Log-Normal

Template:Probability distribution

 | cdf       = \frac12 + \frac12\,\mathrm{erf}\Big[\frac{\ln x-\mu}{\sqrt{2}\sigma}\Big]
 | mean      = e^{\mu+\sigma^2/2}
 | median    = e^{\mu}\,
 | mode      = e^{\mu-\sigma^2}
 | variance  = (e^{\sigma^2}\!\!-1) e^{2\mu+\sigma^2}
 | skewness  = (e^{\sigma^2}\!\!+2) \sqrt{e^{\sigma^2}\!\!-1}
 | kurtosis  = e^{4\sigma^2}\!\! + 2e^{3\sigma^2}\!\! + 3e^{2\sigma^2}\!\! - 6
 | entropy   = \frac12 + \frac12 \ln(2\pi\sigma^2) + \mu
 | mgf       = (defined only on the negative half-axis, see text)
 | char      = representation \sum_{n=0}^{\infty}\frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2} is asymptotically divergent but sufficient for numerical purposes
 | fisher    = \begin{pmatrix}1/\sigma^2&0\\0&1/(2\sigma^4)\end{pmatrix}
 }}

In probability theory, a log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable Y is log-normally distributed, then X = \log(Y) has a normal distribution. Likewise, if X has a normal distribution, then Y = \exp(X) has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values.

Log-normal is also written log normal or lognormal. The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.[1] The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.[1]

A variable might be modeled as log-normal if it can be thought of as the multiplicative product of many independent random variables each of which is positive. (This is justified by considering the central limit theorem in the log-domain.) For example, in finance, the variable could represent the compound return from a sequence of many trades (each expressed as its return + 1); or a long-term discount factor can be derived from the product of short-term discount factors. In wireless communication, the sas caused by shadowing or slow fading from random objects is often assumed to be log-normally distributed: see log-distance path loss model.

The log-normal distribution is the maximum entropy probability distribution for a random variate X for which the mean and variance of \ln(X) are fixed.[2]

μ and σ

In a log-normal distribution X, the parameters denoted μ and σ are, respectively, the mean and standard deviation of the variable’s natural logarithm (by definition, the variable’s logarithm is normally distributed), which means

X=e^{\mu+\sigma Z}

with Z a standard normal variable.

This relationship is true regardless of the base of the logarithmic or exponential function. If loga(Y) is normally distributed, then so is logb(Y), for any two positive numbers ab ≠ 1. Likewise, if e^X is log-normally distributed, then so is a^{X}, where a is a positive number ≠ 1.

On a logarithmic scale, μ and σ can be called the location parameter and the scale parameter, respectively.

In contrast, the mean and standard deviation of the non-logarithmized sample values are denoted m and s.d. in this article.

A log-normal distribution with mean m and variance v has parameters[3]

\mu=\ln\left(\frac{m^2}{\sqrt{v+m^2}}\right), \sigma=\sqrt{\ln(1+\frac{v}{m^2})}

Characterization

Probability density function

The probability density function of a log-normal distribution is:[1]

f_X(x;\mu,\sigma) = \frac{1}{x \sigma \sqrt{2 \pi}}\, e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}},\ \ x>0

This follows by applying the change-of-variables rule on the density function of a normal distribution.

Cumulative distribution function

The cumulative distribution function is

F_X(x;\mu,\sigma) = \frac12 \left[ 1 + \operatorname{erf}\!\left(\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) \right] = \frac12 \operatorname{erfc}\!\left(-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) = \Phi\bigg(\frac{\ln x - \mu}{\sigma}\bigg),

where erfc is the complementary error function, and Φ is the cumulative distribution function of the standard normal distribution.

Characteristic function and moment generating function

All moments of the log-normal distributions exist and it holds that

\operatorname{E}(X^n)=\mathrm{e}^{n\mu+\frac{n^2\sigma^2}{2}}.

However, the moment generating function

\operatorname{E}(e^{t X})=\sum_{n=0}^\infty \frac{t^n}{n!}\operatorname{E}(X^n)

does not converge.

The characteristic function, E[e itX], has a number of representations. The integral itself converges for Im(t) ≤ 0. The simplest representation is obtained by Taylor expanding e itX and using formula for moments below, giving

\varphi(t) = \sum_{n=0}^\infty \frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2}.

This series representation is divergent for Re(σ2) > 0. However, it is sufficient for evaluating the characteristic function numerically at positive \sigma as long as the upper limit in the sum above is kept bounded, n ≤ N, where

\max(|t|,|\mu|) \ll N \ll \frac{2}{\sigma^2}\ln\frac{2}{\sigma^2}

and σ2 < 0.1. To bring the numerical values of parameters μσ into the domain where strong inequality holds true one could use the fact that if X is log-normally distributed then Xm is also log-normally distributed with parameters μmσm. Since \mu\sigma^2 \propto m^3, the inequality could be satisfied for sufficiently small m. The sum of series first converges to the value of φ(t) with arbitrary high accuracy if m is small enough, and left part of the strong inequality is satisfied. If considerably larger number of terms are taken into account the sum eventually diverges when the right part of the strong inequality is no longer valid.

Another useful representation is available[4][5] by means of double Taylor expansion of e(ln x − μ)2/(2σ2).

The moment-generating function for the log-normal distribution does not exist on the domain R, but only exists on the half-interval (−∞, 0].

Properties

Location and scale

For the log-normal distribution, the location and scale properties of the distribution are more readily treated using the geometric mean and geometric standard deviation than the arithmetic mean and standard deviation.

Geometric moments

The geometric mean of the log-normal distribution is e^{\mu}. Because the log of a log-normal variable is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median.[6]

The geometric mean (mg) can alternatively be derived from the arithmetic mean (ma) in a log-normal distribution by:

m_g = m_ae^{-\tfrac{1}{2}\sigma^2}.

Note that the geometric mean is less than the arithmetic mean. This is due to the AM–GM inequality, and corresponds to the logarithm being convex down. The correction term e^{-\tfrac{1}{2}\sigma^2} can accordingly be interpreted as a convexity correction. From the point of view of stochastic calculus, this is the same correction term as in Itō's lemma for geometric Brownian motion.

The geometric standard deviation is equal to e^{\sigma}.

Arithmetic moments

If X is a lognormally distributed variable, its expected value (E – the arithmetic mean), variance (Var), and standard deviation (s.d.) are

\begin{align}
 & \operatorname{E}[X] = e^{\mu + \tfrac{1}{2}\sigma^2}, \\
 & \operatorname{Var}[X] = (e^{\sigma^2} - 1) e^{2\mu + \sigma^2} = (e^{\sigma^2} - 1)(\operatorname{E}[X])^2\\
 & \operatorname{s.d.}[X] = \sqrt{\operatorname{Var}[X]} = e^{\mu + \tfrac{1}{2}\sigma^2}\sqrt{e^{\sigma^2} - 1}.
 \end{align}

Equivalently, parameters μ and σ can be obtained if the expected value and variance are known; it is simpler if σ is computed first:

\begin{align}
 \mu &= \ln(\operatorname{E}[X]) - \frac12 \ln\!\left(1 + \frac{\mathrm{Var}[X]}{(\operatorname{E}[X])^2}\right) = \ln(\operatorname{E}[X]) - \frac12 \sigma^2, \\
 \sigma^2 &= \ln\!\left(1 + \frac{\operatorname{Var}[X]}{(\operatorname{E}[X])^2}\right).
 \end{align}

For any real or complex number s, the sth moment of log-normal X is given by[1]

\operatorname{E}[X^s] = e^{s\mu + \tfrac{1}{2}s^2\sigma^2}.

A log-normal distribution is not uniquely determined by its moments E[Xk] for k ≥ 1, that is, there exists some other distribution with the same moments for all k.[1] In fact, there is a whole family of distributions with the same moments as the log-normal distribution.

Mode and median

The mode is the point of global maximum of the probability density function. In particular, it solves the equation (ln ƒ)′ = 0:

\mathrm{Mode}[X] = e^{\mu - \sigma^2}.

The median is such a point where FX = 1/2:

\mathrm{Med}[X] = e^\mu\,.

Coefficient of variation

The coefficient of variation is the ratio s.d. over m (on the natural scale) and is equal to:

\sqrt{e^{\sigma^2}\!\!-1}

Partial expectation

The partial expectation of a random variable X with respect to a threshold k is defined as g(k) = \int_k^\infty \!xf(x)\, dx where f(x) is the probability density function of X. Alternatively, and using the definition of conditional expectation, it can be written as g(k)=E[X | X > k]*P(X > k). For a log-normal random variable the partial expectation is given by:

g(k) = \int_k^\infty \!xf(x)\, dx
           = e^{\mu+\tfrac{1}{2}\sigma^2}\, \Phi\!\left(\frac{\mu+\sigma^2-\ln k}{\sigma}\right).

The derivation of the formula is provided in the discussion of this World Heritage Encyclopedia entry. The partial expectation formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the Black–Scholes formula.

Other

A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).[7]

The harmonic (H), geometric (G) and arithmetic (A) means of this distribution are related;[8] such relation is given by

H = \frac{G^2}{ A} .

Log-normal distributions are infinitely divisible.[1]

Occurrence

The log-normal distribution is important in the description of natural phenomena. The reason is that for many natural processes of growth, growth rate is independent of size. This is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for firms (companies). It can be shown that a growth process following Gibrat's law will result in entity sizes with a log-normal distribution.[9] Examples include:

  • In biology and medicine,
    • Measures of size of living tissue (length, skin area, weight);[10]
    • For highly communicable epidemics, such as SARS in 2003, if publication intervention is involved, the the number of hospitalized cases is shown to satistfy the lognormal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of maximum rate of entropy production[11] .
    • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth;
    • Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations)[12]


Consequently, reference ranges for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.
  • In hydrology, the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.[13]
  • in social sciences and demographics
    • In economics, there is evidence that the income of 97%–99% of the population is distributed log-normally.[14]
    • In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal[15] (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as Benoît Mandelbrot have argued [16] that log-Lévy distributions which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. Indeed stock price distributions typically exhibit a fat tail.[17]
    • city sizes
  • technology
    • In reliability analysis, the lognormal distribution is often used to model times to repair a maintainable system.
    • In wireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution." [18]
    • It has been proposed that coefficients of friction and wear may be treated as having a lognormal distribution [19]
    • In spray process, such as droplet impact, the size of secondary produced droplet has a lognormal distribution, with the standard deviation :\sigma=\frac{\sqrt{6}}{6} determined by the principle of maximum rate of entropy production[20] If the lognormal distribution is inserted into the Shannon entropy expression and if the rate of entropy production is maximized (principle of maximum rate of entropy production), then σ is given by :\sigma=\frac{1}{\sqrt{6}}[20] and with this parameter the droplet size distribution for spray process is well predicted. It is an open question whether this value of σ has some generality for other cases, though for spreading of communicable epidemics, σ is shown also to take this value.[11]

Maximum likelihood estimation of parameters

For determining the maximum likelihood estimators of the log-normal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. To avoid repetition, we observe that

f_L (x;\mu, \sigma) = \prod_{i=1}^n \left(\frac 1 x_i\right) \, f_N (\ln x; \mu, \sigma)

where by ƒL we denote the probability density function of the log-normal distribution and by ƒN that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

\begin{align} \ell_L (\mu,\sigma | x_1, x_2, \dots, x_n)

 & {} = - \sum _k \ln x_k + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n) \\

& {} = \operatorname {constant} + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n). \end{align}

Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, L and N, reach their maximum with the same μ and σ. Hence, using the formulas for the normal distribution maximum likelihood parameter estimators and the equality above, we deduce that for the log-normal distribution it holds that

\widehat \mu = \frac {\sum_k \ln x_k} n,
       \widehat \sigma^2 = \frac {\sum_k \left( \ln x_k - \widehat \mu \right)^2} {n}.

Multivariate log-normal

If \boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma) is a multivariate normal distribution then \boldsymbol Y=\exp(\boldsymbol X) has a multivariate log-normal distribution[21] with mean

\operatorname{E}[\boldsymbol Y]_i=e^{\mu_i+\frac{1}{2}\Sigma_{ii}} ,

and covariance matrix

\operatorname{Var}[\boldsymbol Y]_{ij}=e^{\mu_i+\mu_j + \frac{1}{2}(\Sigma_{ii}+\Sigma_{jj}) }( e^{\Sigma_{ij}} - 1) .

Generating log-normally distributed random variates

Given a random variate Z drawn from the normal distribution with 0 mean and 1 standard deviation, then the variate

X= e^{\mu + \sigma Z}\,

has a log-normal distribution with parameters \mu and \sigma.

Related distributions

  • If X \sim \mathcal{N}(\mu, \sigma^2) is a normal distribution, then \exp(X) \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2).
  • If X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2) is distributed log-normally, then \ln(X) \sim \mathcal{N}(\mu, \sigma^2) is a normal random variable.
  • If X_j \sim \operatorname{Log-\mathcal{N}}(\mu_j, \sigma_j^2) are n independent log-normally distributed variables, and Y = \textstyle\prod_{j=1}^n X_j, then Y is also distributed log-normally:
Y \sim \operatorname{Log-\mathcal{N}}\Big(\textstyle \sum_{j=1}^n\mu_j,\ \sum_{j=1}^n \sigma_j^2 \Big).
  • Let X_j \sim \operatorname{Log-\mathcal{N}}(\mu_j,\sigma_j^2)\ be independent log-normally distributed variables with possibly varying σ and μ parameters, and Y=\textstyle\sum_{j=1}^n X_j. The distribution of Y has no closed-form expression, but can be reasonably approximated by another log-normal distribution Z at the right tail. Its probability density function at the neighborhood of 0 has been characterized[22] and it does not resemble any log-normal distribution. A commonly used approximation (due to L.F. Fenton, but previously stated by R.I. Wilkinson without mathematical justification[23]) is obtained by matching the mean and variance:
\begin{align}
 \sigma^2_Z &= \log\!\left[ \frac{\sum e^{2\mu_j+\sigma_j^2}(e^{\sigma_j^2}-1)}{(\sum e^{\mu_j+\sigma_j^2/2})^2} + 1\right], \\
 \mu_Z &= \log\!\left[ \sum e^{\mu_j+\sigma_j^2/2} \right] - \frac{\sigma^2_Z}{2}.
 \end{align}

In the case that all X_j have the same variance parameter \sigma_j=\sigma, these formulas simplify to

\begin{align}
 \sigma^2_Z &= \log\!\left[ (e^{\sigma^2}-1)\frac{\sum e^{2\mu_j}}{(\sum e^{\mu_j})^2} + 1\right], \\
 \mu_Z &= \log\!\left[ \sum e^{\mu_j} \right] + \frac{\sigma^2}{2} -  \frac{\sigma^2_Z}{2}.
 \end{align}
  • If X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2), then X + c is said to have a shifted log-normal distribution with support x ∈ (c, +∞). E[X + c] = E[X] + c, Var[X + c] = Var[X].
  • If X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2), then a X \sim \operatorname{Log-\mathcal{N}}( \mu + \ln a,\ \sigma^2).
  • If X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2), then \tfrac{1}{X} \sim \operatorname{Log-\mathcal{N}}(-\mu,\ \sigma^2).
  • If X \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2) then X^a \sim \operatorname{Log-\mathcal{N}}(a\mu,\ a^2 \sigma^2). for a \neq 0\,
  • Lognormal distribution is a special case of semi-bounded Johnson distribution
  • If X|Y \sim \mathrm{Rayleigh}(Y)\, with Y \sim \operatorname{Log-\mathcal{N}}(\mu, \sigma^2), then X \sim \mathrm{Suzuki}(\mu, \sigma)\, (Suzuki distribution)

Similar distributions

A substitute for the log-normal whose integral can be expressed in terms of more elementary functions[24] can be obtained based on the logistic distribution to get an approximation for the CDF

F(x;\mu,\sigma) = \left[\left(\frac{e^\mu}{x}\right)^{\pi/(\sigma \sqrt{3})} +1\right]^{-1}.

This is a log-logistic distribution.

See also

Notes

References

  • Aitchison, J. and Brown, J.A.C. (1957) The Lognormal Distribution, Cambridge University Press.
  • E. Limpert, W. Stahel and M. Abbt (2001) Log-normal Distributions across the Sciences: Keys and Clues, BioScience, 51 (5), 341–352.
  • MathWorld. Electronic document, retrieved October 26, 2006.


  • expand by hand

Further reading

  • Robert Brooks, Jon Corson, and "The Pricing of Index Options When the Underlying Assets All Follow a Lognormal Diffusion", in Advances in Futures and Options Research, volume 7, 1994.

External links

Template:ProbDistributions
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