Log-normal distribution

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Log-normal
Probability density function
Plot of the Lognormal PDF
Some log-normal density functions with identical parameter but differing parameters
Cumulative distribution function
Plot of the Lognormal CDF
Cumulative distribution function of the log-normal distribution (with )
Notation
Parameters ,
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF defined only for numbers with a non-positive real part, see text
CF representation is asymptotically divergent but sufficient for numerical purposes
Fisher information

In probability theory, a log-normal (or lognormal) 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. Likewise, if has a normal distribution, then the exponential function of is has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. 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 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.[2]

Notation[edit]

Given a log-normally distributed random variable and two parameters and that are, respectively, the mean and standard deviation of the variable’s natural logarithm, then the logarithm of is normally distributed, and we can write as

with a standard normal variable.

Relation between normal and lognormal distribution. If is normally distributed, then is lognormally distributed.

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 is a positive number .

The two parameters and are not location and scale parameters for a lognormally distributed random variable X, but they are respectively location and scale parameters for the normally distributed logarithm ln X. The quantity eμ is a scale parameter for the family of lognormal distributions.

In contrast, the mean, standard deviation, and variance of the non-logarithmized sample values are respectively denoted , s.d., and in this article. The two sets of parameters can be related as (see also Arithmetic moments below)[3]

Characterization[edit]

Probability density function[edit]

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(0,1) distribution.

Then we have[1]

Cumulative distribution function[edit]

The cumulative distribution function is

where is the cumulative distribution function of the standard normal distribution (i.e. N(0,1)).

This may also be expressed as follows:

where erfc is the complementary error function.

Characteristic function and moment generating function[edit]

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.[4] 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 t but is not defined for any complex value of t 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.[5] In particular, its Taylor formal series diverges:

However, a number of alternative divergent series representations have been obtained[5][6][7][8]

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[9]

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[edit]

Let denoted the geometric mean, and the geometric standard deviation of the random variable X, while and are as usual the mean, or expected value, and standard deviation.

Geometric moments[edit]

The geometric mean of the log-normal distribution is , and the geometric standard deviation is .[10][11] By analogy with the arithmetic statistics, one can define a geometric variance, , and a geometric coefficient of variation,[10] .

Because the log-transformed variable is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median, .[12]

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. 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[edit]

For any real or complex number n, the n-th moment of a log-normally distributed variable X is given by[1]

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are given by

respectively.

The parameters μ and σ can be obtained if the arithmetic mean and the arithmetic variance are known:

A probability distribution is not uniquely determined by the moments E[Xn] = e + 1/2n2σ2 for n ≥ 1. That is, there exist other distributions with the same set of moments.[1] In fact, there is a whole family of distributions with the same moments as the log-normal distribution.[citation needed]

Mode and median[edit]

Comparison of mean, median and mode of two log-normal distributions with different skewness.

The mode is the point of global maximum of the probability density function. In particular, it solves the equation :

The median is such a point where :

Arithmetic coefficient of variation[edit]

The arithmetic coefficient of variation is the ratio (on the natural scale). For a log-normal distribution it is equal to

Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

Partial expectation[edit]

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 Φ 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[edit]

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

Other[edit]

A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).[13]

The harmonic , geometric and arithmetic means of this distribution are related;[14] such relation is given by

Log-normal distributions are infinitely divisible,[15] but they are not stable distributions, which can be easily drawn from.[16]

Occurrence and applications[edit]

The log-normal distribution is important in the description of natural phenomena. This follows, because 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 (though if the standard deviation is sufficiently small, the normal distribution can be an adequate approximation).

This multiplicative version of the central limit theorem is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies.[17] 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.

Examples include the following:

  • Human behaviors
    • The length of comments posted in Internet discussion forums follows a log-normal distribution.[18]
    • The users' dwell time on the online articles (jokes, news etc.) follows a log-normal distribution.[19]
  • In biology and medicine,
    • Measures of size of living tissue (length, skin area, weight);[20]
    • For highly communicable epidemics, such as SARS in 2003, if publication intervention is involved, the number of hospitalized cases is shown to satisfy 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.[21]
    • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth;[citation needed]
    • Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations)[22]
    • In neuroscience, the distribution of firing rates across a population of neurons is often approximately lognormal. This has been observed in the hippocampus and entorhinal cortex,[23] and elsewhere in the brain.[24][25]
Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see distribution fitting

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.

  • The length of chess games tends to follow a log normal distribution.[35]

Extremal principle of entropy to fix the free parameter [edit]

  • In applications, is a parameter to be determined. In cases that there are no data to determine this parameter, it is possible to evaluate it from some universal principle. One is the entropy method. For growing processes which are governed by production and dissipation, it was shown that one can use some extremal principle of Shannon entropy to determine this parameter to be . This value can then be used to give some scaling relation between the inflexion point and maximum point of the lognormal distribution.[36] It is shown that this relationship is determined by the base of natural logarithm e = 2.718 and exhibits some geometrical similarity to the minimal surface energy principle. These scaling relations are shown to be useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth,swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences,etc) . For instances, the lognormal function with such fits well with the size of secondary produced droplet during droplet impact [37] and the spreading of one epidemic disease.[21]

Maximum likelihood estimation of parameters[edit]

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

where is the density function of the normal distribution . Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

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, 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

Multivariate log-normal[edit]

If is a multivariate normal distribution then has a multivariate log-normal distribution[38][39] with mean

and covariance matrix

Related distributions[edit]

  • If is a normal distribution, then
  • If is distributed log-normally, then is a normal random variable.
  • If are independent log-normally distributed variables, and , then is also distributed log-normally:
  • Let be independent log-normally distributed variables with possibly varying and parameters, and . The distribution of has no closed-form expression, but can be reasonably approximated by another log-normal distribution at the right tail.[40] Its probability density function at the neighborhood of 0 has been characterized[16] 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 and mathematical justified by Marlow[41]) is obtained by matching the mean and variance of another lognormal distribution:
In the case that all have the same variance parameter , these formulas simplify to
  • If then is said to have a shifted log-normal distribution with support . , .
  • If then
  • If then
  • If then for
  • Lognormal distribution is a special case of semi-bounded Johnson distribution
  • If with , then (Suzuki distribution)
  • A substitute for the log-normal whose integral can be expressed in terms of more elementary functions[42] can be obtained based on the logistic distribution to get an approximation for the CDF
This is a log-logistic distribution.

See also[edit]

Notes[edit]

  1. ^ a b c d e Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994), "14: Lognormal Distributions", Continuous univariate distributions. Vol. 1, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-58495-7, MR 1299979 
  2. ^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF). Journal of Econometrics. Elsevier. 150 (2): 219–230. doi:10.1016/j.jeconom.2008.12.014. Retrieved 2011-06-02.  Table 1, p. 221.
  3. ^ "Lognormal mean and variance"
  4. ^ Heyde, CC. (1963), "On a property of the lognormal distribution", Journal of the Royal Statistical Society, Series B (Methodological), 25 (2): 392–393, doi:10.1007/978-1-4419-5823-5_6 
  5. ^ a b Holgate, P. (1989). "The lognormal characteristic function, vol. 18, pp. 4539–4548, 1989". Communications in Statistical – Theory and Methods. 18 (12): 4539–4548. doi:10.1080/03610928908830173. 
  6. ^ Barakat, R. (1976). "Sums of independent lognormally distributed random variables". Journal of the Optical Society of America. 66 (3): 211–216. doi:10.1364/JOSA.66.000211. 
  7. ^ Barouch, E.; Kaufman, GM.; Glasser, ML. (1986). "On sums of lognormal random variables" (PDF). Studies in Applied Mathematics. 75 (1): 37–55. 
  8. ^ Leipnik, Roy B. (January 1991). "On Lognormal Random Variables: I – The Characteristic Function". Journal of the Australian Mathematical Society Series B. 32 (3): 327–347. doi:10.1017/S0334270000006901. 
  9. ^ S. Asmussen, J.L. Jensen, L. Rojas-Nandayapa (2016). "On the Laplace transform of the Lognormal distribution", Methodology and Computing in Applied Probability 18 (2), 441-458. Thiele report 6 (13).
  10. ^ a b Kirkwood, Thomas BL (Dec 1979). "Geometric means and measures of dispersion". Biometrics. 35 (4): 908–9. doi:10.2307/2530139. 
  11. ^ Limpert, E; Stahel, W; Abbt, M (2001). "Lognormal distributions across the sciences: keys and clues". BioScience. 51 (5): 341–352. doi:10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2. 
  12. ^ Daly, Leslie E.; Bourke, Geoffrey Joseph (2000). Interpretation and uses of medical statistics (5th ed.). Wiley-Blackwell. p. 89. ISBN 978-0-632-04763-5. doi:10.1002/9780470696750. 
  13. ^ Damgaard, Christian; Weiner, Jacob (2000). "Describing inequality in plant size or fecundity". Ecology. 81 (4): 1139–1142. doi:10.1890/0012-9658(2000)081[1139:DIIPSO]2.0.CO;2. 
  14. ^ Rossman, Lewis A (July 1990). "Design stream flows based on harmonic means". J Hydraulic Engineering. 116 (7): 946–950. doi:10.1061/(ASCE)0733-9429(1990)116:7(946). 
  15. ^ Thorin, Olof (1977). "On the infinite divisibility of the lognormal distribution". Scandinavian Actuarial Journal. 1977 (3): 121–148. ISSN 0346-1238. doi:10.1080/03461238.1977.10405635. 
  16. ^ a b Gao, X.; Xu, H; Ye, D. (2009), "Asymptotic Behaviors of Tail Density for Sum of Correlated Lognormal Variables". International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 630857. doi:10.1155/2009/630857
  17. ^ Sutton, John (Mar 1997). "Gibrat's Legacy". Journal of Economic Literature. 32 (1): 40–59. JSTOR 2729692. 
  18. ^ Pawel, Sobkowicz; et al. (2013). "Lognormal distributions of user post lengths in Internet discussions - a consequence of the Weber-Fechner law?". EPJ Data Science. 
  19. ^ Yin, Peifeng; Luo, Ping; Lee, Wang-Chien; Wang, Min (2013). Silence is also evidence: interpreting dwell time for recommendation from psychological perspective. ACM International Conference on KDD. 
  20. ^ Huxley, Julian S. (1932). Problems of relative growth. London. ISBN 0-486-61114-0. OCLC 476909537. 
  21. ^ a b Wang, WenBin; Wu, ZiNiu; Wang, ChunFeng; Hu, RuiFeng (2013). "Modelling the spreading rate of controlled communicable epidemics through an entropy-based thermodynamic model". Science China Physics, Mechanics and Astronomy. 56 (11): 2143–2150. ISSN 1674-7348. doi:10.1007/s11433-013-5321-0. 
  22. ^ Makuch, Robert W.; D.H. Freeman; M.F. Johnson (1979). "Justification for the lognormal distribution as a model for blood pressure". Journal of Chronic Diseases. 32 (3): 245–250. doi:10.1016/0021-9681(79)90070-5. Retrieved 27 February 2012. 
  23. ^ Mizuseki, Kenji; Buzsáki, György (2013-09-12). "Preconfigured, skewed distribution of firing rates in the hippocampus and entorhinal cortex". Cell Reports. 4 (5): 1010–1021. ISSN 2211-1247. PMC 3804159Freely accessible. PMID 23994479. doi:10.1016/j.celrep.2013.07.039. 
  24. ^ Buzsáki, György; Mizuseki, Kenji (2017-01-06). "The log-dynamic brain: how skewed distributions affect network operations". Nature Reviews. Neuroscience. 15 (4): 264–278. ISSN 1471-003X. PMC 4051294Freely accessible. PMID 24569488. doi:10.1038/nrn3687. 
  25. ^ Wohrer, Adrien; Humphries, Mark D.; Machens, Christian K. (2013-04-01). "Population-wide distributions of neural activity during perceptual decision-making". Progress in Neurobiology. 103: 156–193. ISSN 1873-5118. PMID 23123501. doi:10.1016/j.pneurobio.2012.09.004. 
  26. ^ Ritzema (ed.), H.P. (1994). Frequency and Regression Analysis (PDF). Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp. 175–224. ISBN 90-70754-33-9. 
  27. ^ Clementi, Fabio; Gallegati, Mauro (2005) "Pareto's law of income distribution: Evidence for Germany, the United Kingdom, and the United States", EconWPA
  28. ^ Wataru, Souma (2002-02-22). "Physics of Personal Income". arXiv:cond-mat/0202388Freely accessible. 
  29. ^ Black, F.; Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy. 81 (3): 637. doi:10.1086/260062. 
  30. ^ Mandelbrot, Benoit (2004). The (mis-)Behaviour of Markets. Basic Books. ISBN 9780465043552. 
  31. ^ Bunchen, P., Advanced Option Pricing, University of Sydney coursebook, 2007
  32. ^ O'Connor, Patrick; Kleyner, Andre (2011). Practical Reliability Engineering. John Wiley & Sons. p. 35. ISBN 978-0-470-97982-2. 
  33. ^ http://wireless.per.nl/reference/chaptr03/shadow/shadow.htm Archived January 13, 2012, at the Wayback Machine.
  34. ^ Gros, C; Kaczor, G.; Markovic, D (2012). "Neuropsychological constraints to human data production on a global scale". The European Physical Journal B. 85 (28). doi:10.1140/epjb/e2011-20581-3. 
  35. ^ http://chess.stackexchange.com/questions/2506/what-is-the-average-length-of-a-game-of-chess/4899#4899
  36. ^ Wu, Ziniu; Li, Juan; Bai, Chenyuan (2017). "Scaling Relations of Lognormal Type Growth Process with an Extremal Principle of Entropy". Entropy. 19 (56): 1–14. doi:10.3390/e19020056. 
  37. ^ Wu, Z.N. (2003), [ "Prediction of the size distribution of secondary ejected droplets by crown splashing of droplets impinging on a solid wall"]. Probabilistic Engineering Mechanics, Volume 18, Issue 3, July 2003, Pages 241–249. doi:10.1016/S0266-8920(03)00028-6
  38. ^ Tarmast, Ghasem (2001). Multivariate Log–Normal Distribution (PDF). ISI Proceedings: 53rd Session. Seoul. 
  39. ^ Halliwell, Leigh (2015). The Lognormal Random Multivariate (PDF). Casualty Actuarial Society E-Forum, Spring 2015. Arlington, VA. 
  40. ^ Asmussen, S.; Rojas-Nandayapa, L. (2008). "Asymptotics of Sums of Lognormal Random Variables with Gaussian Copula". Statistics and Probability Letters. 78 (16): 2709–2714. doi:10.1016/j.spl.2008.03.035. 
  41. ^ Marlow, NA. (Nov 1967). "A normal limit theorem for power sums of independent normal random variables". Bell System Technical Journal. 46 (9): 2081–2089. doi:10.1002/j.1538-7305.1967.tb04244.x. 
  42. ^ Swamee, P. K. (2002). "Near Lognormal Distribution". Journal of Hydrologic Engineering. 7 (6): 441–444. doi:10.1061/(ASCE)1084-0699(2002)7:6(441). 

Further reading[edit]

  • Crow, Edwin L.; Shimizu, Kunio (Editors) (1988), Lognormal Distributions, Theory and Applications, Statistics: Textbooks and Monographs, 88, New York: Marcel Dekker, Inc., pp. xvi+387, ISBN 0-8247-7803-0, MR 0939191, Zbl 0644.62014 
  • Aitchison, J. and Brown, J.A.C. (1957) The Lognormal Distribution, Cambridge University Press.
  • Limpert, E; Stahel, W; Abbt, M (2001). "Lognormal distributions across the sciences: keys and clues". BioScience. 51 (5): 341–352. doi:10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2. 
  • Eric W. Weisstein et al. Log Normal Distribution at MathWorld. Electronic document, retrieved October 26, 2006.
  • Holgate, P. (1989). "The lognormal characteristic function". Communications in Statistics - Theory and Methods. 18 (12): 4539–4548. doi:10.1080/03610928908830173. 
  • Brooks, Robert; Corson, Jon; Donal, Wales (1994). "The Pricing of Index Options When the Underlying Assets All Follow a Lognormal Diffusion". Advances in Futures and Options Research. 7. SSRN 5735Freely accessible. 

External links[edit]