Cumulative Ascending distribution




Format: CumulA(min, max, {x}, {P})

Uses

The cumulative distribution is a non-parametric distribution with a wide variety of applications:

1. modeling expert opinion of extremal events

The cumulative ascending distribution is very useful to model an expert's opinion of a variable whose range covers several orders of magnitude in some sort of exponential way. For example, the size of impact on insurers of a large earthquake, the financial impact of a market crash, or some other extremal event for which that are no relevant data with which to estimate the variable. In such circumstances, it is fruitless to attempt to use a Relative distribution directly.

2. Empirical distribution of data

The Cumulative distribution is useful for converting a set of data values into a first or second order empirical distribution (an ogive). The functions VoseOgive and VoseOgiveU in ModelRisk help construct this distribution.

3. Constructing uncertainty distributions for parameters

The Cumulative distribution can be used to construct uncertainty distributions when using some classical statistical methods.

4. modeling expert opinion

The Cumulative distribution is used in some texts to model expert opinion. The expert is asked for a minimum, maximum and a few percentiles (e.g. 25%, 50%, 75%). However, we have found it largely unsatisfactory because of the insensitivity of its probability scale. A small change in the shape of the Cumulative distribution that would pass unnoticed produces a radical change in the corresponding relative frequency plot that would not be acceptable. The figure below provides an illustration:

A smooth and natural relative frequency plot (A) is converted to a cumulative frequency plot (B) and then altered slightly (C). Converting back to a relative frequency plot (D) shows that the modified distribution is dramatically different to the original, though this would almost certainly not have been appreciated by comparing the cumulative frequency plots. For this reason, we usually prefer to model expert opinion looking at the relative frequency distribution instead.

ModelRisk functions added to Microsoft Excel for the Cumulative Ascending distribution

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

VoseCumulAObject constructs a distribution object for this distribution.

VoseCumulAProb returns the probability density or cumulative distribution function for this distribution.

VoseCumulAProb10 returns the log10 of the probability density or cumulative distribution function. 

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

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

VoseCumulAFitP returns the parameters of this distribution fitted to data.

 

Cumulative Ascending distribution equations

 

 

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