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A note on relationships between moments, central moments and cumulants from multivariate distributions
81'&111111111~ & ELSEVIER
Statistics & Probability Letters 39 (1998) 4 9 - 5 4
A note on relationships between moments, central moments and cumulants from multivariate distributions N. Balakrishnan a,*, Norman L. Johnson b, Samuel Kotz c a Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 b Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA c Department of Management Science and Statistics, University of Maryland, College Park, MD 20742, USA Received October 1995
Abstract In this note, we present some relationships between moments, central moments and cumulants from multivariate distributions. Recently, Smith (1995) presented four simple recursive formulas that translate moments to cumulants and vice versa. Here, we derive similar recursive formulas between the central moments and the cumulants. (~) 1998 Elsevier Science B.V. All rights reserved
Keywords." Central moments;
Cumulants; Moments; Stirling number of the first kind; Stifling number of the second kind
1. Introduction Let X = (XI,X2 . . . . ,Xk)' be a multivariate random vector. For r = @1, r2 . . . . . rk)', denote as usual
[fI rl
/~',(x) = e
(1)
,
Li=l
J
for the rth moment,
I~r)(X ) =
E
""
= E
Li=I
(Xi - r i ) !
(2)
i=1
for the (descending) rth factorial moment, and p,(X) = E
X, - E[X~])"
for the rth central moment, and K,(X) for the rth cumulant o f X. * Corresponding author. 0167-7152/98/$19.00 (~ 1998 Elsevier Science B.V. All rights reserved PH S01 6 7 - 7 1 5 2 ( 9 8 ) 0 0 0 2 7 - 3
(3)
N. Balakrishnan et al. / Statistics & Probability Letters 39 (1998) 49-54
50
In a recent note, Smith (1995) presented four very simple and interesting recursive formulas that translate ff,(X) to xr(X) and vice versa. He mentioned that the purpose of those relationships is to serve as useful tools in computational work rather than in an algebraic approach. With the same goal in mind, we first present some simple relationships between fir(X), #~r)(X) and #,(X). Next, proceeding along the lines of Smith (1995) we present two recursive formulas that translate the central moments #~(X) to the cumulants h-~(X) and vice versa.
2. Moments and central moments From Eq. (3), we have the rth central moment as #r(X) = E
- E[X/]) ri ,
which, upon using binomial expansions, gives
\ , ~.~ ] r l/'~k rk
()... ( )
itrtX,=X-',...~--,t_l~i,+...+i ~ rl i~=o i~=o il
rk ik
X (E[Xl]) i' ... (E[Xk]) i' ]gtr_i(X),
(4)
where i = (ib i2. . . . . ik )t. Similarly, from Eq. (1) we have the rth moment as
]Atr(X) = g
Xt~'i = g
{(X i - g[x/])
q- E [ X / ] } r'
i=1
=~'''~ i, =0
i,=0
(?'1)''' (rk)(E[~f,])i' ""(f[gk])it i~ ik
]lr_i(X ).
(5)
These relationships are useful, particularly one in (4), to determine some central measures. The results for the univariate case can be found in any introductory textbook, while the results for the multivariate case may be found in some books dealing with multivariate distribution theory; see, for example, Johnson et al. (1997).
3. Moments and factorial moments From Eq. (2), we have the rth factorial moment as
It(,)(X)= E
i=l rl
(Xi - ri)! r~
= Z"" il =0
Z
s(rl, il ) " " s(rk, i,) pI(X),
(6)
ik =0
where s(n, i) is the Stirling number of the first kind (Johnson et al., 1992) defined by
x! _ Z (x -- n)!
n
i=0
s(n,i)xi.
(7)
N. Balakrishnan et al. I Statistics & Probability Letters 39 (1998) 4 9 - 5 4
51
Similarly, we may write the rth moment as
= E
rl
rh
= Z""
Z
it = 0
(8)
S(r,, il ) ' " S(rk, ik) pli)(X),
i~ =(}
where S(n, i) is the Stirling number of the second kind (Johnson et al., 1992) defined by n x" = Z S ( n ' i )
x! (x ~ i ) !
(9)
"
i=0
These relationships are useful, particularly one in (8), in cases where the downward factorial moments are easy to determine. For example, in the case of X having a multivariate P61ya-Eggenberger distribution (Johnson et al., 1997) with joint probability mass function
P ( x 1, x 2 . . . . . x k ) = Pr
( X / = xi )
= (~)
ai[X'"]
a[n"l ,
(10)
where ~~=, x, = n, ~-~f=, ai = a, a Ix'el = a(a + c ) . • • (a + x - 1 c) with a [°'c] = 1
(11)
and
., xl !x2 ! - • - xk ! is the multinomial coefficient, it is known that (Johnson et al., 1997) t
]A(r)(X)-
rt(r )
k
l[r,c/al
(12)
H - [ rpj j'c/al" , i=l
in the above equation, r = ~--~=l ri and pj = aj/a. Then, by using the relationship in (8), we readily obtain rt
#;(X) . . . . it = 0
ZS(r,,i,)."S(rk,ik) it = 0
n(i)
k
~ H p ~
i''c/a] ,
(13)
j= 1
where i = ij. The relations (6) and (8) for the univariate case can be found in numerous texts; see, for example, Johnson et al. (1992).
N. Balakrishnan et al. I Statistics & Probability Letters 39 (1998) 49-54
52
4. Central moments and cumulants
Denoting the moment generating function of X by of X by Kx(t) = log ~ox(t), we have oc
Kx(t) . . . .
~Ox(t) =
E[et'X], and the cumulant generating function
k
Z Kr(X) I-[ tr'/ri!'
rl=0
rk=0
(14)
i=1
where x , ( X ) is the rth cumulant of X. Then, the central moment generating function of X can be written as oe
k
~ pr(X)I~ tri/ri!
E[e r(x-u)] . . . . rl=0
r~=0
i=1
= exp{Kx(t) - t'#},
(15)
where # = E[X] = (E[XI] . . . . . E[Xk])'. As in Smith (1995), we shall use ]2i],i2,...,i/to denote [.li,,i2,...,i/,o,...,o , Kihi2,...,i/ to denote Ki],i2,...,i/,o,...,o, D i = d/dti for the differential operator, and D r = dr/dt[. Let us suppose the multivariate cumulants and all central moments of the form Pr,,r~,..,r, are known for 1 ~<(~< k - 1. Consider now the calculation of the central moment Pr,r2, .,r~+~ when re+l ~> 1. We have ]~rl,r2 ........ , : IDa'.
• - D r r. .D. .I + ,1{
e KX(t)-t'~ }]
t=0
= [D?'. "Dr/Dr/+'-' {DF+,(Kx(t))- E[Xe+,]} erX(t)-t'"]t=o ...~
r'+'-'(r,)
=~-~
Z i/+l = 0
il =0
.
. 1. •D/+
- E[Xr+I] IDa' =_~_.~...~
"'"
if
\
. . (Kx(t))" D]'
i/+l
i~+,
O/+l
(
eKX(t)-t'#
)]
t=0
i~r/ F)r/ +l -- I {eXx(t)-t'u}]t= ° • "~/.~,+1
r ~) + ,Z- , (
it =0
(rt)(rf+l-1)
il
r,i, . . . ( i/ \ i/+,
i1+l =0
xxr,-i, ........ , - i , , gi,,...,i,,- E[Xt+l]#r,.r2 ........ r,+,-l.
(16)
Next, let us suppose the central moments and all cumulants of the form ~:r,.r2,...,r~ are known for 1 <~[<<.k-1. Consider the calculation of the cumulant ~cr,.r2........ , when rt+l 1> 1. We have tC,,.r:.....r,, = [D~'...
. . . [D~' . where
~x(t)
DTD~ ~]{log qgx(t)}] t:0 r. . . . . , D:D:+l{l°g~Px(t)+
t' #}],=0 ,
(17)
is the central moment generating function given by
= E [e"(x-')]
L
A
=
e-t'¢.
(18)
N. Balakrishnan et al. I Statistics & Probability Letters 39 (1998) 49 54
53
Note that D/+I log ~Px(t) = (D:+I I P x ( t ) ) ~ x l ( t ) .
(19)
(pxJ(t) = q : ( t ) = e x p ( - K x ( t ) ) ,
(20)
With
(p*(t) can be thought of as a moment generating function corresponding to a distribution with cumulant generating function - K x ( t ) . With/~* denoting the mean vector of that distribution, we may write the corresponding central moment generating function as ~*(t) = q~*(t) e - t ' f , so that (Pxl(t) = ~o*(t) e t'~ = ~P*(t) e t'(p+u*).
(21)
From (17), using (19) and (21), we obtain ~cr, ......... , =
[D~ ~ " " ~.,or,. ,{ ( D / + I /~:+t ~,
r. . . .
,-,(
= Z "'" Z Z il,jl =0 i/,j/=O i/+l,j:+l=O X
(
~x(t))
~*(t)e t (~+~) +/~/+l
r,
).(
il,jl,rl-il--jl
F/+ 1 -- 1
}]
t=o
r/ ""
)
i/,j/,r/-i/-j/
)
\i,'+l,j/+l, r:+l -- 1 -- i:+l -- j/+l /+1
Xllr --i --j ..... r/+ --i,+,--j/.i 12ih...,i/+I H (~i "{- ~ 7 ) j' i=1 + l#+l l { r l = r2 . . . . .
r / = 0, r/+l = 1}.
(22)
In Eq. (22), ~li*hi2,...,i/+, denotes the central moment of a distribution with cumulant generating function - K x ( t ) r or the moment generating function as in (20), I{.} denotes the indicator function, }--~
~(r')( il i2=0
r2 ) i2 Krl--ihr2--i2,1 ~lil,i2 -- E[X3] ]Art,r2,
(23)
N. Balakrishnan et aL I Statistics & Probability Letters 39 (1998) 49-54
54
which can be readily computed. Using this, (16) gives next
~ ~rt,r2,2 :
( r l ) ( r2 ) il i2 Krl--i~,r2--i2,2flit,i2
it=0 i2=0 r, r2 ( ) ( ) q- ~ ~ rlil r2i2 bCrl-il'r2-i2'll'liL'i2't-E[X3]l~lr'r'l" ""
(24)
i1=0 i2:0
One can proceed similarly to determine all the higher-order central moments. References
Johnson, N.L., Kotz, S., Balakrishnan, N., 1997. Discrete Multivariate Distributions. Wiley, New York. Johnson, N.L., Kotz, S., Kemp, A.W., 1992. Univariate Discrete Distributions, 2nd ed. Wiley, New York. Smith, P.J., 1995. A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. The Amer. Statist. 49, 217-218.
Statistics & Probability Letters 39 (1998) 4 9 - 5 4
A note on relationships between moments, central moments and cumulants from multivariate distributions N. Balakrishnan a,*, Norman L. Johnson b, Samuel Kotz c a Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1 b Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA c Department of Management Science and Statistics, University of Maryland, College Park, MD 20742, USA Received October 1995
Abstract In this note, we present some relationships between moments, central moments and cumulants from multivariate distributions. Recently, Smith (1995) presented four simple recursive formulas that translate moments to cumulants and vice versa. Here, we derive similar recursive formulas between the central moments and the cumulants. (~) 1998 Elsevier Science B.V. All rights reserved
Keywords." Central moments;
Cumulants; Moments; Stirling number of the first kind; Stifling number of the second kind
1. Introduction Let X = (XI,X2 . . . . ,Xk)' be a multivariate random vector. For r = @1, r2 . . . . . rk)', denote as usual
[fI rl
/~',(x) = e
(1)
,
Li=l
J
for the rth moment,
I~r)(X ) =
E
""
= E
Li=I
(Xi - r i ) !
(2)
i=1
for the (descending) rth factorial moment, and p,(X) = E
X, - E[X~])"
for the rth central moment, and K,(X) for the rth cumulant o f X. * Corresponding author. 0167-7152/98/$19.00 (~ 1998 Elsevier Science B.V. All rights reserved PH S01 6 7 - 7 1 5 2 ( 9 8 ) 0 0 0 2 7 - 3
(3)
N. Balakrishnan et al. / Statistics & Probability Letters 39 (1998) 49-54
50
In a recent note, Smith (1995) presented four very simple and interesting recursive formulas that translate ff,(X) to xr(X) and vice versa. He mentioned that the purpose of those relationships is to serve as useful tools in computational work rather than in an algebraic approach. With the same goal in mind, we first present some simple relationships between fir(X), #~r)(X) and #,(X). Next, proceeding along the lines of Smith (1995) we present two recursive formulas that translate the central moments #~(X) to the cumulants h-~(X) and vice versa.
2. Moments and central moments From Eq. (3), we have the rth central moment as #r(X) = E
- E[X/]) ri ,
which, upon using binomial expansions, gives
\ , ~.~ ] r l/'~k rk
()... ( )
itrtX,=X-',...~--,t_l~i,+...+i ~ rl i~=o i~=o il
rk ik
X (E[Xl]) i' ... (E[Xk]) i' ]gtr_i(X),
(4)
where i = (ib i2. . . . . ik )t. Similarly, from Eq. (1) we have the rth moment as
]Atr(X) = g
Xt~'i = g
{(X i - g[x/])
q- E [ X / ] } r'
i=1
=~'''~ i, =0
i,=0
(?'1)''' (rk)(E[~f,])i' ""(f[gk])it i~ ik
]lr_i(X ).
(5)
These relationships are useful, particularly one in (4), to determine some central measures. The results for the univariate case can be found in any introductory textbook, while the results for the multivariate case may be found in some books dealing with multivariate distribution theory; see, for example, Johnson et al. (1997).
3. Moments and factorial moments From Eq. (2), we have the rth factorial moment as
It(,)(X)= E
i=l rl
(Xi - ri)! r~
= Z"" il =0
Z
s(rl, il ) " " s(rk, i,) pI(X),
(6)
ik =0
where s(n, i) is the Stirling number of the first kind (Johnson et al., 1992) defined by
x! _ Z (x -- n)!
n
i=0
s(n,i)xi.
(7)
N. Balakrishnan et al. I Statistics & Probability Letters 39 (1998) 4 9 - 5 4
51
Similarly, we may write the rth moment as
= E
rl
rh
= Z""
Z
it = 0
(8)
S(r,, il ) ' " S(rk, ik) pli)(X),
i~ =(}
where S(n, i) is the Stirling number of the second kind (Johnson et al., 1992) defined by n x" = Z S ( n ' i )
x! (x ~ i ) !
(9)
"
i=0
These relationships are useful, particularly one in (8), in cases where the downward factorial moments are easy to determine. For example, in the case of X having a multivariate P61ya-Eggenberger distribution (Johnson et al., 1997) with joint probability mass function
P ( x 1, x 2 . . . . . x k ) = Pr
( X / = xi )
= (~)
ai[X'"]
a[n"l ,
(10)
where ~~=, x, = n, ~-~f=, ai = a, a Ix'el = a(a + c ) . • • (a + x - 1 c) with a [°'c] = 1
(11)
and
., xl !x2 ! - • - xk ! is the multinomial coefficient, it is known that (Johnson et al., 1997) t
]A(r)(X)-
rt(r )
k
l[r,c/al
(12)
H - [ rpj j'c/al" , i=l
in the above equation, r = ~--~=l ri and pj = aj/a. Then, by using the relationship in (8), we readily obtain rt
#;(X) . . . . it = 0
ZS(r,,i,)."S(rk,ik) it = 0
n(i)
k
~ H p ~
i''c/a] ,
(13)
j= 1
where i = ij. The relations (6) and (8) for the univariate case can be found in numerous texts; see, for example, Johnson et al. (1992).
N. Balakrishnan et al. I Statistics & Probability Letters 39 (1998) 49-54
52
4. Central moments and cumulants
Denoting the moment generating function of X by of X by Kx(t) = log ~ox(t), we have oc
Kx(t) . . . .
~Ox(t) =
E[et'X], and the cumulant generating function
k
Z Kr(X) I-[ tr'/ri!'
rl=0
rk=0
(14)
i=1
where x , ( X ) is the rth cumulant of X. Then, the central moment generating function of X can be written as oe
k
~ pr(X)I~ tri/ri!
E[e r(x-u)] . . . . rl=0
r~=0
i=1
= exp{Kx(t) - t'#},
(15)
where # = E[X] = (E[XI] . . . . . E[Xk])'. As in Smith (1995), we shall use ]2i],i2,...,i/to denote [.li,,i2,...,i/,o,...,o , Kihi2,...,i/ to denote Ki],i2,...,i/,o,...,o, D i = d/dti for the differential operator, and D r = dr/dt[. Let us suppose the multivariate cumulants and all central moments of the form Pr,,r~,..,r, are known for 1 ~<(~< k - 1. Consider now the calculation of the central moment Pr,r2, .,r~+~ when re+l ~> 1. We have ]~rl,r2 ........ , : IDa'.
• - D r r. .D. .I + ,1{
e KX(t)-t'~ }]
t=0
= [D?'. "Dr/Dr/+'-' {DF+,(Kx(t))- E[Xe+,]} erX(t)-t'"]t=o ...~
r'+'-'(r,)
=~-~
Z i/+l = 0
il =0
.
. 1. •D/+
- E[Xr+I] IDa' =_~_.~...~
"'"
if
\
. . (Kx(t))" D]'
i/+l
i~+,
O/+l
(
eKX(t)-t'#
)]
t=0
i~r/ F)r/ +l -- I {eXx(t)-t'u}]t= ° • "~/.~,+1
r ~) + ,Z- , (
it =0
(rt)(rf+l-1)
il
r,i, . . . ( i/ \ i/+,
i1+l =0
xxr,-i, ........ , - i , , gi,,...,i,,- E[Xt+l]#r,.r2 ........ r,+,-l.
(16)
Next, let us suppose the central moments and all cumulants of the form ~:r,.r2,...,r~ are known for 1 <~[<<.k-1. Consider the calculation of the cumulant ~cr,.r2........ , when rt+l 1> 1. We have tC,,.r:.....r,, = [D~'...
. . . [D~' . where
~x(t)
DTD~ ~]{log qgx(t)}] t:0 r. . . . . , D:D:+l{l°g~Px(t)+
t' #}],=0 ,
(17)
is the central moment generating function given by
= E [e"(x-')]
L
A
=
e-t'¢.
(18)
N. Balakrishnan et al. I Statistics & Probability Letters 39 (1998) 49 54
53
Note that D/+I log ~Px(t) = (D:+I I P x ( t ) ) ~ x l ( t ) .
(19)
(pxJ(t) = q : ( t ) = e x p ( - K x ( t ) ) ,
(20)
With
(p*(t) can be thought of as a moment generating function corresponding to a distribution with cumulant generating function - K x ( t ) . With/~* denoting the mean vector of that distribution, we may write the corresponding central moment generating function as ~*(t) = q~*(t) e - t ' f , so that (Pxl(t) = ~o*(t) e t'~ = ~P*(t) e t'(p+u*).
(21)
From (17), using (19) and (21), we obtain ~cr, ......... , =
[D~ ~ " " ~.,or,. ,{ ( D / + I /~:+t ~,
r. . . .
,-,(
= Z "'" Z Z il,jl =0 i/,j/=O i/+l,j:+l=O X
(
~x(t))
~*(t)e t (~+~) +/~/+l
r,
).(
il,jl,rl-il--jl
F/+ 1 -- 1
}]
t=o
r/ ""
)
i/,j/,r/-i/-j/
)
\i,'+l,j/+l, r:+l -- 1 -- i:+l -- j/+l /+1
Xllr --i --j ..... r/+ --i,+,--j/.i 12ih...,i/+I H (~i "{- ~ 7 ) j' i=1 + l#+l l { r l = r2 . . . . .
r / = 0, r/+l = 1}.
(22)
In Eq. (22), ~li*hi2,...,i/+, denotes the central moment of a distribution with cumulant generating function - K x ( t ) r or the moment generating function as in (20), I{.} denotes the indicator function, }--~
~(r')( il i2=0
r2 ) i2 Krl--ihr2--i2,1 ~lil,i2 -- E[X3] ]Art,r2,
(23)
N. Balakrishnan et aL I Statistics & Probability Letters 39 (1998) 49-54
54
which can be readily computed. Using this, (16) gives next
~ ~rt,r2,2 :
( r l ) ( r2 ) il i2 Krl--i~,r2--i2,2flit,i2
it=0 i2=0 r, r2 ( ) ( ) q- ~ ~ rlil r2i2 bCrl-il'r2-i2'll'liL'i2't-E[X3]l~lr'r'l" ""
(24)
i1=0 i2:0
One can proceed similarly to determine all the higher-order central moments. References
Johnson, N.L., Kotz, S., Balakrishnan, N., 1997. Discrete Multivariate Distributions. Wiley, New York. Johnson, N.L., Kotz, S., Kemp, A.W., 1992. Univariate Discrete Distributions, 2nd ed. Wiley, New York. Smith, P.J., 1995. A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. The Amer. Statist. 49, 217-218.