variable. In particular,

Compare these equations with and The sequence of momentsof X increases even more rapidly than that of and (similarly to the lognormal distribution) the distribution of X is also not determined by itsmoments. Any normal distribution say, can be obtained as a degenerate limiting case of the convolution, letting a 0, andkeeping the mean and the variance equal to and respectively.

Unlike the three-parameter lognormal distribution, the likelihood of thelognormal-normal convolution has no singularities, and the distribution isinfinitely differentiable with respect to all its parameters. However, it is stillpossible that data may conform better to a normal than to a lognormal-normal distribution. (1991) finds that his nondegeneracy test (see Section 4) is equally applicable in the lognormal-normal case with re-placed by

Moment estimation is straightforward. Let = and S 2= -

and let = - and = then utilizing equa-tion or its explicit solution when estimating moments of the three-parameter lognormal, we have

= log

BIBLIOGRAPHY 249

However, (1991) finds that this modified method of moments in thecase of lognormal-normal distribution is by far less effective than the conven-tional one. Moment estimators turn out to be highly biased, but still they mayprovide a useful starting point for maximum likelihood estimation. This approach is a very time-consuming routine operation for thenormal distribution, since it involves numerous evaluations of the density and its derivatives for each value in the sample. Especially undesirable aresituations where the initial is large and the initial close to zero (casesclose to degeneracy). recommends a hybrid algorithm behaving likesteepest descent method in the initial steps and like Newton's method close to the optimum.

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