Page 63 - Contributed Paper Session (CPS) - Volume 2

CPS1428 Enobong F. U.
2. Burr XII Activity Duration Distribution: Parameter Elicitation
A random variable X is said to follow a 3 parameter Burr XII distribution
with scale () and shape parameters c and k if the probability density function
is given as
−1 −(+1)
() = (1 + ( ) ) ; ≥ 0, > 0, > 0, > 0 (2)

(see Burr, 1942)

The cumulative density function is
−

() = 1 − (1 + ( ) ) (3)

The moment is given as
ℎ

⁄
⁄

( ) = Γ(− )Γ( +1) ; >
Γ(+1)
Hence, the mean is
1
1

⁄
Γ(− ⁄ )Γ( +1)

() = ; > 1 (4)
Γ(+1)
Variance of is defined as
() = ( ) − [()]
2
2
2
2
1
1
2
⁄
⁄
Γ(− ⁄ )Γ( +1) Γ(− ⁄ )Γ( +1) 2

() = − ( ) (5)
Γ(+1) Γ(+1)

From equation (3)
1 1
= [(1 − ()) − 1] (6)

Equation (6) is the q quantile of the 3P-Burr XII distribution.
th
Let , and represent the lower, median and upper expert judgment
percentiles of a 3-P Burr XII distribution. Consequently, the lower, median and
upper quantiles of a 3P Burr XII distribution are respectively.
1 1
= [(1 − ( )) − 1] (7)

1 1
= [(1 − ( )) − 1] (8)

1 1
= [(1 − ( )) − 1] (9)

Note that the choice of the median quantile is based on the fact that the
median as a measure of location is not affected by outliers in the data; hence
it is well suited for positively skewed distribution. To estimate , and , we
solved equations (7), (8) and (9) simultaneously to obtain the following results:
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