Let X_={X_,t^1∈N},..., X_={X_, t^d∈N} be independent sequences of i.i.d. real-valued random variables and let S_t=S_+…+S_ where t=(t^1,...,t^d) and S_=Σ_?t^i-μ_i)/σ_i), i=1,...,d. A sequential sampling plan determines the way of taking one observation from one of the processes X_,...,X_, according to the previous sampled data. We show that the random sum of observations under any sequential sampling scheme is asymptotically normal. An easy application of the normality yields the classical result of sequential interval estimation of means of two polulations

Shikimi Takuhisa

Let X_={X_,t^1∈N},..., X_={X_, t^d∈N} be independent sequences of i.i.d. real-valued random variables and let S_t=S_+…+S_ where t=(t^1,...,t^d) and S_=Σ_≤t^i-μ_i)/σ_i), i=1,...,d. A sequential sampling plan determines the way of taking one observation from one of the processes X_,...,X_, according to the previous sampled data. We show that the random sum of observations under any sequential sampling scheme is asymptotically normal. An easy application of the normality yields the classical result of sequential interval estimation of means of two polulations

Shikimi, Takuhisa

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Asymptotic normality for sums along data-dependent sampling schemes

Let X_={X_,t^1∈N},..., X_={X_, t^d∈N} be independent sequences of i.i.d. real-valued random variables and let S_t=S_+…+S_ where t=(t^1,...,t^d) and S_=Σ_?t^i-μ_i)/σ_i), i=1,...,d. A sequential sampling plan determines the way of taking one observation from one of the processes X_,...,X_, according to the previous sampled data. We show that the random sum of observations under any sequential sampling scheme is asymptotically normal. An easy application of the normality yields the classical result of sequential interval estimation of means of two polulations.長崎大学経済学部研究年報. vol.21, p.29-37; 2005departmental bulletin pape

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29Asymptotic normality for sums alongdata-dependent sampling schemesTakuhisa ShikimiAbstractLet XCI) = {X1,tl,t1EN},. .., Xed) = {Xd,td, tdE N} be independent sequences of i.i.d. real-valued ran-dom variables and let St= SUI + ... +SdY, where t= (t l, ... ,td) and Si,ti= Lsi,,;,ti( (Xi,Si - Pi) / ai), i= 1, ... ,d. A sequential sampling plan determines the way of taking one observation fromone of the processes XO), ...,X(d), according to the previous sampled data. We show thatthe random sum of observations under any sequential sampling scheme is asymptotical-ly normal. An easy application of the normality yields the classical result of sequentialinterval estimation of means of two polulations.Key words: asymptotic normality; sequential sampling plan; sequential interval estima-tion.IntroductionThis note intends to prove asymptotic normality for random sums of random variables in whichthe summands are selected in a predictable manner from two or more independent sequences ofi.i.d. real-valued random variables. Let X(!) = {Xl,tl,t 1 EN}, ... , Xed) = {Xd,td,tdEN} be indepen-dent sequences of Li.d random variables with EXi,l = Pi, VXi,l = a7 E (0,00), i = 1, ...,d. For t= (t1,... ,td), let St=Sl,tl+"'+Sd,td, where Si,ti=Lsi,,;,ti((Xi,Si-Pi)/ai) (i=l, ... ,d). A sequential sam-pling scheme determines which population to be observed based on the previous observations,and therefore is represented by a sequence of Nd-valued stopping points (rn) satisfying apredictable condition. Asymptotic normality of Sr. is shown by a straightforward application of amartingale central limit theorem (Brown(197l), McLeish (1974) , Chow and Teicher (1988)or Billingsley (1995)). This result will be applied to sequential estimation of the defference ofmeans of two populations.2 NotationThe set N is the set of positive integers, N* the set N U {a} and d is a positive integer. Through-out, the set N*d will be denoted by r, the set N*d - {O} by I, where 0 = (0, ... ,0). Ele-30ments of r are denoted by the letters s,t, ... If s= (SI, ... ,Sd) and t= (tl, ...,td) are elements in rands i ~ t i for all i = 1, ... ,d, then we say that s is less than or equal to t (or t is greater than or equal tos). This relation between sand t is denoted by s~ t, and this relation forms an partial order in theset r. If s~ t and s* t, we write s< t. For t= (tl, ...,td ) Er,we set I tl = t l + ... + t d• The directsuccessors of t E r are the elements u E r such that t < u and 1 u - t I = 1. The set of direct succes-sors of t is denoted by Dt• The direct predecessors of tE I are the elements s E r such that s < tand 1t - s 1= 1. If tn are elements in I, we will write tn ----+lX1 to express that I tn1----+ 00, and we usethe symbol I* to denote the set I U {lX1}. The order structure of r is extended to I* by setting t<lX1. We agree by convention that 1 lX11 = 00.Let (Q, gT, p) be a probalility space. A filtration {gTt, tE r} is a family of sub- (J -algebras of gTsuch that gTs C 'fftfor s < t. A function T: Q ----+I* is called a stopping point if {T = t} E gTt for all tE r. For a stopping point T relative to {gTt, t E r}, gTT is the class of subsets F in gT tx:i = (J (gTt,tEr) such that Fn {T= t} E gTt for all tEr. A sequence of stopping points r = {rn, nEN*} issaid to be a predictable increasing path if( i ) ro = 0 a.s.,(ii) rn+1 EDr• a.s.,(iii) rn+1 is gTrn -measurable.A predictable increasing path can be thought of as a sampling strategy. Namely, it provides ateach stage of the observation a way of deciding which population to be observed, depending onthe previous data. A predictable increasing path naturally arises in sequential sampling or mul-tiarmed bandits problems (Cairoli and Dalang (1996)). Clearly, in dimension 1 (d= 1), the con-cept of a predictable increasing path is of little interest since in this case there is only onepredictable increasing path, namely, the deterministic increasing path defined by rn = n for all nE N*. For all predictable increasing paths r, we agree by convention that roo = l><J. We will writegTJ instead of gTr•. For precesses X indexed by I and predictable increasing paths r = {rn}, XJdenotes X r•. If {Xt , tE n is a martingale relative to {gTt, tE nand r is a predictable increasingpath, then {XJ, tEN} forms a martingale relative to the filtration {gT~, nEN}.3 The resultLet XCi) = {Xu, ti EN} (i = l, ... ,d) be a sequence of independent and identically distributed real-valued random variables. We assume that X(j), ,X(d) are independent sequences. Further sym-bols and assumptions are as follows: for i= l, ,d,gTi,o={0, Q}.31For t= (t1, ... ,td) EI*, letSt= SUI + +Sd,t d ,fiTt= fiTUl V V fiTd,fd.(3.1)(3.2)Let {fiTt ,tE I*} denote the filtration defined by (3.2) and let r denote any predictable increas-ing path relative to {fiTt, t E I*}. Clearly S = {St, tEn is a martingale (note that the sum St hasalready been normalized) , and hence so is {Sf, j E N} as was mentioned in the preceeding sec-tion. DefineY/ = S/ - Sil (j?:: 1),Z~j=Y/ l,[n (j=l, ... ,n).(3.3)(3.4)It is evident that the triangular array {Z~ , fiT/ ' j = 1, ...,n}, n E N} is a martingale difference ar-ray. Thus applying the martingale central limit theorem yields the following result.Theorem 3.1. Let S= {St, tE I} be the process defined by (3.1) and r any predictable increasingpath. Then we have~~N(O,1) (l~n-+oo), (3.5)where ~ denotes the convergence in distribution.proof By the martingale central limit theorem, it suffices to show that as n-+ oon1:E[(Z,;j)21 fiTJ-l] ~ 1, (3.6)j=ln1:E[(Z~j)21{IZkl~d] -+ 0,j=l(3.7)where ~ denotes the convergence in probability and 1A stands for the indicator function of theset A E fiT. The latter condition is called the Lindeberg condition. We assume without loss ofgenerality that f-Li= 0 and (Ji= 1, i= 1, ... ,d. We shall denote by P t the set of direct predecessors oftEl. For j?::l, we haveIt follows from the predictability of r that {n= t} n {rj-l =s} E fiTs. We see therefore the last dis-play is equal to32L L f{=t}n{. =s} (S/ -S/-1)2dPt:ltl=jsEPt rJ rJ-l= L L P(n=t, rj-l =S)EXY,1 = 1t: It 1 =j SEPtThe last equality follows from the independence of l{n=t,n-l}=s} and (St-Ss)2(note that {n=t}n {n-l =s} E grs)· Consequently, we obtain thatn 1 nL E[ (ZJj) 21 gr/-IJ = ~ L E[ (Y/ ) 21 gr/-IJ = 1.j=l nj=lThe proof of (3.7) will be done similarly. Since for each £ > 0= L L P{n=t} n {n-l =s} f (St-Ss)21{ISt-Ssl~£.;n}dPt: It 1 =j SEPtSince the integral of the last display converges to 0 as n --+ 00 ,n,L E [ (ZJj) 21 { Iz~ I~ d ]J= 11 n=-L L L P({n=t}n{rj-l=S}) f(St-Ss)21{ISt-Ssl~£.;n}dP--+O(n--+ oo ).nj=l t:lt!=jsEP tThis completes the proof of the theorem.4 An application to sequential estimationIn this section we will consider an application of the result in the previous section to sequentialestimation for the difference of means of two independent populations, with the specified sam-pling plan introduced by Robbins, Simons and Starr (1967). Before proceeding to discussingthe application, we need to prepare a simple extension of Anscombe's theorem to martingalecases. Anscombe's theorem asserts that randomly stopped sums are asymptotically normallydistributed. The central idea used in the proof of Anscombe's theorem is the notion of uniformcontinuity in probability. A sequence {Yn, n E N} of random variables is said to be uniformlycontinuous in probability iff for all £ > 0, there exists an 0> 0 for whichsup p( max IYn+k- Ynl ~d < E.n {O~k~na} .33(4.1)Anscombe's theorem is obtained by using the following lemma, which will be also applied toprove its generalization to the martingale case stated below. The proof of the lemma can befound in Ghosh, Mukhopadhyay and Sen (1997).Lemma 4.1. Suppose that{ Yn, n E N}is uniformly continuous in probability. Let 'fb (b > 0) be apositive integer-valued random variable such that 'fbib converges in probability to it constant c E (0,00 ). If Y n converges in distribution to a random variable Yas n ----+ 00 , then Y r bconverges in distri-bution to Y as b ----+ 00 •An adapted process {Yn} is called an L2-martingale if it is a martingale and E Y~< 00 foreach n.Theorem 4.2. Suppose that{YnJFn, nEN} is an L2-martingale with mean o. Let Zn= Yn-Yn- 1(n ~ 2), Zl = Y 1and assume that E~ is bounded. Let 'fb (b > 0) be a positive integer-valuedrandom variable such that 'fbib converges in probability to a constant cE (0, 00 ). If Ynl jii con-verges in distribution to N(O, 1) as n ----+00, then as b ----+00?} ~N(O,1),'V 'fb~} ~ N(O,1),(4.2)(4.3)Proof It suffices to prove (4.2) because YrJJbC= Yrb('fblbc)1/2/~. In proving that YrJ~converges in distribution to N(O, 1), by virtue of Lemma 4.1, we need only verify the uniformcontinuity in probability of Ynl jii. Let us begin with showing that {Ynl jii, n EN} is uniformlyintegrable. Since {Zn} are uncorrelated and EZn= 0 (n E N), it follows from the boundedness ofE~ that(say) ,whence suPnE[ (n- 1/2Yn) 2J < 00. It follows therefore that Ynl jii is uniformly integralbe.Writing p(0) = 1- (11 (1 + 0)) 1/2 for 0 > 0, we haveas 0 ----+ O. (4.4)For n~ 1, k~ 1, we have34I;n:~ - ~I = I;n:~ - J:~k+J:~k - ~I~~I Yn+k~ Ynl + 1(n:l' ~111~1~Jnl Yn+k- Ynl +-(n:krJ I~IIf 0> 0 and k~ no ,It follows thereby thatsup p[ max 1 Yn+k - Ynl >£]n;'l O~k~na In+k In~sup p[ max 1Yn+k - Ynl >£In/2J+sup p[p(o) I~I >£/2].n;'l O~k~na n;'l 'Vn(4.5)From (4. 4) and (4. 5) it remains only to show that the first term of (4. 5) converges to 0 as 0--+O. It should be noted that for each nEN, (Yn+k- Yn,9Tn+k,kEN) is a martingale. Hence, us-ing the maximal inequality for martingales, we obtainp[ max 1 Yn+k - Yn1 > £In/2] = p[ max I Yn+k - Yn1 > £In/2]O~k~na O~k~[naJ4 I 124 I n + [naJ 12~-2E Yn+[no] - Yn =-2E L Zjn£ n£ j=n+ 1whencesup p[ max 1 Yn+k - Ynl > £In/2J ~4~0 --+ 0n;'l O~k~na £(0--+0).This completes the proof of the theorem.By this theorem, if {SJ, n E N} is the process defined by (3.1), thenS~ SN(O 1)JbC "35provided that 'fbi b converges in probability to a positive constant c.Let us introduce two independent sequences of i.i.d. real-valued random variables, say Xco= {XI,t1,tIEN} and X(2) = {X2,t2 ,t2E N}, where ,uiEXi,1 and O<a7=VXi,I(i=1,2) areunknown. The problem is interval estimation of ,u= ,ul- ,u2' Set :Jfr t= gfl,t1 V gf2,t2, for t= (tl ,t2),where g-;',ti= a (Xi,si,Si~ti) and gfi,O= {0, Q}. Define, for t i;:;2,whereXi,ti=(1lti)L.si~tiXi,siCi=1,2). Let r be a predictable increasing path withd=2. Weintroduce a stopping rule relative to {gfJ ,nEN}, which is one of the stopping rules proposedby Robbins, Simons and Starr (1967): for b> 0(4.6)where m;:; 2 is the initial sample size of observations on Xl and X2. Defining three more or lessequivalent stopping rules including (4.6), Robbins, Simons and Starr (1967) studied the se-quential interval estimation of the difference of the means of two populations, where the vari-ances are unknown. More precisely, with a sampling rule and several stopping rules Robbins,Simons and Starr (1967) found a approximate confidence· interval for ,u of fixed width and ofpreassigned coverage probality. Their procedure consists of (i) a sampling scheme that tells usat each stage whether the next observation, if needed, would be an Xco or X(2) and (ii) a stop-ping rule, e.g., (4.6). Their sampling strategies is as follows: we take m(;:;2) observations onXCI) and X(2) to begin with. Then, inductively define r by_ + I(d-l ~ Urk-l) + I(rh-l> Urk-l)rn-rn-l el 2 '" V e2 2 V 'rn-l r~-l rn-l r~-l(4.7)where el = (1 ,0) , and e2 = (0 , 1)' . Hereafter r denotes the predictable increasing path de-fined (4.7). With the sampling scheme (4. 7) and the stopping rule (4.6), we will show thatXl, rL - X 2,r;. - ,u has asymptotically normal distribution with mean 0 and variance b- I for largeb. This consequence implies the result of Robbins, Simons and Starr (1967).For rand 'fb' they showed that as b-+ ooanda.s., (4.8)36Set for t= (t l , t2)a.s. (4.9)U sing Theorem 3. 1, (4 . 9) and Theorem 4. 2, we haveand so by (4. 9) and (4. 10) we haveBy (4.8) and (4.9) we haveLikewise, as b --+002ri/ --+ a2 (al +a2) .Therefore applying the central limit theorem and Anscombe's theorem yieldFurther, note that by (4. 8) as b --+ 00(al +a2hL --+ 1-a(rb(4.10)(4.11)(4. 12)(4.13)(4.14)Since by (4. 12) and (4. 13) /b (Xi, rL - PI) converges in distribution, it follows from (4. 14) thatas b --+0037Consequently, by (4.11) we obtain the following asymptotic normality:This implies the result of Robbins, Simons and Starr (1967).References[l]Billingsley, P. (1995). Probability and Measure, 3rd ed., ]. Wiley & Sons, New York.[2]Brown, B. M. (1971). Martingale central limit theorems, Ann. Math. Statist., 42,59-66.[3]Cairoli, R. and Dalang, R.C. (1996). Sequential Stochastic Optimization, J. Wiley & Sons,New York.[4]Chow, Y. S. and Teicher, H. (1988). Probability Theory: Independence, Interchangability,Martingales, 2nd ed., Springer-Verlag, New York.[5]item Ghosh, M., Mukhopadhyay, N., and Sen, P.K (1997). Sequential Estimation, ]. Wiley& Sons, New York.[6]McLeish, D. L. (1974). Dependent central limit theorems and invariance principles, Ann.Prob, 2,620-628.[7]Robbins, H., Simons, G., and Starr, N. (1967). A sequential analogue of the Behrens-Fisherproblem, Ann. Math. Statist., 38,1384-1391.

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Asymptotic normality for sums along data-dependent sampling schemes

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