How adding new information modifies the estimation of the mean and the variance in PERT: a maximum entropy distribution approach

Abstract

This paper presents an alternative method to estimate the mean and the variance when using the program evaluation and review technique (PERT). Different levels of information, provided by an expert, are considered in the PERT scenario to obtain the values of the mean and the variance by means of a maximum entropy distribution approach. In our opinion, the information to be taken into account should be only that supplied by the expert. We also perform a numerical analysis to examine how the estimates vary if new information is added. From this, we conclude that the inclusion of new information (even for analytic purposes) produces significant changes in the estimates proposed.

Appendix

The results shown in this Appendix are from “Maximum entropy distributions with position and unimodality restrictions”, an unpublished working paper, University of Las Palmas de Gran Canaria (2009) available upon request to authors.

Given two distributions with probability density function (pdf) $f(x)$ and $g(x)$, the Kullback–Leibler divergence or relative entropy of $f$ with respect to $g$ is defined by $D_{KL} \left[ {f:g} \right] = \smallint f(x) \cdot log\frac{f(x)}{g(x)} dx$, where $g(x)$ is the reference distribution. In our case, in a bounded domain the reference distribution is $Uniform_{{\left[ {a,b} \right]}}$ distribution. If we consider restrictions written as $E_{f} \left[ {g_{k} (t)} \right] = \mu_{k}$, for $k = 1, \ldots ,m$, where $g_{k} (t)$ are known functions and $\mu_{k}$ are fixed and known values. Then, $\mathop {\hbox{max} }\limits_{f} \left\{ { - D_{KL} [f:g]} \right\}$, subject to $E_{f} \left[ {g_{k} (t)} \right] = \mu_{k}$, for $k = 1, \ldots ,m$; has the solution $f\left( t \right) \propto exp\left\{ {\mathop \sum \limits_{1}^{m} \lambda_{k} g_{k} (t)} \right\}$, where the constant of proportionality $f(t)$ is a density function and $\lambda_{k}$ are given by the restrictions.

The lowest divergence from the $Uniform_{[a,b]}$ distribution is the least possible information subject to the restriction of belonging to class $Q_{**}$.

• A.1. Obtaining$f^{(3)} (t)$ In this case the assumptions are $g_{0} (t) = 1_{[a,b]} (t)$ with $\mu_{0}$ = 1, and, $g_{1} (t) = 1_{{[a_{1} ,b_{1} ]}} (t)$ with $\mu_{1} = \varepsilon .$

• A.2. Obtaining$f^{(2)} (t)$.To obtain $f^{(2)} (t)$ the following affirmations made (without proof) are necessary.

The next Proposition 1 is known as the Khintchine characterisation.

Proposition 1

(see Feller 1968). A random variable$T$is unimodal with mode at$M$if and only if T − M = U.Z where$U$is a random variable that follows a$Uniform$distribution at$\left[ {0,1} \right]$and Z is a random variable independent of$U$.

Henceforth, $g(z)$ denotes the unknown density function of the random variable $Z$. The following proposition connects the pdf of $T$ with that of $Z$.

Proposition 2

Let$T$be a random variable, unimodal with mode at$M$, then$f(t)$can be written as:

$$ f(t) = \left\{ {\begin{array}{*{20}l} {\mathop \int \limits_{t - M}^{\infty } g(z) \cdot \frac{1}{z} dz;} \hfill & {if\; t \ge M} \hfill \\ {\mathop \int \limits_{ - \infty }^{t - M} g(z) \cdot \frac{1}{ - z} dz;} \hfill & {if\; t < M} \hfill \\ \end{array} } \right.. $$

Proposition 3, below, then allows us to specify restrictions on $T$ as restrictions on $Z$.

Proposition 3

Let$T$be a random variable, unimodal with mode at$M$, i.e., T − M = U.Z as in Proposition 1. Hence, $E[h(t)] = \mu$, where$h( \cdot )$is a known function and$\mu$is a known constant, if and only if$E[h^{*} (z)] = \mu$, where

$$ h^{*} (z) = \left\{ {\begin{array}{*{20}l} {\frac{1}{z}\mathop \int \limits_{0}^{z} h(M + u)du;} \hfill & {z > 0} \hfill \\ {\frac{1}{ - z}\mathop \int \limits_{z}^{0} h(M + u)du;} \hfill & {z < 0} \hfill \\ \end{array} } \right. $$

From these propositions we can obtain maximum entropy distributions with restrictions of unimodality with known mode. The method for doing so consists of using Proposition 3 and then specifying the maximum entropy distribution for $Z$ under the restrictions. From this distribution and using Proposition 2 we obtain the distribution of $T$. Specifically, to obtain $f^{(2)} (t)$ the restriction on ${\text{T}}$ is given by $h(t) = 1_{[a,b]} (t)$ and $\mu$ = 1 which becomes a restriction on Z given by $h^{*} (z) = 1_{[a - M, b - M]} (z)$.