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1 Math-Net.Ru All Russia mathematical portal H. G. Tucker, A Estimate of the Compoudig Distributio of a Compoud Poisso Distributio, Teor. Veroyatost. i Primee., 1963, Volume 8, Issue 2, Use of the all-russia mathematical portal Math-Net.Ru implies that you have read ad agreed to these terms of use Dowload details: IP: November 22, 217, 11:8:25
2 A Estimate of the Compoudig Distributio of a Compoud Poisso Distributio 211 НЕКОТОРЫЕ СВОЙСТВА РАЗЛОЖЕНИЯ ВОЛЬДА СТАЦИОНАРНЫХ СЛУЧАЙНЫХ ПРОЦЕССОВ Е. А. РОБИНСОН (УПСАЛА, ШВЕЦИЯ) (Резюме) Оснвные результаты рабты (теремы 11 13) касаются впрса представимсти величин x t + a (являющихся наилучшим пргнзм значений x t + a стацинарнг в ширкм смысле прцесса п значениям x s, s^t) в виде ряда с *7+а ~ 2 k s X ts> so где кэффициенты k s удвлетвряют услвию: 2 I k s I 2 < Предварительн привдятся нектрые свйства пследвательнстей {w t }, 2 I w t\ 2 AN ESTIMATE OF THE COMPOUNDING DISTRIBUTION OF A COMPOUND* POISSON DISTRIBUTION* HOWARD O. TUCKER 1. Statemet of the Problem. A radom variable X is said to have a compoud Poisso distributio if there exists a distributio fuctio G for which G (+ ) ad such that oo P{X }\^ e~ K (X /\)dg(x) (1) for, 1,2,.... The distributio fuctio G is called the compoudig distributio^ The compoud Poisso distributio is kow to be idetifiable i G, i. е., if oo oo ^ e~ K (X /\) dg x (X) ^ e~? (X /\)dg 2 (X) for, 1,2,..., the G x G 2 (see [5]). Thus it makes sese to cosider the problem^ of estimatig G whe oe has idepedet observatios o X. Let Х ъ X 2,..., X,... deote a sequece of idepedet, idetically distributed radom* variables with commo compoud Poisso distributio (1). The problem cosidered here is that of costructig a fuctio G N (X) of \Х Ъ..., X N, A,, for every positive iteger N ad every X >, which coverges with probability oe to G (X) as N-> oo at all values of X at which G is cotiuous. The explicit formula for G v (X) ad the proof of cosistecy is give i sectio Notatio ad Prelimiary Results.' I this sectio we give some ecessary backgroud o the Stieltjes momet problem ad two lemmas which will be eeded i the costructio of the estimate. First we establish some otatio. The letter N will always deote the size of the sample. For, 1, 2,..., we shall deote p () P [X ] ad 1 N il * This research was supported i part by a research grat (R. G. 843) from the Natioal Istitutes of Health. 6*.
3 ! 212 H. G. Tucker i. е., p N () is the proportio of times X i the first N observatios. It is clear that EPJV ( N ) ~ P ( я ) * a^ г ac * N а ш * PN ( л ) ~* P( N ) as T V o o with probability oe for every /г. Further, if a (а ъ a 2,...,a,...) is a ifiite sequece of real umbers, the we shall deote Д П (a) I a i+]? I /, Д' 1» (a) a i+}+1? /. for, 1,2,..., where a 1. For coveiece, ay symbol with subscript cosidered equal to oe. A sequece of real umbers p, (fx x, ji 2,.. ;\ р, л,...) will be called a Stieltjes sequece if there is a probability distributio fuctio T 7 such that F(-\-) ad i $ x4f(x) zero will be momet for л, 1, 2,... We shall eed to refer to the followig result which characterizes Stieltjes momet sequeces (see [4], theorem 1.3): A ecessary ad sufficiet coditio that p be a Stieltjes momet sequece is that for, 1, 2,... // [i /s, a Stieltjes А,гМ> ad A >(II)> (2) momet sequece, the for every iteger, the determiats А Я ad A^ (p) are both positive or both zero (because of our assumptio that F (+ ) ). // they are both zero for the first time whe, the they are both zero for all > /г, ad F is determied ad discrete with positive poits at which jumps occur. We shall also eed to refer to the so-called algebraic momet problerri: give 2 real umbers p.,..., p 2 _ v to fid distict real umbers х ъ..., x ad positive reals Pi> P2> i P such that 2 Pi x^^k> k > 1 '"-> 2/г-1. I a paper by Mammaa (see [3]), a proof is give of a theorem which states that a ecessary ad sufficiet coditio that the algebraic momet problem have a solutio is that A; (V>) > for < i < 1. This solutio is ecessarily uique. The values of x lt..., x are obtaied by fidig the distict roots of 1 X 2 x Ho Hi M<2 Hi H2 Из Hvi+i. MTI-1 H 2 «I The positive umbers p l f..., p are the obtaied by solvig 2 p i x^ H*> 6, 1,...,л-1. We shall eed the followig two lemmas. Lemma 1. Let С я + 1 be a closed, covex set i ( + i)-dimesioal Euclidea space (i+i) f a / z d i e t c be the projectio o the first coordiates ito E {). Assume that C
4 A Estimate of the Compoudig Distributio of a Compoud Poisso Distributio 213 is closed. Let f be ay fuctio defied o C such that for every x (x b..., x ) EE C the (x u..., x, f(x)) e C +1. Fially, assume that for every (x lt..., x ) o the boudary of C there is oe ad oly oe real umber х п _^г such that (х ъ..., x, x ^_ ± ) &l ^ The f is cotiuous o the boudary of C. Proof: Let x be a poit o the boudary of C, ad assume to the cotrary that / is ot cotiuous at x. The there exists a sequece of poits {x m } i C ad a e > such that x m -> x ad \f (x m ) / (x) > 8 for all m. The for] ifiitely may values of m, f (x m ) > / (x) + e or / (x m )< / (x) e. We may, without loss of geerality (ad for simplicity), assume that /(x m )^/(x) + e for all values of m. Sice C l1 _ Ll is covex, this meas that the lie segmet L m edpoits are determied by the two poits (x, /(x)) ad (x m, /(x m )) is i C ^_ v deote the lie segmet whose edpoits are (x, f (x)) ad (x, / (x) + o)- Sice for all m, ad sice C +1 whose Let L ^mdc _^x is closed, this [implies that L С C + V This however cotradicts the last assumptio i the hypothesis, ad the lemma is proved. Lemma 2. Let p. ad v be two Stieltjes momet sequeces such that А л (p.) A (v) ad A^ (pt) A^1} (v) for, 1, 2,... The \i v for all. Proof: Case (i). Suppose Д л (р,)> for all. I this case it is easy to prove by iductio that p v for all. Case (ii). Suppose A m (ji) > ad A m + 1 (p.) for some m^l. The, i the iductive maer used i case (i) oe ca prove that \x ~ for 1 ^ < 2m + 3. Hece by the above theorem givig ecessary ad sufficiet coditios that a sequece be a Stieltjes momet sequece, we kow that both p. ad v are Stieltjes momet sequeces of distributios with exactly m + 1 poits of icrease. However, by the uiqueess property of the theorem proved i Mammaa's paper quoted above, there is oly oe such distributio which has the first 2m -f- 1 momets p v, Hi v l f..., H 2m _ _ 1 v 2m _ _ r Hece»x v, ad the lemma is proved. 3. Costructio of the Estimate of G. I this sectio a estimate G N (K) of G (Я) is costructed ad its cosistecy is proved. This will hold for arbitrary G for which G(+)[. For ay sequece of real umbers a (а ъ a 2,..., a,...) we defie a trasformatio 7, Та (Txa, T 2 a,..., Г а,...) as follows. If A (a) > ad A^ (a) > for all, we defie Т л а a for all. If, however, there is a iteger m such that A 1 (a)>,..., A m (a)> ad we the^defie Ai 1 * (a) >,..., A<J) (a) > A m + 1 (a)< or A ^ d X O, Г я а a for < < 2m + 1 ad the defie Т д а for > 2m + 1 so that Д ш + / г (Га) A^(^ a ) f o r kl,2,... Such values of T a certaily do exist, because if we fid a distributio fuctio which has exactly m-\- 1 poits of icrease ad whose first 2m + 1 momets (after the zero-th) are а ъ a 2,..., «2 m+i» ^ e ' ^or «> 2m+ 1, oe value ^of Г п а is the -ih momet of this distributio fuctio. By lemma 2 we obtai that \T & is uiquely defied for /2>2m + l. The followig lemma is of fudametal importace. Lemma 3. Let ^ (u. x,..., u.;,...) be a Stieltjes momet sequece, ad let {p } be a sequece of sequeces of real umbers, where we deote \i (ц 1,..., \i.,...).
5 214 H. G. Tucker Suppose that \i i -» \x i as -» oo for i 1, 2,.... The T i^ -> as л- -> oo /or ееп/ i (Note: р д is ot ecessarily a Stieltjes momet sequece, but Tp always is.) Proof: Case (i). Suppose A (p) > ad AJ^ (p) > for all. For ay iteger m ad for all sufficietly large values of л, АДр я) > ad AJ 1 * (р Л) > for <t<m. Hece, for these values of л, which establishes the lemma i this case. Case (ii). Suppose that, for some m, A i (p) >, AJ 1 * (p) > for < i < m ad A f. (p) for all / > m. By the same proof as i case (i) we obtai that for sufficietly large л, T P Pi, ~* a s -* for <i<2m + l. The oly problem is to prove that T y. -> ^ as л -» oo for all / > 2m + 1. For ay o-egative iteger r ad ay sequece of real umbers a we may write A r + 1 (a) ad A^ 1(a) i the form A r+i ( a ) A r ( a ) a 2r+2 + fr ( a i > - fl 2T+l)» ( a ) A ^ (a) + *,(*!.. Sr + 2 )> where / r ad g r are defied by the equalities i which they are ivolved. Sice ^ 2m+2, -* H 2 m + 2 R a s " - > o o, sice [A m (p r t) -> A w (p) > as л o o, ad sice A m + 1 (р Л) -> A m^.j_ (p) as л -» с, it follows that H 2 m + 2 lim ( - / w ( H 1,,. -. ^ 2 m + 1, ) / A m ( p J ) lim (- f m (u. 1? u, 2 m + 1 )/ A m fti)). But for large values ot л, 2/л +2 «* Я -//поч Л> H 2 m + 1,«)/A m(la r t) or ^2m-)-2, * Hece Г 2 т + 2 р я jx 2 m + 2 as л - oo. I the same way oe ca prove T 2m _j_ 3 p rt -» Р 2/Г ц_ 3 as л->. For coordiates after 2ra + 3 we prove the lemma (i this case) by iductio. Suppose we have proved that Tp ^ as л * oo for < i < 2m + 2& + 1, where /5 > 1. Sice ДЭД_ л+1 (p) ad A ^ (p), it follows that g m+k ([i lf..., H 2 m + 2^ + 2 ), ad, if \1 2т + 2к + 2 фр 2т + 2к +*, the (px,..., \i> 2 m + 2 k + v H 2 m + 2^ + 2 )<. Now, g m J r k is a cotiuous fuctio i its 2m + 2fc + 2 argumets^ad is a secod degree polyomial i its last argumet which has a double positive root whe the values of the first 2m + 2k + 1 argumets are the first 2m terms of a Stieltjes momet sequece v for which A^^(v). This positive root is a cotiuous fuctio i the first 2m + 2& + 1 terms of such Stieltjes momet sequeces; call this fuctio h(v lt... - v 2m+2M-i)- T h u s Ь У eductio hypothesis, if A m -H(p )< or А ^ ^ Д О for all sufficietly large values of л, we have lim h (Тф п,..., T 2 m p ) h (p x,..., и- 2 Ш + 2 / г + 1 ) M^/+^s* Я-*-СО It also might occur that for ifiitely for all values of ) that may values of л (ad for simplicity we assume Al (fi) > Д т+* <«*»> > > A l' <«) > A + * > '
6 A Estimate of the Compoudig Distributio of a-compoud Poisso Distributio 215 but The (,i )< or 4 ( '» W l W<. ^2m+2^+2^«~/m+/e(Hi,» ^2^+2^+1, Jv/m+& Obi)' To this occurrece we shall apply lemma 1. Let C t deote the set of all i-tuples i E^ which are the first i terms of a Stieltjes which is closed i the relative topology of E^ momet sequece. The set C t is a covex set with the origi deleted. Further, the poit (Hi. > H 2 m ) is a boudary poit of C 2 m + 2 k + v because..., H 2 W + 2 &. и') S ^ ^2w+2^-f 1 ^ o r а п У I х ' Ф H 2 /+2fe+1* ^lsd, Г 2ш4-2^+2^' s u c h t h a t o w ' * h e r e i s o ^ (Hi,, 2i + 2 k+v ^гт+г^+г! 1 ) e ^т+гй-н* We ow have the hypotheses of lemma 1 satisfied, ad thus o e u m ber, Tzm+zk+bP l i m (~ fm+k (&1,п> * * ' ^2т + 2 к+1,п)1^т+к W) M^m-f 2*4-2 l i m -»oo -^oo 1 1» Sice for each, I^2m+2^+2,M o r ~~ fm+k^u' *» l X 2m+2^+i,/z^Am+^ amely each with the proper pre-coditios i its subsequece, it follows that T 2m+2k+2U- ~* ^2/14-2*4-2 a s ~* * This same type of argumet works for provig that T 2m+2k _^3\i И 2т 4-2 &4-з as л с by cosiderig the fuctios f m j rk j rl ad g m + k respectively. This completes the proof of the lemma, Now let рдг {\ p N ()/p N ()}, ad let {m N } be a sequece of odd itegers for which m N -> oc as N. Let K N (x) deote the distributio fuctio (obtaied as the solutio of the algebraic momet problem) whose first m N momets are T \i N, ^ < m N. We ext defie Я for every positive X. This is our estimate of G, ad we ow prove that for almost all X it is a cosistet estimate. Theorem. At all values of X > at which G is cotiuous, G N (X) -> G (X) as N -* with probability oe. Proof: Let us deote The equatio (1) implies that X H(X) ^ e~ x dg(x). \p() ^ X dh(x) for, 1, 2,.... Now p() H () >, so H (X)/p () is a probability distributio
7 216 Г. Г. Тьюккер i X. Let К(Х) Н (Х)/р (). It is clear that the power series is absolutely coverget for zj<l. Hece, by a well-kow theorem (see [1], page 176) it follows that К (X) is uiquely determied by its momets, these momets beig {\ p ()/p (),, 1,...}. By lemma 3 ad the fact that m N oo as N oo it follows that 1% N () ->\p ()/p () u. () as Noo with probability oe for each fixed, и е., CO X dk N (X) -> ^ A/ 1 d/c W as -> oo. Sice /((A,) is uiquely determied by its momets we obtai from the Frechet Shohat theorem (see [2], page 185) that at every X at which К is cotiuous, K N (? -» К (X) as iv *-> oo with probability oe. Hece рдг () at all X at which Я is cotiuous. We easily ote that G(A) ^ e x dh(x). By the Helly theorem, at all X at which Я is cotiuous, (Я) -+ H (X) as N -> с, with probability oe, G N (X) p N () $ e T dtf* ( T ) - ^ e x dh (%) G (X) as iv -* oo with probability oe. Now G ad Я both have the same poits of cotiuity. Thus the theorem is proved. Uiversity of Califoria, Riverside Пступила в редакцию 2Л.61 REFERENCES [1] H. Cramer Mathematical Methods of Statistics, Priceto Uiversity Press, Priceto, N. J., [2] M. Loeve, Probability Theory, D. Va Npstrad, Priceto (Secod Editio). I96. [3] C. Ma mm a a, Sul problema algebraic dei momet!, A. Scuola, Norm. Sup., Pisa, 8 (1954), [4] J. A. Shohat ad J. D. T a m a г к i, The Problem of Momets, Mathematical Surveys No. I, Amer. Math. Soc, New York, T943. [5] H. Teicher, Idetifiabilitv of Mixtures, A. Math. Stat., 32 (1961), ОЦЕНКА ВЕСОВОЙ ФУНКЦИИ СЛОЖНОГО ПУАССОНОВСКОГО РАСПРЕДЕЛЕНИЯ Г. Г. ТЬЮККЕР (РИВЕРСАЙД, КАЛИФОРНИЯ) (Резюме) Распределение верятнстей случайнй величины X называется слжным пуасснвским, если с О где я1, 2,... и G (X) функция распределения (весвая функция) такая, чт G(+). Пусть Х ъ..., X N взаимн независимые случайные величины, пдчиняющиеся слжнму пуасснвскму распределению. В рабте устанавливается связь прблемы мментв с задачей ценки весвй функции G (X) и стрится алгритм, пзвляющий кнструирвать такую выбрчную ценку G N (X), ктрая зависит лишь т Х ъ X N и Х\ если N -> т G N (X) t с верятнстью единица, слаб схдится к неизвестнй весвй функции G (X).
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