t
This work was supported in part by the U.S. Army Research Office-Durham
under Contract No. DAHC04-7l-C-0042.
1.
University of North Carolina at Chapel Hill.
2. Temple University, Philadelphia, and University of North Carolina at
Chapel Hill.
EXTENDED AND MULTIVARIATE TUKEY LAMBDA DISTRIBUTIONS t
N. L. Johnson l
and
S. Kotz 2
Institute of Statistics Mimeo Series No. 831
Department of Statistios
University of North Carolina at Chapel Hill
June, 1972
EXTENDED AND MULTIVARIATE TUKEY LAMBDA DISTRIBUTIONS t
N. L. Johnson l
S. Kotz 2
and
SUf,lt,1ARY
The symmetrical Tukey lambda distributions are the distributions of
[X
A
1 (l_X)A] x A-I
to 1).
where
X has a standard uniform distribution (range
°
Systems of multivariate distributions can be formed by applying trans-
formations
A.
Y
i
when
(i = 1, ... ,m)
X.
1.
I~=l
~
Xj
=
[X.
1.
1.
-
A
-1
(I-Xi) i] x Ai
have a joint Dirichlet distribution (with
Since no more than one of the
1).
bution, though all have beta distributions,
of
[X
A
- (l-X)A]
when
~.,e
XIS
~
~
Xi
°
can have a uniform distri-
are led to study distributions
X has a general beta distribution on
[0,1].
These
distributions are termed extended Tukey lambda distributions. Properties of
these distributions are studied.
described and a
Properties of the multivariate ones are also
numerical illustration is presented.
1. INTRODUCTION
The Tukey lambda distributions [2,7,8] have been found to be useful in a
number of univariate sampling problems.
Y
(1)
where
=
a > 0,
[aT
and
A
T
These are distributions of
A
- (l-T) ]/A
has a standard uniform distribution, with density
(0 < t < 1).
The transformation (1) is monotonic increasing i.f
A ~ 0.
If
a
= 1,
the distribution of
A
>
0,
Y is symmetrical.
and decreasing if
Some properties
of such symmetricaZ Zambda distributions have been described by Joiner and
Rosenblatt [5].
2
If it is desired to form a multivariate distribution with variables
4It
which can be obtained by some such transformation as (1), it is natural to
consider a joint Dirichlet distribution for
r[ m
L8
(2)
T (tl,···,t )
1"'" m
m
\m
8
i
- 8.
j
J
j
m
(m
IT t. 8.-1](
J
1
r<8.)
lT
j=O
J
The marginal distribution of
L.i=O
·=0
=
PT
J
T.
J
Tl, •.• ,T ,
m
j=l J
.
-
with density
m
Lt. )80-1
j=l J
is standard beta, with parameters
Clearly, no more than one of the
m variables
T , ..• ,T
can
1
m
have a uniform or indeed a symmetrical distribution.
We are thus led to consider the distribution of
a standard beta distribution with general parameters
We shall refer to r.v.
bution as an extended
Y in (1) when
T
Y in (1) when T has
8 (> 0)
and
~
(> 0).
has a standard beta distri-
~ambda variab~e.
Since we can now obtain skew distributions even with
restrict ourselves to this case.
a
= 1,
we will
In the next section, we will describe some
properties of this family of distributions, and in the final section we will
describe multivariate distributions with members of this family as marginals.
2.
EXTENDED TUKEY LAMBDA DISTRIBUTIONS
If
the range of variation of
A > 0
is from
-1
A
we take
Y
to
A < 0
If
= 10g[T/(1-T)]
The density function of
=
Corresponding to
which also has unlimited range of variation
(see [3]).
(3)
it is unlimited.
Y is
A = 0,
3
where each
t
has to be expressed in terms of
~
(A simple explicit form for
to satisfy the relation
[t A _ (l_t)A]A- l •
=
y
y
Py(y)
is not available.)
The r-th moment about zero is
r
L
(4)
j=O
In the case
A
= 0,
(-l)j~ (j-l) (cp),
iance,
lSI
(~) (-l)j B(6 + (r-j)A, cp + jA).
J
the j-th cumulant of
(see [3]).
and
of
6
2
Y is
K.(Y)
J
= ~(j-l)(e) +
From (4) it is possible to compute the mean, var-
Y.
Some values of
lSI
and
S2
are given in
Table 1.
Consider now the variation in shape of the distribution of
changes in
A,
the parameters
Y
-
A
= 2.
cp
having fixed values.
Since
2
= 2Y - 1 = y - (1_y)2
(l-Y)
we see that the values of
and
8
Y with
62 must be the same for
and
lSI
A
= 1 and
In each one of these two cases we have a beta distribution with the
same parameters (though different range) as that of the original variables.
In virtue of the continuity of the function involved,
mum or minimum value for some value of
a minimum when
value of
e = cp,
but can be a maximum when
for some values of
(8,cp,A)
8
A
By considering neighboring loci of
with slightly different values of
lSI
and
S2
e
must take a maxi-
A between 1 and 2.
A in a remarkably narrow range about
a few examples.
S2
and
cp
#
cp,
= 1.45
It is, in fact,
and occurs for a
-- see Table 2 for
(/S ,S2) (for A varying)
l
it can be seen that, at least
there will be more than one set of values
giving these values for the shape parameters.
4
Table 1
e
e
Moment-ratios for extended Tukey lambda distributions
A=0.5
e
¢
1(31
0.25
0.25
0.25
0.25
0.25
0.25
0.5
1.0
5.0
10.0
0
0.59
1.09
1.67
1.72
0.5
0.5
0.5
0.5
1.0
0.5
1.0
5.0
10.0
1.0
a
1.0
1.0
5.0
5.0
10.0
5.0
10.0
5.0
10.0
10.0
Table 2
e
(*
A=1,2
13 2
1.42
1.98
3.23
5.99
6.35
lSI
13 2
lSI
0
0.68
1.38
2.93
3.38
1. 29
1.91
3.79
0
0.59
1.03
1.90
2.45
0.48
1.08
1.14
0
1. 72
2.20
3.96
4.29
2.08
0
0.64
1.93
2.31
0
1.50
2.14
7.25
9.94
1.80
0.40
1.05
1.52
0
1.83
2.27
3.28
5.00
2.45
0.64
0.73
0
0.16
0
3.08
3.38
2.73
2.84
2.86
1.18
1.52
0
0.33
0
4.20
5.78
2.54
2.82
2.74
0.35
0.77
0
-0.39
0
2.24
2.91
3.70
3.12
3.61
Values of
13.59
18.36
A giving maximum or minimum
<j>
A
113 1
0.25
0.25
0.25
0.25
0.25
0.25
0.5
1.0
5.0
10.0
1.43
1.45
1.43
1.42
1.30
0
0.691
1.425
3.179
3.718
1. 268*
1.897*
3.868
15.64
22.00
0.5
0.5
0.5
0.5
0.5
1.0
5.0
10.0
1.44
1.47
1.43
1.41
0
0.662
2.109
2.549
1. 469*
2.134*
8.171
11. 69
= minimum)
A=4
13 2
e
a
13
2
1.44
2.03
3.12
6.23
9.63
13 2
A
113 1
13 2
1
5
5
1
5
10
5
10
1.45
1.45
1.43
1.48
1.51
0
1.299
1.688
0
0.380
1. 753
4.549
6.573
2.489*
2.836
10
10
1.49
0
2.705*
1
1
<j>
5
This might be expected since there are three parameters available to give
~
(lSI and
two specified values
S2)'
is to consider what happens when
known that for
A = 1,
Another way of looking at the situation,
A is fixed, but
e
the line (Johnson and Kotz [4]) corresponding to
latter is approached as
y,
according as
It is
S2 - (/S 1 )2 - 1 = 0
is covered. For A > 0, genS2 - (/1\)2 - 1 = 0 and
erally, the region covered is that bet,,7een the line
Py(y) +
vary.
~
the region between the boundary
and the Type III line
Clearly,
and
~ +
a
or
PT(t) + 0
with
00
8
(Type III variable)A.
The
remaining constant.
at the extremes of the range of variation of
00
or
00,
for
A > 1.
Other modal values (if any) will correspond to solutions of the equation
= o
(5)
=
(note that
dpy(y)
dt
• dy
dt
and
dt
dy
:f: 0).
Since
A-2
A-2
(A-1)(t
- (l-t)
)
A
1
t - + (l_t)A-l
=
(5) is equivalent to
e - 1 - (~-l)u - (A-l)u(u A- 2 - 1) (u A-
(5)'
with
u = t(l-t).
from
0
to
+ 1)-1 = a
increases from
u
Note that
1
0
to
as
00
increases
t
1.
From (5) , , we have
(5) "
u
A-I
(p-A)U - (e -1)
e - A - (~-l)u
=
=
1 - AU + (pu-el
=
u - A - (~u-e )
g (u),
say.
We consider a few special cases.
For
~
from
A
<
1
< min(e,~),
(e-l)/(~-A) to
g(u)
increases from
a
to
00
as
u
increases
(e-A)/(~-l), while u A- 1 decreases as u increases.
6
Equation (5)" thus has just one root (since for
~
u
outside this range,
g(u)
is negative).
A similar situation (in reverse) holds if
¢
< min(l,A)
e
and
>
max(l,A)
and (5)" has no solution.
as
y + -1
is that of
and
T.
6 < min(l,A).
slope of
Py(y)
+
g(u)
e
> 1
and
The density function of
<jJ > 1
and
6 > 1,
=0
Py(y)
at
t
<P
Y
< 1,
u
=0
0)
>
Py(y) + 0
is J-shaped, as
¢ > max(l,A)
A similar situation (in reverse) holds i f
For
If
is always negative (for
In this case, since
y+ 1.
as
00
then
min(6,~).
1 < A<
or 1,
and
hence the
is also zero at these points.
A
An interesting special case corresponds to
= 6+~-1.
Then (5)"
becomes
( ~-lJ
also note that in the symmetrical case
u A- 1
whence
t
expected.
=~
-1
1
=
u
~ve
6-1 (6+<jJ-2)
e
= ~,
(5)" is satisfied by
= 1
and the corresponding modal value for
Y is
0,
as is to be
The distribution can, however, be bimodal (with antimode at 0).
This is so, for example, if
A
>
1
e =¢
and
<
~(A2_A+2).
3. METHODS OF FITTING
Although the values of
¢ and
lSI and
8
2
do not determine the values of
A uniquely, it is possible to fit the extended lambda distributions
by monents, using the first, second and third sample moments.
of variation is finite, it needs to be known, also.)
however, very convenient, at any rate at present.
using a trial value of
~
y's
e,
A to produce values of
(If the range
The procedure is not,
An iterative procedure
T
corresponding to observed
(either using tables or a computing machine) can be employed.
"observed T's" values of
e
and
<jJ
From the
can be obtained by equating sample mean
7
T) to the corresponding values (8(8+~)-1, 8~(8+~)-Z(8+~+1)-1
and variance (of
~
respectively) for a standard beta distribution.
~,
a new value for
Using these values of
8
and
A can be obtained by solving the equation
Sample mean
Y
Ar (8+~)
=
rr (8+A)
_ r (8+A)f
r(8+¢H) [r(8)
r(¢)j·
The process is then repeated.
Alternatively a form of fitting to specified percentile points might be
employed.
Whatever method is used, tables enabling the transformation
y
= tA -
(l-t)A
to be easily inverted are useful.
4. MULTIVARIATE DISTRIBUTION
Consider the joint distribution of
A
Y.
1
and the joint density of
1 -
I~=lTa
h
1
(i = 1, ... ,m)
1
Tl,TZ, ... ,T
is given by (2).
m
Ta , ... , Ta
given any subset
T ,
j
l
0eta with parameters
to
A
[T. i _ (l-T.) i]A.- l
=
1
tribution of
Yl' .•• 'Ym where
'\1<:.
/..._]8
8.,
J
1-.
a
i
The conditional dis-
of the remaining
T's,
is
and range of variation
o
k
- 8.
J
.
Y.
The corresponding conditional distribution of
lambda distribution.
is not an extended
J
It is, one might say, a "generalized extended lambda dis-
tribution", being the distribution of
A.
A
1
[ (cT') J - (l-cT ') j] A. J
when
eters
c
=1
8j ,
- Lhk_1T
-
I~=o8i
-
ah
,
and
L~=18a.
1
T'
has a standard beta distribution with paramand range of variation
- 8j
The range of variation of
o
to
1
Y.
J
is thus from
-1
to
A.
-1
•
A. J
A
A.- [c J-(l-c) j ],
J
Although this distribution is undoubtedly of somewhat complicated form"-simple
8
expressions, even for moments, not being available -- we can get quite a good
idea of the joint distribution by considering percentile points of the array
distributions.
The median regression of
Med (Y.IY
J
on
Y.
J
Y
Y,
aI' ••. , a k
Y)
a l ,· • " a k
=
m
k
for example is
A.
A
(cT I) J - (l-cTiJ j
~
~
where
L e.
IT I (e.,
~
i=O
J
Le
-
i=l a i
1
1
- e.)
2
J
k
=
c
1 -
L
h=l
(Note that
T'
~
T'
h
Y
does not depend on
~
(based on
T .
a '
(Mean-Mode)
~
An approximate formula for
Y.)
a l ,···, a k
3(Mean-Median»
is
Other percentile points can be obtained in a similar way, but special
care must be taken to allow for the sign of
A ••
J
The effect of a change in the values of
to
yl
yl
aI' .•. , a k
is easily assessed.
Since
m
It(e.,
L e.
J i=O
Prey. < yly
J
aI' .. "
y]
ak
from
Y
Y
aI' .•• , a k
k
-
1
Le
i=l a i
- e.)
=
m
k
1 - I (e., L e. - L e - e.)
t J i=O 1
i=1 a i
J
where
A.
y
=
[(ct) J
A.
(l-ct) J]A.-
1
J
with
A
k
c
=
1 -
L
i=l
if
J
t
a.
1
A
(A -l[t h _ (1-t ) h]
h
h
h
=
if
A.
J
<
0
9
we have
Pr [Y
where
j
I aI' ..• , ya,
< y y,
]
if
m
k
1 - I t/ ,(e j , L ei - L e -e.)
c c
i=O
i=l a i J
if
A. > 0
J
=
k
c' = 1-\~ It' ,
L~= a.
m
k
I t/ ,(e., L e.- L 8 -e.)
c c
J i=O ~ i=l a i J
provided of course that
y
lies within the conditional
~
limits of variation of
Y.
in each case.
J
-1
-1
A < 0,
respectively, if
A
A.
If
0,
>
J
A•
(l-c) J]A.-
these are
1
to
[c j -
to
A.
A.
1
[c' J - (l-c') J]A.-
J
J
they are
j
A•
11..- 1 [c J J
1
A.
11..- [c' J J
A•
(I-c) J]
00,
to
A.
(l-c') J]
00,
to
respec tively.
Tables of percentile points of the beta distribution [1, 6] can be used
to construct regions of various kinds containing specified proportions of the
joint distribution.
5. NUMERICAL ILLUSTRATION
As a simple example, we shall consider the case
Then
Y.
J
where
-
T.
J
=
2[/T. - I(l-T.)]
J
J
(j
= 1,2),
has a beta distribution with parameters land 3.
moment ratios of
Y.
J
E[Y. ]
J
The moments and
are
=
4
45
=
-0.089;
val' (Y.)
J
=
0.1093;
10
(Sl,S2)
(The
tribution of
point is in the Pearson Type I region.)
Y ,
2
given
Pr[Y 2
where
y
<
Y ,
l
ylY l
k
k
= 2[(ct)2-(1-ct)2]
The conditional dis-
has cumulative distribution function
= Yl] =
with
c
I
t
(1,2)
= l-t 1
1- (l-t) 2),
(=
and
= 2[tl~-(1-tl)~]
Yl
(so
Y ~ 0
l
that according as
Constants needed for calculation of a few percentiles of this conditional
distribution are set out below:
P
Then
=
Pr[Y
< ylY
2
l
0.01
0.05
0.25
0.50
0.75
0.95
0.99
= Yl ]
t
=
1 - l(l-p)
0.00513
0.02532
0.13397
0.29289
0.50000
0.77639
0.90000.
k
k
Y = 2[{t(1-t l )}2 - U-(1-t l )t}2]
where
t
l
is given in the fol-
lowing table
Yl
-0.5
-0.4
-0.3
-0.2
-0.1
0
t
Thus for example, the median of
2[{0.29289 x
t
Yl
0.1
0.2
0.3
0.4
0.5
l
0.32600
0.36000
0.39453
0.42947
0.46467
0.50000.
YZ'
l
0.53533
0.57053
0.60547
0.64000
0.67400
when
0.39453}~ - {1 - 0.29289
x
Y = 0.3
l
0.39453}~]
is
= -
1.201
11
For
Y
l
= -0.3,
the median is
0.60547}~
2[{0.28289 x
Note that since
t
O.60547}~]
- {I - 0.29289 x
= 0.29289 for P = 0.5,
= -
0.986.
the conditional median is not
greater than
2[(O.29289)~ - (O.707ll)~]
whatever be the value of
= -
0.547
Y •
l
6. CONCLUDING REMARKS
Although the preceeding results are mathematically not very elegant, the
distributions do have some practical appeal.
In particular, computation of
probabilities, is relatively straightforward.
The original Dirichlet distri-
bution determines the mathematical structure of the results.
We have not so
far been able to find a natural parent distribution giving formulae which are
any simpler.
By comparison with Dirichlet distributions, the extra parameters
A ,A , •.. ,A
introduce added flexibility.
m
l 2
value of this feature.
Evidence is lacking on the general
It is likely that in some, perhaps most, cases only a
few of the variables will need transformation (i.e., most
equivalently, 2».
A's
will be 1 (or,
In such cases, the untransformed variates will, of course,
still have conditional beta distributions (with nonstandard range) just as in
Dirichlet distributions.
12
REFERENCES
[1]
New TabZes of the IncompZete Garrona-Function Ratio
and of Percentage Points of the Chi-square and Beta Distributions~
Harter, B.L. (1964)
Aerospace Res. Lab., Wright-Patterson AFB, Ohio.
[2]
Hastings, C., Mosteller, F., Tukey, J.W. and Winsor, C.P. (1947) Low
moments for small samples: a comparative study of order statistics,
Ann. Math. Statist.~ 18, 413-426.
[3]
Johnson, N.L. (1949) Systens of frequency curves generated by methods of
translation, Biometrika~ 36, 149-176.
[4]
Johnson, N.L. and Kotz, S. (1972) Power transformations of gamma
variables, Biometrika~ 59, 226-229.
[5]
Joiner, B.L. and Rosenblatt, Joan R. (1971) Some properties of the range
in samples from TUkey's symmetric lambda distributions,
J. Amer. Statist. Assoc.~ 66, 394-399.
[6]
Pearson, E.S. and Hartley, H.O. (1964) Biometrika TabZes for
Statisticians~ 1, Cambridge University Press.
[7]
Shapiro, S.S. and Wilk, M.B. (1965)
normality (complete samples),
[8]
Tukey, J.W. (1962)
33, 1-67.
An analysis of variance test for
Biometrika~ 52, 591-611.
The future of data analysis,
Ann. Math.
Statist.~
FOOTNOTES
Some Key Words: Tukey lambda distributions; Beta distributions; Dirichlet
distributions; Transformations; Median regression.
t
This work was supported in part by the U.S. Army Research Office-Durham
under Contract No. DAHC04-7l-C-0042.
1.
University of North Carolina at Chapel Hill.
2.
Temple University, Philadelphia, and University of North Carolina at
Chapel HilL