Beta rectangular distribution: Difference between revisions

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{{Short description|Concept in statistics}}

{{Probability distribution

| name = Beta Rectangular

| type = density

| pdf_image = No[[File:Beta imagerect availablepdf.svg|352px|The support interval is [0,1].]]

| cdf_image = No[[File:Beta imagerect availablecdf.svg|352px|The support interval is [0,1].]]

| notation =

| parameters = <math>\alpha > 0</math> [[shape parameter|shape]] ([[real number|real]])<br /><math>\beta > 0</math> [[shape parameter|shape]] ([[real number|real]]) <br /> <math>1>0<\theta ><1 0</math> mixture parameter <br />

| support =<math>x \in (a, b)\!</math>|

| pdf = <math>\begin{cases}

\frac{\theta \Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}

\frac{(x - a)^{\alpha-1} (b-x)^{\beta - 1} }{(b - a)^{\alpha + \beta +- 1}}

+ \frac{1 - \theta}{b-a}

Line 35 ⟶ 36:

}}

In [[probability theory]] and [[statistics]], the '''beta rectangular distribution''' is a [[probability distribution]] that is a finite [[mixture distribution]] of the [[beta distribution]] and the [[uniform distribution (continuous)|continuous uniform distribution]]. The support is of the distribution is indicated by the parameters ''a'' and ''b'', which are the minimum and maximum values respectively. The distribution provides an alternative to the beta distribution such that it allows more density to be placed at the extremes of the bounded interval of support.<ref>{{cite journal |last1=Hahn |first1=E. D. | year=2008 |title=Mixture densities for project management activity times: A robust approach to PERT |journal=European Journal of Operational Research |volume= 188 |issue= 2|pages=450–459 |publisher=Elsevier |doi= 10.1016/j.ejor.2007.04.032|accessdate=}}</ref> Thus it is a bounded distribution that allows for [[outliers]] to have a greater chance of occurring than does the beta distribution.

==Definition==

=== Probability density function ===

If parameters of the beta distribution are ''&alpha;'' and ''&beta;'', and if the mixture parameter is ''&theta;'', then the beta rectangular distribution has [[probability density function]]{{cncitation needed|date=April 2012}}

:<math>

p(x|\alpha, \beta, \theta)=\begin{cases}

\frac{\theta \Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}

\frac{(x - a)^{\alpha-1} (b-x)^{\beta - 1} }{(b - a)^{\alpha + \beta +- 1}}

+ \frac{1 - \theta}{b-a}

& \mathrm{for}\ a \le x \le b, \\[8pt]

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=== Cumulative distribution function ===

The [[cumulative distribution function]] is{{cncitation needed|date=April 2012}}

:<math>

F(x|\alpha, \beta, \theta)=

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==Applications==

===Project management===

The [[PERT distribution]] variation of the [[beta distribution]] is frequently used in [[PERT]], [[critical path method]] (CPM) and other [[project management]] methodologies to characterize the distribution of an activity’sactivity's time to completion.<ref>{{cite journal |last1=Malcolm|first1=D. G. | last2=Roseboom |first2=J. H. | last3=Clark|first3=C. E. | last4=Fazar|first4=W. | year=1959 |title=Application of a technique for research and development program evaluation |journal=Operations Research |volume= 7|issue=5 |pages=646&ndash;669 | doi=10.1287/opre.7.5.646}}</ref>.

In PERT, restrictions on the betaPERT distribution parameters lead to shorthand computations for the mean and standard deviation of the beta distribution:

:<math> \begin{align}

E(x) & {} = \frac{a + 4m + b}{6} \\

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\end{align}

</math>

where ''a'' is the minimum, ''b'' is the maximum, and ''m'' is the mode or most likely value. However, the variance is seen to be a constant conditional on the range. As a result, there is no scope for expressing differing levels of uncertainty that the project manager might have about the activity time.

Eliciting the beta rectangular’srectangular's certainty parameter ''&theta;'' allows the project manager to incorporate the rectangular distribution and increase uncertainty by specifiyingspecifying ''&theta;'' is less than 1. The above expectation formula then becomes

:<math>

E(x) = \frac{\theta(a+4m+b) + 3(1-\theta)(a+b)}{6} .

</math>

If the project manager assumes the beta distribution is symmetric under the standard PERT conditions then the variance is

:<math>

\operatorname{Var} (x) = \frac{(b-a)^2 (3-2\theta)}{36} ,

</math>

while for the asymmetric case it is

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\operatorname{Var} (x) = \frac{(b-a)^2(3-2\theta^2)}{36}.

</math>

The variance can now be increased when uncertainty is larger. However, the beta distribution may still apply depending on the project manager’smanager's judgment.

The beta rectangular has been compared to the uniform-two sided power distribution and the uniform-generalized biparabolic distribution in the context of project management. The beta rectangular exhibited larger variance and smaller kurtosis by comparison.<ref>{{cite journal |last1=López Martín|first1=M. M. | last2=García García |first2=C. B. | last3=García Pérez |first3=J. | last4=Sánchez Granero |first3first4=M. A. | year=2012 |title= An alternative for robust estimation in project management |journal= European Journal of Operational Research|volume= in press220|issue=2 |pages=443–451 |publisher=Elsevier |doi= 10.1016/j.ejor.2012.01.058 |accessdate=}}</ref>.

The beta rectangular has been compared to the uniform-two sided power distribution and the uniform-generalized biparabolic distribution in the context of project management. The beta rectangular exhibited larger variance and smaller kurtosis by comparison<ref>{{cite journal |last1=López Martín|first1=M. M. | last2=García García |first2=C. B. | last3=García Pérez |first3=J. | last4=Sánchez Granero |first3=M. A. | year=2012 |title= An alternative for robust estimation in project management |journal= European Journal of Operational Research|volume= in press|issue= |pages= |publisher=Elsevier |doi= 10.1016/j.ejor.2012.01.058 |accessdate=}}</ref>.

===Income distributions===

The beta rectangular distribution has been compared to the elevated two-sided power distribution in fitting U.S. income data.<ref>{{cite journal |last1=García |first1=C.B. | last2=García Pérez |first2=J. | last3=van Dorp |first3=J.R. | year=2011 |title= Modeling heavy-tailed, skewed and peaked uncertainty phenomena with bounded support |journal= Statistical Methods and Applications|volume= 20 |issue= 4|pages=463–486 |publisher=Springer |doi= 10.1007/s10260-011-0173-0|accessdates2cid=3648290 }}</ref>. The 5-parameter elevated two-sided power distribution was found to have a better fit for some subpopulations, while the 3-parameter beta rectangular was found to have a better fit for other subpopulations.

==References==

{{reflist}}

{{ProbDistributions|continuous-bounded}}

[[Category:Continuous distributions]]

[[Category:Compound probability distributions]]

[[fr:Loi bêta rectangulaire]]