Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials
Kirtee Kiran Kamalja, Ramkrishna Lahu Shinde Department of Statistics, School of Mathematical Sciences, North Maharashtra University, Jalgaon, India
E-mail: [email protected] Received May 13, 2011; revised June 2, 2011; accepted June 8, 2011
Abstract In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in
an outcome success (0) and failure (1) i.e. we have a sequence , 0n
n
Xn
Y
of - 0 0 1 1
, , ,0 1 0 1
S
valued Markov dependent bivariate trials. By using the method of conditional probability generating func-
tions (pgfs), we derive the pgf of joint distribution of 1 1 2 20 1 0 1
0 1 0 1
, , , ,, ; ,
n k n k n k n kX X Y Y where for , de- 0,1i 1, i
i
n kX
notes the number of occurrences of i-runs of length in the first component and 2, in k denotes the number
of occurrences of i-runs of length k in the second component of Markov dependent bivariate trials. Further we consider two patterns
1ik iY
2i
1 and 2 o lengths 1k and 2k respectively and obtain the pgf of joint dis-tribution of 1 2, ,X g method of conditional probability generating functions where
f ,n nY usin 1 2, ,n nX Y
number of occurrences of pattern denotes the 1 2 of length 1 2k in the second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distribu-tions are studied using the joint distribution of runs.
k first (
Keywords: Markov Dependent Bivariate Trials, Conditional Probability Generating Function,
Joint Distribution
1. Introduction The distributions of several run statistics are used in various areas such as reliability theory, testing of statis-tical hypothesis, DNA sequencing, psychology [1], start up demonstration tests [2] etc. There are various count-ing schemes of runs. Some of the most popular counting schemes of runs are non-overlapping success runs of length [3], overlapping success runs of length [4], success runs of length at least , - overlapping suc-cess runs of length [5], success runs of exact length
[6].
k kk
kk
The probability distribution of various run statistics associated with the above counting schemes have been studied extensively in the literature in different situations such as independent Bernoulli trials (BT), non-identical BT, Markov dependent BT (MBT), higher order MBT, binary sequence of order , multi-state trials etc. But very little work is found on the distribution theory of run statistics in case of bivariate trials which has applications
in different areas such as start up demonstration tests with regard to simultaneous start ups of two equipment, reliability theory of two dimensional consecutive
k
,k r out of 1,k n : F -Lattice system etc as specified by [7]. [7] have studied the distribution of sooner and later waiting time problems for runs in Markov dependent bivariate trials by giving system of linear equations of the conditional pgfs of the waiting times. The distribution of number of occurrences of runs in the two components of bivariate sequence of trials and their joint distributions are still unknown to the literature.
Consider a sequence , 0nX n
a a
of -valued trials where is set of all possible outcomes of trials under study. The simple pattern is composed of specified sequence of states i.e. 1 2 k
SS
k a where
1 2, ,a a ka S . The number of occurrences of patterns can be counted according to the non-overlapping or overlapping counting scheme. The non-overlapping cou- nting scheme starts recounting of the pattern immediately after the occurrence of the pattern while the overlapping
K. K. KAMALJA ET AL. 116
counting scheme of patterns allows an overlap of pre- specified fixed length in the successive occurrences of patterns.
Recently the study of distributions of different statis-tics based on patterns has become a focus area for many researchers due to its wide applicability area. Distribu-tion of , the waiting time for the occurrence of pattern of length in the sequence of multistate trials is studied by [1,8,9]. [10] considered the sequence
1 2 generated by Polya’s urn scheme and study the waiting time distribution of for .
, ,r kW
,
thr
r
k
,X X
, ,r k
Joint distribution of number of occurrences of pattern
1 of length 1k and pattern 2 of length 2 in n Markov dependent multi-state trials is studied by [11]. [12] considered a sequence
W 1
k
, 1, 2,iX i of - dimensional i.i.d. Random column vectors whose entries are -valued i.i.d. random variables and obtain the waiting time distribution of two dimensional patterns with general shape. The general method, which is an extension of method of conditional pgfs, is used to study these distributions by [12].
m
0,1
Even though the distribution of waiting time of the pattern of general shape in the sequence of multi-variate trials with i.i.d. components has been done, the joint distribution of number of occurrences of patterns
in the sequence of component of the -variate trials
, 1, 2, ,i i m
m thi
1 2, , , nX X X is still unknown. Here we derive the pgf of joint distribution of number of occurrences of runs in both the components of the bivariate trials and generalize this study to the distribu-tion of number of occurrences of patterns in both com-ponents of the bivariate trials.
In this paper we consider the sequence
of -valued Markov dependent bivariate trials. In Sec-
trials. In Section
. The Joint Distribution of Number of
Let
, 0n
nY
nX
Stion 2, we obtain the pgf of joint distribution of number of occurrences of i -runs of length 1
ik in first compo-nents and i -runs of length 2
ik in t second compo-nents of the bivariate trials 0,1i . We study this joint distribution of runs under th verlapping counting scheme of runs by using the method of conditional pgfs. Further in section 3, we study the joint distribution of number of occurrences of pattern 1 of length 1k in the first component and number of o rrences of pattern
2 of length 2k in the second component of bivariate 4, we develop an algorithm to evaluate
the exact probability distributions of the random vari-ables under study. As an application of the derived joint distributions, in Section 5, we obtain distributions of several waiting times associated with the runs and pat-terns in bivariate trials. In Section 6 we present some numerical work based on distribution of runs and pat-terns. Finally in Section 7, we discuss an application and
generalization of the studied distributions.
he
cu
e non-o
c
2Occurrences of Runs
0 0 1 1, , ,
0 1 0 1S
.
, 0n
n
n
Consider the sequence
X
Y
of S -valued Markov dependent bivari-
ate trials with the transition probabilities,
1i iX Xx u u ,
1
,
1, 2, ,
uv xyi i
xP p S
Y Yy v v y
i n
and the initial probabilities
0
0
πxyPY y
X x xS
y
.
for
Let
1 2, ,i i
i i
n k n kX Y
1
be the number of -runs ( = 0,1) of
le
i i
ngth ik ik mpon
2 i te n n trials associa d with the first (second ent bivariate trials (i.e. number of i -runs in
) co of 0 1 0 1, , , , , ,n nX X X Y Y Y . In this section
e derive thew joint distribution of
1 1 2 20 1 0 1, ; ,X X Y Y .
0 1 0 1, , , ,n k n k n k n k
0 1 0 1, ; ,n t t s s be the pgf of distribution of Let
1 1 2 20 1 0 1
1
, , , ,, ; ,
n k n k n k n kX X Y Y0 . Assume that for a non-nega1 0 tive
integer c n , we hav
have obs c ). We
fine
e observed until thn c trial
(i.e. we erved iX de- , 0,1, ,
i
i nY
i j 0 1 0 1, ; ,i j
c c t t s s l distribution of number of i -run
1 2 1 2, ; , , ; ,i m j m i m j m as pgf of condi-
tiona s of length 1ik in
1, ,n c nX X and number of j runs of length - 2jk in
ven that we have bserved
n cX ently h
1, , nY Y gi oXn c
0 1
0 1
, , ,n c
X
Y Y
1
Y
and curr ave i -run of
length in first component and -runs of length im j2jm in second component of bivariate trials is ob-
ed, 1 11, 2, ,i im k , 2 21, 2, ,the
serv j jm k , , 0,1.i j We define 1ca a for 0,1a . Now by ass 00 1uming π , we have,
(2.1)
Also we have,
0,0;0,00 1 0, ; ,t t s t t 1 0 1 0 1, ; ,n ns s s
1 2, ; ,i ji m j m
t t 1 10 0 1 0 1
2 2
, ; , 1 for 1, 2,..., ,
1, 2,..., and , 0,1
i i
j j
s s m k
m k i j
(2.2)
Conditioning on the first trial we have the following system of recurrent relation of conditional pgfs for
terns of lengths 1k and 2k respectively where , 0,1i ja b , 1 21, 2, , ; 1, 2, ,i k j k . We co
non-overlapping counting scthehiw b
urs. in a e of has to restart counting from scratch each time the pattern
1,n no verlapping oc-currences of patterns 1 in the first component of n bivariate trials (i.e. in 1 2, , , n
given sequenc
X de tes th
n trials, one
mber of non-oocc
Let e nu
X X X ) and 2,nY d
notes the number of non-ov lapping occurrences of pat-terns 2 in the second component of n bi riate tri-als ( 1 2, , , nY Y Y ). Consider the random vector 1 2, ,,n nX Y . In this section we obtain the pgf of joi distribution of 1 2, ,,n nX Y ethod of ndi-tional pgfs.
Let of joint distribution of 1 2, ,,n nX Y be
e-
nt
er
using m
vai.e. in
co
pgf 1 2,n t t t .n Assum
, we have observed until
. For c n , we define, the fol-
lowing conditionLet
e that for a non-negative integer th
c trial i.e.
1 2, ,X X
Y Y
c n n
1 2 n cY al pgfs.
, n cX
( , )i jc t be pgf of condition ribution of
r terns , ,al dist
number of occu rences of pat in 1 1n c nX X and number of occurrences of pattern 2 in
1n c n
at thn c trial) we ha
, ,Y Y of bivariate trials give obser no
ven that currently (i.e. ne of the sub- ved
patterns of 1 and 2 and n c
n c
X i
Y j
, where
iS
. jSimilarly let,
1
,,i j
c be pg nal distrition of num er of
t f of conditio bu-b occurrences of patterns 1 in
1, ,n c nX X bivariate trials giveserved the suco
and patterns in of n that at
b-pattern of first one of
2 n c
h i
1n c trial we in
, , nY Yhave ob-the
th
lengt of 1
mponent and n the sub pattern of 2 is ob-served in the second component of bivariate trials and
n c i
n cY j
X a
where 1, 2i k and 0,1j .
Let
1, ,
2
,,i j
c t be pgf of conditional distri tion of number of occurrences of patterns 1 in 1, ,n c n
buX X
and patterns in Y Y of bivariate trials 2
thn c e obse
1, ,n c trial we h
in the fi
n
avrst com
given that at the sub-pattern of pon
pattern of the
e ob d
rved none of ent and a sub 1
length j of 2 in the second compo-
nent of bivariat trials and n cis servejn c bY
where 0,1i
iX
and 21,2, ,k j .
Also let 1 2
,, ,i j
c t as pgf of onditional distribution of cpattern number of occurren in ces of 1 1, ,n c nX X
and pattern 2 in 1, ,n c nY Y of bivari
of leng f 1
ate trials givenhe sub-pattern that at obse ved t th
n c o
th i
trial we have r coi hen t
and
first mponent and a sub pa
iat s
ttern of the length j of 2 in the second compo-
nent of bivar e trial in c
jn c
aX
bY
where
11, 2, ,i k and 21,2, ,j k . Let,
1ca ai i , i k11, 2, , and 1c
j jb b , 1, 2,j 2,k .
For c n , we as esum
i
1ci
that at
1
If
thn c
observed i
ondition
trial, the
su of s n the first
compone riate trials. we c on the next
trial as
b-pattern of
nt of biva
n c
n c
X
Y
length i
1 a
1 1n cy
i.e. of leng-
th i of 1
the sub-pattern
observ d at thn c trial breaks at
1
e st
n c trial then to check whether any sub-pattern
of 1 of length r r i 1
has occurred, we define the
indicator fu ion irnct as,
1 2 2 3 1 1, ,..., , ci r i r r i r ia a a a a a a
,1 20, , 1 for 1, 2, , and 1, ,i j t i k j k . (3.2)
Now for each denotes the number of trials remaining to observe) we condition on the nex to obtain the recurrent relations of conditional pgfs as follows.
2,
1 c n ( c t trial
1 2 211
11 1
11 1 1 1
, 1,,
,1,
11,, ,
c
c cc
c c c
c ij ij a b
a bb
ccij a b ij a b
t p t p t
p t p t
1
1 1
,11,1,, 1,
cai ja b c c
if 1, 0,i j ,
1 11 1 2
2 1
1 2 211
2 1 1 1
111
1
11 1
, 1,1,, 1, ,
,1 ,1
1 11, , 1,,
, ,
1 1 11,,
1,
1,,
1
i i
i ci
ci i
i c c ci
c ci i
c
ci i
i j ia j a bc c
u ai ic ca j a b
u b a bi icca j a b
i b
ca j a b
t p t
p t u t u
p t u t u
p t
if
1
11,2, , 2; ,1i k j 0
1 11 1 2
2 1
2 211
1 12 1
211
1
11 1
, 1,1, 1, 1, ,
,1 ,1
1 11, 1,,
,( , )1 1 11,,
1,
11,,
1
1
i i
i ci
ci i
c ci c i
c ci i
c
ci i
i j ia j a bc c
u aic ca j a b
a bu b i icca j a b
i b
ca j a b
t p t t
p t u t
p t u t
p t t
iu
u
if
1 1; 0,i k j 1
1
1 11 1 2 11
11 1
11 1 1 1
,1, 1,1,, 1, , ,
,1,
11,, ,
c
ci i
c cc
c c ci i
ai ja j a bc c ca j a b
a bb
cca j a b a j a b
t p t p t
p t p
21,
t
if 1
Similarly we have recurrent relations of the condi-tional pgfs
velop an algorithm to the exact probability distri-
bution of
beapp ility distribution of
rve that t
involves matrix polynomial in
1 1 2 20 0 1
0 1
, , , ,,
n k n k n kX X
1n k
obtain
1 1 2 20 1 0 1
0 1 0
, , , ,, ;
n k n k n k n kX X Y
lied to obtain the exact probab
1 2 from its pgf.
1,Y which can further
, ,,n nX Y
Obse he pgf of joint distribution of
1 2 21 0 1
0 1 0 1
, , ,, ; ,
n k n k n kX X Y Y as obtained in (2.5) in general 1
0,n k
0 1 0 1, ; ,nP t t s s 0t , 1t , 0s and 1s of order n where
0 1 0 1
1 1 12 12
0 0 0 0
, ; ,n
n
i i j j ij i ji j i j
P t t s s
11A B t B s B t s
.
Hen e joint probability distribution can be ob-din
ce thtained by expan g the polynom respect to ial with
0t , 1t , 0s and 1s . That is,
2 20 1
1
0 1 0 10, ,
0 1 0 1 0 1 0 1
, ; ,
coefficient of in , ; ,
n k n k
yn
P X x X x Y y Y y
t t s s t t s s
1 10 1
0 01
0 1, ,n k n k
x yx
1
i.e.
0 1 0 1, , , ,
co cient of in , ; ,
n k n k n k n k
x yt s t t s s (4.1)
10 , ;P X y1 2 2
1 0 1,
effi n
X x Y Y
edious since tive operation ting recurrent relations are fo
0 1 0 1
Exact formula to obtain the coefficient matrix is quiet t multiplication of matrices is not commuta-
. But the interesund between these coefficient matrices. Let
; y be the coefficient matrix of 0 1 0 1, ; ,n nC x x y y C xx yt s in the expansion of ma l 1 0 1; ,nP t s s and for n
trix polynomia, let 0 , t 1
; , 0,1, , for 0,1n i iD x y x y n i . The following
Lemma gives the recurrent relations of the coefficient matrices of ;nC x y with 1 ;nC x y .
Lemma 4.1 Let ;nC x y be the coefficient matrix
of x yt s in the expansion of the matrix polynomial
0 0 1, ; ,nP t t s s Then 1 ;nC x y satisfies the following
re lation. current re
1
11
1 1 10
1
i
21 1 1
0
1 112
1 1 1 1 10 0
; ;
; 0
; 0
; 0;
n n
n ii i
n jj jj
n iji j i ji j
C x y C x y A
C x e y B I x e
C x y e B I y e
C x e y e B I x e y
0e
(4.2) with 1 0;0C A , 1
1 1;0 iiC e B , 11 10; jjC e B
and 121 1 1; iji jC e e B for , 0,1i j . Here ie is the
thi row of the identity matrix of order 2. Proof Obviously for 1n , we have,
1
1if ; 0, 0,1iB x e y i
21 1
if
; if 0; ,
se
i
j j
A x
C x y B x y e
O
w atrix of
21 1if ; , 0,1; 0,1
otherwi
ij i iB x e y e i j
0; 0y
0,1j
here O is the null m same order as that of A. For 2n , observe that 2 ,C x y satisfies (4.2). Assu at E (4 ) is trume th quation e for some .2 2r r n .
Here we note that it is quiet difficult to study the later waiting time distribution between 1
0F , 20F 1
1F , 21F ,
particularly when one is interested ining time distribution for the later otween more than two events. For thbution of later occurring event betw the components of the bivariate trialterns in the two components of bivacan be developed in general. For thderive the waiting time distributi event between the two events
stuccu
e waitineens or
is we reon of l
dyingrring
g tim any twfor th
riate trials thfer [1
ate
the even
e do run
e twoe th3]
r occu
wait-t be-istri-s in
pateory
who rring
-
0F and 1F where iF
0,1i for the dependentof occurreanof
is the event that -rfirst time in the sequence er
BT. [13] use the nnces of 1-runs (i.e
f occurren s of ( )
5.2. Waiting Time Distribution for Runs Let
i
joi.
ce
unof
t di succes
0-
of lehighstributs-runsruns
ngth orde
ion o) i.e.
ik ocr Markf nu
of length failure
cursov
mber
1krunsd the number o
length k . 0
,i
jr iW
un of lbe the waiting time for the ccurrence of
-r ength
thir o
i jik for the com t of bivariate
thj ponentrials, 0,1i ; 1, 2j . We obtain the distribution of
,i
jr iW using distr tion of
,ibu j
i
j
n kX number of occur-
rences of run of length i - jik
1 ; stribu
for the com nt of bi ate trials . T pgf
of jo
(2.5) is as
follows.
thjhe
pone
0 1 0 1, ; ,n t t s s f n vari 0,i
int di1
1, 2j tion o
1 1 2 20 1 0 1
0 1
, , , ,, ; ,
n k n k n k n kX X X as obtained in 0X
0 1 0 1
1 1 1 11 2 12
0 0 0 0
'
n
i i j j ij ji j i j
p A B t B s B
In particular pgf of marginal distribution of 10
1
,n kX is
obtained by setting 1 0 1
, ; ,
1
n
i
t t s s
t s
1t s s in the pgf
0 1 0 1, ; ,n t t s s o f 1 1 2 20 1 0 1
0 1 0 1
, ,0 , ,1 , ,0 , ,1, ; ,
n k n k n k n kX X Y Y a n d
is as follows.
0
11 1 2 12 12 12 120 0 1 00 0 01 0 10 11
0
,1;1,1
'
n
n
jj
t
p A B t B B B t B t B
Let pgf of ,
1
ji
j
n kX be ,
j jn i it for 0,1i ; 1, 2j .
Then pgf of 10
1
,n kX in (5.7) can be expressed in a simp
fied form as follows.
li-
1 1 1 1 1 1,0 0 0 0 0 0 0,1;1,1 'n nt t t p A D t
1 1 2 2 12 12
1n
here w 0 1 0 1 10 11A A B B B B B 1
a rib
and
d from emma 4
1 12 120 0 00 01D B B B . The exact prob bility dist ution of 1
0
1
,n kX can be
obtaine L .1 as follows.
11 1P X r p r (5.8)
0, nn k
where
'C
nC r is the c ix of 10
rt
mial 1 1 10 0 0
noefficient matr in the
matrix
B
(5.7)
polyno A D t and in for general2 m n mC r satisfies the recurrent relation
1m D 1 1C r1 0 1m mA 0
with
s ma
C r C r
10
11 0
if 0
if 1
otherwise
A i
C i D i
O
where O is the null matrix of order same a trix 10A and 1
0D . The pro 5.8) can be wri as follows.
bability in ( tten
10,n k
P X 1 1 11 0 1 0' 1 1n nr p C r A C r D
The above components of can be in-terpreted as follows.
1 2, ,,n nX Y is 10n us-heorem 4.1. Th d described in ng
cation and Extension
[14] introduced the two-dimensional engineering system co of nents arranged in
-rows and -colum system and its compo- can be n wo g or failed state. The system
components fails. Particularly, for ulate the states of the comp as a sequence of
me that the failed state
state 0
ingjoinTa
the algorithm given in T 1 2, ,,n nX Y is
e evaluatethe followit pmf of
ble 1.
7. Appli
nsisting of a grid n compo mns. Therkin
1, we cahis sy
mnentsfai
indcomand in
n either i
ls if and only if a grid of size r sn form
stem
1, 2,
m r onents in t
otherwise. For
n thi ependent bivariate trials. We assu
ponent in a column is in state 1 if it is in ,i n we define,
1 if first r components in column are ile
wise
th
i
iX
in fa d state
0 otherand
1 if last r component s in column arein failed stateth
i
iY
0 otherwiseThe reliability of consecutive- ,r s -out-of- 1,r n :
F-Lattice system can now be obtained simply by using the joint distribution of as, P(consecutive -o Fworks) =
1 2, ,,n nX Y
ut-of- 1,r n :- ,r s -Lattice system 0 ,
1 2, ,n n
where 1 is 1-run of length 0,P X Y
s first component and 2 is 1-run of leng
in theth s in the second component
of the bivariate trials. Extending the concept of bivariate trials to mu vari-
ate trials, the joint distributions of num er of occuof patterns of length
ltib rrences
i ik 1,2, ,i m ials can be used
m.
in the compone -variate t to
out-o
Tab
thi get the nt of
f-
m
,m n
r
: F-Lattice systereliability of general two-dimensional consecutive-
le 1. Distribution of
r s -
1 210, 10,, X Y
210,
.
Y
110,X
0 1 2 3 Sum
0 0.00257 0.00666 0.00179 3.5E–05 0.01106
1 0.04765 0.08133 0.02006 0.00044 0.14949
2 0.16288 0.21551 4793 0.00113 0.42744
3 0.14927 0.15937 0.03184 0.00075 0.34122
4 0.03465 0.0294 0.0049 9.9E–05 0.06912
5 0.00101 7.3E–05 1.2E–06 0.00167
Sum 0.39803 0.49285 0.10666 0.00245 1.00000
0.0
7
0.00058
The pgf 1 2, , ,n mt t t int distributio
in t of jo n
of , , 1, 2, ,inX i m
patterns i
, the number of occurrences of of length
-variate trials obtained in ik in the component of general by ng the method
nditional pgfs is of form,
thi usim
of co
12...1 , 1
n i ii i j
1 2' ... ... 1
n
n m
ij i j k kt p A B t B t t B t t t
e Le for thebility dist ibuti of
i j (7.1)
Extending th mma 4.1 pgf in (7.1) exact joint proba r on , , 1, 2, ,
i
inX i m can
i
be obta d. Th bove study of runs a d patterns can be extended
in another direction by ge ralizing the sequence of M
nee a n
ne
ri o deriving th is
arkov dependent bivariate trials to the sequence of Markov dependent multivariate trials. In the following, we discuss b efly the meth d of e joint d -
tribution of 2 2 21 2 21
, , , , , ,, , ,
r rn k n k n kY Y Y
where 2, , iin k
Y
,
1i two dim
2i
, 2, , r otes the number o den f occurrences of f rectangu shape ensional patterns o lar
i r 1, 2, , multivariate trials.
in the sequence of Markov dependent
Let , m'2
1 2 , , 0,1 1, 2,m m iS X x x x
so that
, x ; i
2mS# m2 . Consid q e
a d er the se uence of m -variat
2mS -v lue ovd Mark depen ent trials 0
dependent trials ,iX i . Let the
of these Markov transition probabilities 0 be, ,iX i
riving joint distribution of number of occurrences of pat-terns i 1, 2, ,i r in the
X . We obtain the joint
sequence of multi-state trials . For this consider the following transf ence of -variate sequence o
ndent t
,n
as dorm
-val
one by [11]ation of sequ
ued Markov depe m
rials f
2mS , 0iX i to the
f Ma dependent trials univariate se , 0
quence o.
rkov
i
Define the function 2 10: m m
Z ig S S such that
12m
i
1iig x x where
10S g nce 2
m mx x S . He
10 0,1,2, , 2 1mmS . Each x in 2
mS can be treated as a m -digit binary number. The function g x con-verts this m -digit binary number into a unique equiva-lent decimal number in 10
mS . Now corresponding to the m -variate sequence of
2mS -valued Markov dependent trials , 0iX i e , w
the univariate sequence -valued Markov ndent tri
havedepe
of 10Sm
als , 0iZ i where iiZ g X . The transition probabilities for the sequence of trials , 0iX i and
follows. for the converted sequence of trials
, 0iZ i are related as
1r r x yP X y X x P
P Z
where
1
2, , 1
r r ij
m
g y Z g x p
x y S r
g x i and g y j . Convert the pattern 2
1 2 ii i i ika a a 1,2, ,i r into atte a p rn 10
1 2 ii i i ikb b b where ijb g ija for
in t ce of
1, 2, ij . , kNow the original problem of studying the joint distribu-
tion of 2 2, rk
he sequen
2mS -valued Markov dependent trials
2 2 211
, , , , ,, , ,
rn k n n kY Y Y
1 2, , , nX X X problem of studying distribution of reduces to the
10 10 101 2 21
, , , , , ,, , ,
r rn k n k n kY Y Y
for the sequence of
10mS -valued Markov dependent trials 1 2, , , nZ Z Z . [15]
obtain e tion of number of occurrences of i -runs 0,1, 2, ,i m of length ik , while [11] ob-tain the joint distribution of num er of occurrences of p erns in the sequence of
th joint distribu
batt -valued M0,1, , m arkov
dependent trials using the method of conditional pgfs. d wa e distributions can also be studied The relate iting tim
fo cess as in Se e caMarkov dependent m ltivariate trials. 7. References
. quen
. 78
llowing the same pro ction 5 in thu
se of
[1] S J. Schwager, “Run Probabilities in Se ces of
Markov-Dependent Trials,” Journal of American Statis-tical Association, Vol , No. 381, 1983, pp. 168-175. doi:10.2307/2287125
N. Balakrishnan and P. S. Chan, “Start- tration Tests with Rejection of Units upon Observing d Fail-ures,” Annals of Institute of Statistic Mathematics, Vol. 52, No. 1, 2000, pp. 184-196.
[2] up Demons
al
doi:10.1023/A:1004101402897
ller, “An Introduction to Probability Theory and Its cations,” 3rd Edition, John
[3] W. FeAppli Wiley & Sons, Hoboken, 1968.
[4] K. D. Ling, “On Binomial Distributions of Order k,” Sta-tistics & Probability Letters, Vol. 6, No. 1988, pp. 247-250. doi:10.1016/0167-7152(88)90069-7
4,
[5] S. Aki and K. Hirano, “Number of Success-Runs of ecified Le til Certain
Generalized B nomial Distribu Annals of Institute o l Mathema l. 52, No. 4,
Sp ngth un Stopping Time Rules and i tions of Order k,”
f Statistica tics, Vo 2000, pp. 767-777. doi:10.1023/A:1017585512412
[6] A. M. Mood, “The Distribution Theory of Runs,” Annals Mathematical Statistics, Vol .
als,” Annals of Institute of s, Vol. 51, No. 1, 1999, pp. 17-29
[7] S. Aki and K. Hirano, “Sooner and Later Waitin Problems for Runs in Markov Dependent Bivaria
Statistical Mathematic.
doi:10.1023/A:1003874900507
[8] M. Uchida, “On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables,” Annals of Institute of Statistical Mathematics, Vol. 50, No. 4, 1998, pp. 655-671. doi:10.1023/A:1003756712643
[9] D. L. Antzoulakos, “Waiting Times for Patterns in a Se-quence of Multistate Trials,” Journal of Applied Prob-ability, Vol. 38, No. 2, 2001, pp. 508-518. doi:10.1239/jap/996986759
[10] K. Inoue and S. Aki, “Genlems Associat
eralized Waiting Time Prob-ed with Pattern in Polya’s Urn Scheme,”
Annals of Institute of Statistical Mathematics, Vol. 54, No. 3, 2002, pp. 681-688. doi:10.1023/A:1022431631697
[11] K. S. Kotwal and R. L. Shinde, “Joint Distributions ofPatterns in the Seque
nce of Markov Dependent Multi-
, 2004, 169-182.
State Trials,” 2011 Submitted.
[12] S. Aki and K. Hirano, “Waiting Time Problems for a Two-Dimensional Pattern,” Annals of Institute of Statis-tical Mathematics, Vol. 56, No. 1doi:10.1007/BF02530530
[13] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Runs in a Sequence of Higher-Order Two-State Markov trials,” Annals of Institute of Statistical Mathematics, Vol. 58, No. 1, 2006, pp. 537-554. doi:10.1007/s10463-005-0024-6
[14] A. A. Salvia and W. C. Lasher, “2-Dimensional Consecu-tive k-out-of-n: F Models,” IEEE Transactions on Reli-ability, Vol. R-39, No. 3, 1990, pp. 382-385. doi:10.1109/24.103023
