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On the joint distribution of runs in the sequence of Markov-dependent multi-state trials
ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 1065–1074 www.elsevier.com/locate/stapro
On the joint distribution of runs in the sequence of Markov-dependent multi-state trials R.L. Shinde, K.S. Kotwal Department of Statistics, School of Mathematical Sciences, North Maharashtra University, Jalgaon 425 001, India Received 23 June 2004; received in revised form 5 May 2005 Available online 27 December 2005
Abstract Under different popular counting schemes of runs we have obtained probability-generating function (pgf) of the joint distribution of runs in a sequence of Markov-dependent multi-state trials. A method of conditional pgf’s is used to obtain these joint distributions. Expanding the matrix polynomial of degree n, involved in the pgf, a recurrent formula for evaluation of exact probability distribution is obtained. To show the feasibility and compatibility of the results obtained, the joint distributions are numerically evaluated. r 2005 Elsevier B.V. All rights reserved. Keywords: Runs; Markov-dependent multi-state trials; Conditional probability generating functions
1. Introduction During the last six decades, the distributions of run statistics in the sequence of Bernoulli trials (BT) have been extensively studied by many researchers using different methods. Various counting schemes of success runs with respect to BT are also considered in these studies. The most common counting schemes of runs and the corresponding random variables with respect to usual BT are as follows:
N n;k : number of non-overlapping consecutive k successes in the sense of Feller’s (1968) counting, M n;k : number of overlapping consecutive k successes in the sense of Ling’s (1988) counting, G n;k : number of success runs of size greater than or equal to k, X n;k;‘ : the number of ‘-overlapping ð0p‘pk 1Þ success runs of length k in n trials according to Aki and Hirano (2000), i.e. the number of success runs of length k which may have overlapping part of length ‘ with the previous success run of length k that has been counted, E n;k : number of success runs of exact length k followed by failure in the sense of Mood’s (1940) counting.
In the last two decades, researchers have extended their study of distributions of runs from BT to more generalized trials such as non-identical BT, Markov-dependent BT, m-dependent Markov BT, binary sequences of order k, binary sequences of order ðk; rÞ, etc. In few studies generalized multi-state trials also considered. Corresponding author.
E-mail address: [email protected] (R.L. Shinde). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.12.005
ARTICLE IN PRESS R.L. Shinde, K.S. Kotwal / Statistics & Probability Letters 76 (2006) 1065–1074
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Mood (1940) obtained distribution of number of runs of elements of ith kind in sequence of k kinds of elements a1 ; a2 ; . . . ; ak by using combinatorial arguments. Since combinatorial method becomes more complex in case of multi-state trials or dependent trials, new methods were developed to study the distributions of run statistics such as Markov chain imbedding technique, conditional pgf method, induced Markov chain method etc. Schwager (1983) studied the probability of occurrences of a run (a run consists of any specified sequence of outcomes) in a sequence of n multi-state trials with vðvX2Þ possible outcomes by using recursive method of renewal equations. The probability of v outcomes was assumed to be depending on the trial index and also on the L-preceding outcomes. Departing from the traditional combinatorial approach, Fu and Koutras (1994) developed a Markov chain approach to obtain the distribution of number of runs in n non-identical BT. Based on this approach Fu (1996) introduced a ‘forward and backward principle’ for method of Markov chain imbedding to study exact joint distributions of runs and patterns in a sequence of multi-state trials. Doi and Yamamoto (1998) obtained joint distribution of c kinds of runs in the sequence of ðc þ 1Þ-state trials by using a finite Markov chain method proposed by Fu and Koutras (1994) under all the above counting schemes except for ‘-overlapping counting scheme. Koutras and Alexandrou (1995) refined a finite Markov chain imbedding approach by introducing Markov chain imbeddable variable of binomial type (MVB) to remove the dependence of order of transition probability matrices of imbedded Markov chain on number of trials ðnÞ. Han and Aki (1999) extended the concept of MVB to Markov chain imbeddable variable of multinomial type (MVM) to study joint distribution of runs in a sequence of multi-state trials. They have developed a recursive method for the evaluation of joint distribution of runs. Further they have introduced Markov chain imbeddable variable of returnable type to study the distribution of random variable E n;k in case of BT and the concept is extended to study joint distribution of runs of exact length ki in multi-state trials. Readers may also refer the recent books by Balakrishnan and Koutras (2002) and Fu and Lou (2003) for the study of distributions of runs, scans and patterns. In this paper we consider a time homogeneous sequence fX n ; nX0g of Markov chain with state space f0; 1; . . . ; mg. Let X in;ki ði ¼ 0; 1; . . . ; mÞ, in general denote the number of i-runs of length ki ð1pki pnÞ in n trials, and X n ¼ ðX 0n;k0 ; X 1n;k1 ; . . . ; X m n;km Þ. Then corresponding to the five counting schemes of runs, we have the 0 1 0 1 m five random vectors N n ¼ ðN 0n;k0 ; N 1n;k1 ; . . . ; N m n;km Þ, M n ¼ ðM n;k0 ; M n;k1 ; . . . ; M n;km Þ, G n ¼ ðG n;k0 ; G n;k1 ; . . . ; i 0 1 m 0 1 m i i Gm n;km Þ, E n ¼ ðE n;k0 ; E n;k1 ; . . . ; E n;km Þ and X n;‘ ¼ ðX n;k0 ;‘0 ; X n;k1 ;‘1 ; . . . ; X n;km ;‘m Þ. If l n ¼ maxfx jPðX n;ki ¼ xi Þ40g, i ¼ 0; 1; . . . ; m, then ‘in takes the values ½n=ki , n ki þ 1, ½ðn þ 1Þ=ðki þ 1Þ, ½ðn þ 1Þðki þ 1Þ and ½ðn ‘i Þ=ðki ‘i Þ when X in;ki is N in;ki , M in;ki , G in;ki , E in;ki and X in;ki ;‘i respectively. We study the distributions of N n ; M n ; G n ; E n and X n;‘ by using the method of conditional pgfs. The pgf of distribution of X n is in general obtained by using the recurrence relations of conditional pgfs. A recurrent formula for evaluation of exact probability distribution of X n is given. In short the article is organized as follows. Section 2 involves derivation of pgf of distribution of X n under the five counting schemes introduced above. In Section 3, a method of obtaining exact probability distribution from the pgf of distribution of X n is developed. The formula obtained is interpreted in terms of joint distribution of ðX n ; X n1 Þ. Finally in Section 4, we present the numerical study of joint distributions obtained. 2. The joint distribution of runs Let X 0 ; X 1 ; . . . ; X n be f0; 1; 2; . . . ; mg-valued Markov-dependent trials with transition probabilities, PðX r ¼ jjX r1 ¼ iÞ ¼ pij
i; j ¼ 0; 1; . . . ; m and r ¼ 1; 2; . . . ; n 3
m X
pij ¼ 1
i ¼ 0; 1; . . . ; m
j¼0
and PðX 0 ¼ iÞ ¼ pi i ¼ 0; 1; . . . ; m. Let fn ðt0 ; t1 ; . . . ; tm Þ ¼ fn ðtÞ be the pgf of distribution of X n . Assume that for a non-negative integer cpn, we have observed X 0 ; X 1 ; . . . ; X nc . We define fcði;jÞ ðtÞ ¼ fcði;jÞ as pgf of conditional joint distribution of number of uruns ðu ¼ 0; 1; . . . ; mÞ of length ku in X ncþ1 ; . . . ; X n given that currently (at ðn cÞth trial) we have i-run of length j, j ¼ 1; 2; . . . ; ki and i ¼ 0; 1; ; . . . ; m. Assuming PðX 0 ¼ 0Þ ¼ 1, we have, fn ðtÞ ¼ fð0;0Þ n ðtÞ. We also have, fði;jÞ 0 ðtÞ ¼ 1
8ði; jÞ.
(1)
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Conditioning on the first trial we have ð0;1Þ fð0;0Þ n ðtÞ ¼ p00 fn1 ðtÞ þ
m X
p0r fðr;1Þ n1 ðtÞ.
r¼1
Conditioning on the next trial from each stage, we have the following recurrent relations of conditional pgfs for each c ¼ 1; 2; . . . ; n and i ¼ 0; 1; . . . ; m, ði;jþ1Þ fði;jÞ c ðtÞ ¼ pii fc1 ðtÞ þ
m X
pir fðr;1Þ c1 ðtÞ;
j ¼ 1; 2; . . . ; ki 2.
r¼0 rai
In the following subsections we give the recurrent relations of conditional pgfs fcði;ki 1Þ ðtÞ and fcði;ki Þ ðtÞ, i ¼ 0; 1; . . . ; m to derive the pgfs of X n;‘ , Gn and E n . Here we note that the random vectors N n and M n are particular cases of the random vector X n;‘ with ‘i ¼ 0 for i ¼ 0; 1; . . . ; m, and ‘i ¼ ki 1 for i ¼ 0; 1; . . . ; m, respectively. Therefore, we have not studied the distributions of N n and M n separately. 2.1. Distribution of X n;‘ In the generalized ‘i -overlapping ð0p‘i pki 1Þ counting scheme associated with the random variable X in;ki ;‘i ði ¼ 0; 1; . . . ; mÞ, the number of ‘i -overlapping i-runs of length ki are counted in n trials, i.e. an i-run may have an overlapping part of length ‘i with the previous run that has occurred. Conditioning on the next trial i 1Þ from each stage, we have following recurrent relations of conditional pgfs fði;k ðtÞ and c ði;ki Þ fc ðtÞ; i ¼ 0; 1; . . . ; m, for each c ¼ 1; 2; . . . ; n. ði;ki Þ fcði;ki 1Þ ðtÞ ¼ pii ti fc1 ðtÞ þ
m X
pir fðr;1Þ c1 ðtÞ,
r¼0 rai
fcði;ki Þ ðtÞ ¼
8 m P ði;‘i þ1Þ > > pir fðr;1Þ > pii fc1 ðtÞ þ c1 ðtÞ > > r¼0 < rai
if ‘i ¼ 0; 1; 2; . . . ; ki 2;
m P > ði;ki Þ > pii ti fc1 ðtÞ þ pir fðr;1Þ > c1 ðtÞ > > r¼0 :
if ‘i ¼ ki 1:
rai
The above system of recurrent relations of conditional pgf can be written as ! m X fc ðtÞ ¼ A þ Bi ti fc1 ðtÞ; c ¼ 1; 2; . . . ; n, i¼0 1Þ where fc ðtÞ ¼ ðfð0;0Þ ðtÞ; fcð0;1Þ ðtÞ fcð0;k0 Þ ðtÞ; fð1;1Þ ðtÞ fð1;k ðtÞ fðm;1Þ ðtÞ fcðm;km Þ ðtÞÞ0 , c c c c
2
0
6 60 6 6 6 A ¼ 60 6 6 .. 6. 4 0
0
0
p00 e1
p01 e1
A00
A01
A10
A11
.. .
.. .
..
Am0
Am1
.
0
p0m e1
3
7 A0m 7 7 7 A1m 7 7 7 .. 7 . 7 5 Amm Pm 1þ
k i¼0 i
,
Pm 1þ
k i¼0 i
(2)
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2
0
6 60 6 m 6 X 6 Bi ¼ 6 0 6 i¼0 6 .. 6. 4 0
0
0
0
0
B00
B01
B10
B11
.. .
.. .
..
Bm0
Bm1
.
0
0
3
7 B0m 7 7 7 B1m 7 7 7 .. 7 . 7 5 Bmm Pm 1þ
i¼0
ð3Þ
Pm
ki 1þ
k i¼0 i
e1 ¼ ð1 0 0 0Þ0 and Aij , Bij ði; j ¼ 0; 1; . . . ; mÞ are matrices of order ki kj . For 0p‘i pki 2 the elements of matrices Aii and Bii ; i ¼ 0; 1; . . . ; m are as follows: 8 > < pii if x ¼ 1; 2; . . . ; ki 2 and y ¼ x þ 1; or x ¼ ki and y ¼ ‘i þ 1; ðAii Þxy ¼ > : 0 otherwise; ðBii Þxy ¼
pii
if x ¼ ki 1 and y ¼ ki ;
0
otherwise:
While for ‘i ¼ ki 1, ( pii if x ¼ 1; 2; . . . ; ki 2 and y ¼ x þ 1; ðAii Þxy ¼ 0 otherwise;
ðBii Þxy ¼
8 > < pii > :
0
if x ¼ ki 1 and y ¼ ki ; or x ¼ ki and y ¼ x;
i ¼ 0; 1; . . . ; m,
i ¼ 0; 1; . . . ; m,
otherwise:
For ‘i ¼ 0; 1; . . . ; ki 1, Bij ¼ O ðnull matrixÞ and the elements of matrix Aij are as follows: pij if x ¼ 1; 2; . . . ; ki and y ¼ 1; ðAij Þxy ¼ 8iaj i; j ¼ 0; 1; . . . ; m. (4) 0 otherwise P We note that the matrix Bi ði ¼ 0; 1; . . . ; mÞ is the matrix m i¼0 Bi with all Blk except Bii replaced by the null matrix. Using (2) recurrently with (1), we get the pgf of X n;‘ as follows: !n m X 0 fn ðtÞ ¼ p A þ Bi ti 1 , (5) i¼0
where 1 is the column vector with all elements equal to 1 and p0 ¼ ½1 0 0 0 0. 2.2. Distribution of G n The random variable G in;ki ði ¼ 0; 1; . . . ; mÞ in G n counts the number of i-runs of length at least ki in n trials. Hence we get the following system of recurrent relations of conditional pgfs fcði;ki 1Þ ðtÞ and fcði;ki Þ ðtÞ for i ¼ 0; 1; . . . ; m: ði;ki Þ fcði;ki 1Þ ðtÞ ¼ pii ti fc1 ðtÞ þ
m X
pir fðr;1Þ c1 ðtÞ,
r¼0 rai
ði;ki Þ fcði;ki Þ ðtÞ ¼ pii fc1 ðtÞ þ
m X r¼0 rai
pir fðr;1Þ c1 ðtÞ.
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Following the same steps as in Section 2.1, we obtain the pgf of distribution of G n as in (5) with matrices A and Pm B having the same structure as in (3). Additionally matrices Aij , iaj, Bij , i; j ¼ 0; 1; . . . ; m involved in A i¼0P i and m B are same as in (4) and the elements of matrix Aii , i ¼ 0; 1; . . . ; m are as follows: i¼0 i 8 > < pii if x ¼ 1; 2; . . . ; ki 2 and y ¼ x þ 1; or if x ¼ ki and y ¼ x; ðAii Þxy ¼ > : 0 otherwise: 2.3. Distribution of E n The random variable E in;ki ði ¼ 0; 1; . . . ; mÞ in E n counts the number of i-runs of exact length ki followed by i þ1Þ ðtÞ as the pgf of conditional joint distribution of number of u-runs an outcome other than i. We define fði;k c ðu ¼ 0; 1; . . . ; mÞ of length ku in X ncþ1 ; . . . ; X n given that currently (at ðn cÞth trial) we have i-run of length greater than ki , i ¼ 0; 1; . . . ; m. Conditioning on the next trial from each stage, we have following recurrent i 1Þ relations of conditional pgfs fði;k ðtÞ, fcði;ki Þ ðtÞ and fcði;ki þ1Þ ðtÞ for i ¼ 0; 1; . . . ; m: c 8 m P ði;ki Þ > > pii fc1 ðtÞ þ pir fðr;1Þ if c ¼ 2; 3; . . . ; n; > c1 ðtÞ > > r¼0 < rai fcði;ki 1Þ ðtÞ ¼ m P > ði;ki Þ > pii ti fc1 ðtÞ þ pir fðr;1Þ > c1 ðtÞ if c ¼ 1; > > r¼0 :
rai
ði;ki þ1Þ fcði;ki Þ ðtÞ ¼ pii fc1 ðtÞ þ
m X
pir ti fðr;1Þ c1 ðtÞ;
c ¼ 1; 2; . . . ; n,
r¼0 rai
ði;ki þ1Þ fcði;ki þ1Þ ðtÞ ¼ pii fc1 ðtÞ þ
m X
pir fðr;1Þ c1 ðtÞ;
c ¼ 1; 2; . . . ; n.
r¼0 rai
The above recurrent relations of conditional pgfs can be written as 8 m P > > B t if c ¼ 2; 3; . . . ; n 1; A þ i i fc1 ðtÞ > < i¼0 fc ðtÞ ¼ m P > ðnÞ ðnÞ > > þ B t A i i fc1 ðtÞ if c ¼ 1; :
(6)
i¼0
1 þ1Þ fc ðtÞ ¼ ðfð0;0Þ ; fð0;1Þ fcð0;k0 þ1Þ ; fð1;1Þ fð1;k fðm;1Þ fcðm;km þ1Þ Þ0 , for c ¼ 1; 2; . . . ; n; c c c c c P P ðnÞ m m have the same structure as in (3) except that Aij replaced by AðnÞ A; AðnÞ , ij and Bij i¼0 Bi and i¼0 Bi Pm ðnÞ ðnÞ ðnÞ ðnÞ replaced by Bij ði; j ¼ 0; 1; . . . ; mÞ in case of A and i¼0 Bi , respectively. The matrices Aij , Bij , Aij and BðnÞ ij ; i; j ¼ 0; 1; . . . ; m are of orderðki þ 1Þ ðkj þ 1Þ with their elements as follows: 8 > < pii x ¼ 1; 2; . . . ; ki and y ¼ x þ 1; or x ¼ ki þ 1 and y ¼ x; i ¼ 0; 1; . . . ; m; ðAii Þxy ¼ > : 0 otherwise;
where
( ðAij Þxy ¼
pij 0
x ¼ 1; 2; . . . ; ki 1; ki þ 1 and y ¼ 1; 8iaj; otherwise;
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( ðBij Þxy ¼
pij 0
if x ¼ ki and y ¼ 1; 8iaj; otherwise;
Bii ¼ Oðnull matrixÞ; i ¼ 0; 1; . . . ; m; 8 > < pii x ¼ 1; 2; . . . ; ki 2; ki and y ¼ x þ 1; ðnÞ or x ¼ ki þ 1 and y ¼ x; i ¼ 0; 1; . . . ; m; ðAii Þxy ¼ > : 0 otherwise; ( ðBðnÞ ii Þxy
¼
pii
if x ¼ ki 1 and y ¼ x þ 1;
0
otherwise;
i ¼ 0; 1; . . . ; m;
ðnÞ AðnÞ ij ¼ Aij and Bij ¼ Bij 8iaj.
Note that the matrix Bi ðBðnÞ i Þ is the matrix except the matrices of E n : fn ðtÞ ¼ p
0
Pm
Pm
i¼0 Bi ð ðnÞ ðnÞ Bi0 ; Bi1 ; . . . ; Bim ðBi0 ; Bi1 ; . . . ; BðnÞ im Þ
Aþ
m X
!n1 Bi ti
A
ðnÞ
þ
i¼0
m X
ðnÞ i¼0 Bi Þ
with all Blk ðBðnÞ lk Þ replaced by the null matrix
for i ¼ 0; 1; . . . ; m. Using (6) recurrently, we get the pgf
! BðnÞ i ti
1.
(7)
i¼0
Remark 1. Marginal distribution of any sub-vector of X n can be obtained from the joint pgf fn ðtÞ, by replacing ti by 1 corresponding to the random variables X in;ki not included in the sub-vector. For example, the pgf of marginal distribution of number of i-runs of length ki in n trials (i.e. X in;ki ) is given by 0 B fn ðti Þ ¼ p0 @A þ
m X r¼0 rai
1n C Br þ Bi ti A 1 .
(8)
3. Exact distribution of X n From the previous n we observed that the pgfs of N n , M n , G n , E n and X n;‘ in general involves matrix P section, polynomial A þ m in t0 ; t1 ; . . . ; tm of order n. Hence the joint probability distribution can be i¼0 Bi ti obtained by expanding the polynomial with respect to t0 ; t1 ; . . . ; tm , i.e. PðX n ¼ xÞ ¼ coefficient of tx0 0 tx1 1 . . . txmm in fn ðtÞ,
(9)
where x ¼ ðx0 ; x1 ; . . . ; xm Þ. P Let Dr ¼ fx ¼ ðx0 ; x1 ; . . . ; xm Þj m r ¼ 1; 2; . . . ; n. The following lemma gives thePrecurrent i¼0 xi prg; n relations of the coefficient matrices of tx0 0 tx1 1 . . . txmm for x 2 Dn in the polynomial expansion of ðA þ m i¼0 Bi ti Þ . P n Lemma 1. Let C n ðxÞ be the coefficient matrix of tx0 0 tx1 1 . . . txmm in the polynomial expansion of ðA þ m i¼0 Bi ti Þ . Then C n ðxÞ satisfies the recurrent relation C n ðxÞ ¼ C n1 ðxÞA þ
m X
C n1 ðx ei ÞBi I fxi 140g ;
x 2 Dn
(10)
i¼0
with C 1 ð0Þ ¼ A and C 1 ðei Þ ¼ Bi for i ¼ 0; 1; . . . ; m. Proof. We prove this lemma by induction. For n ¼ 1, we have, C 1 ð0Þ ¼ A and C 1 ðei Þ ¼ Bi for i ¼ 0; 1; . . . ; m. Also for n ¼ 2, P C 2 ðxÞ satisfiesP(10) for all x 2 D2 . Assume that Eq. (10) is true for Pmsome rð2pronÞ. Hence x0 x 1 r xm we have, ðA þ m i¼0 Bi ti Þ ¼ i¼0 C r1 ðx ei ÞBi I fxi 140g . x2Dr C r ðxÞt0 t1 . . . tm where C r ðxÞ ¼ C r1 ðxÞA þ
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Then Aþ
m X
!rþ1 Bi ti
Aþ
¼
i¼0
0 ¼@
m X
!r Bi ti
Aþ
m X
i¼0
X
! B i ti
i¼0
1
C r ðxÞtx0 0 tx1 1
. . . txmm A
Aþ
x2Dr
¼
X
m X
! Bi ti
i¼0
( C r ðxÞA þ
x2Drþ1
m X
)
C r ðx ei ÞBi I fxi 140g tx0 0 tx1 1 . . . txmm .
i¼0
Thus (10) is true for r þ 1. Hence the lemma is proved.
&
Theorem 1. The exact probability distribution for variables X n;‘ , N n , M n , and G n , is in general given by PðX n ¼ xÞ ¼ p0 C n ðxÞ 1 ,
(11)
where C n ðxÞ is the coefficient matrix of tx0 0 tx1 1 . . . txmm in the matrix polynomial ðA þ Proof. The proof follows from (9) and Lemma 1.
Pm
i¼0 Bi ti Þ
n
and satisfies (10).
&
Theorem 2. The exact probability distribution for E n is given by ( ) m X ðnÞ ðnÞ 0 PðE n ¼ xÞ ¼ p C n1 ðxÞA þ C n1 ðx ei ÞBi I fxi 1X0g 1 ,
(12)
i¼0
where C n1 ðxÞ is the coefficient matrix of tx0 0 tx1 1 . . . txmm in matrix polynomial ðA þ Proof. The proof follows from (7), (9) and Lemma 1.
Pm
n1 i¼0 Bi ti Þ
and satisfies (10).
&
Let the double pgf of X n in case of N n , Gn , M n and X n;‘ be Fðt; zÞ. i.e. Fðt; zÞ ¼
1 X
fn ðtÞzn ¼
1 X X
PðX n ¼ xÞtx zn
n¼0 x2Dn
n¼0
where tx ¼ tx0 0 tx1 1 . . . txmm . Simplifying the above expression, we get the double pgf of X n : " !#1 m X 0 B i ti 1; 0ozo1 Fðt; zÞ ¼ p I z A þ i¼0
with matrices A; B0 ; B1 ; . . . ; Bm corresponding to the random variables N n , G n , M n and X n;‘ as obtained in Section 2 and I the identity matrix. The expected number of i-runs of length ki in n trials is EðX in;ki Þ ¼
d fð1; 1; . . . ; 1; ti ; 1; . . . 1Þjti ¼1 ; dti
i ¼ 0; 1; . . . ; m.
On simplifying this expression for random variables N in;ki , M in;ki , G in;ki and X in;ki ;‘i we have !j¼1 n m X X i 0 Aþ Br Bi 1 . EðX n;ki Þ ¼ p j¼1
r¼0
Remark 2. The exact probability distribution of X n given in (11) can be expressed as ( ) m X 0 PðX n ¼ xÞ ¼ p C n1 ðxÞA þ C n1 ðx ei ÞBi I fxi 1X0g 1 . i¼0
(13)
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The components of the above expression can be interpreted: p0 fC n1 ðxÞAg 1 ¼ PðX n ¼ x; X n1 ¼ xÞ and p0 fC n1 ðx ei ÞBi I fxi 1X0g g 1 ¼ PðX n ¼ x; X n1 ¼ x ei Þ i ¼ 0; 1; . . . ; m,
(14)
so that PðX n ¼ xÞ ¼ PðX n ¼ x; X n1 ¼ xÞ þ
m X
PðX n ¼ x; X n1 ¼ x ei Þ.
i¼0
These interpretations are useful for deriving different waiting time distributions. In case of E n , expressions (13) and (14) can be obtained similarly by making appropriate changes in the matrices involved. 4. Numerical study In this section, distributions obtained in Section 2 are numerically evaluated using the method developed in Section 3. We consider f0; 1; 2g-valued Markov chain with transition probability matrix 0 0 0:3 6 P ¼ 1 4 0:5 2 0:4 2
1 0:4 0:4 0:5
2 3 0:3 7. 0:1 5 0:1
Tables 1–4 give the exact probability distributions of random variables N n , X n;‘ , G n and E n , respectively, for n ¼ 10, ðk0 ; k1 ; k2 Þ ¼ ð3; 2; 3Þ and the overlap parameters ð‘0 ; ‘1 ; ‘2 Þ ¼ ð1; 0; 1Þ.
Table 1 Exact distribution of N 10 ¼ ðX ¼ N 010;3 ; Y ¼ N 110;2 ; Z ¼ N 210;3 Þ
Y ¼0
X ¼0
X ¼1
X ¼2
X ¼3
0.0070 2.7E05
0.0001
Sum
Z Z Z Z Z
¼0 ¼1 ¼2 ¼3 ¼4
0.1817 0.0052 3.0E05 1.5E08 0
0.0739 0.0011 1.6E06
Z Z Z Z
¼0 ¼1 ¼2 ¼3
0.3316 0.0055 1.1E05 0
0.0842 0.0005
Z¼0 Z¼1 Z¼2
0.2123 0.0016 0.0000
0.0274 0.0000
Z¼0 Z¼1
0.0562 0.0001
0.0023
Y ¼4
Z¼0
0.0055
0.0055
Y ¼5
Z¼0
0.0001
0.0001
Y ¼1
Y ¼2
Y ¼3
Sum
0.7998
0.2690 0.0034
0.4252 0.0002 0.2415 0.0587
0.1895
0.0106
0.0001
1.0000
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Table 2 Exact distribution of X 10;‘ ¼ ðX ¼ X 010;3;1 ; Y ¼ N 110;2;0 ; Z ¼ X 210;3;1 Þ
Y ¼0
X ¼0
X ¼1
X ¼2
X ¼3
X ¼4
0.0113 0.0001 1.7E07
0.0010 1.5E06
4.1E05
Sum
Z Z Z Z Z
¼0 ¼1 ¼2 ¼3 ¼4
0.1817 0.0051 8.1E05 8.0E07 4.3E09
0.0687 0.0010 7.7E06 1.9E08
Z Z Z Z
¼0 ¼1 ¼2 ¼3
0.3316 0.0054 4.7E05 1.8E07
0.0800 0.0005 9.2E07
Z¼0 Z¼1 Z¼2
0.2123 0.0016 5.0E06
0.0267 2.8E05
Z¼0 Z¼1
0.0562 0.0001
0.0023
Y ¼4
Z¼0
0.0055
0.0055
Y ¼5
Z¼0
0.0001
0.0001
Y ¼1
Y ¼2
Y ¼3
Sum
0.7998
0.2690 0.0074 8.3E06
0.0003
0.4252 0.0009 0.2415 0.0587
0.1793
0.0196
0.0012
4.1E05
1.0000
Table 3 Exact distribution of G10 ¼ ðX ¼ G010;3 ; Y ¼ G110;2 ; Z ¼ G210;3 Þ
Y ¼0
Y ¼1
Y ¼2
Y ¼3 Sum
X ¼0
X ¼1
X ¼2
Z¼0 Z¼1 Z¼2
0.181671 0.005173 2.45E05
0.076170 0.001103 1.16E06
0.004835 0.000015
Z¼0 Z¼1 Z¼2
0.403527 0.006126 0.000009
0.095673 0.000516
0.002250
Z¼0 Z¼1 Z¼2
0.188408 0.001052 1.52E07
0.019331 0.000013
Z¼0 Z¼1
0.013783 0.000006
0.000230
0.799779
0.193037
Sum
0.26899
0.50810 0.000083 0.20889 0.01402 0.007183
1.00000
The row (column) sums in the above tables show the marginal distribution of X 1n;k1 ðX 0n;k0 Þ, in general. Observe that PðN in;ki ¼ 0Þ, PðM in;ki ¼ 0Þ, PðG in;ki ¼ 0Þ and PðX in;ki ;‘i ¼ 0Þ ð0p‘i pki 1Þ are same for i ¼ 0; 1; 2. Also PðX 1n;ki ;0 ¼ jÞ ¼ PðN 1n;ki ¼ jÞ, j ¼ 0; 1; . . . ½kn1 .
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Table 4 Exact distribution of E 10 ¼ ðX ¼ E 010;3 ; Y ¼ E 110;2 ; Z ¼ E 210;3 Þ
Y ¼0
Y ¼1
Y ¼2
Y ¼3 Sum
X ¼0
X ¼1
X ¼2
Z¼0 Z¼1 Z¼2
0.405238 0.006839 2.24E05
0.087127 0.000951 1.09E06
0.002780 0.000010
Z¼0 Z¼1 Z¼2
0.336711 0.004101 0.000007
0.049564 0.000298
0.000650
Z¼0 Z¼1 Z¼2
0.091021 0.000618 1.52E07
0.006645 5.76E06
Z¼0 Z¼1
0.007403 6.40E06 0.851967
Sum
0.50297
0.39133
0.09830 0.00741 0.144592
0.003441
1.00000
Acknowledgements The authors are thankful to the referee for some useful suggestions and comments. The second author would also like to thank the Council of Scientific and Industrial Research (CSIR), New Delhi, India, for awarding Junior Research Fellowship. References Aki, S., Hirano, K., 2000. Number of success runs of specified length until certain stopping time rules and generalized binomial distributions of order k. Ann. Inst. Statist. Math. 52, 767–777. Balakrishnan, N., Koutras, M.V., 2002. Runs with Scans and Applications. Wiley-Interscience, New York. Doi, M., Yamamoto, E., 1998. On the joint distribution of runs in a sequence of multi-state trials. Stat. Probab. Lett. 39, 133–141. Feller, W., 1968. An Introduction to Probability Theory and Its Applications, vol. I, third ed. Wiley, New York. Fu, J.C., 1996. Distribution theory of runs and patterns associated with a sequence of multi-state trials. Stat. Sinica 6, 957–974. Fu, J.C., Koutras, M.V., 1994. Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058. Fu, J.C., Lou, W.Y.W., 2003. Distribution Theory of Runs and Patterns and its Applications (A Finite Markov chain Imbedding Approach). World Scientific, Singapore. Han, Q., Aki, S., 1999. Joint distributions of runs in a sequence of multi-state trials. Ann. Inst. Statist. Math. 51, 419–447. Koutras, M.V., Alexandrou, V.A., 1995. Runs, scans and urn model distributions: a unified Markov chain approach. Ann. Inst. Statist. Math. 47, 743–766. Ling, K.D., 1988. On binomial distributions of order k. Stat. Probab. Lett. 6, 247–250. Mood, A.M., 1940. The Distribution Theory of Runs. Ann. Math. Statist. 11, 367–392. Schwager, S.J., 1983. Run probability in sequences of Markov-dependent trials. J. Am. Stat. Assoc. 78, 168–175.
Statistics & Probability Letters 76 (2006) 1065–1074 www.elsevier.com/locate/stapro
On the joint distribution of runs in the sequence of Markov-dependent multi-state trials R.L. Shinde, K.S. Kotwal Department of Statistics, School of Mathematical Sciences, North Maharashtra University, Jalgaon 425 001, India Received 23 June 2004; received in revised form 5 May 2005 Available online 27 December 2005
Abstract Under different popular counting schemes of runs we have obtained probability-generating function (pgf) of the joint distribution of runs in a sequence of Markov-dependent multi-state trials. A method of conditional pgf’s is used to obtain these joint distributions. Expanding the matrix polynomial of degree n, involved in the pgf, a recurrent formula for evaluation of exact probability distribution is obtained. To show the feasibility and compatibility of the results obtained, the joint distributions are numerically evaluated. r 2005 Elsevier B.V. All rights reserved. Keywords: Runs; Markov-dependent multi-state trials; Conditional probability generating functions
1. Introduction During the last six decades, the distributions of run statistics in the sequence of Bernoulli trials (BT) have been extensively studied by many researchers using different methods. Various counting schemes of success runs with respect to BT are also considered in these studies. The most common counting schemes of runs and the corresponding random variables with respect to usual BT are as follows:
N n;k : number of non-overlapping consecutive k successes in the sense of Feller’s (1968) counting, M n;k : number of overlapping consecutive k successes in the sense of Ling’s (1988) counting, G n;k : number of success runs of size greater than or equal to k, X n;k;‘ : the number of ‘-overlapping ð0p‘pk 1Þ success runs of length k in n trials according to Aki and Hirano (2000), i.e. the number of success runs of length k which may have overlapping part of length ‘ with the previous success run of length k that has been counted, E n;k : number of success runs of exact length k followed by failure in the sense of Mood’s (1940) counting.
In the last two decades, researchers have extended their study of distributions of runs from BT to more generalized trials such as non-identical BT, Markov-dependent BT, m-dependent Markov BT, binary sequences of order k, binary sequences of order ðk; rÞ, etc. In few studies generalized multi-state trials also considered. Corresponding author.
E-mail address: [email protected] (R.L. Shinde). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.12.005
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Mood (1940) obtained distribution of number of runs of elements of ith kind in sequence of k kinds of elements a1 ; a2 ; . . . ; ak by using combinatorial arguments. Since combinatorial method becomes more complex in case of multi-state trials or dependent trials, new methods were developed to study the distributions of run statistics such as Markov chain imbedding technique, conditional pgf method, induced Markov chain method etc. Schwager (1983) studied the probability of occurrences of a run (a run consists of any specified sequence of outcomes) in a sequence of n multi-state trials with vðvX2Þ possible outcomes by using recursive method of renewal equations. The probability of v outcomes was assumed to be depending on the trial index and also on the L-preceding outcomes. Departing from the traditional combinatorial approach, Fu and Koutras (1994) developed a Markov chain approach to obtain the distribution of number of runs in n non-identical BT. Based on this approach Fu (1996) introduced a ‘forward and backward principle’ for method of Markov chain imbedding to study exact joint distributions of runs and patterns in a sequence of multi-state trials. Doi and Yamamoto (1998) obtained joint distribution of c kinds of runs in the sequence of ðc þ 1Þ-state trials by using a finite Markov chain method proposed by Fu and Koutras (1994) under all the above counting schemes except for ‘-overlapping counting scheme. Koutras and Alexandrou (1995) refined a finite Markov chain imbedding approach by introducing Markov chain imbeddable variable of binomial type (MVB) to remove the dependence of order of transition probability matrices of imbedded Markov chain on number of trials ðnÞ. Han and Aki (1999) extended the concept of MVB to Markov chain imbeddable variable of multinomial type (MVM) to study joint distribution of runs in a sequence of multi-state trials. They have developed a recursive method for the evaluation of joint distribution of runs. Further they have introduced Markov chain imbeddable variable of returnable type to study the distribution of random variable E n;k in case of BT and the concept is extended to study joint distribution of runs of exact length ki in multi-state trials. Readers may also refer the recent books by Balakrishnan and Koutras (2002) and Fu and Lou (2003) for the study of distributions of runs, scans and patterns. In this paper we consider a time homogeneous sequence fX n ; nX0g of Markov chain with state space f0; 1; . . . ; mg. Let X in;ki ði ¼ 0; 1; . . . ; mÞ, in general denote the number of i-runs of length ki ð1pki pnÞ in n trials, and X n ¼ ðX 0n;k0 ; X 1n;k1 ; . . . ; X m n;km Þ. Then corresponding to the five counting schemes of runs, we have the 0 1 0 1 m five random vectors N n ¼ ðN 0n;k0 ; N 1n;k1 ; . . . ; N m n;km Þ, M n ¼ ðM n;k0 ; M n;k1 ; . . . ; M n;km Þ, G n ¼ ðG n;k0 ; G n;k1 ; . . . ; i 0 1 m 0 1 m i i Gm n;km Þ, E n ¼ ðE n;k0 ; E n;k1 ; . . . ; E n;km Þ and X n;‘ ¼ ðX n;k0 ;‘0 ; X n;k1 ;‘1 ; . . . ; X n;km ;‘m Þ. If l n ¼ maxfx jPðX n;ki ¼ xi Þ40g, i ¼ 0; 1; . . . ; m, then ‘in takes the values ½n=ki , n ki þ 1, ½ðn þ 1Þ=ðki þ 1Þ, ½ðn þ 1Þðki þ 1Þ and ½ðn ‘i Þ=ðki ‘i Þ when X in;ki is N in;ki , M in;ki , G in;ki , E in;ki and X in;ki ;‘i respectively. We study the distributions of N n ; M n ; G n ; E n and X n;‘ by using the method of conditional pgfs. The pgf of distribution of X n is in general obtained by using the recurrence relations of conditional pgfs. A recurrent formula for evaluation of exact probability distribution of X n is given. In short the article is organized as follows. Section 2 involves derivation of pgf of distribution of X n under the five counting schemes introduced above. In Section 3, a method of obtaining exact probability distribution from the pgf of distribution of X n is developed. The formula obtained is interpreted in terms of joint distribution of ðX n ; X n1 Þ. Finally in Section 4, we present the numerical study of joint distributions obtained. 2. The joint distribution of runs Let X 0 ; X 1 ; . . . ; X n be f0; 1; 2; . . . ; mg-valued Markov-dependent trials with transition probabilities, PðX r ¼ jjX r1 ¼ iÞ ¼ pij
i; j ¼ 0; 1; . . . ; m and r ¼ 1; 2; . . . ; n 3
m X
pij ¼ 1
i ¼ 0; 1; . . . ; m
j¼0
and PðX 0 ¼ iÞ ¼ pi i ¼ 0; 1; . . . ; m. Let fn ðt0 ; t1 ; . . . ; tm Þ ¼ fn ðtÞ be the pgf of distribution of X n . Assume that for a non-negative integer cpn, we have observed X 0 ; X 1 ; . . . ; X nc . We define fcði;jÞ ðtÞ ¼ fcði;jÞ as pgf of conditional joint distribution of number of uruns ðu ¼ 0; 1; . . . ; mÞ of length ku in X ncþ1 ; . . . ; X n given that currently (at ðn cÞth trial) we have i-run of length j, j ¼ 1; 2; . . . ; ki and i ¼ 0; 1; ; . . . ; m. Assuming PðX 0 ¼ 0Þ ¼ 1, we have, fn ðtÞ ¼ fð0;0Þ n ðtÞ. We also have, fði;jÞ 0 ðtÞ ¼ 1
8ði; jÞ.
(1)
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Conditioning on the first trial we have ð0;1Þ fð0;0Þ n ðtÞ ¼ p00 fn1 ðtÞ þ
m X
p0r fðr;1Þ n1 ðtÞ.
r¼1
Conditioning on the next trial from each stage, we have the following recurrent relations of conditional pgfs for each c ¼ 1; 2; . . . ; n and i ¼ 0; 1; . . . ; m, ði;jþ1Þ fði;jÞ c ðtÞ ¼ pii fc1 ðtÞ þ
m X
pir fðr;1Þ c1 ðtÞ;
j ¼ 1; 2; . . . ; ki 2.
r¼0 rai
In the following subsections we give the recurrent relations of conditional pgfs fcði;ki 1Þ ðtÞ and fcði;ki Þ ðtÞ, i ¼ 0; 1; . . . ; m to derive the pgfs of X n;‘ , Gn and E n . Here we note that the random vectors N n and M n are particular cases of the random vector X n;‘ with ‘i ¼ 0 for i ¼ 0; 1; . . . ; m, and ‘i ¼ ki 1 for i ¼ 0; 1; . . . ; m, respectively. Therefore, we have not studied the distributions of N n and M n separately. 2.1. Distribution of X n;‘ In the generalized ‘i -overlapping ð0p‘i pki 1Þ counting scheme associated with the random variable X in;ki ;‘i ði ¼ 0; 1; . . . ; mÞ, the number of ‘i -overlapping i-runs of length ki are counted in n trials, i.e. an i-run may have an overlapping part of length ‘i with the previous run that has occurred. Conditioning on the next trial i 1Þ from each stage, we have following recurrent relations of conditional pgfs fði;k ðtÞ and c ði;ki Þ fc ðtÞ; i ¼ 0; 1; . . . ; m, for each c ¼ 1; 2; . . . ; n. ði;ki Þ fcði;ki 1Þ ðtÞ ¼ pii ti fc1 ðtÞ þ
m X
pir fðr;1Þ c1 ðtÞ,
r¼0 rai
fcði;ki Þ ðtÞ ¼
8 m P ði;‘i þ1Þ > > pir fðr;1Þ > pii fc1 ðtÞ þ c1 ðtÞ > > r¼0 < rai
if ‘i ¼ 0; 1; 2; . . . ; ki 2;
m P > ði;ki Þ > pii ti fc1 ðtÞ þ pir fðr;1Þ > c1 ðtÞ > > r¼0 :
if ‘i ¼ ki 1:
rai
The above system of recurrent relations of conditional pgf can be written as ! m X fc ðtÞ ¼ A þ Bi ti fc1 ðtÞ; c ¼ 1; 2; . . . ; n, i¼0 1Þ where fc ðtÞ ¼ ðfð0;0Þ ðtÞ; fcð0;1Þ ðtÞ fcð0;k0 Þ ðtÞ; fð1;1Þ ðtÞ fð1;k ðtÞ fðm;1Þ ðtÞ fcðm;km Þ ðtÞÞ0 , c c c c
2
0
6 60 6 6 6 A ¼ 60 6 6 .. 6. 4 0
0
0
p00 e1
p01 e1
A00
A01
A10
A11
.. .
.. .
..
Am0
Am1
.
0
p0m e1
3
7 A0m 7 7 7 A1m 7 7 7 .. 7 . 7 5 Amm Pm 1þ
k i¼0 i
,
Pm 1þ
k i¼0 i
(2)
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2
0
6 60 6 m 6 X 6 Bi ¼ 6 0 6 i¼0 6 .. 6. 4 0
0
0
0
0
B00
B01
B10
B11
.. .
.. .
..
Bm0
Bm1
.
0
0
3
7 B0m 7 7 7 B1m 7 7 7 .. 7 . 7 5 Bmm Pm 1þ
i¼0
ð3Þ
Pm
ki 1þ
k i¼0 i
e1 ¼ ð1 0 0 0Þ0 and Aij , Bij ði; j ¼ 0; 1; . . . ; mÞ are matrices of order ki kj . For 0p‘i pki 2 the elements of matrices Aii and Bii ; i ¼ 0; 1; . . . ; m are as follows: 8 > < pii if x ¼ 1; 2; . . . ; ki 2 and y ¼ x þ 1; or x ¼ ki and y ¼ ‘i þ 1; ðAii Þxy ¼ > : 0 otherwise; ðBii Þxy ¼
pii
if x ¼ ki 1 and y ¼ ki ;
0
otherwise:
While for ‘i ¼ ki 1, ( pii if x ¼ 1; 2; . . . ; ki 2 and y ¼ x þ 1; ðAii Þxy ¼ 0 otherwise;
ðBii Þxy ¼
8 > < pii > :
0
if x ¼ ki 1 and y ¼ ki ; or x ¼ ki and y ¼ x;
i ¼ 0; 1; . . . ; m,
i ¼ 0; 1; . . . ; m,
otherwise:
For ‘i ¼ 0; 1; . . . ; ki 1, Bij ¼ O ðnull matrixÞ and the elements of matrix Aij are as follows: pij if x ¼ 1; 2; . . . ; ki and y ¼ 1; ðAij Þxy ¼ 8iaj i; j ¼ 0; 1; . . . ; m. (4) 0 otherwise P We note that the matrix Bi ði ¼ 0; 1; . . . ; mÞ is the matrix m i¼0 Bi with all Blk except Bii replaced by the null matrix. Using (2) recurrently with (1), we get the pgf of X n;‘ as follows: !n m X 0 fn ðtÞ ¼ p A þ Bi ti 1 , (5) i¼0
where 1 is the column vector with all elements equal to 1 and p0 ¼ ½1 0 0 0 0. 2.2. Distribution of G n The random variable G in;ki ði ¼ 0; 1; . . . ; mÞ in G n counts the number of i-runs of length at least ki in n trials. Hence we get the following system of recurrent relations of conditional pgfs fcði;ki 1Þ ðtÞ and fcði;ki Þ ðtÞ for i ¼ 0; 1; . . . ; m: ði;ki Þ fcði;ki 1Þ ðtÞ ¼ pii ti fc1 ðtÞ þ
m X
pir fðr;1Þ c1 ðtÞ,
r¼0 rai
ði;ki Þ fcði;ki Þ ðtÞ ¼ pii fc1 ðtÞ þ
m X r¼0 rai
pir fðr;1Þ c1 ðtÞ.
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Following the same steps as in Section 2.1, we obtain the pgf of distribution of G n as in (5) with matrices A and Pm B having the same structure as in (3). Additionally matrices Aij , iaj, Bij , i; j ¼ 0; 1; . . . ; m involved in A i¼0P i and m B are same as in (4) and the elements of matrix Aii , i ¼ 0; 1; . . . ; m are as follows: i¼0 i 8 > < pii if x ¼ 1; 2; . . . ; ki 2 and y ¼ x þ 1; or if x ¼ ki and y ¼ x; ðAii Þxy ¼ > : 0 otherwise: 2.3. Distribution of E n The random variable E in;ki ði ¼ 0; 1; . . . ; mÞ in E n counts the number of i-runs of exact length ki followed by i þ1Þ ðtÞ as the pgf of conditional joint distribution of number of u-runs an outcome other than i. We define fði;k c ðu ¼ 0; 1; . . . ; mÞ of length ku in X ncþ1 ; . . . ; X n given that currently (at ðn cÞth trial) we have i-run of length greater than ki , i ¼ 0; 1; . . . ; m. Conditioning on the next trial from each stage, we have following recurrent i 1Þ relations of conditional pgfs fði;k ðtÞ, fcði;ki Þ ðtÞ and fcði;ki þ1Þ ðtÞ for i ¼ 0; 1; . . . ; m: c 8 m P ði;ki Þ > > pii fc1 ðtÞ þ pir fðr;1Þ if c ¼ 2; 3; . . . ; n; > c1 ðtÞ > > r¼0 < rai fcði;ki 1Þ ðtÞ ¼ m P > ði;ki Þ > pii ti fc1 ðtÞ þ pir fðr;1Þ > c1 ðtÞ if c ¼ 1; > > r¼0 :
rai
ði;ki þ1Þ fcði;ki Þ ðtÞ ¼ pii fc1 ðtÞ þ
m X
pir ti fðr;1Þ c1 ðtÞ;
c ¼ 1; 2; . . . ; n,
r¼0 rai
ði;ki þ1Þ fcði;ki þ1Þ ðtÞ ¼ pii fc1 ðtÞ þ
m X
pir fðr;1Þ c1 ðtÞ;
c ¼ 1; 2; . . . ; n.
r¼0 rai
The above recurrent relations of conditional pgfs can be written as 8 m P > > B t if c ¼ 2; 3; . . . ; n 1; A þ i i fc1 ðtÞ > < i¼0 fc ðtÞ ¼ m P > ðnÞ ðnÞ > > þ B t A i i fc1 ðtÞ if c ¼ 1; :
(6)
i¼0
1 þ1Þ fc ðtÞ ¼ ðfð0;0Þ ; fð0;1Þ fcð0;k0 þ1Þ ; fð1;1Þ fð1;k fðm;1Þ fcðm;km þ1Þ Þ0 , for c ¼ 1; 2; . . . ; n; c c c c c P P ðnÞ m m have the same structure as in (3) except that Aij replaced by AðnÞ A; AðnÞ , ij and Bij i¼0 Bi and i¼0 Bi Pm ðnÞ ðnÞ ðnÞ ðnÞ replaced by Bij ði; j ¼ 0; 1; . . . ; mÞ in case of A and i¼0 Bi , respectively. The matrices Aij , Bij , Aij and BðnÞ ij ; i; j ¼ 0; 1; . . . ; m are of orderðki þ 1Þ ðkj þ 1Þ with their elements as follows: 8 > < pii x ¼ 1; 2; . . . ; ki and y ¼ x þ 1; or x ¼ ki þ 1 and y ¼ x; i ¼ 0; 1; . . . ; m; ðAii Þxy ¼ > : 0 otherwise;
where
( ðAij Þxy ¼
pij 0
x ¼ 1; 2; . . . ; ki 1; ki þ 1 and y ¼ 1; 8iaj; otherwise;
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( ðBij Þxy ¼
pij 0
if x ¼ ki and y ¼ 1; 8iaj; otherwise;
Bii ¼ Oðnull matrixÞ; i ¼ 0; 1; . . . ; m; 8 > < pii x ¼ 1; 2; . . . ; ki 2; ki and y ¼ x þ 1; ðnÞ or x ¼ ki þ 1 and y ¼ x; i ¼ 0; 1; . . . ; m; ðAii Þxy ¼ > : 0 otherwise; ( ðBðnÞ ii Þxy
¼
pii
if x ¼ ki 1 and y ¼ x þ 1;
0
otherwise;
i ¼ 0; 1; . . . ; m;
ðnÞ AðnÞ ij ¼ Aij and Bij ¼ Bij 8iaj.
Note that the matrix Bi ðBðnÞ i Þ is the matrix except the matrices of E n : fn ðtÞ ¼ p
0
Pm
Pm
i¼0 Bi ð ðnÞ ðnÞ Bi0 ; Bi1 ; . . . ; Bim ðBi0 ; Bi1 ; . . . ; BðnÞ im Þ
Aþ
m X
!n1 Bi ti
A
ðnÞ
þ
i¼0
m X
ðnÞ i¼0 Bi Þ
with all Blk ðBðnÞ lk Þ replaced by the null matrix
for i ¼ 0; 1; . . . ; m. Using (6) recurrently, we get the pgf
! BðnÞ i ti
1.
(7)
i¼0
Remark 1. Marginal distribution of any sub-vector of X n can be obtained from the joint pgf fn ðtÞ, by replacing ti by 1 corresponding to the random variables X in;ki not included in the sub-vector. For example, the pgf of marginal distribution of number of i-runs of length ki in n trials (i.e. X in;ki ) is given by 0 B fn ðti Þ ¼ p0 @A þ
m X r¼0 rai
1n C Br þ Bi ti A 1 .
(8)
3. Exact distribution of X n From the previous n we observed that the pgfs of N n , M n , G n , E n and X n;‘ in general involves matrix P section, polynomial A þ m in t0 ; t1 ; . . . ; tm of order n. Hence the joint probability distribution can be i¼0 Bi ti obtained by expanding the polynomial with respect to t0 ; t1 ; . . . ; tm , i.e. PðX n ¼ xÞ ¼ coefficient of tx0 0 tx1 1 . . . txmm in fn ðtÞ,
(9)
where x ¼ ðx0 ; x1 ; . . . ; xm Þ. P Let Dr ¼ fx ¼ ðx0 ; x1 ; . . . ; xm Þj m r ¼ 1; 2; . . . ; n. The following lemma gives thePrecurrent i¼0 xi prg; n relations of the coefficient matrices of tx0 0 tx1 1 . . . txmm for x 2 Dn in the polynomial expansion of ðA þ m i¼0 Bi ti Þ . P n Lemma 1. Let C n ðxÞ be the coefficient matrix of tx0 0 tx1 1 . . . txmm in the polynomial expansion of ðA þ m i¼0 Bi ti Þ . Then C n ðxÞ satisfies the recurrent relation C n ðxÞ ¼ C n1 ðxÞA þ
m X
C n1 ðx ei ÞBi I fxi 140g ;
x 2 Dn
(10)
i¼0
with C 1 ð0Þ ¼ A and C 1 ðei Þ ¼ Bi for i ¼ 0; 1; . . . ; m. Proof. We prove this lemma by induction. For n ¼ 1, we have, C 1 ð0Þ ¼ A and C 1 ðei Þ ¼ Bi for i ¼ 0; 1; . . . ; m. Also for n ¼ 2, P C 2 ðxÞ satisfiesP(10) for all x 2 D2 . Assume that Eq. (10) is true for Pmsome rð2pronÞ. Hence x0 x 1 r xm we have, ðA þ m i¼0 Bi ti Þ ¼ i¼0 C r1 ðx ei ÞBi I fxi 140g . x2Dr C r ðxÞt0 t1 . . . tm where C r ðxÞ ¼ C r1 ðxÞA þ
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Then Aþ
m X
!rþ1 Bi ti
Aþ
¼
i¼0
0 ¼@
m X
!r Bi ti
Aþ
m X
i¼0
X
! B i ti
i¼0
1
C r ðxÞtx0 0 tx1 1
. . . txmm A
Aþ
x2Dr
¼
X
m X
! Bi ti
i¼0
( C r ðxÞA þ
x2Drþ1
m X
)
C r ðx ei ÞBi I fxi 140g tx0 0 tx1 1 . . . txmm .
i¼0
Thus (10) is true for r þ 1. Hence the lemma is proved.
&
Theorem 1. The exact probability distribution for variables X n;‘ , N n , M n , and G n , is in general given by PðX n ¼ xÞ ¼ p0 C n ðxÞ 1 ,
(11)
where C n ðxÞ is the coefficient matrix of tx0 0 tx1 1 . . . txmm in the matrix polynomial ðA þ Proof. The proof follows from (9) and Lemma 1.
Pm
i¼0 Bi ti Þ
n
and satisfies (10).
&
Theorem 2. The exact probability distribution for E n is given by ( ) m X ðnÞ ðnÞ 0 PðE n ¼ xÞ ¼ p C n1 ðxÞA þ C n1 ðx ei ÞBi I fxi 1X0g 1 ,
(12)
i¼0
where C n1 ðxÞ is the coefficient matrix of tx0 0 tx1 1 . . . txmm in matrix polynomial ðA þ Proof. The proof follows from (7), (9) and Lemma 1.
Pm
n1 i¼0 Bi ti Þ
and satisfies (10).
&
Let the double pgf of X n in case of N n , Gn , M n and X n;‘ be Fðt; zÞ. i.e. Fðt; zÞ ¼
1 X
fn ðtÞzn ¼
1 X X
PðX n ¼ xÞtx zn
n¼0 x2Dn
n¼0
where tx ¼ tx0 0 tx1 1 . . . txmm . Simplifying the above expression, we get the double pgf of X n : " !#1 m X 0 B i ti 1; 0ozo1 Fðt; zÞ ¼ p I z A þ i¼0
with matrices A; B0 ; B1 ; . . . ; Bm corresponding to the random variables N n , G n , M n and X n;‘ as obtained in Section 2 and I the identity matrix. The expected number of i-runs of length ki in n trials is EðX in;ki Þ ¼
d fð1; 1; . . . ; 1; ti ; 1; . . . 1Þjti ¼1 ; dti
i ¼ 0; 1; . . . ; m.
On simplifying this expression for random variables N in;ki , M in;ki , G in;ki and X in;ki ;‘i we have !j¼1 n m X X i 0 Aþ Br Bi 1 . EðX n;ki Þ ¼ p j¼1
r¼0
Remark 2. The exact probability distribution of X n given in (11) can be expressed as ( ) m X 0 PðX n ¼ xÞ ¼ p C n1 ðxÞA þ C n1 ðx ei ÞBi I fxi 1X0g 1 . i¼0
(13)
ARTICLE IN PRESS R.L. Shinde, K.S. Kotwal / Statistics & Probability Letters 76 (2006) 1065–1074
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The components of the above expression can be interpreted: p0 fC n1 ðxÞAg 1 ¼ PðX n ¼ x; X n1 ¼ xÞ and p0 fC n1 ðx ei ÞBi I fxi 1X0g g 1 ¼ PðX n ¼ x; X n1 ¼ x ei Þ i ¼ 0; 1; . . . ; m,
(14)
so that PðX n ¼ xÞ ¼ PðX n ¼ x; X n1 ¼ xÞ þ
m X
PðX n ¼ x; X n1 ¼ x ei Þ.
i¼0
These interpretations are useful for deriving different waiting time distributions. In case of E n , expressions (13) and (14) can be obtained similarly by making appropriate changes in the matrices involved. 4. Numerical study In this section, distributions obtained in Section 2 are numerically evaluated using the method developed in Section 3. We consider f0; 1; 2g-valued Markov chain with transition probability matrix 0 0 0:3 6 P ¼ 1 4 0:5 2 0:4 2
1 0:4 0:4 0:5
2 3 0:3 7. 0:1 5 0:1
Tables 1–4 give the exact probability distributions of random variables N n , X n;‘ , G n and E n , respectively, for n ¼ 10, ðk0 ; k1 ; k2 Þ ¼ ð3; 2; 3Þ and the overlap parameters ð‘0 ; ‘1 ; ‘2 Þ ¼ ð1; 0; 1Þ.
Table 1 Exact distribution of N 10 ¼ ðX ¼ N 010;3 ; Y ¼ N 110;2 ; Z ¼ N 210;3 Þ
Y ¼0
X ¼0
X ¼1
X ¼2
X ¼3
0.0070 2.7E05
0.0001
Sum
Z Z Z Z Z
¼0 ¼1 ¼2 ¼3 ¼4
0.1817 0.0052 3.0E05 1.5E08 0
0.0739 0.0011 1.6E06
Z Z Z Z
¼0 ¼1 ¼2 ¼3
0.3316 0.0055 1.1E05 0
0.0842 0.0005
Z¼0 Z¼1 Z¼2
0.2123 0.0016 0.0000
0.0274 0.0000
Z¼0 Z¼1
0.0562 0.0001
0.0023
Y ¼4
Z¼0
0.0055
0.0055
Y ¼5
Z¼0
0.0001
0.0001
Y ¼1
Y ¼2
Y ¼3
Sum
0.7998
0.2690 0.0034
0.4252 0.0002 0.2415 0.0587
0.1895
0.0106
0.0001
1.0000
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Table 2 Exact distribution of X 10;‘ ¼ ðX ¼ X 010;3;1 ; Y ¼ N 110;2;0 ; Z ¼ X 210;3;1 Þ
Y ¼0
X ¼0
X ¼1
X ¼2
X ¼3
X ¼4
0.0113 0.0001 1.7E07
0.0010 1.5E06
4.1E05
Sum
Z Z Z Z Z
¼0 ¼1 ¼2 ¼3 ¼4
0.1817 0.0051 8.1E05 8.0E07 4.3E09
0.0687 0.0010 7.7E06 1.9E08
Z Z Z Z
¼0 ¼1 ¼2 ¼3
0.3316 0.0054 4.7E05 1.8E07
0.0800 0.0005 9.2E07
Z¼0 Z¼1 Z¼2
0.2123 0.0016 5.0E06
0.0267 2.8E05
Z¼0 Z¼1
0.0562 0.0001
0.0023
Y ¼4
Z¼0
0.0055
0.0055
Y ¼5
Z¼0
0.0001
0.0001
Y ¼1
Y ¼2
Y ¼3
Sum
0.7998
0.2690 0.0074 8.3E06
0.0003
0.4252 0.0009 0.2415 0.0587
0.1793
0.0196
0.0012
4.1E05
1.0000
Table 3 Exact distribution of G10 ¼ ðX ¼ G010;3 ; Y ¼ G110;2 ; Z ¼ G210;3 Þ
Y ¼0
Y ¼1
Y ¼2
Y ¼3 Sum
X ¼0
X ¼1
X ¼2
Z¼0 Z¼1 Z¼2
0.181671 0.005173 2.45E05
0.076170 0.001103 1.16E06
0.004835 0.000015
Z¼0 Z¼1 Z¼2
0.403527 0.006126 0.000009
0.095673 0.000516
0.002250
Z¼0 Z¼1 Z¼2
0.188408 0.001052 1.52E07
0.019331 0.000013
Z¼0 Z¼1
0.013783 0.000006
0.000230
0.799779
0.193037
Sum
0.26899
0.50810 0.000083 0.20889 0.01402 0.007183
1.00000
The row (column) sums in the above tables show the marginal distribution of X 1n;k1 ðX 0n;k0 Þ, in general. Observe that PðN in;ki ¼ 0Þ, PðM in;ki ¼ 0Þ, PðG in;ki ¼ 0Þ and PðX in;ki ;‘i ¼ 0Þ ð0p‘i pki 1Þ are same for i ¼ 0; 1; 2. Also PðX 1n;ki ;0 ¼ jÞ ¼ PðN 1n;ki ¼ jÞ, j ¼ 0; 1; . . . ½kn1 .
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Table 4 Exact distribution of E 10 ¼ ðX ¼ E 010;3 ; Y ¼ E 110;2 ; Z ¼ E 210;3 Þ
Y ¼0
Y ¼1
Y ¼2
Y ¼3 Sum
X ¼0
X ¼1
X ¼2
Z¼0 Z¼1 Z¼2
0.405238 0.006839 2.24E05
0.087127 0.000951 1.09E06
0.002780 0.000010
Z¼0 Z¼1 Z¼2
0.336711 0.004101 0.000007
0.049564 0.000298
0.000650
Z¼0 Z¼1 Z¼2
0.091021 0.000618 1.52E07
0.006645 5.76E06
Z¼0 Z¼1
0.007403 6.40E06 0.851967
Sum
0.50297
0.39133
0.09830 0.00741 0.144592
0.003441
1.00000
Acknowledgements The authors are thankful to the referee for some useful suggestions and comments. The second author would also like to thank the Council of Scientific and Industrial Research (CSIR), New Delhi, India, for awarding Junior Research Fellowship. References Aki, S., Hirano, K., 2000. Number of success runs of specified length until certain stopping time rules and generalized binomial distributions of order k. Ann. Inst. Statist. Math. 52, 767–777. Balakrishnan, N., Koutras, M.V., 2002. Runs with Scans and Applications. Wiley-Interscience, New York. Doi, M., Yamamoto, E., 1998. On the joint distribution of runs in a sequence of multi-state trials. Stat. Probab. Lett. 39, 133–141. Feller, W., 1968. An Introduction to Probability Theory and Its Applications, vol. I, third ed. Wiley, New York. Fu, J.C., 1996. Distribution theory of runs and patterns associated with a sequence of multi-state trials. Stat. Sinica 6, 957–974. Fu, J.C., Koutras, M.V., 1994. Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058. Fu, J.C., Lou, W.Y.W., 2003. Distribution Theory of Runs and Patterns and its Applications (A Finite Markov chain Imbedding Approach). World Scientific, Singapore. Han, Q., Aki, S., 1999. Joint distributions of runs in a sequence of multi-state trials. Ann. Inst. Statist. Math. 51, 419–447. Koutras, M.V., Alexandrou, V.A., 1995. Runs, scans and urn model distributions: a unified Markov chain approach. Ann. Inst. Statist. Math. 47, 743–766. Ling, K.D., 1988. On binomial distributions of order k. Stat. Probab. Lett. 6, 247–250. Mood, A.M., 1940. The Distribution Theory of Runs. Ann. Math. Statist. 11, 367–392. Schwager, S.J., 1983. Run probability in sequences of Markov-dependent trials. J. Am. Stat. Assoc. 78, 168–175.