# BenktanderGibratDistribution

represents a Benktander distribution of type I with parameters a and b.

# Details • The probability density for value in a BenktanderGibrat distribution is proportional to for .
• BenktanderGibratDistribution allows and to be any positive real numbers such that .
• BenktanderGibratDistribution allows a and b to be dimensionless quantities. »
• BenktanderGibratDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• represents a continuous statistical distribution defined over the interval and parametrized by two positive values a, b that satisfy and that determine the overall behavior of the probability density function (PDF). Depending on the values of a and b, the PDF of the BenktanderGibrat distribution may be monotonic decreasing or may be unimodal, with potential singularities approaching the leftmost boundary of its domain. The tails of the PDF of BenktanderGibratDistribution are "heavy" (in the sense that the PDF decreases more slowly than exponentially for large values ), though the distribution is considered "subexponential" rather than "fat-tailed". (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
• BenktanderGibrat distributions were formulated by Swedish actuary Gunnar Benktander, who noted that empirical mean excess functions suggest the need for distributions that are "intermediate" between the Pareto and exponential distributions. Most commonly, BenktanderGibrat distributions are used to model asset loss within actuarial science applications. Moreover, subexponential distributions such as the BenktanderGibrat are also useful in studying the properties of random walks. BenktanderGibrat distributions are sometimes called type I Benktander distributions, with the second type of distribution often known as the BenktanderWeibull distribution and implemented as BenktanderWeibullDistribution in the Wolfram Language. Both are considered to be "close to" log-normal distributions, in the sense that the PDFs of the two distributions are qualitatively similar.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a BenktanderGibrat distribution. Distributed[x,BenktanderGibratDistribution[a,b]], written more concisely as xBenktanderGibratDistribution[a,b], can be used to assert that a random variable x is distributed according to a BenktanderGibrat distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[BenktanderGibratDistribution[a,b],x] and CDF[BenktanderGibratDistribution[a,b],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a BenktanderGibrat distribution, EstimatedDistribution to estimate a BenktanderGibrat parametric distribution from given data, and FindDistributionParameters to fit data to a BenktanderGibrat distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic BenktanderGibrat distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic BenktanderGibrat distribution.
• TransformedDistribution can be used to represent a transformed BenktanderGibrat distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a BenktanderGibrat distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving BenktanderGibrat distributions.
• BenktanderGibratDistribution is related to a number of other statistical distributions. For example, as previously noted, both BenktanderGibratDistribution and BenktanderWeibullDistribution belong to the same "family" of distributions, namely those that are considered intermediate between ParetoDistribution and ExponentialDistribution. Qualitatively, the PDF of BenktanderGibratDistribution is similar to that of LogNormalDistribution, thus inducing (qualitative) relationships between BenktanderGibratDistribution and both NormalDistribution and JohnsonDistribution. ParetoDistribution is considered a limiting case of BenktanderGibratDistribution, in the sense that the PDF of tends to that of ParetoDistribution[1,a+1] whenever b tends to 0. This fact implies that the PDFs of both and approach the same limiting function as b tends to 0. Furthermore, the stationary renewal distribution associated with a BenktanderGibrat distribution has precisely the same formula on its domain as the PDF of BeniniDistribution[a,b,1].

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance:

The median can be found numerically:

## Scope(8)

Generate a sample of pseudorandom numbers from a BenktanderGibrat distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness:

Kurtosis:

Different moments with closed forms as functions of parameters:

Closed form for symbolic order:

Hazard function:

Quantile function:

Use dimensionless Quantity to define BenktanderGibratDistribution:

## Applications(3)

An insurance company finds its claim sizes in units of deductibles follow a Benktander type I distribution with parameters and . Find the probability that claim sizes will exceed 2:

Compute the mean excess function for a Benktander type I distribution:

For large , it approaches that of log-normal, also known as Gibrat distribution:

Now replace the complimentary error function with its asymptotics at large arguments:

Find stationary renewal distribution associated with a Benktander type I distribution:

Compare it with BeniniDistribution:

## Properties & Relations(3)

Relationships to other distributions:

A ParetoDistribution is the limiting case of BenktanderGibratDistribution:

Benktander type I distribution is subexponential:

## Neat Examples(1)

PDFs for different a values with CDF contours: