Log-normal distribution


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 is log-normally distributed, then has a normal distribution. Equivalently, if has a normal distribution, then the exponential function of,, has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences as well as medicine, economics and other fields, e.g. for energies, concentrations, lengths, financial returns and other amounts.
The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.
A log-normal process is the statistical realization of 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. The log-normal distribution is the maximum entropy probability distribution for a random variate for which the mean and variance of are specified.

Definitions

Generation and parameters

Let be a standard normal variable, and let and be two real numbers. Then, the distribution of the random variable
is called the log-normal distribution with parameters and. These are the expected value and standard deviation of the variable's natural logarithm, not the expectation and standard deviation of itself.
This relationship is true regardless of the base of the logarithmic or exponential function. If is normally distributed, then so is, for any two positive numbers. Likewise, if is log-normally distributed, then so is, where.
In order to produce a distribution with desired mean and variance, one uses
and
Alternatively, the "multiplicative" or "geometric" parameters and can be used. They have a more direct interpretation: is the median of the distribution, and is useful for determining "scatter" intervals, see below.

Probability density function

A positive random variable X is log-normally distributed if the logarithm of X is normally distributed,
Let and be respectively the cumulative probability distribution function and the probability density function of the N distribution.
Then we have

Cumulative distribution function

The cumulative distribution function is
where is the cumulative distribution function of the standard normal distribution.
This may also be expressed as follows:
where erfc is the complementary error function.

Multivariate log-normal

If is a multivariate normal distribution then has a multivariate log-normal distribution with mean
and covariance matrix
Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the univariate distribution.

Characteristic function and moment generating function

All moments of the log-normal distribution exist and
This can be derived by letting within the integral. However, the expected value is not defined for any positive value of the argument as the defining integral diverges. In consequence the moment generating function is not defined. The last is related to the fact that the lognormal distribution is not uniquely determined by its moments.
The characteristic function is defined for real values of but is not defined for any complex value of that has a negative imaginary part, and therefore the characteristic function is not analytic at the origin. In consequence, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. In particular, its Taylor formal series diverges:
However, a number of alternative divergent series representations have been obtained.
A closed-form formula for the characteristic function with in the domain of convergence is not known. A relatively simple approximating formula is available in closed form and given by
where is the Lambert W function. This approximation is derived via an asymptotic method but it stays sharp all over the domain of convergence of.

Properties

Geometric or multiplicative moments

The geometric or multiplicative mean of the log-normal distribution is. It equals the median. The geometric or multiplicative standard deviation is. By analogy with the arithmetic statistics, one can define a geometric variance,, and a geometric coefficient of variation, has been proposed. This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of itself.
Note that the geometric mean is smaller than the arithmetic mean. This is due to the AM–GM inequality, and corresponds to the logarithm being convex down. In fact,
In finance the term is sometimes 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.

Arithmetic moments

For any real or complex number, the -th moment of a log-normally distributed variable is given by
Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable are given by
respectively.
The arithmetic coefficient of variation is the ratio. For a log-normal distribution it is equal to
This estimate is sometimes referred to as the "geometric CV" because of its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.
The parameters and can be obtained if the arithmetic mean and the arithmetic variance are known:
A probability distribution is not uniquely determined by the moments for. That is, there exist other distributions with the same set of moments. In fact, there is a whole family of distributions with the same moments as the log-normal distribution.

Mode, median, quantiles

The mode is the point of global maximum of the probability density function. In particular, it solves the equation :
Since the log-transformed variable has a normal distribution and quantiles are preserved under monotonic transformations, the quantiles of are
where is the quantile of the standard normal distribution.
Specifically, the median of a log-normal distribution is equal to its multiplicative mean,

Partial expectation

The partial expectation of a random variable with respect to a threshold is defined as
Alternatively, and using the definition of conditional expectation, it can be written as. For a log-normal random variable the partial expectation is given by:
where is the normal cumulative distribution function. The derivation of the formula is provided in the discussion of this Wikipedia 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.

Conditional expectation

The conditional expectation of a lognormal random variable with respect to a threshold is its partial expectation divided by the cumulative probability of being in that range:

Alternative parameterizations

In addition to the characterization by or, here are multiple ways how the lognormal distribution can be parameterized. ProbOnto, the knowledge base and ontology of probability distributions lists seven such forms:
Consider the situation when one would like to run a model using two different optimal design tools, e.g. PFIM and PopED. The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results.
For the transition following formulas hold
For the transition following formulas hold
All remaining re-parameterisation formulas can be found in the specification document on the project website.

Multiple, Reciprocal, Power

If two independent, log-normal variables and are multiplied , the product is again log-normal, with parameters and, where. This is easily generalized to the product of such variables.
More generally, if are independent, log-normally distributed variables, then

Multiplicative Central Limit Theorem

The geometric or multiplicative mean of independent, identically distributed, positive random variables shows, for approximately a log-normal distribution with parameters and, as the usual Central Limit Theorem, applied to the log-transformed variables, proves. That distribution approaches a Gaussian distribution, since decreases to 0.

Other

A set of data that arises from the log-normal distribution has a symmetric Lorenz curve.
The harmonic, geometric and arithmetic means of this distribution are related; such relation is given by
Log-normal distributions are infinitely divisible, but they are not stable distributions, which can be easily drawn from.

Related distributions

For a more accurate approximation, one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and the right tail.

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. Note that
where is the density function of the normal distribution. Therefore, the log-likelihood function is
Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, and, reach their maximum with the same and. Hence, the maximum likelihood estimators are identical to those for a normal distribution for the observations,
For finite n, these estimators are biased. Whereas the bias for is negligible, a less biased estimator for is obtained as for the normal distribution by replacing the denominator n by n-1 in the equation for.
When the individual values are not available, but the sample's mean and standard deviation s is, then the corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation and variance for and :

Statistics

The most efficient way to analyze log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate.

Scatter intervals

A basic example is given by scatter intervals: For the normal distribution, the interval contains approximately two thirds of the probability, and contain 95 %. Therefore, for a log-normal distribution,
of the probability. Using estimated parameters, the approximately the same percentages of the data should be contained in these intervals.

Confidence interval for \mu^*

Using the principle, note that a confidence interval for is, where is the standard error and q is the 97.5 % quantile of a t distribution with n-1 degrees of freedom. Back-transformation leads to a confidence interval for,

Extremal principle of entropy to fix the free parameter \sigma

The log-normal distribution is important in the description of natural phenomena. In a prototype case, a justification runs as follows: Many natural growth processes are driven by the accumulation of many small percentage changes. These become additive on a log scale. If the effect of any one change is negligible, the central limit theorem says that the distribution of their sum is more nearly normal than that of the summands. When back-transformed onto the original scale, it makes the distribution of sizes approximately log-normal.
This multiplicative version of the central limit theorem is also known as Gibrat's law, after Robert Gibrat who formulated it for companies. If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if that's not true, the size distributions at any age of things that grow over time tends to be log-normal.
A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables. Examples are the simple gravitation law connecting masses and distance with the resulting force, or the formula for equilibrium concentrations of chemicals in a solution that connects concentrations of educts and products. Assuming log-normal distributions of the variables involved leads to consistent models in these cases.
Even if none of these justifications apply, the log-normal distribution is often a plausible and empirically adequate model. Examples include the following:
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.