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A note on characterizations of the geometric distribution
ELSEVIER
Journal of S|atistical Planning and Inference 67 (1998} 187 190
journalof statistical planning and inference
A note on characterizations of the geometric distribution Henrik Cobbers', Udo Kamps ~'b'* ~'h~stitute olStatistics, Aachen ~.hdrersitr of Tt't'hooiog)', 52056 Aachen. Ge/'manr ~'Department ol'Statistics, Unicc'rsitr of Dormmnd. 44221 Dortmund, Getvnanr Received 24 October 199o
Abstract Well-known characterizations of the geometric distribution via the independerlce of sortie contrast and the minimurn in a sample of i.i.d, random variables arc illustrated and supplemented, q" 1998 Elsevier Science B.V. All rights reserved. A M S ('/ass!li('atiopl: primary 62E10; secondary 62G30 Keywords: Geomctric distribution: Ch,'Lracterizalion: Order statistics; Contrast
i. Introduction Several characterizations of the geometric dislribulion can bc I'ourld in the literature by means o1' the independence of I11c minimum X t.,, and a contrast Xh.,, ....... X ~.,, for some 2 <~ k ~ m where X ~.,, ~.~ ... ~ X , . , denote the order statistics corresponding to i.i.d, random variables X~ . . . . . X , . X t is said to have a geometric distribution if l ) ( X I = j) = p(I -- p)J Ibr allj c ~t, ~= I0. I . . . . ~j and some i) e (0. I). It is well known that X ~.,, a n d ( X a., -- X ~.,, . . . . . X,,.,, - X , ~.,,)are independent, if X ~ has a geometric distribution. Ferguson (1965} considered the case , = k = 2. Wc give a graphical summary of subsequent results, in which the full independence is not needed, in their simplitied versions, assuming lhe support of X t to be contained in the nonnegative integers and dropping the respective conditions, the characterizations of the geometric distribution can be illustrated by horizontal and vertical lilacs. For example. Srivastava (19741 requires X ~.,, to be independent of the event I X , , . , , - X ~., = O}j.
*Correspondence address: Institute of Statistics. Aachen [Jnivcrsity of Tcchnoh)gy. 52()5() Ailchcn, Germany. 0378-3758/98/$19.00 ~ 1998 Elsevier Science B.V. All rights reserved. Pll S 0 3 7 8 - 3 7 5 8 1 9 7 ) 0 0 1 2 I - 3
H. Cohbers. U. Kanq~s/Journal olStatistical Phnming and h!lbrence 67 (1998) 187, 190
188
2. Results As indicated by dotted lines in Fig. 1 it is shown in this note that the geometric distribution can be characterized by 'diagonals', too. Moreover, under additional conditions, the independence of events corresponding to shifted diagonals turns out to be a characteristic property. Let Xt . . . . ,X,, n>t2, be i.i.d, random variables and p ~ = P ( X ~ = i ) , qi = P ( X t > i), i e Mo, and q - t = 1. Theorem I. Let supp(Xt) c I%1o, m ~ N o be a f i x e d integer, po, Pro+ l, P,,,+ ,_ > O, and let q~ = q~+ t .for all 0 <~j <<.m. I f the events [ X t.,, = m + i] and ~X2.,,~ - X t . , , = . j ] are imlependent f o r i = j, j + i, j + 2 and fiJr allj ~ ~ o , then P ( X t = ./) = !'o( I - po)i.lbr all
Proof. In the independence condition P(Xt.,,=m+i,
X2.,,-Xt.,,=j)-P(Xt.,=m+i}P{X2.,,-Xt,,,=j)
(,)
we have
..n- 1
P I X t . , , = m + i , X , ,, - Xt.,, = j ) =
q~,+~-t - q',',,+i - n P.,+iq,,,+i. n-
" "
1
n P,,,+i(q,,, +i+,-t
_n-
1
- q,,,.;+ j),
,J = 0.
.j e NI,
Xt.,, ! ! I.,. t f O°
! !
.÷
• ,0 ~£
t_ . . . . . I |
I
o°
o
,'
0 . |'
'
0
-
O
I
,• ,0
I
0° o
°
I
0
'
,0* • .'
I .
t:
,0
°"
'' .. o~
°'~
00
,°
| •• |,° l~
09
"
.:. _ _.
•
I
|
,0
--"
9 .O*
I O' .i °
I
.D
I
°
t
,'
| t
••
i I
2
m
X~..-Xu. (2 s k g nl
Fig, I, ( ' h a r a c t e r i ~ a t i o . of tile geometl'ic distribution: ( J Arnold {1980k k = 2. see Becker (19•4. p, 94) for X~. ~,, ,-,~ .\'~,,: ( ........ ~ Sreehari {19831, k = 2 {see also Alzaid el ai,, 1998k Srivaslava 119,',16L k = n. ~, ~ l~ + I. N a g a r a j a and S r i v a s t a v a (1987), !~ = 0. i, = I; ( . . . . . . . } Srivastava {1974). k = n (see also ( ; a l a n l b o s 1975. Srivastava 1981}. EI-Ncv,cihi and ( ; o v i n d a r a j u l u (1979}. N a g a r a j a a n d Srivastava (1987}: ( ' ' ' ) EI-Neweihi and G o v m d a r a j u l u 11979k k = 2, G o v i n d a r a j u l u (1980).
H, Cohhers. U. h~unps/Journal of Statistical Planning am/h!/i, rence 67 0998) 16'7-190
189
and ,!
P ( X , . , = m + i) = q',',,+ i - l - q ,,, + i.
Identity (.) with (i) i =.j = 0 a n d i = I . . / = 0 and (ii) i = j = 0 and i = 2 , j = 0 leads to the e q u a t i o n aq~-l--bq~=c
witha=q~(1-q~),
.,n(k+ !)- i
and c = no
b=
l-q~-I
(I - qo),
where k = m + i in case (i) a n d m + 2 in case (ii). T h e p o l y n o m i a l f (qk) = a q'd- ~ -- h q~ -- c has two zeros in the unit interval (0. 1), which are given by qk = q~÷t a n d qk = q,~. However, the latter leads to a contradiction, since then pk = q k - t - q k = q~,-q,~ = 0. H e n c e qk is given by q~+~, k = m + 1, m + 2. T h e assertion n o w follows by i n d u c t i o n o n j, noticing that, for some fixed .J, Eq. (.) yields ,,-,
_ ..,,-,
qm + 2j + ! ~
q',',, + i -
~lm + 2,i - -
q",, + .~ + ,
P,,, + j
,,-,
(q:,,-+ ~j-1 ~
q',',,+ j - , - q',',,+ .~ P,,, + .~+,
,
qm + 2jl,
putting i =.j a n d i = j + i. and ,,- t q,,,,
.,,.,
2i+.,
=
q,,.,
q',',, +.~, l -
q',',, + s + 2
P,,, + s
~
,.j + t
q',',, + . i - ~ -
q',',, +.i
P,,,, i +
,.,,-,
-,
(qm+2j-
_,,- ! I --
qm+2.p,
putting i = j a n d i = j + 2. T h e d o t t e d lines in the a b o v e Iigurc illustrate the case m = 0 o1" Tlaeorcm I. In it similar way we o b t a i n the following theorem, where the 'characterizing diagonals' arc 'shifted IO the right'. Theorem
2. L e t s u p p ( X i1 ~ ~ . , m e ~ , he a.li.\'ed i n t e g e r , p . c (0. I}, a n d h't q i = q[}' !
l o t all 0 ~ j <~ m + I. ! / ' t h e e r e n t s I X 1.. =il
and ~Xa.,,-Xi.,,=m
+ I jI
a r e i n d e p e m h ' n t f o r i = O. 1, a n d ![ f
~X i X , . , , - X I . , , = m + j + ,.j "~ I I . n = i~j a n d a r e i n d e p e n d e n t /i~r i = j , j + I,j + 2 a n d lbr all j ~ NN., t h e n
P(Xt =,it
= p.(I
-
pot i
.lot aU./ ~ ~,,.
Proof. P u t t i n g i = 0 and i = I in P(Xt.,, = i.X,.,,-
Xt.,, = m + 1 ) = P I X I . , , = i ) P ( X a . , , -
X t . , , - - - m + 1)
yields q,,, + z = q'~'~+ 3 By a n a l o g y with the p r o o f of T h e o r e m I, the assertion follows by i n d u c t i o n on .j.
190
H. Cohhers. U. Kantps /Journal o[Statistical Phlntting att,t btfercplCe 67 (1998) 187 190
References Alzaid, A.A., La,:, K.S., Rao, C.R., Shanbhag, D.N., 1988. Solution of Deny convolution equation restricted to a half line via a random walk approach. J. Multivar. Anal. 24, 309329. Arnold, B.C., 1980. Two characterizations of the geometric distribution. J. Appi. Probab. 17, 570 ~573. Becket, A., 1984. Charakterisierungen diskreter Verteilungen durch Verteilungseigenschaften yon Ordnungsstatistiken. Dissertation, Aachen University of Technology. EI-Neweihi, E., Govindarajulu, Z., 1979. Characterizations of geometric distribution and discrete IFR (DFR) distributions using order statistics. J. Statist. Plann. Inference 3, 85-90, Ferguson, T.S., 1965. A characterization of the geometric distribution. Amer. Math. Monthly 72, 256 260. Galambos, J., 1975. Characterizations of probability distributions by properties of order statistics II. In: Patil, G.P. et ai. (Eds.i, Statistical Distributions in Scientific Work, vol. 3. Reidel, Dordrecht, pp. 89 101. Govindarajuh,, Z.. 1980. Characterization of the geometric distribution using properties of order statistics. J. Statist. Phmn. Inference 4, 237 247. Nagaraja, H.N., Srivastava, R.C.. 1987. Some characterizations of geometric type distributions based on order statistics. J. Statist. Piann. Inference 17, 181 191. Sreehari, M.. 1983. A characterization of the geometric distribution. J. Appl, Probab. 20, 209 212. Srivastava, R.C., 1974. Two characterizations of the geometric distribution, J. Amer. Statist, Assoc, 69, 267 26t). Srivastava, R.C. 1981. On some characterizations of the geometric distribution. In: Taillie, C. et al. IEds.J, Statistical Distributions in Scientific Work, vol. 4. Reidel, Dordrecht. pp. 349 355. Sri~,astava, R.C.. 1986. On characterizations of the geometric distribution by independence of fimctions of order statistics. J. Appl. Probab. 23, 227 232.
Journal of S|atistical Planning and Inference 67 (1998} 187 190
journalof statistical planning and inference
A note on characterizations of the geometric distribution Henrik Cobbers', Udo Kamps ~'b'* ~'h~stitute olStatistics, Aachen ~.hdrersitr of Tt't'hooiog)', 52056 Aachen. Ge/'manr ~'Department ol'Statistics, Unicc'rsitr of Dormmnd. 44221 Dortmund, Getvnanr Received 24 October 199o
Abstract Well-known characterizations of the geometric distribution via the independerlce of sortie contrast and the minimurn in a sample of i.i.d, random variables arc illustrated and supplemented, q" 1998 Elsevier Science B.V. All rights reserved. A M S ('/ass!li('atiopl: primary 62E10; secondary 62G30 Keywords: Geomctric distribution: Ch,'Lracterizalion: Order statistics; Contrast
i. Introduction Several characterizations of the geometric dislribulion can bc I'ourld in the literature by means o1' the independence of I11c minimum X t.,, and a contrast Xh.,, ....... X ~.,, for some 2 <~ k ~ m where X ~.,, ~.~ ... ~ X , . , denote the order statistics corresponding to i.i.d, random variables X~ . . . . . X , . X t is said to have a geometric distribution if l ) ( X I = j) = p(I -- p)J Ibr allj c ~t, ~= I0. I . . . . ~j and some i) e (0. I). It is well known that X ~.,, a n d ( X a., -- X ~.,, . . . . . X,,.,, - X , ~.,,)are independent, if X ~ has a geometric distribution. Ferguson (1965} considered the case , = k = 2. Wc give a graphical summary of subsequent results, in which the full independence is not needed, in their simplitied versions, assuming lhe support of X t to be contained in the nonnegative integers and dropping the respective conditions, the characterizations of the geometric distribution can be illustrated by horizontal and vertical lilacs. For example. Srivastava (19741 requires X ~.,, to be independent of the event I X , , . , , - X ~., = O}j.
*Correspondence address: Institute of Statistics. Aachen [Jnivcrsity of Tcchnoh)gy. 52()5() Ailchcn, Germany. 0378-3758/98/$19.00 ~ 1998 Elsevier Science B.V. All rights reserved. Pll S 0 3 7 8 - 3 7 5 8 1 9 7 ) 0 0 1 2 I - 3
H. Cohbers. U. Kanq~s/Journal olStatistical Phnming and h!lbrence 67 (1998) 187, 190
188
2. Results As indicated by dotted lines in Fig. 1 it is shown in this note that the geometric distribution can be characterized by 'diagonals', too. Moreover, under additional conditions, the independence of events corresponding to shifted diagonals turns out to be a characteristic property. Let Xt . . . . ,X,, n>t2, be i.i.d, random variables and p ~ = P ( X ~ = i ) , qi = P ( X t > i), i e Mo, and q - t = 1. Theorem I. Let supp(Xt) c I%1o, m ~ N o be a f i x e d integer, po, Pro+ l, P,,,+ ,_ > O, and let q~ = q~+ t .for all 0 <~j <<.m. I f the events [ X t.,, = m + i] and ~X2.,,~ - X t . , , = . j ] are imlependent f o r i = j, j + i, j + 2 and fiJr allj ~ ~ o , then P ( X t = ./) = !'o( I - po)i.lbr all
Proof. In the independence condition P(Xt.,,=m+i,
X2.,,-Xt.,,=j)-P(Xt.,=m+i}P{X2.,,-Xt,,,=j)
(,)
we have
..n- 1
P I X t . , , = m + i , X , ,, - Xt.,, = j ) =
q~,+~-t - q',',,+i - n P.,+iq,,,+i. n-
" "
1
n P,,,+i(q,,, +i+,-t
_n-
1
- q,,,.;+ j),
,J = 0.
.j e NI,
Xt.,, ! ! I.,. t f O°
! !
.÷
• ,0 ~£
t_ . . . . . I |
I
o°
o
,'
0 . |'
'
0
-
O
I
,• ,0
I
0° o
°
I
0
'
,0* • .'
I .
t:
,0
°"
'' .. o~
°'~
00
,°
| •• |,° l~
09
"
.:. _ _.
•
I
|
,0
--"
9 .O*
I O' .i °
I
.D
I
°
t
,'
| t
••
i I
2
m
X~..-Xu. (2 s k g nl
Fig, I, ( ' h a r a c t e r i ~ a t i o . of tile geometl'ic distribution: ( J Arnold {1980k k = 2. see Becker (19•4. p, 94) for X~. ~,, ,-,~ .\'~,,: ( ........ ~ Sreehari {19831, k = 2 {see also Alzaid el ai,, 1998k Srivaslava 119,',16L k = n. ~, ~ l~ + I. N a g a r a j a and S r i v a s t a v a (1987), !~ = 0. i, = I; ( . . . . . . . } Srivastava {1974). k = n (see also ( ; a l a n l b o s 1975. Srivastava 1981}. EI-Ncv,cihi and ( ; o v i n d a r a j u l u (1979}. N a g a r a j a a n d Srivastava (1987}: ( ' ' ' ) EI-Neweihi and G o v m d a r a j u l u 11979k k = 2, G o v i n d a r a j u l u (1980).
H, Cohhers. U. h~unps/Journal of Statistical Planning am/h!/i, rence 67 0998) 16'7-190
189
and ,!
P ( X , . , = m + i) = q',',,+ i - l - q ,,, + i.
Identity (.) with (i) i =.j = 0 a n d i = I . . / = 0 and (ii) i = j = 0 and i = 2 , j = 0 leads to the e q u a t i o n aq~-l--bq~=c
witha=q~(1-q~),
.,n(k+ !)- i
and c = no
b=
l-q~-I
(I - qo),
where k = m + i in case (i) a n d m + 2 in case (ii). T h e p o l y n o m i a l f (qk) = a q'd- ~ -- h q~ -- c has two zeros in the unit interval (0. 1), which are given by qk = q~÷t a n d qk = q,~. However, the latter leads to a contradiction, since then pk = q k - t - q k = q~,-q,~ = 0. H e n c e qk is given by q~+~, k = m + 1, m + 2. T h e assertion n o w follows by i n d u c t i o n o n j, noticing that, for some fixed .J, Eq. (.) yields ,,-,
_ ..,,-,
qm + 2j + ! ~
q',',, + i -
~lm + 2,i - -
q",, + .~ + ,
P,,, + j
,,-,
(q:,,-+ ~j-1 ~
q',',,+ j - , - q',',,+ .~ P,,, + .~+,
,
qm + 2jl,
putting i =.j a n d i = j + i. and ,,- t q,,,,
.,,.,
2i+.,
=
q,,.,
q',',, +.~, l -
q',',, + s + 2
P,,, + s
~
,.j + t
q',',, + . i - ~ -
q',',, +.i
P,,,, i +
,.,,-,
-,
(qm+2j-
_,,- ! I --
qm+2.p,
putting i = j a n d i = j + 2. T h e d o t t e d lines in the a b o v e Iigurc illustrate the case m = 0 o1" Tlaeorcm I. In it similar way we o b t a i n the following theorem, where the 'characterizing diagonals' arc 'shifted IO the right'. Theorem
2. L e t s u p p ( X i1 ~ ~ . , m e ~ , he a.li.\'ed i n t e g e r , p . c (0. I}, a n d h't q i = q[}' !
l o t all 0 ~ j <~ m + I. ! / ' t h e e r e n t s I X 1.. =il
and ~Xa.,,-Xi.,,=m
+ I jI
a r e i n d e p e m h ' n t f o r i = O. 1, a n d ![ f
~X i X , . , , - X I . , , = m + j + ,.j "~ I I . n = i~j a n d a r e i n d e p e n d e n t /i~r i = j , j + I,j + 2 a n d lbr all j ~ NN., t h e n
P(Xt =,it
= p.(I
-
pot i
.lot aU./ ~ ~,,.
Proof. P u t t i n g i = 0 and i = I in P(Xt.,, = i.X,.,,-
Xt.,, = m + 1 ) = P I X I . , , = i ) P ( X a . , , -
X t . , , - - - m + 1)
yields q,,, + z = q'~'~+ 3 By a n a l o g y with the p r o o f of T h e o r e m I, the assertion follows by i n d u c t i o n on .j.
190
H. Cohhers. U. Kantps /Journal o[Statistical Phlntting att,t btfercplCe 67 (1998) 187 190
References Alzaid, A.A., La,:, K.S., Rao, C.R., Shanbhag, D.N., 1988. Solution of Deny convolution equation restricted to a half line via a random walk approach. J. Multivar. Anal. 24, 309329. Arnold, B.C., 1980. Two characterizations of the geometric distribution. J. Appi. Probab. 17, 570 ~573. Becket, A., 1984. Charakterisierungen diskreter Verteilungen durch Verteilungseigenschaften yon Ordnungsstatistiken. Dissertation, Aachen University of Technology. EI-Neweihi, E., Govindarajulu, Z., 1979. Characterizations of geometric distribution and discrete IFR (DFR) distributions using order statistics. J. Statist. Plann. Inference 3, 85-90, Ferguson, T.S., 1965. A characterization of the geometric distribution. Amer. Math. Monthly 72, 256 260. Galambos, J., 1975. Characterizations of probability distributions by properties of order statistics II. In: Patil, G.P. et ai. (Eds.i, Statistical Distributions in Scientific Work, vol. 3. Reidel, Dordrecht, pp. 89 101. Govindarajuh,, Z.. 1980. Characterization of the geometric distribution using properties of order statistics. J. Statist. Phmn. Inference 4, 237 247. Nagaraja, H.N., Srivastava, R.C.. 1987. Some characterizations of geometric type distributions based on order statistics. J. Statist. Piann. Inference 17, 181 191. Sreehari, M.. 1983. A characterization of the geometric distribution. J. Appl, Probab. 20, 209 212. Srivastava, R.C., 1974. Two characterizations of the geometric distribution, J. Amer. Statist, Assoc, 69, 267 26t). Srivastava, R.C. 1981. On some characterizations of the geometric distribution. In: Taillie, C. et al. IEds.J, Statistical Distributions in Scientific Work, vol. 4. Reidel, Dordrecht. pp. 349 355. Sri~,astava, R.C.. 1986. On characterizations of the geometric distribution by independence of fimctions of order statistics. J. Appl. Probab. 23, 227 232.