Generalized random walk and distributions of some rank order statistics

Journal

of Statistical

Planning

and Inference

119

24 (1990) 119-133

North-Holland

GENERALIZED RANDOM WALK AND DISTRIBUTIONS SOME RANK ORDER STATISTICS Jagdish

SARAN

and Sarita

OF

RAN1

Department of Mathemafical Statistics, University of Delhi, Delhi 110007, India

Received

7 December

Recommended

1987; revised

manuscript

Abstract: This paper deals with the derivation of some rank order statistics the extended journs number

above

received

30 September

1988

by I. Vincze

Dwass technique. height r, number

of sojourns

at height

of the joint and marginal

related to the generalized The rank order statistics of crossings r from above

random

probability

distributions

walk with steps 1 and -,LJ by using

considered

of height r, number

include total length of all soof sojourns

at height r and the

(r>O).

AIMS Subject Classification: 62630. Key words andphrases: Extended Dwass technique; generalized random walk; sojourns r; sojourns at height r from above; total length of all sojourns above height r; crossings r - at lattice and non-lattice points.

at height at height

1. Introduction Mohanty and Handa (1970) have extended Dwass’ technique (1967) to the case when one sample size is an integer multiple of the other (i.e. m=yn) and derived the distributions of some rank order statistics. Later Saran and Sen (1979) and Sen and Saran (1983) derived the joint distributions of some rank order statistics related to the maxima of the generalized random walk with steps 1 and -p and the distribution of ND:,(r), the number of crossings of height r (rr 0), respectively, using the extended Dwass technique. Further Sen and Kaul (1985) have derived joint and marginal distributions of LP,n, the length of positive sojourns, NP,n, the total number of sojourns, NV:,,, the number of positive sojourns, and N&, the number of crossings of the origin. They have also derived the joint and marginal distributions of L,,,, R,,., Rln and NPTn, where R,,. and R& denote, respectively, the number of reflections and the number of positive reflections. In this paper we consider the above mentioned generalized random walk with steps 1 and -p and derive, for r> 0, the joint and marginal distributions of the following rank order statistics: (i) L,,,(r), the total length of all sojourns above height r = total number of steps above height r, 0378-3758/90/$3.50 0 1990, Elsevier Science Publishers

B.V. (North-Holland)

120

J. Saran. S. Rani / Generalized random walk

(ii) N,,.(r), the total number of sojourns at height r, (iii) NfiTn(r), the number of sojourns at height r from above, and (iv) NpTn(r), the number of crossings of height r, by using the extended Dwass technique, thus generalizing and extending of Aneja

(1975), Sen and Kaul (1985) and Sen and Saran

the work

(1983).

2. The method In order to derive the distributions of rank order statistics we shall use the extended Dwass technique given by Mohanty and Handa (1970). They considered the generalized random walk (Sj: Sj = Cf= i Wi, S, = W0 = 0} generated by a sequence { Wi} of independent random variables with common probability distribution P(Wi=+l)=p,

p being a positive integer. The main theorem used for finding quoted

from Mohanty

15i
P(Wi=-~)=q=l-p,

and Handa

the distributions

of rank

order

statistics

is

(1970):

1. Suppose Vp,n is a rank order statistic for every n and VP is the corresponding function defined on the random walk which is completely determined by W,, W,, . . . . W,and does not depend on WT+,, Wr+2, . . . . whenever T>O (where T is the time for the last return to zero in the random walk). Define

Theorem

h(p)=E(V&

(1)

a<@(~++).

Then we have the following power series (in powers of ppq) expansion

h(p) l-(p+l)PPqYh

(2)

(PP’4)“, n=O

where for y see (ii) and (iii) in Section 3 below.

3. Some auxiliary

results

The basic results needed in the sequel (1970) and Sen and Kaul (1985). (i) For any (Y and p,

5 &(a,

k=O

where Ak@98)=$j$j

a)@ = xa

are quoted

from

Mohanty

and Handa

(3)

121

J. Saran, S. Rani / Generalized random walk

the last inequality assuring the convergence (ii) The probability generating function

of the series. (PGF) for the first return

to the origin

is F(t) = (,Lf+ l)ppqxV+’

(4)

where PV4f

/I+1 _ x-1 -x”+1

(iii) The probability

and

(t(~‘+‘p~q<~~/(LL+l)~+l.

of never returning

6 = l-F(l)

to the origin

is

= 1-(p+l)pPqyP

(5)

where y is the value of x when t = 1. (iv) The PGF for the time to reach k is G(t,k) = (px#,

(v) The probability

k=

1,2 ,....

of ever reaching

G(l,k)=(~y)~,

k=

(6) k is

1,2 ,....

(7)

(vi) The contribution a to the PGF of Lp by a positive return at a lattice point or a positive sojourn which is such that Sk, = 0, Sj> 0 and Sk, = 0 for all k,
(8)

(vii) The contribution /3 to the PGF of Lp by a negative return which is always at a lattice point or a negative sojourn such that in the path segment defining it Sk, = 0, Sj< 0 and Sk, = 0 for all kl < j < k, is given by P = PV’4YP’.

(9)

(viii) The contribution to the PGF of Lp by the pair of positive and negative journs involved in a crossing at the lattice or non-lattice point is given by

so-

where y=l+-

t p-1

xt

aj=i

Y

j

cC-F

(ix) The segment of the path {L$} between the two consecutive indices j for which Sj_, = -1, Sj=O and Sj+l = +l will be called a section. Path segments between the origin and the first such index j where St = +l and also between the last such indexj and the last return to the origin from the negative side will also be con-

122

J. Saran, S. Rani / Generalized random walk

sidered to be sections. Let the i-th section in the path {A”] comprise mi positive sojourns (returns) preceding a pair of positive and negative sojourns (not necessarily returns) involved in a crossing followed by ni negative sojourns (returns). Then the contribution to the PGF of 15~ by the i-th section is

(11) (x) The following

power

series expansions

are also used in the sequel: (12)

where (z) is the smallest

integer

greater

than

or equal to z. Further

(5)(i)(‘c:-‘)(~,~)j(*~,~)j+i+~(~-l), j=O

whence

k=O

(13)

i=O

when t = 1, x= y and a =p and we have

IfIt=, = wPYY~o

(-l)'(;)(PY)'-'

= (l@YY(PY - 1Y.

4. Joint distribution

of L,(r),

(14)

Np(r), Np+(r) and N,*(r), r > 0

Let E(tL”(“; N,(r) = a, N,+(r) = b, N,*(r) = 2c) denote the PGF of the length L,(r) of all sojourns above height r when the number N,(r) of total sojourns at height r is a, the number Np+(r) of positive sojourns at height r from above is b and the number N,*(r) of crossings of height r is 2c in the generalized random walk {A”, 0 ~j< m}. Then for r>O, E(t’fl@); N,(r) = a, NV+(r)= b, N,*(r) = 2c) = ; P&(r) = (py)’

= g, N,(r) = a, N;(r)

= 6, N:(r)

= 2c)tg

Aabpa-b-‘yc-lq(l-(py)~‘)+Babpa~b-IyCq(l-(py)~) +L4&‘pa-b-‘yC-‘qt

M-1 ,;, (pxt)S(l-

(py)fl”-“)

1

(1%

where

To establish (15), there arise the following four mutually exclusive cases: (i) the first and the last points of contact of height rare return points (Figure I), (ii) the first point of contact of height r is a crossing and the last point of contact of height r is a return point (Figure 2),

123

J. Saran, S. Rani / Generalized random walk

(iii) the first point of contact contact of height r is a crossing

of height r is a return (Figures 3 and 4),

point

and the last point

of

(iv) the first and the last points of contact of height Y are crossings. If there is a crossing at the last point of contact of height r (as in (iii) and (iv) above) then again there are two cases: (a) when the crossing at the last point of contact takes place at a lattice point, (b) when the crossing at the last point of contact takes place at a non-lattice point. Let M and D be the first and last return points of height r, respectively. Then the path {S,} comprises three segments, viz. from 0 to M, M to D and D to 03. Of these, the first segment from 0 to M occurs with probability (py)‘, by (7). The probability of the last segment from D to 03 equals q(l - (p~)~) provided the last point of contact of height r is a lattice point.

\

Fig. 1.

In case (i), taking M as a new origin, the path segment NU = a - 1, Nti’ = b, Nfl*= 2c (see Figure 1) and the contribution MD to the requisite PGF is

MD entails Si =-p, of the path segment

by (13b) of Sen and Kaul (1985). Thus in case (i), the PGF of L,(r),

when N,(r)

= a,

N,i(r) = b, N,*(r) = 2c is given

by

Similarly, in case (ii), on transferring the origin to M, the path segment MD entails S, = +l, Nfi =a- 1, N,’ =6, N,*=2c1 (see Figure 2). Then on using (14) of

-

\

v Fig. 2.

J. Saran, S. Rani / Generalized random walk

124

Sen and Kaul (1985), the PGF of L,(r) when N,(r) = a, Np+(r) = b, N:(r) = 2c, in case (ii), is given by (17) In case (iii)(a), the last point of contact of height r is a crossing occurring at a lattice point (see Figure 3). Arguing in a similar manner as in case (i), the PGF of L,(r) when N,(r) = a, NU+(r)= b, N,*(r) = 2c, in case (iii)(a), is given by

(18)

\

Fig. 3.

In case (iii)(b), there is a crossing at the last point of contact of height r occurring at a non-lattice point (see Figure 4). Taking M as a new origin, the path segment MD has S, = -p, Nti = a - 2, N,’ = b - 1, N;” = 2c- 1. Again, the path segment MD comprises a sub-segment MA, which contains, say n > 0 negative returns, c - 1 sections A,A2, A,A3, . . . , A,_,A, and a sub-segment A,D which contains, say m 20 positive returns. Excluding c - 1 pairs of positive and negative sojourns which may or may not be returns involved in c- 1 downward crossings (one appearing in each of the c - 1 sections), there are b - 1 - c + 1 = b - c positive and II - b - c negative sojourns which are returns also. The number of arrangements of b-c positive soSimilarly, the number journs into c sub-segments A,A,,A2A3, . . . , A,D is (:I:). of arrangements of a - b - c negative sojourns into c sub-segments MA,, A, AZ,. . . , A,_,A, with the sub-segment MA, comprising at least one sojourn is (“J!;‘). Therefore, the total number of arrangements of b - c positive and a - b - c negative sojourns is (~I{)(“,“~“). Let the i-th section contain mi positive sojourns preceding

I

‘5

Fig. 4.

.I. Saran, S. Ram’ / Generalized random walk

125

the pair of positive and negative sojourns constituting a crossing followed by nj negative sojourns. The path segment from D to 00 (Figure 4) is such that it starts from height r with a positive step, then crosses height r only once at a non-lattice point and thereafter it does not reach height r. Suppose it crosses height r at a non-lattice point from the point P such that the ordinate of the point P is s (O
by using a similar argument as used by Sen and Saran (1983, Lemma 3). Thus in case (iii)(b), the PGF of L,(r) when N,(r) = a, $(r) = b, N,*(r) = 2c is given by

using

(8), (9) and (ll),

and this equals

b-rpa-b~‘yC-rqt~~r (PYY(~I:)(a;~;2)a

P-1

(JJxf)s(l-(py)@-s)

since Cfl: mifm = b-c and J$cr: ni+n=a-b-c. Likewise, it can be shown that the PGF of L,(r) when N,(r) =a, NT(r) = 2c in cases (iv)(a) and (iv)(b), respectively, are given by

(19)

N,f(r) = b,

and P-1

b-‘pa-b-lyc-lqt s;l (pxt)S(l(PYY( ;I:)( “r”,‘)cf The result

(15) follows

on adding

(p_JJ)~‘-“).

(21)

(16) to (21).

5. Deductions (i) Summing I$&(‘);

(1.5) over a, we get Np+(r) = 6, Np*(r) = 2c)

+ab-lpc-lycyc-l

V-1 El

mwq~(l-

(PY)p-“)

. (22)

J. Saran, S. Rani / Generalized random walk

126

(ii) Summing

(22) over b, we get

E(&@); =

N,*(r) = 2c)

(PY)’

t

@x)cPc-lYcrc-lq(l - (PYYW+-Pvv> M-1 +,c-l,cpc-lycvc-1

(iii) Summing

El

wosq~u

- h?wS)

I

(23)

1

(15) over c, we get

E(W); N,(r) = a, Np+(r) = b) abD”pb-lq(l-(PY)“)

(24) (iv) Summing E(t”“‘;

(24) over a, we get abyq(l

N,+(r) = b) = (py)’

+j3yy)b(l

- (PY)~)

i

P-1 SC] wf)v

+CPYU +PYy)b-kt (v) Summing E(W);

N,(r)

- (~~1fi-9

1

.

(25)

(24) over 6, we get = a)

.bpOpb-‘yCq(l

- (pyp) v-1

c

.

s=l (26)

(vi) Summing

E(t Lq

(25) over b, we get

= (py)‘(l-a-&J’)-’

i\

q(l-(py)“) +

where y’= 1+ t c j=l

(xt/y)j.

w

P-1

c (PxtY(l-

s=l

(PYY")

I

(27)

J. Saran,

When t=l, a=/$ x=Y reduce, respectively, to (vii)

P(N,(r)

and

S. Rani / Generalized

random

walk

127

y’=yy,

the results

(15) and (22) to (26)

say, then

= a, NO+(r) = 6, N,*(r) = 2c)

+(“91)

P

+

YF4(1-

pII-2Y;-1q

(PY)P)

P-1 c (PYRl - (PYF”)

* (28) 1

s=l

(viii)

P(NP+(r) = 6, N,*(r) = 2c) b+cP lYlcPIYC(l +PwM1-

h?w~

P-2

+P b+c-2Y;-1Ycq El (PY>“U-(PYY”> (ix)

P(N,*(r)

= 2c) O-2 EO (PY)P-s-P(PY)P+l

= (PY)‘(PY-l)c-lPcm’Yc+lq [ (equivalent

60

(xi>

to (26) in Sen and Saran P&(r)

I

.

(29)

1

(30)

(1983)).

= a, Np+(r) = b)

Wfl+W= 6) = (MPbrq (1+PYIY)~(~ - (PYY') P-1

+(l+DYIY)b-l (xii)

P&(r)

1

sgl (PY)W (PYY”) .

(32)

= a)

When p= 1, a=+[1-(1-4pqt2)1’2],

/?=p and y=y’=l,

then the results

(1.5) and

J. Saran, S. Rani / Generalized random walk

128

(22) to (33) reduce,

respectively,

to the corresponding

results

of Aneja

(1975,

Ch. II).

6. Probability

distributions

The following distributions can be derived by using Theorem PGF’s coefficients of tg(ppq)“((p+ lb ) from the corresponding previous section and usingnresults (12), (13) and (14).

(cl+l>n

(

>p(LP,n(r)= g>N,,.(r) = q&((r) = (EJjo f, ,jo ~=~,~)(-l)*(i)(‘+i-l)~~(~~+l) a,

n

=

1 and collecting derived in the

b, A&(r)

= 2c)

.[(“;“r’)(c;1)+(“-y’>(5)] (p+l)m-r-l

*I(

A,,(r-H+~(a-j-l),~+l)

m-l

>

(~+l)m-r

-

A,#-H+M-j),fi+l) >

(

1

+(:::),;,r’)x

1,

iO ;;:(-l)x

. (‘l’)(~)~+i-l)a~,(H+s-~,~+~~ (~+l)rq-r-s-1 ii *L m1=((r+Wp) (

ml-l

A,,(r-H+~(a-j-l),~+l) >

(p+l)m,-r-p-1

-

E m2=((r+Wll)

mz-1

(

A,,(r-s-H+~(a-j),~+l) >

1 ,

(34)

where H=j+i-k+p(k+b-j), H+b+(~+l)~=H+b+(~+1)~2,+s-~=g, l+v,+a-j-l+m=A+v*+a-j+m=A,+v,+m,+a-j-2 = A,+v,+m2+a-j-2

(P + lb

(

> =(91;>i n

W,,

Ar)

= g, N,In(r)

= 6, Iv&(~)

= n.

= 24

..

i E .i o*(/k)(/+:_I)A*(H,p+l)

j=O k=O i=O

m=(r/p)

J. Saran, S. Rani / Generalized random walk

4

(,u+l)m-r-l

129

C-l

m-l

A&-H+c+p(b+c-j-l),p+l)

j

I(

>

+0 C

A&-H+c+l+,n(b-j+c),p+l)

j

_

(jY+l)m-r

(

c-l

m

A,,(r-H+c+p(b-j+c),p+l)

j

>I( +

I

>

0

‘: A,,@-H+c+l+p((b--j+c+l),p+l) 1

c-l ~cr’i~~~~+~-ll~*,~~+~-~,~+lI + jFo

,go

;go

1::

(-lJk

(p+l)m,-r-s-l f

-L

ml=(v+WP) _

A,,(r+c-H+p((b-j+c-l),p+l)

m,-1

(

ii m2=((r+LWjl) (

1

(p+l)m,-r-p-l ,

A,,(r+c-H-s+,u(b-j+c),p+l)

m,-1

>

(35)

I

where H+b+(/f+l)A=H+b+(p+l)A,+s-p=g, L+v,+b+c-j+m-1

=A+v,+b+c-j+m=A+v,+b-j+c+m =A+v,+b-j+c+l+m =b+c-j-2+A,+v,+m, =I+v,+b+c-j-2+m,=n.

= f: i f j=O k=O i=O

i m=(r/b)

(_l)k

-I (~)(i+:I)A;(l,~-ti)

(,u+l)m,-r-s-l

(p+l)m,-r-p-l (36)

130

J. Saran, S. Rani / Generalized random walk

where

c+j+i-k+p(k+c-j),

I=

I+(p+l)i

J= c+r-j-i+k+p(c-l-k),

=I+s-,u+(,B+l)A,

=g,

A,+v,+2c-j-1+m=A+vz+2c-j+m=A+vg+2c-jfm =A+v,+2c-j+l+m=A,+v,+2c-j-2++ =A,+v,+2c-j-2+m,=n. P(-$, n(r) = g, &Ft, (r) = b)

(p+l)m,-r-s-l

(p+l)m,-r-p-l 4,(M+p

-.%P + 1)

where

r+I+l-j-i+k+p(I-k) H+b+(p+l)A

=M,

=H+s+l+(p+l)A,

=g,

A+v,+b-j+I+m=A+v2+b-j+l+l+m=A,+vv,+b-j+l+m, = ~i,+v3+b-j+I+m2

(P + lb

(

n

=

= n.

>

PC&,n (4 = 8)

E i ; h E (-l)k u=O I=0

j=O

k=O

;=O

i

m=(r/fl>

’ (J(;>c)(“:-‘> (p+l)m-r-l

K

m-l

>

4,(MP

+ 1)

(37)

J. Saran, S. Rani / Generalized random walk

131

-((~+~-r)A”*(M+lr,a+l)] p-1

c

+

(p+l)m,-r-s-l

A,,(k+s+pu,p+l)

S=l

I

Ii _

m,-1

(

ml = ((r+s)/p)

>

A”,(MP+

1)

(p+l)m,-r-p-l

(

ii mz=((r+JtV,u)

.A”,w+P--s,P++l)

m,-1

)

II

(38)

7

where k=j+i-k+p(t!-j+k), k-tu+(p+l)A

M=r+/+l-j-i+k+p((I-k), =k+u+s+2+pu+(p+l)A,

=g,

A+v,+v+I+m-j=A+v,+v-j+I+l+m=A,+v,+v+l-j+f+m, =A,+v,+v+I+l-j+m,=n.

(P + lb

(

>

Wy,, n (4 = a, y,(r)

= 6, N&(r)

= 2c)

_;:‘I:l(““r’~~~~(-l)~~c~l~~~~l~;

. [,n=i,pl

[ (

(p+l)m-r-l m_l (p+l)m-r

_

m

( P-1 +c s=l

L m, _

)4(r++~~~-c).~+1)

E = ((r+s)/fl)

)

A>.,@-j+&-c+l),p+l)

I

(p+l)m,-r-s-1 C

m,-1

(,n+l)m,-r-p-l E m,-1 W=((r+Pc)//c) (

A&-j+s+,u(a-c-l),p+l) )

)

Ai@-j+&-c),p+

1) 11

(p+l)m-r-1

-K _

m-l

(p+l)m-r m

>

Aiz(r-j+P(@-c-

A2(r-j+,da-c),p+l)

lX,L+ 1)

I

,

where A+a-c+m=A,+a-c+l+m=A,+a-c-l+m =A,+a-c-l+m,

=A+a-c-l+m,=n.

(39)

132

J. Saran, S. Rani / Generalized random watk

. L1;;mJ,p) ((~+l)m-r-rl)Ai(c-j+r+s+B(bl),ptl) m-l m -Lc

c mI=(v+P)/P) (

(p+l)m,-r-p-l >

m,-1

where A+&l+m=At+mt+b-1

A,,(c+r-j+l+pb,/J+1)

1 )

(40)

=n.

(p+l)n n

(

>

P($,(r)

c-l = i;O (-1)’

= 2c)

C;l

$-l-i

( -

L

1:;

> ((iu~1)~-~+s)A,~x_,((~+2)c+r-i-s~+l)

J_

-,n,,~~:,,, (equivalent

.

i

5;

(‘P+1’:-r-1)n,~Ir~((~+2)c+r-i+l.p+l)]

to (25) in Sen and Saran

m_,zsj,p,

(p+l)mt-r-l A,_,-,,(r+f+2_j+~ub,~+l)

m,-1

mr =cCr.//l)K

>

(y+l)m,-r

_ ( a,

(1983)).

((e+l)~I:s-l~A,,_h~~(~+~+~+l-~+~~,~+l)

cc +Ll

(41)

ml

A.~,,_,(r+[+2-j+~((b+l),~+l) >

(p+l)m,-r-p-1

1

. (42) 3 Probability distributions corresponding to (24), (26), (31) and (33) can be obtained from (34) and (39) by summing over appropriate variables. -fi

c

m2=

((r+pO/p)

m,-1

A,_,_,,(r+p+l+l-j+@,p+l)

>

J. Saran,

S. Rani / Generalized

random

walk

133

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K.G. (1975). Random

Statistics, Dwass,

walk and rank order statistics.

of Delhi,

M. (1967). Simple random

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University

S.G. and B.R. Sci. Math.

Handa

Hungar.

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statistics.

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Ann.

related

Math.

Statist.

to a generalised

38, 1042-1053. random

walk.

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related

to a generalized

random

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31, 121-135.

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