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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.