UncertainLMH distribution

UncertainLMH( xLow, xMedian, xHigh, pLow, lb, ub)

Also the Analytic functions (see #Analytic distribution functions below):

CumUncertainLMH( x, xLow, xMedian, xHigh, pLow, lb, ub)

CumUncertainLMHInv( p, xLow, xMedian, xHigh, pLow, lb, ub)

DensUncertainLMH( x, xLow, xMedian, xHigh, pLow, lb, ub)

Specifies a smooth continuous distribution from «xLow», «xMedian» and «xHigh» estimates. By default these are the 10-50-90 percentile estimates. You can specify the optional «pLow» percentile level as a positive number less than 0.5 to use different percentile levels. For example, setting «p» to 0.25 treats «xLow»-«xMedian»-«xHigh» as 25-50-75 percentile estimates.

When «lb» (lower bound) and «ub» (upper bound) are not specified, the distribution is unbounded (tails in both directions). When «lb» is specified, the distribution is bounded from below by «lb». When «ub» is specified, the distribution is bounded from above by «ub». Hence, this distribution function can be used to specify unbounded, semi-bounded and fully-bounded smooth continuous distributions.

The distribution return is called a Keelin MetaLog SPT distribution" (Symmetric Percentile Triplet), or a Keelin distribution. This distribution was introduced in

Thomas W. Keelin (Nov. 2016), "The Metalog Distribution", Decision Analytics, 13(4):243-277,

The distribution used by this function is a 3-term Keelin MetaLog.

Feasibility

It is required that the parameters specified are ordered as «lb» < «xLow» < «xMedian» < «xHigh» < «ub». However, not all ordered combinations of percentile estimates result in a valid 3-term Keelin distribution. Those combinations that are valid are called feasible, and parameter combinations that cannot be fit exactly are called infeasible. Infeasible combinations are typically very extreme. When an infeasible combination of parameters is encountered, the error message identifies the range of possible values for «xMedian» that would result in a feasible distribution given the other parameters.

For the unbounded case, the parameter combination is feasible when (Keelin 2016, Proposition 2)

[math]\displaystyle{ k \lt r \lt 1-k }[/math]

where

[math]\displaystyle{ k = {1\over 2} ( 1 - 1.66711 ({1\over 2} - p_{low} ) }[/math]

[math]\displaystyle{ r = {{x_{median} - x_{low}} \over { x_{high} - x_{low} } } }[/math]

k = 0.16658 in the 10-50-90 case.

Examples

An unbounded smooth continuous distribution with a 10% probability of being <= 8, a 50% probability of being <= 29, and a 90% probability of being less that 44. In other words, the 10-50-90 estimates are 8,29,44:

UncertainLMH(8, 29, 44) →

A positive-only (semi-bounded) distribution with 10-50-90 percentiles of 8, 29 and 44:

UncertainLMH(8, 29, 44, lb:0)→

A semi-bounded from above distribution with 10-50-90 percentiles of 8, 29, 44 and an upper bound of 60

UncertainLMH(8, 29, 44, ub:60)→

A full bounded (between 0 and 60) distribution with 10-50-90 percentiles of 8, 29 and 44.

UncertainLMH(8, 29, 44, lb:0 ub:60)→

Comparison of three unbounded distributions with different percentile levels. 5-50-95, 10-50-90 and 25-50-75. Notice that the CDF's value at X=8 is equal to the percentile level, that all three CDF curves pass through (29, 0.5), and that the CDF at x=44 is 1 minus the percentile level.

Index percentile := [5%, 10%, 25%] Do UncertainLMH(8, 29, 44, percentile)→

Analytic distribution functions

The analytic distribution functions compute the exact metric at a point on a distribution without any sampling error.

CumUncertainLMH( x, xLow, xMedian, xHigh, pLow, lb, ub)

The cumulative probability function, computes the probability that the true value (or a value sampled from the distribution) is less than or equal to «x»

CumUncertainLMHInv( p, xLow, xMedian, xHigh, pLow, lb, ub)

The inverse cumulative probability function, aka called the quantile function. Returns the value x where there is a «p» probability of the true value (or a value sampled randomly from the distribution) being less than or equal to x.

DensUncertainLMH( x, xLow, xMedian, xHigh, pLow, lb, ub)

The probability density function. Returns the probability density at «x».