_{1}

^{*}

The Beta Distribution is almost exclusively used for situations, after range normalization, wherein a continuous random variable is defined on the closed range [0, 1]. Since the beta distribution is intrinsically a two parameter distribution, this creates problems in some applications where specification of more than one parameter is difficult. In this note, two new classes of single parameter continuous probability distributions on a closed interval are introduced. These distributions remove some of the theoretical and practical problems of using the Beta Distribution for applications. The Burr Type XI Distribution has desirable characteristics for many applications especially when there is ambiguity in the definition of the specified parameter.

The standard two parameter Beta Distribution is the most widely used distribution for situations wherein a continuous random variable is confined to a bounded interval. After appropriate normalization to the interval, the Beta Distribution provides a flexible family of probability density functions capable of modeling a wide variety of natural phenomena. There are situations, however, where the Beta Distribution cannot model natural phenomena or its use is problematic. In their recent book, Kotz and van Dorp [

Exponential tilting is a well known method (See Davison [

(1).

Define the moment generation function of as,

then

is also a probability density function on the closed interval. If has no parameters, then defines a one parameter family on with parameter t, where t can range over the interval to. The probability density function has several desirable properties. First, the moment generating function of is given by so that the mean and variance of the density function are given by the equations

from which explicit formulae for the mean and variance can be obtained. The function is the cumulant generating function of the density. Secondly, the form of (3) implies that is in the exponential family of distributions. This family of distributions has been well studied and has several desirable properties. For example, in this case, if one has a random sample from, then one can obtain the maximum likelihood estimate of t by simply setting the sample mean equal to and solving for t. The asymptotic variance of this estimate if given by. Finally, if

is a concave function, then t can be uniquely related to the mode, , through the relationship,

where is the first derivative of with respect to x, evaluated at. This means that one can specify a distribution by specifying the mode, use (5) to find t, and then use (4) to determine the basic statistical properties of the distribution.

The choice of is extremely broad, but as pointed out by one of the referees, it is best to start with which is symmetric about the point to guarantee that any modal value between 0 and 1 can be modeled As examples of the above, consider three simple choices for on the closed interval. The first is the Beta (2, 2) distribution with probability density function,

The second is the Gilbert distribution (Edwards [

The third is a translated version of the Raised Cosine distribution (Proakis [

These probability density functions are shown in

As can seen, the tilted Beta and Gilbert densities are

almost indistinguishable and quite distinct from the tilted Burr density. This similarity between the tilted Gilbert and tilted Beta distributions seems to be typical for both low and high values of the mode.

The skewness of the distributions might be viewed as a desirable characteristic if one believed that experts tend to underestimate the probability of rare events. (In a reliability situation with a mode say of 0.99, the tilted densities would be left skewed and model a situation wherein experts tend to overestimate the reliability.)

Again, there is almost no difference between the tilted Beta and tilted Gilbert distributions over the whole range of the mode. Further the probabilities of being less than the mode for the tilted Burr distribution are uniformly closer to 0.5 than the other two distributions. However, the deviation from 0.5 is large even for the tilted Burr distribution. Accordingly, if one agreed with Trout [

probabilities since the modes and medians do not seem close.

Let be the cumulative probability distribution function corresponding to the probability density function on, i.e.

Then for, is also a cumulative probability distribution function on the range. Indeed, if K is an integer, n, then is the cumulative distribution function of the maximum value of X from a random sample of size n where X has probability density function. Define,

then is a probability density function on. Further, is also a probability density function, the rotated image of about the fixed point. If and are both concave, then has a unique mode on. Under these conditions, it is straightforward to show that the relationship between K and is given the by the equations,

where is the first derivative of with respect to x. Accordingly, one can generate a probability distribution by specifying the mode, use (11) to find K and if the mode is greater than 0.5 use (10) to find the density function. If the mode is less than 0.5, then one uses (10) with x replaced by. Percentiles x_{p} for the density can easily be found by solving the equation

Unfortunately, the form of (10) does not lend itself to closed form solutions for moments.

If one has a random sample of size n from, then the maximum likelihood of K is,

with asymptotic variance. A form of this distribution was discussed by Topp and Leone [

As can be seen, as in the case of the Tilted distributions, the BetaMAX and GilbertMAX densities are indistinguishable and quite distinct from the BurrMAX distribution. Both the BetaMAX and GilbertMAX distributions show right skew (they would be left skewed if) while the BurrMAX is almost symmetric. In contrast to the Tilted distributions shown in

Accordingly, in

When is the Burr distribution, (11) can be written as

With the appropriate value of K, the cumulative distribution function of the Burr Type XI distribution is given by

The first equation in (15) was first given by Burr [

Since (14) are mixed trigonometric equations, there is no closed form equation to find the mode, , given K .

The median, , or indeed any percentile x_{p}, for, can be obtained by solving the equation

which, since it is a mixed trigonometric equation, has no closed form solution. Equation (17), however, can be solved using numerical procedures. If, then to find x_{p}, replace p with in (54), solve, and subtract the solution from 1.

No closed form solution or general numerical solutions can be found for the mean and standard deviation of the Burr Type XI distribution. Accordingly, direct numerical integration of the integrals defining the first and second central moments was performed to obtain the resulting means and standard deviations as given in

In risk situations where one estimates very small probabilities, or in a reliability context when one estimates probabilities very close to 1, use of a Max distribution becomes problematic as the values of K become very large. One can capitalize, however, on the fact that for K = n, an integer, all of the Max distributions can be thought of as representing the distributions of the maximum (if) or the minimum (if) of n random samples taken from the appropriate distribution. Accordingly, one can use the theory of extreme values and extend the results from integer values, n, to a continuous value.

Following the discussion in Johnson et al. [

If the value of can be determined, then the above distribution becomes a function of alone and is a one parameter distribution. In Johnson et al. [

For the Burr distribution, (19) yields the value so that for or the theory of Extreme Value Distributions indicates that the Burr Type

XI distribution can be closely approximated to nine decimal places by a Weibull extreme value distribution on the range. Specifically, the cumulative distribution function is given by

with corresponding probability density functions,

If, we have from the properties of the Weibull distribution [Johnson et al. [

For, the corresponding results are,

For the BetaMAX and GilbetMAX distributions, (19) indicates that so that for λ ≤ 0.1473 or λ ≥ 0.8527 both distributions can be closely approximated to nine decimal places by a Weibull extreme value distribution on the range which coincides with a form of the Rayleigh distribution Johnson et al. (1995), Chapter 18, Section [

It is clear from the previous discussion that the Gilbert distribution and Beta (2, 2) distributions yield Tilted, MAX and Extreme Value distributions which are essentially numerically indistinguishable. Accordingly, I see no applications for any form of the Gilbert distribution. In risk or reliability studies, where would be expected to be close to 0 or close to 1 respectively, the Extreme value distributions would seem to be most useful. They have a relatively simple form and one can obtain good approximations to their moments using (20) and (21). Further one can obtain percentiles of the extreme value distributions using the formulas,

By replacing p or 1 − p in (24) with a uniformly distributed random variable U, one can easily simulate samples of any size from these distributions. The choice of whether to use hinges on whether one believes that experts are truly estimating the mode and not the median. If they are, then choosing would seem to be preferred since it allows for expert under estimation of probabilities in the case of risk applications, or over estimation of probabilities in reliability applications. On the other hand, if there is ambiguity as to whether experts are estimating the mode or median when asked for the “most likely value”, then using the Extreme Value distribution with would seem most appropriate since for this distribution the median and mode are almost identical.

In PERT or stochastic CPM applications where would not be expected to be either very small or very large, the Extreme Value distributions would not be appropriate. Typically either the Triangle distribution with mode or the Beta distribution with mean and standard deviation of have been used in this situation. The standard deviations of these two distributions as well as the BetaMax and Burr Type XI distributions are shown in

It seems clear that the BetaMax and Burr Type XI distributions are better than both the usual Beta approximation model and Triangular distribution since for these

two distributions the variability is substantially lower as one moves away from the middle of the modal range. If one was worried about the ambiguity of the term “most likely value”, then one would use the Burr Type XI distribution instead of the Usual PERT model based on

Perhaps the most useful application of these one parameter distributions is to allow experts with limited backgrounds in probability to more accurately specify their uncertainties about the situations they are working with. For example consider the problem of estimating the chances of a failure in a power system. The expert needs only to come up with one estimate, say 0.001, and the distributions discussed in this paper would automatically generate a plausible distribution for the uncertainty in this figure. Given that many risk assessment studies and complicated projects consist of hundreds to thousands of uncertain steps, the reduction in difficulty by using one parameter families should greatly ease the problem of assigning reasonable uncertainty to the myriad steps.