Logarithmic distribution



Format: Logarithmic(q)

 

The logarithmic distribution (sometimes known as the Logarithmic Series distribution) is a discrete, positive distribution, peaking at x = 1, with one parameter and a long right tail. The figures below show two examples of the Logarithmic distribution.

 

Uses

The logarithmic distribution is quite popular in insurance for modeling a claim frequency. It has been used to describe, for example: the number of items purchased by a consumer in a particular period; the number of bird and plant species in an area; and the number of parasites per host. There is some theory that relates the latter two to an observation by Newcomb (1881) that the frequency of use of different digits in natural numbers followed a Logarithmic distribution.

The logarithmic distribution has also been used in microbiology to model the number of organisms in a cluster. The principal motivation for doing this is that if the number of clumps in a sample are Poisson distributed, then the total number of organisms in the sample follows a Pólya distribution.

Zero-modified version

When modeling or analyzing counting data, it is often desirable to modify probability of zero of the discrete distribution we use, to more accurately model the probability of "no event occurring". We can make two types of modifications to our distribution for this:

  • Zero-inflated model - we increase the probability of zero.

  • Zero-truncated model - as the probability at zero of an unmodified Logarithmic distribution is already zero by itself, there is no need for a separate zero-truncated model.

ModelRisk functions added to Microsoft Excel for the Logarithmic distribution

VoseLogarithmic generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseLogarithmicObject constructs a distribution object for this distribution.

VoseLogarithmicProb returns the probability mass or cumulative distribution function for this distribution.

VoseLogarithmicProb10 returns the log10 of the probability mass or cumulative distribution function.  

VoseLogarithmicFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseLogarithmicFitObject constructs a distribution object of this distribution fitted to data.

VoseLogarithmicFitP returns the parameters of this distribution fitted to data.

 

ModelRisk functions added to Microsoft Excel for Zero-Inflated Logarithmic distribution

VoseZILogarithmic generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZILogarithmicObject constructs a distribution object for this distribution.

VoseZILogarithmicProb returns the probability mass or cumulative distribution function for this distribution.

VoseZILogarithmicProb10 returns the log10 of the probability mass or cumulative distribution function.  

VoseZILogarithmicFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZILogarithmicFitObject constructs a distribution object of this distribution fitted to data.

VoseZILogarithmicFitP returns the parameters of this distribution fitted to data.

 

Logarithmic distribution equations

Zero-Inflated Logarithmic distribution equations

 

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