Download Exact distribution of positive linear combinations of

Statistics & Probability Letters 56 (2002) 45 – 50
Exact distribution of positive linear combinations of inverted
chi-square random variables with odd degrees of freedom Viktor Witkovsk)y
Institute of Measurement Science, Slovak Academy of Sciences, Dubravska cesta 9, 84219 Bratislava, Slovak Republic
Received April 2001; received in revised form July 2001; accepted August 2001
Abstract
The exact distribution of a linear combination with positive coe0cients of inverted chi-square variables with odd degrees
of freedom is derived. This distribution function could be expressed as another linear combination of distribution functions
c 2002 Elsevier Science B.V. All rights reserved
of chi-square random variables. MSC: primary 62E15
Keywords: Inverted chi-square distribution; Inverted gamma distribution; Distribution function; Characteristic function
1. Introduction
Witkovsk)y (2001a) derived the characteristic function of inverted gamma distribution and suggested the
calculation of the exact distribution of a linear combination of independent inverted gamma variables by
using the inversion formula of Gil-Pelaez (1951), which leads to one-dimensional numerical integration. In
this paper, we derive the exact distribution of a linear combination with positive coe0cients of inverted
chi-square variables with odd degrees of freedom. The distribution function of such a linear combination of
random variables could be expressed as another linear combination of distribution functions of chi-square
random variables. The basic idea is that used by Walker and Saw (1978) who derived the exact distribution
for a linear combination of Student’s t random variables with odd degrees of freedom, which relies on the
fact that the characteristic functions of such variables are proportional to the polynomial functions. The exact
distribution of a general linear combination of independent t variables could be evaluated by one-dimensional
numerical integration, for more details see Witkovsk)y (2001b).
The research has been supported by the grant VEGA 1=7295=20 from the Scienti=c Grant Agency of the Slovak Republic.
E-mail address: [email protected] (V. Witkovsk)y).
c 2002 Elsevier Science B.V. All rights reserved
0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 1 6 5 - 1
46
V. Witkovsky / Statistics & Probability Letters 56 (2002) 45 – 50
2. Inverted gamma variables
Let Z ∼ G(; ) denote a gamma random variable with the shape parameter ¿ 0 and the scale parameter
¿ 0. Then the inverted gamma variable Y = 1=Z, Y ∼ IG(; ), has its probability density function fY (y)
de=ned for y ¿ 0 by
fY (y) =
1
()
+1
1
1
:
exp −
y
y
(1)
We notice that for = =2; = 1; 2; : : : ; and = 2 the random variable Z ∼ G(; ) has the chi-square
distribution with degrees of freedom, and hence the random variable Y ∼ IG(; ) has the inverted chi-square
distribution with degrees of freedom.
Theorem 1. Let Y ∼ IG(; ) be an inverted gamma random variable with its probability density function
fY (y) given by (1). Then the characteristic function of Y is
Y (t) = E(eitY ) =
2(−it)(1=2) K {(2=)(−it)1=2 }
;
()
(2)
where K (z) denotes the modi8ed Bessel function of second kind.
Proof. Witkovsk)y (2001a).
Corollary 1. Let Yn ∼ IG(n ; ) be an inverted gamma random variable with n = n + 1=2 and ¿ 0 for
n = 0; 1; 2; : : : . Let w = (2=)(−2it)1=2 . Then the characteristic function n (t) of Yn is
0 (t) = exp{−w};
1 (t) = exp{−w}(1 + w);
2 (t) = exp{−w}(1 + w + 13 w2 )
(3)
and for n ¿ 2; n+1 (t) is given by the recurrence relation
n+1 (t) =
w2
n−1 (t) + n (t):
(2n + 1)(2n − 1)
Proof. Eq. (10:2:17) in Abramowitz and Stegun (1965, p. 444) states that
1=2
K1=2 (z) =
exp{−z};
2z
1=2
K3=2 (z) =
exp{−z}(1 + z −1 );
2z
1=2
K5=2 (z) =
exp{−z}(1 + 3z −1 + 3z −2 ):
2z
(4)
(5)
De=ne
fn (z) = (−1)n+1
1=2
Kn+1=2 (z);
2z
(6)
V. Witkovsky / Statistics & Probability Letters 56 (2002) 45 – 50
47
then, according to Eq. (10:2:18) in Abramowitz and Stegun (1965)
fn−1 (z) − fn+1 (z) = (2n + 1)z −1 fn (z):
From (2) we observe that for n ¿ 1
1=2
Kn+1=2 (w) = [2(n − 1) + 1]!!
w−n n (t);
2w
(7)
(8)
where w = (2=)(−2it)1=2 ; [2(n − 1) + 1]!! = 1 × 3 × 5 × · · · × (2(n − 1) + 1), and we obtain the required
result.
Let Y(1 ;1 ) ; : : : ; Y(m ;m ) denote a set of independent inverted gamma variables, where Y(k ;k ) ∼ IG(k ; k ),
with k ¿ 0 and k ¿ 0; k = 1; : : : ; m. Let us de=ne a general linear combination of such variables, say
L=
m
k Y(k ;k )
(9)
k=1
with real coe0cients k . If k (t) denotes the characteristic function of the random variable Y(k ;k ) , then L (t),
the characteristic function of L, is
L (t) = 1 (1 t) · · · m (m t):
(10)
The exact value of the distribution function FL (x) = Pr(L 6 x) could be evaluated by using the inversion
formula of Gil-Pelaez (1951) which leads to one-dimensional numerical integration, namely
−itx
1 1 ∞
e L (t)
FL (x) = −
dt:
(11)
Im
2 0
t
3. Linear combination of inverted chi-square variables
We consider now a positive linear combination of inverted chi-square random variables with odd degrees
of freedom, i.e.
L=
m
k X k ;
(12)
k=1
where Xk ∼ 1=2k , k ¿ 0 for all k = 1; : : : ; m and k ∈ {1; 3; 5; : : :}. Note, that the characteristic function of
random variable Xk is that of Y(k ;k ) with k = k =2 and k = 2. For odd degrees of freedom k the parameter
k = nk + 1=2, where nk = 0; 1; 2; : : : . Hence, the characteristic functions of the variables Xk are of the form
(3) and (4) given by Corollary 1, with w = − (2it)1=2 .
In what follows, we will proceed in accordance with Walker and
√ Saw (1978). For any =xed integer n and
for any positive constant c ¿ 0 we will denote by n (c) and Wn ( c) the n + 1 dimensional vectors




1
0 (ct)

 √
 (ct) 
 ( cw) 
 1


 √




√


( cw)2  :
(13)
n (c) =  2 (ct)  and Wn ( c) = 


 . 


.
 . 
..


 . 


√
n (ct)
( cw)n
48
V. Witkovsky / Statistics & Probability Letters 56 (2002) 45 – 50
Table 1
The lower-triangular elements of the matrix Q9
Column
n
1
2
3
0
1
2
1
1
1
4
5
6
7
8
9
1
1
1
3
3
1
1
2
5
1
15
4
1
1
3
7
2
21
1
105
5
1
1
4
9
1
9
1
63
1
945
6
1
1
5
11
4
33
2
99
1
495
1
10395
7
1
1
6
13
5
39
10
429
2
715
4
19305
1
135135
8
1
1
7
15
2
15
1
39
2
585
2
6435
4
225225
1
2027025
9
1
1
8
17
7
51
7
255
1
255
4
9945
2
69615
1
765765
10
1
34459425
Table 2
The non-zero elements of the matrix Q9−1
Column
n
1
0
1
2
3
4
5
6
1
−1
2
3
1
−3
3
3
−18
45
−45
4
5
6
7
15
−150
675
−1575
105
−1575
11 025
945
−19 845
10 395
Column
n
4
7
8
9
1575
5
−4410
99 225
−99 225
6
7
8
198 450
−1 190 700
4 465 125
−291 060
3 929 310
−32 744 250
135 135
−4 864 860
85 135 050
9
2 027 025
−91 216 125
10
34 459 425
Further, let Qn denote the (n + 1) × (n + 1) lower triangular matrix of the (polynomial) coe0cients of the
characteristic functions 0 ; 1 ; : : : ; n . The elements of the matrix Qn could be determined by recurrence
relation (4). Namely, the elements {Qn }i; j of Qn are given by the following relation: First, set {Qn }1; 1 = 1,
and {Qn }i; 1 = {Qn }i; 2 = 1 for i = 2; : : : ; n + 1, and note that {Qn }i; j = 0 for j ¿ i. Then
{Qn }i−2; j−2
{Qn }i; j = {Qn }i−1; j +
(14)
(2i − 3)(2i − 5)
for i = 3; : : : ; n + 1, and j = 3; : : : ; i. For illustration, Table 1 presents the elements of the matrix Q9 and
Table 2 presents the elements of the matrix Q9−1 . Based on that, we may write
√
√
(15)
n (c) = exp{− cw} Qn Wn ( c);
where w = − (2it)1=2 .
V. Witkovsky / Statistics & Probability Letters 56 (2002) 45 – 50
49
m
m √
For the linear combination L = k=1 k Xk , we will denote = k=1 k .
Then the characteristic function L (t) of L could be written as
L (t) = 1 (1 t) · · · m (m t)
= exp{−w}
m
{Qn }nk +1 Wn ( k ) = exp{−w}# Wn ();
(16)
k=1
where n represents the appropriate maximum order of the polynomial of the characteristic function L (t);
{Qn }nk +1 represents the (nk + 1)st row of the matrix Qn (row with the index nk ) and # represents the suitably
normalized convolution coe0cients. Since Qn is a non-singular matrix, we may therefore (based on (15))
write
Qn−1 n (2 ) = exp{−w}Wn ()
(17)
and using (16) we obtain
L (t) = exp{−w}# Wn () = # Qn−1 n (2 ) = $ n (2 );
(18)
m
where $ = # Qn−1 . That is, the characteristic function of L = k=1 k Xk , the positive linear combination of
independent inverted chi-square variables with odd degrees of freedom, is a linear combination of the characteristic functions of the random variables 2 Xj , j = 1; 3; 5; : : : ; 2n+1. Based on that we can state the following
theorem:
m
Theorem 2. Let L = k=1 k Xk be a positive linear combination of independent inverted chi-square variables
with odd degrees of freedom; where k ¿ 0; Xk ∼ 1=2k ; and k ∈ {1; 3; 5; : : :} for all k = 1; : : : ; m. Let us
m √
denote = k=1 k . Then the distribution function FL (x) = Pr(L 6 x) of L is
2 n
FL (x) =
;
(19)
$j 1 − Fj
x
j=0
where $ = ($0 ; $1 ; : : : ; $n ) is the vector of coe:cients given by (18) and Fj (z) is the cumulative distribution
function of the chi-square random variable 2j with j = 2j + 1 degrees of freedom.
Proof. From (18), the cdf of L is given by
FL (x) = Pr(L 6 x) =
n
2
$j Pr( X2j+1 6 x) =
j=0
=
n
j=0
n
j=0
$j Pr
2
2
2j+1
2 n
2
2
=
;
$j Pr 2j+1
¿
$j 1 − F j
x
x
6x
(20)
j=0
2
where (for simplicity of notation) we denote by 2j+1
a random variable which has chi-square distribution
with 2j + 1 degrees of freedom.
Example. Let us consider the linear combination
L = X1 + X3 + X5 ∼
1
1
1
+ 2 + 2;
12
3
5
50
V. Witkovsky / Statistics & Probability Letters 56 (2002) 45 – 50
with =
√
1+
√
1+
√
1 = 3. The characteristic function L (t) of L is
L (t) = exp{−w}(exp{−w}(1 + w))(exp{−w}(1 + w + 13 w2 ))
= exp{−3w}(1 + 2w + 43 w2 + 13 w3 )
4 1
; 81 )W3 (3) = exp{−3w}# W3 (3);
= exp{−3w}(1; 23 ; 27
with W3 (3) = (1; 3w; 9w2 ; 27w3 ) and w = − (2it)1=2 . Then, according to (18)
7 2 5
$ = # Q3−1 = ( 13 ; 27
; 9 ; 27 ):
The exact cdf FL (x) of L according to (19) is given as
7
2
5
9
9
9
9
1
1 − F0
+
1 − F1
+
1 − F2
+
1 − F3
;
FL (x) =
3
x
27
x
9
x
27
x
2
where Fj (z) is cdf of 2j+1
.
References
Abramowitz, M., Stegun, I.A., 1965. Handbook of Mathematical Functions. Dover, New York.
Gil-Pelaez, J., 1951. Note on the inversion theorem. Biometrika 38, 481–482.
Walker, G.A., Saw, J.G., 1978. The distribution of linear combinations of t-variables. J. Amer. Statist. Assoc. 73 (364), 876–878.
Witkovsk)y , V., 2001a. Computing the distribution of a linear combination of inverted gamma variables. Kybernetika 37 (1), 79–90.
Witkovsk)y , V., 2001b. On the exact computation of the density and of the quantiles of linear combinations of t and F random variables.
J. Statist. Plann. Inference 94, 1–13.