A GUIDE TO THE BURR TYPE XII DISTRIBUTIONS
by
Robert N. Rodriguez*
Department of Statistios
University of North CaroUna at ChapeZ HiU
Institute of Statistics
I~meo
Series No. 1064
April? 1976
* This
research was supported in part by a National Science Foundation
Graduate Fellowship and by the U. S. Arffry Research Office under Grant
No. DAHC04-74-C-0030.
A Guide to the Burr Type XII Distributions
by
Robert N. Rodriguez*
Diagrams of areas in the
(1f31~
13 2) plane corresponding to the
Burr Type XII distributions are presented.
-e
Some key words:
Burr Type XII distribution; beta coefficient diagram;
Weibull distribution.
* This
research ,vas supported in part by a National Science F01.mdation
Graduate Fellowship and by the U. S. Amy Research Office under Grant
No. DAHC04-74-C-0030 .
.e
2
Suppose that Z is a positive random variable with probability
density function
k
(Z > 0)
PZ(Z) = (l+z)k+l
where k is a positive parameter.
TIlen for c > 0, X = Zl/c has the
cumulative distribution function
(x > 0).
X is said to have the Burr distribution of Type XII with parameters
c and k.
The purpose of this exercise is to develop a diagram in the
(is!,
82) plane corresponding to distributions of Type XII variables
(and their powers). The treatment follows the approach used by Jolmson
·e
and Kotz [8] for power transfonnations of gamma variables.
Burr [1] introduced a system of distributions
distribution functions F(x)
ii'1
1942 by considering
satisfying the differential equation
g£ = A(F)
g(x).
This is a generalization of the Pearson equation; the object was to fit
a distribution function, rather than a density, to data and then obtain
the density by differentiation.
In the special case where A(F)
=
F(l-F),
the solution of Burr's equation is
1
F(x) = l+exp{-G(x)}
where G(x)
= I~oo
g(t)dt.
The Type XII distribution is a particular
fom of this solution and was discussed in some detail by Burr, who noticed
.e
3
that the family gives rise to a useful range of values of skewness (a )
3
and kurtosis
(a ).
He recomrilended fitting by a method of cumulative
4
moments and presented a short table of a 3 and a 4 as functions of
c and k.
Burr Type XII distributions were ftlTther investigated by Hatke [6]
in 1949.
the
She presented a diagram of the moment_ ratio ncoverageY9 using
(a~~ 0) coordinate system due to Craig [5].
It appears that the
Type XII family was not studied again until 1968 ~ when Burr [2] and Burr
and Cislak [4] examined the distributions of sample median and range.
By then the computation of moments had been siraplified by the use of
electronic computers.
.e
Burr CL'1d Cislak pointed out that the coverage
is actually much greater than that claimed by Hatke.
a revised
(a; p 0)
the Pearson system.
They presented
diagram which also relates the coverage to that of
A similar chart is to be fomd in a short communication
by Burr [3] in which c
and k values giving corrbinations of a3 and (14
are tabled.
However use of
coordinates does not provide the clearest
exposition of the Type XII family.
bounds are not
apparent~
Ord [9] presented a
(B p
In particular the upper and lower
(nor have they been derived analytically).
13 2)
version of the chart constructed by Burr
and Cislak which does not appear to clarify the region of coverage.
fact~
bound.
In
there is some d..oubt as to the validity of the shape of the lower
Finally it should be noted that none of the diagrams published
so far indicates the effect of rodifying
c
and k.
4
The rth fooment about the origin of X = ZI/c is
(r < ck)
so that the fourth ror;lent is finite if ck > 4.
the restrictions c
~
1 and k
~
1 in
[l]~
Burr originally used
possibly since the density
function corresponding to X
f.,,(x)
.J\.
kcxc - 1
= (l+xc) 1+1
(
(x > 0)
is l~imodal at x = [(c-I)/(ck+I)]l/C if c > 1 and L-shaped if c = 1.
In wllat follows k and c are taken to be positive with ck > 4.
The parametric equations for
IB:l
and 13 Z are:
i
2
r (k) A3-3r(k) AZAl+ 2A
IS:"=--~--;or-. . . . . .,--~
1
[r(k)Az-AiJ3/2
-e
3
2
2
~
r Ck)A4 -4r Ck)A3Al+6rCk)AZAI-3AI
13 - ---------...,.~-r;;-2[rCk) AZ-Ai]3/Z
where Aj
= rC~l)rCk-t)~
j
= 1~Z~3~4.
Tables of the moment-ratios
"Jere obtained for fixed values of the power parameter 1/c.
These nmnerical
results extended the table of cx,3 and cx,4 constructed originally by BuTI'
in [1] and were used in the preparation of Figures 1 and Z which are the
aTlalogrtes of the diagrams in Jolmson and Kotz [8].
Clearly a
(1Sl~
8Z)
coordinate system is the best choice for displaying the results of the
calculations.
.e
Figure 1 shows the loci of Cv\3l, 13 2) points for constant c with
k varylllg. T.~e Type XII Burr distributions occupy a region s~~ped like
5
FIG. 1.
S2)
FOR THE TYPE XII BURR DISTRIBUTIONS
0.5
-0.5
-1. 0
-1. 5
(181,
VALUES OF
1.5
1.0
N
R
L
EV
E
nonnal
rectangular
logistic
extreme value
exponential
H
H
R
2
PEARSON
TYPE I
·e
... , Po.
I
I
1
I
~ I
/:;'1
fpl
E',
~,
l
'I
t'
,,/
"
I
,
i/
i/
I
,,
~' "
~
I
6
I
~,
.e
PEARSON
I'
TYPE IV
I
,I
I
8
k=l
6
FIG. 2.
-1. 5
-1. 0
EXISTENCE REGIONS FOR THE TYPE XII BURR DISTRIBUTIONS
0.5
-0.5
·e
Ruled sector indicates region
of single occupancy; crosshatched sector indicates region
of double occupancy.
.e
6
8
1.0
1.5
7
the prow of an ancient Roman galley.
As 1< varies from
4/c to
+00
the c-constant curve moves in a counter-clockwise direction toward the
tip of the !9prow".
with positive
For c < 3.6 the c-constant lines tenninate at endpoints
1r31.
For c > 3.6 the c-constant lines tenninate at
endpoints with negative
lSI.
These endpoints fonn the lower boundt of the Type XII region.
Their
parametric equations are
lim IS:"" k~ 1 -
rC~1)-3r(~1)r(hl)+2r3cl..l)
c
c
c
c
[rcb.l )-r 2c1.1)]3!2
c
c
rc!t.l ) -4r C~l)rcl..l)+6rC~1)r 2ChI) -3r 4 ChI)
lim 8 = c
c
c
c
c
c.
k~ 2
[rct.l)-r2chl)]2
c
c
These limits are easily obtained from the parametric equations for
and
13 2 by applying Stirling's fonnula.
Ial
The lower bound for the Type
XII region fonns part of the Weibull curve in the
CI61?
13 2)
plane.
Values for the moment ratios of the Weibull distributions are tabulated
in Chapter 20 of Johnson and Kotz [7].
2. 71 when c; 3.35.
point
c
= 1.
82 has a minimum value of about
The Weibu11 curve passes through the exponential
CISI = 2?
82 = 9)? which is the endpoint for the Burr curve with
The Weibull density ftmctions are tmimoda1; for 0 < c ~ 1 the
mode is at zero, and for
c
>
1 the mode is located to the right of
the origin.
The identification of the lower bound with the Weibu11 family can
t We are following the convention of presenting moment ratio diagrams
upside-down. Thus the upper bounds in Figures 1 and 2 are referred to
as 1I1ower bounds" and conversely.
.e
8
be explained as follows:
PiX s (fc)l/C y}
c
=1
- (1 + t-)-k
=1
- exp{-k log(l + ~Yc}
=
c
c 2
1 - exp{ -k[f - ~(t-) + ••• ]}
+
1 - e
_yC
k
as
+
00.
On the other hand 9 the Type XII distribution can be obtained as a smooth
r1ixture of Weibull distributions
(6)0; w>O)
e~
compounded with respect to
-e
where
e
has a standard gamma distribution
with parameter k.
As c
+
00
the endpoints of the c-constant Burr curves approach
a point on the Weibull curve.
In order to derive the limiting coordinates
we use the fact that the ganuna function is analytic in the positive half-
plane, so that
r(l+z)
= r(l)
2 rH(l) + ~ r(3) (1)
2
+ zr' (1) +
+
h
where the coefficients needed are:
tel)
riel)
r"(l)
r(3)(I)
,e
and
= I'
= Wei) = -y
=y
2
+ l;(2)
=y
2
TI
2
+"6
= _y3 - 3y~(2)- 2~(3)
r(4) (1) + ... ,
9
Here
~(z)
and 1;;(z)
is the digammA function,
y
is the Riemann zeta function.
= 0.57722 is Euleris constant,
(See Whittaker and Watson [10].)
It can be shown that
o
5.35.
The limiting Burr curve c = +00 fonns part of the upper bound for
the Burr Type XII region, passing through the logistic point
'4It
(lSi =
0,
82 = 4.2) and approaching the Weibull curve asymptotically as k + 0+ .
82) corresponds
In the Burr region below the curve c = co each point
to a unique pair (c,k) and hence a unique Type XII distribution. (See
(v'S!'
Figure 2.) Above the curve c = co there are two Type XII distributions
for each
(Ial,
82) point, one with c > 3.6 and one with c < 3.6.
The upper bound for the Burr region in the positive
half-plane
Ii3l
presents a special problem.
Type XII family.
It is clearly not a c-limiting member of the
Initially it was thought that this section of the upper
bound might represent the moment ratios of some generalization of the
logistic
distribution~
and several known generalizations were considered
For example, a natural l icandidate i1 is the Burr Type II
family with distribution function (e- y + l)-k. Tabulation of moment
unsuccessfully.
ratios shows that the Type II curve lies well within the Type XII region.
10
Actually the upper botmd corresponds to the Burr Type XII distributions
=1
for which Ie
and c > 4.
This can be shown by applying the method of
~
Lagrange multipliers in order to minimize
13 2
= constant
~
4.2.
subject to the constraint
Using the parametric equations for
lSI
and 13 2 ~
one obtains
[
a/3l ] [a/3 2 ] -1 = [a/3l ] [ d/3Z ]-1
dC
ac
aK
ak
.
This is an extremely complicated
are
omitted~
equation~
but it can be shown that
a simpler theoretical deriv.ation can be
substantiated numerically.)
k
and the algebraic details
=1
given~
necessarily.
(Undoubtedly
although the result is
Consequently the upper bound for
IS:l > 0
corresponds to the distributions
P{X$x}
=
where c
00
=1
_ ___1_
l+xc
(x>O; c>4) ~
corresponds to the logistic distribution.
As indicated by Figure 1 the Burr Type XII region covers sections
of the areas corresponding to the main Pearson Types I ~ IV ~ and VI.
The section of the Type III line joining the nomal and exponential points
is also
included~
as are those Type VII distributions represented by the
line segment joining the normal and logistic points.
Part of the Type
II line segrn.ent is included in the Burr region, but not the rectangular
endpoint
(1i31 = 0 ~
the other
hand~
13 2
= 1. 8)
as claimed by Burr and Cislak [4].
the extreme value point is included.
The Burr region contains points in the heterotypic region.
.e
On
The
lognormal curve (omitted for simplicity) lies between the Type III and
11
Type V curves? so it is clear that the Burr region covers areas corresponding
to each of the three families in the Jolmson system.
ACKNOHlEDGEMENT
The author wishes to thank Professor Nonnan Johnson for suggesting
.~
this problem.
12
REFERHJCES
[1] Burr, loW. (1942). Cumulative frequency functions. Annals of Mathematiaal
Statistias$ 13, 215-232.
[2] Burr, I.W. (1968). On a general system of distributions III. The sample
range. Journal of the Ame~iaan Statistiaal Assoaiation$ 63, 636-643.
[3] Burr, I.W. (1973). Parameters for a general system of distributions to
match a grid of as and a~. Communiaations in Statistias, 2, 1-21.
[4] Burr, I.W. and Cislak, P.J. (1968). On a general system of distributions,
I. Its curve-shape characteristics, II. The sample median. Journal
of the Ameriaan Statistical Association, 63, 627-635.
[5] Cra,ig, C.C. (1936). A new exposition and chart for the Pearson system
of frequency curves. Annals of Mathematiaal Statistias, ?, 16-28.
[6] Hatke, M.A. (1949). A certain cumulative probability ftmction. Annals
of Mathematiaal Statistias$ 20, 461-463.
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[7] Jolmson, N.L. and Kotz, S. (1970). Continuous Univariate
2 vo1s., Boston: Houghton Mifflin Company.
Dist~ibutions.
[8] Jolmson, N.L. and Kotz, S. (1972). Power transformations of gamma variables.
Biometrika, 59, 226-229.
[9] Ord, J.K. (1972). Families of Frequenay Distributions. New York: Hafner
Publishing Company.
[10]
.e
~lliittaker,
E.T. and Watson, G.N. (1927). A
London: Cambridge University Press .
Co~se
of MOdern Analysis.