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ASymptotic Distribution of Sample Size for Certain Methods of Sequential Sampling .L. L. Lasman a.nd E. J. Williams ! ~ ... l' f. Instiillte of Sta:ti.stics,.i Mimeo Series No. 203 June, 1958 k· • , t 4 _ TABLE OF CONTENTS Ohapter I II ... INTRODUC TION • • • • • LITERATURE REVIDl o • • • • • 1. 2. Acceptance Sampling • • • • • • • • • • • • Estimation of p in a Binomial Population •• 3 • Es timation of f.l. in a Normal Population • 4. Asymptotic'1)leory. • • • • • • , • A Recent Application. • • • • • • • • nI SEQUENTIAL STOP RULE9. • • • • • Rectangular Stop Rule • 2 • Linear Stop Rule. • • • • • 3 • Number of Paths to a Line • 4• Hyperbolic Stop Rule. • • IV • oJ . .. .. .. .. . . ....... .. EXTENSION OF RFSULTS • • • • • • • • 0 • • • • • e 1. 'General Function in, Two Dimensions. 2. Functions in s Dimensions • • • • v Sl.1MMARY • • • BIBLIOGRAPHY • r . 0(1.0 It • • • •••• 9.0 ••••• .. , , 1 7 12 14 18 20 • 1. r "",.,'" ' ., 20 28 . 39· 46 ,2 ,2 ,4 ,9 61 . Chapter I INTRODUCTION World War II stimulated the imagination of many great people and this resulted in new ideas that were destined to change the ways of the world. One of the more important statistical ideas was in connection with sequen tial theory. To say that sequential analysis has become important as a statistical tool would be to understate the obvious. Almost every practicing statistician has used sequential methods in one form or another. A dictionary definition of the word Itsequentialtt is: IlSucceeding or follow::ing in order. It of in statistics. And exactly in this sense should it be thought Not every sequential experiment was planned with sequential theory in mind, but any experiment whose observations are dependent on the previous observations, are taken one at a time, or taken in more than one group is a sequential experiment. Many people have been using sequential methods all the time withou t knowing it. These were common-sense procedures such as engineer- ing inspectors used when a first sample drawn was not decisive. Sequential theory makes precise these common-sense procedures. E:xamples of sequential sampling can be found in many fields of application. Industrial concerns use sequential methods extensively, and people in agriculture may soon find uses for the method. The first important departure from the fixed-sample-size approach came in the field of industrial quality control, where double-sampling methods were used. Double-sampling schemes have been developed to solve many • 2 industrial problems, and the extension to sequential sampling as it is known tDday is essentially an outgrowth of these schemes. But even in industry, sequential sampling is not confined to its role in quality control inspection. situations. It is used in many decision-m~ing The continual collection and ana:qsis of data gradually remove doubts existing in many problems. This leads one into the field of sequential experimentation in which one experiment helps the experimenter decide on the procedures for the next. Work in developing design of sequential experiments has only recently begun (Fisher, 1952, and Box, 1957) • However, sequential. techniques are not always practicable. If the time between successive samples is long, and conditions have changed drastically, the initial results may be unusable. difficulty of compll ta tion may be too great. In other cases, the Or yet, to use an often quoted situation, we may be experimenting with the growth of trees, and it may take years for a tree to mature. In this case it would be ridiculous tD grow one tree first, see what happened, plant another one, etc. There are, of course, some instances where sequential sampling is necessary" For example, in' obtaining data on a certain illness Which is not prevalent, the experimenter must wait until the patients go to a hospital for trea:tment, one by one, to collect these data. A main advantage of' sequential procedures is that when they are applicable, they require, in general, fewer observations than the more standard approa.ch in obtaining a desired result. Thus the actual cost .. 3 of experimentation can be greatly reduced and much of the exper;1menters I time saved 0 Most of what has been done up to now concerns seqaential methods as applied to the general' area of hypothesis testing 0 In more recent ;rears, a new look has been given at these methods, and the problem of estimation by means of sequential sampling has been approached. Much of this work has dealt with the estimation of p in a binomial population, and there is some on estimating the mean of a normal population and the difference between two means. This dissertation is concerned mainly with the problem of estimation using sequential sampling. One of the examples' of the theory which is developed concerns the estimation of the difference between two means, so that this estimate will have a specified degree of precision 0 More generally, this dissertation is concerned with finding the asymptotic distribution of the sample size required to reach a certain fixed boundary when the sampling is of a sequential nature. The general theory will consider s populations mixed in unknown proportions. Observations from this mixture will be drawn one at a time, and can be classified correctly 'by population at that time. When two populations are concerned, we get the usual binomial situation, and most of the work of this thesis will deal with this case. "The asymptotic distribu- tion of the sample size is obtained when the boundary is of the form f(:x:t, 42' 000, X s ) ... 0 where x.J. is the number of observations from the ith population and 4 where f has derivatives with respec t to darivativas are of the same sign ~, ••• , x s ' and all the 0 It is shown that the problem of estimating the difference between two means with a specified degree of precision yields the boundary, 1 11 -+--.,. x c , 2 a rectangular hyperbola having the desired properties, and the distribu tion of the sample size for this case is derived. Another example of a boundary is a negatively sloping line in tne ~,x2-p1ane and again the asymptotic distribution of the sample size is given. When the boundary does not have the required properties, the gener~l formulae givan cannot be applied. For the case where the boundary is formed by the two straight lines exact formuJ.ae for the mean and variance of the sample size are obtained. • Ohapter II LITERATURE REVIM In this review, much of the work in sequential theory will be discussed, and the cases where the expected value and variance of the sample size was derived will be pointed. out. In a few cases the asymp- totic distribution of the sample size was obtained, and the conditions will be given for these particular cases. Work in sequential methods, whether it be hypothesis testing or estimation, is of little value until there is some idea of what the sample size will be. n .1. Acceptance Sampling In Ohapter I it was stated that sequential analysis was an outgrowth of World War n'. This is true but for one notable exception. Dodge and Romig (1929) really had the first sequential test procedure when they constructed a double-sampling procedure. Their idea was that the decision whether or not a second sample should be drawn depends on the outcome of the obserwtions in the first s~ple. The advantage of this and other schemes that were to follow is a reduction in inspection costs j for the expected value of the number of observations where samples were taken in more than one group was less than the number necessary in a, single-sample inspec tion plan. In another attempt to reduce costs, Ba.rtky (1943) applied a multipIe sampling technique to estimate the fraction p of defectives in an infinite population. * His procedure is to take an initial sample of nO An excellent review of the literature together with a. rather complete bibliography is found in Anscombe (1953). This chapter is based to a. large extent on that pa.per. 6 units and all additional samples are to be of size n. If the number of defectives in the initial sample is at most c, the lot is accepted. this number exceeds c+k, the lot is rejected. is taken. In general, if, after taking I' If Otherwise the second sample additional samples of size n, the total number of defectives equals e+r, the lot is accepted. If this total exceeds c+r+k, the lot is rejected. Bartky proved that the process terminates with probability one, gave a general formula for the probability of acceptance and for the " expected sample size required, and gave these values more explicitly under 10 different conditions on his sampling scheme. 'What is known today as Sequential AnalYsis was begun about the same time in the United States as it was in England. The pioneers of this work were Wald in this country and Barnard in England • Because of the secrecy connected with this classified work, it was only near the end of the war that any theory was published. However, W'ald did submit a restricted report to the National Defense Research Oommittee in September, 194.3. In this report, W'ald devised his sequential prob- ability ratio test. Somewhat later 'Wald (1944) published a general theory of cumulative S'UnlS which gave the operating characteristic curve of any sequential probability ratio test. In May of 1945, after the restricted classif1cai!;Lon was removed, Wald published the material of the report together lft-eli'the new ad.. -- vances he had made since 1943, in the article, taBequenti~Tests of \ Statistical Hypotheses ti , (1945). book, Sequential ,Analysis (1947). " This paper was the basis of Wald's By use of his now famous identity, \ • 7 W'ald gave the expected value of the sample size under his sequentia.l approach, and for this special case, derived the asymptotic distribution of the sample size. In each of the non-single-sampling plans in acceptance sampling, it is the mean and variance of the sample size that are important when the problem is one of reducing costs. Meanwhile, in England, Barnard (1946) was working on similar problems. He considered first a simple inspection problem, then sequential tests in general and finaJ.ly general inspection problems, and on this last point gave the distribution of the sample size as a function of the particular process curve, which is obtained by plotting fractions defective against frequency of batches having this fraction defective. Barnard also gave formulae for the mean sample size and the variance of the sample size. Working with Barnard, Burman (1946) developed workable formulae for the operating characteristic and average sample size when considering a binom:i.a1 population. 11.2. Estimation of p in a Binomial Population As far as testing hypotheses was concerned, Wald as book covered the most general of situations. these results. There was little to be done to extend Next, attention was turned to the problem of estimation, and many authors considetoed the problem of estimating p in a binomial population. About the time of Wald's 1945 paper, Haldane (1945) first considered this problem. He considered a population of which a certain proportion p possessed a particular attribute. If' one samples se- quentially until a number c of the observations were found having . 8 this attribute, and if N is the total number of observations taken, then an unbiased estimate of p is 1\ P 0-1 N-I ' a- and where q llll I-p 0 Haldane pointed out that if' p is sm.aJ.l, it is often desirable to have the standard error of p roughly proportional to p rather than proportional to <1 p fP 0 For example, when n ... 1000, if' p == 001, then .0031, so that we could not distinguish between populations where go p was 001 and 0005. for c ~ The procedure described above accomplishes this 30 . An unbiased estimate of v(~) was shown by Finney (1949) to be 1'. AI. 1\) !llL v(p "" N-2 • Tweedie (1945)· considered a method of finding the relative frequencies of one type' of blood cell, in which counting is stopped when a fixed number of this type has been recorded. This is the procedure described above, and Tweedie gave it the name Itinverse binomial sampling 0 It The name stems from the fact that the estimate of p obtained by direct and by inverse sampling have cumulant generating functions which are inverses of each othero An extension of inverse sampling to multinomial variables was also given by Tweedie (1952) 0 Here he considered sampling until some linear function of the variables reaches a predetermined value. With * 9 N groups in a population, let Pi be the probability that any one observation will fall into group 1. Then for the simple stop-rule in which sampling ceases when the number of observations in a certain set of g;.roups (i fill 1, •• 0, k, say) becomes equal to some previously fixed number, c, Tweedie showed that the expectation of the sample size is c k 2p· 1 1. and its variance is This paper is discussed further in section II.4. Girshick, Mosteller anti Savage (1946) presented some theorems concerning the unbiased estimation of p for samples drawn from a binOmial population using arbitrary stop rules. They gave applica- tions to single sampling, curtailed single sampling -- with some rejection number, c--curtailed double sampling, and the sequential probability ratio test. The most important of these theorems for our purposes concerns the estimation of p from the ttsample path. lt Consider the positiva quadran t in the x, y-plane .. Let x be the . total number of observations taken ani let y be the number of observa- tions so far in the sample which possess the attribute under study. Each observation can then be regarded as a lattice point -- that is, a point with integral coordinates -- on the diagram. The sequence of 10 points is called a ttsample path." ;rixed bound~, When this sample path reaches a sampling ceases and the estimation is made. The authors show that an unbiased estimate of p is $ (:xl'Yl) , k(J\].'Y1 ) where $(J\].'Yl) is the munber of possible paths from the point (1,1) to the point (J\].'Yl ) on the boundary, and k(J\].'Y1) is the number of possible paths from the origin to this poin t. This theorem is readily generalized to the following situation as stated by Anscombe (1953)g - - Let (t+u, t) be any ua.ccessib1e" point in the diagram, Leo, one tha.t, can be reached by a possible sample path from the origin which does not previously meet the boundary Then i f when sampling ceases the sample path has reached the point (N,R), an unbiased estimate of pt(l-p)U is 0 $~N,R~ k N,R ' where k(N,R) is the number of possible sample paths from the origin to (N,R), while $(N,R) is the number of possible sample paths from (t+u, t) to (N,R). Relating this work to the problem of inverse binomial sampling, 'we have for the fixed boundary x origin to (N,c) is =r The number of paths from the 0 (::i) , and from (1,1) to (N,c) is (::~). Their ratio is c-l N-1 the result of Haldane , 0 For estimation, the number of paths to points on various boundaries becomes important 0 Plackett (1948) presented a method for 11 determining the number of admissible paths from the origin to a point on a boundary 0 This boundary is charac terized by the fae ts that no more than n observations can be taken, and the boundary is to have n+l points where a path may cross ito Plackett showed that if N(x,y) is the number of paths from the origin to the point (x,y) on the boundary, B, then where x and yare the numbers of successes and failures, and where p corresponds to events causing the x coo rdinate to increase and q ... l=p. By equating coefficients of pi, the values of N(x,y) can be obtained for various boundaries with the given properties 0 He also stated that the average sample size can be given in terms of the quantities, N(x,y), which can be done by evaluating Wolfowitz Savage 0 2 (xty) N(x,y) pX cr. (1946) generalized the work of Girshick, Mosteller and He showed that their estimate is the only proper unbiased estimate for sequential tests defined by doubly simple regions, where a proper estimate p(c<) of p is one for which 0 ~ p(o<) ~ 1, for all ..( that are points on the boundary, and where a doubly simple region is f (x) and x ... f (y), where f (x) is non1 2 l decreasing for all x~ 0 and f (y) is non=decreasing for all y ~ 0, 2 one between two curves y lOS and fl(O), f 2 (0) > 00 Further generalizations have been given by Blackwell Blackwell and Girshick (1947), and (1947), where a lower bound was obtained for the variance of an unbiased sequential estimate 12 of a parameter 8, where :lC.l is the first observation and W = :lC.l+x2+·' .+xn is a sufficient statistic for e. This was shown to be The lower bound is attained in the classical case of samples of constant size N, since and this is the only case where the lower bound is attained • .A recent article by Lindley (1957) gave methods of sampling a binomial population to estimate p with a prescribed accuracy. The basis of his approach is the amount of information about p provided from a sample. The rule he proposed is to continue sampling until some measure of this amount of information reaches some prescribed value. In general, a sampling scheme ad,opted to obtain a prescribed amount of . information about a monotone function of p will be different from a scheme regarding the information for p. Lindley applied his results to p and to two , functions of p: I := 2- arcsin fP and 't' = In 1 ; ' These three cases yield approximate equations for three different stop ru1es.o IIi a previous article (1956), Lindley introduced these ideas as a consequence of the concept of information due to Shannon (1948). In neither article did Lindley make an attempt to estimate the sample size required for information to reach the desired leyel. II.3. Estimation of \J: in a Normal Popu1ation s While work was being done on inverse binomial sampling, Stein 13 (194.5) .c0Ilsidered estimation from the standpoint of double sampling. By a two sample test he. solved the problem of determining confidence ." .. intervals of prescribed length and confidence coefficient for the mean, 11, of a normal distribltion with unknown variance (i. SteinVs method corsisted of drawing a sample of size no' and then a further sample of N-n o observations. N is determined by N lOl max {[S2/z1 + 1, no] where s2 denotes the usual estimated variance from the first sample of no observations and z is defined by t iii t.,(.JZ , .e is the desired length of the confidence interval, t.,( is such that P ( -t.,( ~ x ~ t) :0 1-.,( , where x has StudentVs distribution with n -1 degrees of freedom, and o 1-.,( is the desired confidence coefficient. [s2/z ] denotes the greatest integer less than s2/z • = After all N observations are taken, the mean Y of these observations is computed. Then = (Y - 21 t 1 ~11 ~ Y + 2 t) is the required confidence interval, and the confidence coefficient is ~ ~ l-..{. ~ will be close to 1-..< if no on the expected sample size. < ci/z. Stein obtained bounds These bounds are a function of cl and can be evaluated from tables of the incomplete Gamma function, but using 2 E(n) ~ ;- is a fair approximation provided (i > zno • Chapman (19.50) extended Stein IS method and tested the ratio of the means of two normal 14 populations. Wald (1947) considered this problem of est:i:mating the mean of a normal population with unknown variance by means of a confidence interval of given length and confidence coefficient, but he did not obtain a useful solution to the .problem. Stein and Wald (1947) obtained confidence intervals of prescribed length and confidence coefficient for the mean of a normal population wi th known variance by use of sequential procedures. They showed that in order to have the expected number of observations as small as possible,' a fixed sample size should be used. n .4. Asymptotic Theory tlLarge Sample Theory of Sequential EstimationI' is the title of two papers by Anscombe (1949), (1952), who seems to be one of the first to attempt to apply asymptotic theory to sequential estimation. In his first paper, Anscombe stated that the general problem of sequential estimation is formulating a rule of sampling such that an unknown parameter can be estimated with specified accuracy and with He considered estimating a single minimum expected sample size. unknown parameter 9 using only statistics Z which are the sum of m m observations z1' where E(zi) := e and V(zi) := v(~) = v~e} v(e). Then , for large n, whether n is fixed or determined by a sequential procedure, i.e 0' n is the smalles t value of m for which an inequality of the form zm ~ k(m) or Z ~ k(m) is satisfied. m 15' If the boundary satisfies the conditions It is approximately linear in the neighborhood of n , where o nou is the abscissa of the point where the mean path intersects the boundary, (ii) The possibility that it can be crossed elsewhere than in the neighborhood of no can be ignored, (iii) It crosses the mean path at a non~zero angle, then Anscombe showed heuristically that the sample size is asymptotically normally distributed with mean n and variance n vee) [8-k'(n ) f2, 0 0 0 A and hence that a . is asymptotically distributed with mean a and variance vea} no' In the 195'2 paper Anscombe showed that under certain conditions fixed-sample-size formulae are valid for sequential sampling, provided the sample size is large 0 Oox (195'2a) gave a general treatment of double sampling using asymptotic methods 0 He desired to estimate a parameter accuracy using as few observations as possible 0 a with assigned Applying large sample theory, he obtained an estimate with bias O(n~2), and gave applications to the normal distribution with variance known and unknown, and to the binomial distribution 0 He considered the problem of estimating with given standard error the difference between two means of normal populations. Cox showed what the expected sample size would be using his method, gave an inequality for the expected sample size if the procedure had been done sequentially, and conclu4ed that, 16 except where the preliminary sample size is small, the best double-sampling procedure has an expected sample size only slightly greatar than for the best sequential procedure 0 Tweedie (1952) used an extension of Waldls (1945) identity and thereby gave the asymptotic mean and variance of the sample size under the rule that sampling stops when some linear function of the observation ceases to lie between two specified limits. Cox (1952b) derived the bias of Anscombe IS 1949 estimate to the term in n-1 when the unknown parameter is the population mean 0 He used the method of plotting the cumulative sum of the observations against the number of observations until a boundary is reached. Cox presented a new estimate n 2zi where to'" 1 n and b l n is the slope of the boundary at the point at which it is crossed, and he showed that the bias is O(n-2) 0 'Wald (1945), as mentioned previously, derived the asymptotic distribution of the sample size for his particular cases. In another paper, Anscombe (1953) made further strides in the asymptotic theory of sequential estimation and obtained the secondorder asymptotic distribution of the sample size fo,r a certain general type of sampling procedure 0 Second order asymptotic theory is devel- oped by adding a correction term to the asymptotic distribution. of the estimate, so that the error is of lower order when n is large 0 He set confidence limits on' the mean, ~, of a normal populatic n with unknown 17 variance by first making the transformatiom , f xjJ i 1\ • id+1) [ i Xi +1 - ' 2' i . 1, 2, • 0 0' 11-1 · His first order stopping rule then becomes8 n is the least N for which where e is small, and the desired length of the confidence interval, supposed 1-~ is the desired confidence coefficient. It 'is shown that the second-order asymptotic procedure for this problem modifies 'the rule tog n is the least N greater than 3 for which N-1 e2 z 2 ~ u. ~-2 N(N - 2.676 - -2,) J. 4'Z ..( ~ • J.= The right hand side is not unique, in that the multiplier of e2 ~ may 4z..( be modified by a.dding a function of N which is O( (N) for N large. For this rule the expected sample size is approximately The first term on the right hand side is the number of observations 2 ' that would have been needed if (] were known, and the rest is the , . average cost in observations of not knowing (J2. knscombe found the asymptotic distribution of n by considering the sum of a. sequence of independent ·random variables also applied to betwe~ So 0 These results are simple birth-death process, so that the difference the birth rate and the death rate is estimated with given 18 standard error. Anscombe investigated numerically the accuracy of his approximations. A follow-up of some of the results of Anscombe is given in a recent article by Ray (1957), who modified the procedures slightly and attempted some results on small sample theory. t instead of the normal deviate. one change is the use of Studen t' s His sampling procedure is first to take three observations and then sample in pairs. His results are applied to the estimation of a mean with given standard error and estimation of the difference' between two means with given standard error. Ray gave the dis·tribtltion of the sample size as the probability of stopping at the (2rn+1Jst observation and thus the mean and variance of the sample size in-terms of an infinite series, none of which seems to be of much use in practice. II. 2• A Recent Application . The applications of sequential methods may profitably be applied to another field of statistical theory, that of discrimination, although to this writer's know1e4ge nothing yet has been published in this field. ~ In an unpublished paper, Williams (1957) did consider the discrimination of two groups where he assumed that members are drawn one by one at random from the popu1ati,on and then classified into one of two groups. Sampling is continued until a sample with the required prop- erties is obtained. Two such properties might be the following (both of which are considered in the text of this thesis): 1. Sample until at least n members from each group are obtained, in which case 19 E(n) ~ n . .min(p,q)' where p and q are the proportions of groups I and II in the populations. 2. Sample until the difference of the means for the two groups is d~termine<i with a given accuracy. ference is proportional to 1- + 1-, sampling is continued until . nl _ n 2 the condition is satisfied. Since the variance of the dif- Chapter III SECJTENTIAL-STOP RULES In this chapter and the next various sequential stop rules as ~efined earlier in Chapter II are considered, and the mean and variance of the sample size under each rule is obtained. The most general rule is: Given.s populations mixed in unknown proportions and a function of s variables f(~, ••• , x s }' in the nonnegative part of the s-space, sample observations one at a time, evaluate the function after each observation is drawn, and cease sampling when the func tion first changes sign. This rule is equivalent to plotting the function and the sample . path and. continuing sampling until this path crosses the curve of the func tion. ~he sample path is a sequence of points with integral co- ordinates, the coordinate x. at any time being the numbers of observa.- J tions f~om population j thus far in the sample. The main type of function considered is one whose derivatives with respect to:x:t.' •• 0, X s are all of the same sign, so that once the boun- dary is crossed, it cannot be crossed again. TIlo1. Rectangular Stop Rule The first situation to be considered is for the simple binomial case; 8=2, and for -a func tion which will be called the rec tangula.r stop rule. ~re sampling continues until at least ~ observations from the first and a.t least m2 from the second population are obtained. A 21 graphic representation of this situation appears in Figure 1. Figure 1 Rectangular Stop Rule x2 / / // " - ; - - - - - - - - - - x..t (~,O) With the exception of the point (~,m2)' points on the line ~=Inl can be approached only from the left, while points on the line ~=m2 can be approached only from below. Let the proportion of population I in the mixture be p and that of population II be q=l-p. The probability of stopping at the nth trial is the probability that at the (n-l) th trial we have u ~ ~ successes (Le. observations from population I) and v at the nth trial .QI. u =: := m -l fa.ilures (population II) and a failure 2 ~ -1 successes and v ~ m2 failures and a success on the nth trial. If there are"m 2-1 failures, then there are n-m successes in n-l 2 trials. If ml-l successes then n-ml failures. Hence the probability of stopping in n trials is 22 since the probability of 1. m2=1 fai,lur.es in n-l trials is Cn-f)! . n-m 2 m2-l . (n-m2~! m-l)! p , q ., 2 2 • ~-1 successes in n-l trials is' 3 • failure at nth trial is q, 4. success at nth trial is p. NoW, since ~+m2 is the minimum possible sample size. Thus If the sUIllD'lation were from zero to infinity, 'this expectation would be ~ +~ , q P. but with the term must be subtracted. ~ange' m.. +m2 .L to infinity.a correc tion This' is the pair. of sums up to ~+m2. Hence 23 +" ~.... ~ p ~p ~ n~ +m -1 In the first su.mmation let n ... s+m Then the sums are for s ... 0, 1, ••• , n! . n-zrt:t (n~)!~! q 2 2 • and in the second n ... ~ -1, and s = 0, 1, ••• , respectively. The first two terms of E(n) become To evaluate this quantity it is convenient to consider M s+~. -l 2 24 Summing this last expression for i out except the last for i ... l!lIl 0,1, ••• , ntr-l, all terms cancel ~-1- Therefore and where B(u, v} refers to the Beta distribution with parameters u and v. Hence the first two terms reduce to Similar results hold for the remainder of the expression, so that !l!l B ~ m J [ l. (puI!1.-1 (l-u) In.2 + 1 }l u~ (l-u) m.-l 2 du du q <5 ~,m2 p p In a similar marmer, an expression for E(n 2) may be obtained and hence V(n). The first of these 'tnrns out to be 25 which reduces to The expected value and. variance of n can be obtained numerically, bu~ ~o~ large values of involved IIJ. 2 the calculus becomes tedious and Various special cases are next considered and the results 0 and II1 summ.a.rizedo bo For ~Il!l Co For ~ 2, ... 3, For the same oases, i f p := q = t 26 V(n~ ... 2m2 [ 1 + E(n) lllI m <t> mJ 2 + 6(t> .2 - 1'm 2m2 + (2) 2 [4+m2] 4(t) 2m 2 0 0 yen) .. 2m + (t)m2 [20-2m22-3m2] - (t)2m2 2 (4+~)2 0 • t) In all cases, when pDq, as m increases, 2 E(n} ----> 2m2+ yen) decreases until m2 lllI In.., .L for which yen) m2 • ~ • V(:Il) m2 • 1 ~- 'Whim m .. 0, we have the inverse binomial s~ling situation men2 tioned in Chapter II, and the results obtained here agree with those obtained by applying that method directly. Another special case considered is for m1 = m2 • m and p • q • t 0 o. • 27 The sample size is then given by n = m+t, t ~ (Wt-IP 2 ~ m. (~-1)1 Then 2 ~l tIm-I! tJl'"t + m! m-l! 22m = . where the first term is for t > m and 1 the second for t = m. The factor of 2 is introduced since the sample path can cross the boundary from below or from the left. Replacing m by m+l in (1), 2 22m+2 • = 1 , hence (2m+l) ! (m-l)! (m+l)! 2 .. 22m+l 2m , so that E(n) or E(n) = 2m + (2m)! mHin-l}t 1 22m-l 'V 2m + 4~ l!!..211 ' for by Stirlingls approximation we can show that This result can be Verified in the special cases given previously. • 28 ill.2. Linear Stop Rule An example of a negatively sloping function is, for s=2, k:x:t+~ .. m, k > 0, as in Figure 2. Figure 2 Linear Stop Rule: k:x:t+x2 =m ~ (O,m) .(m/k,O) '.XJ. Lattioe points on the -boundary can be approached either from the left or from below, while points which are not lattice points must be approached from the left. Let p and q be defined as in the last section • Then. the proba- bi1i ty of the sample path reaching a particular lattice point, (ll, v) say, in the ~, x 2 -plane is (u+v) ~ u v ~ u p q u.v. - For the present, let k be an integer,then for c!ach integral value of :IJ., there are k integral values of x 2 where the line may be crossed by a sample path. Let PO(r) be the number of paths which e~d by crossing 29 the line at JI]. = r, PI (r) be the number of paths which end by crossing the line at :x:L=r- 1 k, Pk - l (r) be the number of paths which end by crossing the line at k-l ' x.. =r-. .L k" " r - l + ; a k • Then is the total number of paths crossing the line for r-l <x 1 ~ r. The probability of reaching the point (r, m-kr) is then Similarly, the probability of crossing the line at the 0 ther k-l values of x are 2 = (~-(k-l)r) r-l , ]\-1 ( r ) p r qm-kr+k-l , P k-l (r)'" (m-<k-l) r+k-2) r-l since the number of paths crossing the line at a point which is not 30 a lattice point is equal to the number of paths reaching the lattice point just to the left of the line, however, one more step in the ~ direction is needed to cross the line. Hence the probability that the sample path crosses the line in the < Xl ~. r unit interval r-1 is = Lk , say. For k=l, L 1 III r! <:!r)! prqm-r, the ordinary binomial result. Similarly r m-2r L2 = P q In order to obtain the expected value and variance of the sample size we replace L. by a continuous approximation using Stir1ing i s J. formula and the general method of approximation to a binomial. To find the appropriate transformation, we consider the Itmodal lt line, Le. the line around which the sample paths will vary. 'modal line is one from the origin to a point on the boundary. point is The This 31 or This is a form of an average of r. In order to transform to a standardized variable z, we use the transformation r I: mp!'zm qfokp , that is, we expect the variation about this line to be o( assumption which is later verified fi), an 0 since r takes unit steps, and of course ~ r..J - 1 ~ 2npq m e 2 -z 12pq .1 -z dr =.~. e v 2npq the usual normal appro:x:i.mation to the binomial. 2 12pa • 'UZ , Hence z is approximately normally distributed with zero mean and variance pq. Now, Taking logarithms, applying StirlingVs formula, and collecting terms, - log L 2 rv (m-2r ),log [ (m-r)gl m-2r J . ...+ r log [(m-r)p] r + log [21fI'1;:2r Using the transformation 1 >J. 2 + log (1 ... m-~;+l) • 32 r == mp+zu . 2 qt'p we have log L mg-2z rm (mg-zg .fm, 2,v qt2p log mq-2z f1il'J The firs t two terms of 10 g L 2 can be approxima. ted . by 2 + mptzu log [1 ( ..... 2p)(..L. ,",..L qt2p . ':\" {iiiP.r; 2 " mp )J. ~ -mg-2z rm; r(.q+.2) I ...lL + ..i.) _ (qt2 ).2 (..L + ~) 2J q+2p L:. p. \ tmq .fijiq2 p. '-/1fiq rmq2 (mptz fi) [(qt2P) q+2p (..L - -l) (Mp 2 + (q+2p) 2 (..L. - ..:1.) J. fIiip .r.: vmp 2 .tw vmp so that 1 g+2p ) 2 log L ~ log ( 2nmpq 2 2 ) - z 2(0+2p) + log ( qt2p pq or (q+2p) • 33 To obtain the differential element we note that dr ~ q+2p Ill! • Hence ! = ( ~.)2 2npq i 2 (g+2p) 2pq e dz. That is, z is normally distribu.ted with zero mean and variance . and E(r) • -!!m... VCr) "" q,+2p , !92' ~.p , , mV{z) (q+2p) 2 so that r is normally distribu.ted with mean and variance given above. Now n Ill! m-r, hence tV n ( ) N.l!!mpg q+2p , (q+2p) 3' • This method is readily extendable to the general k, where n '"' m-(k-l) r and where \: can be written as 2 1 + . rg + r(m-{k=1}r+llg ~~.-:;w.;~ [ m-kr+l Cm-kr+l) \m=kr+2) + r[m-(k-l)r+l )[m-(k-l)r+2 }s!3 (m-kr+l) (m-1a*2) (m-kr+3J. 34 An argument similar to the one used for L is used on the factors 2 outside of the braces, with m"';kr in place of m-2r and m-(k-l)r in place of m=r. Using StirlingUs formula and expanding the logs this first part becomes 1 - z2(g+kp) (r:~) 2 e 2pq mg-kz iii z {i Replacing r, m=kr and m- (k-l) r by mp+ o/kp , qtkp m-(k-l)z o/kp so tba t in the limit and rm The general term of the series in braces is then approximately ... p , I, so'~'that the factor in braces is l+p+.oo+P ... l+(k-l)p ... qtkp. Again vn dr "'" dz q+kp and 3S and the distribution of n j;s then asymptotically normal with mean :kp_ - ' 2 and variance ma Ck- l ) ':l' - 0 (q+kp)3 The above analysis has been done With the assumption that k is an integer; however, it is.clear that the result holds for any k proof of this statement follows, * when > 00 A in fact the line in question is of the form tet the probability of obtaining an observation from population I be p and from population II be q, and let n l and n 2 be the number of observations from the two populations in a final sample of size n ... n +n o l 2 From elementary theory we know that n l and n 2 have a bivaria.te distribution with means np and nq, variance npq and covariance -npqo Let Y1 and y2 have zero means and variances pq. The line can now be written as or since Yi+ Y2 * l!lil 0, this is I am indebted to Dr this proofo 0 • Go S. Watson for some of the features of \ 36 Hence, where the final term on the right is to allow for discontinuities at the boundary. Taking e:xpec tations , and , where n is replaced by its expected value. obtained previously when f l 0= k and f 2 := These are the results 1- This proof is of a heuristic nature, in that it treats n as if it were fixed and not a random variable. proof follows. A mathematically rigorous This proof makes use of an identity due toWald (1945). For the present purposes, it can be written as { l!(~,t2) ] -n • ~nl+t2"2}. 1 , where ~(\, t ) is the moment generating function of the binomial dis2 tri bl tion, that is The boundary restriction is 37 and by taking t, proportiona.l tof1.' , i.e. 1. the identity becomes E S ~e r. tfl+qe L tf'J -n } E [ e -tc , 2 = e -tc , or equivalently . e·~KJ "Ill where K is the cumulant function, i.e. log ~ = K. By denoting the moment generating function of n by 'fl(t), the a.bove equation becomes \fI(-K) .. e -tc , • and we must solve for t in terms of K. ... Also... 2 sK Then, to a first approximation and = l+K+ 1L + 2 ... 38 Now, or K2pq(f -f ) 2 2 l 2(f P+f q) 3 2 l Substituting this value of t on the right side of the identity and replacing -K by K , as the moment generating function of the sample size, and hence the desired results are obtained. If the approx::i..mation is carried further, it can be shoWl'l that the skewness coefficient goes to zero, for then e K 2 t =.1+t(fl p+f2q)+ '2 + t- (f 3P+f 3q) + 1 2 = l+K+ -K2 2 K3 + - 6 + and the third order approx::i..mation becomes (f 2 l 0.0 p+f 2 2 q) 39 e. so that the third cumula.:nt will also be of order c. ~ and will The skewness is then be proportional to c;/2 = c-1/2 and thus will tend to zero as c increases. nI.J. Number of Paths to a Line' There exists some interesting:m.athematics in connection with ~ind~ng th~ number of pa ths from the origin to the line considered in the last section This becomes useful when the probability of any path 0 finally crossing the line is the same for all paths. Let P(k,m) be the tot;al number of paths to the line kXJ.+x2:~ m. Wh~n m ~- P(l,m) ~ k=l, we again have the usual binomial situation, and llO 2 • '!'he ratio between the successive number of paths as m is . ~ increased in steps of one is 2. For k=2, by direct enumeration we have P(2,O) = 1 P(2,1) = 2 . P(2,2) l!!l 3 P(2,3) l!!l 5 _. ~ P(2,4) "'" B P(2,5) =13 40 P(2,6) == 21, etc. These 'numbers form the ordinary Fibonacci sequence, for which the recurrence relation is P(2,m) == P(2,m-l)+P(2,m-2), m > 1. The ratio of the succeeding terms approaches a constant ~ ~ ::::::; 1.618. is such that for then the recurrence relation for the sequence is which reduces to the above equation. Note that the roots of the equa- tion are t.. and _~ -1. Hence we obtain Hardy and Wright (1938), i.e. the number of paths is a linear combination of a power of the roots of the basic equation. For large m, (_t..)-(m+l) approaches zero, so that t.., the root greater than unity, becomes the only one of importance. Considering the case for k=3, (Figure 3), let P (3,m) be the o number of paths from the origin which cross the boundary at a lattice point, P (3,m) be the number of paths crossing the boundary one unit l from a lattice point in the ~ direction, and P2(3,m), two units in the x 2 direction. Then P (3,m) o 0: P (3,m-l) + P (3,m-l) o l 41 Pl{3,m} P2(3,m~1) P (3,m} 2 P (3,m-l) O 2 . P (3,m) ... fP i (3,m) =PO(3,m-l) ~ + P(3,m-l) ~ ... P (3,m-2) + P (3,m-2) + P(3,m~1) O l lOll Po (3,m-3)+ P (3,m-3) + P2 (3,m-3) + P(3,m-l) l := P(3,m-l) + P(3,m-3) • Figure 3 Proof of the Recurrence Relation for k=3 x2 I I (0, m) (O;m-1) ~ 1\\ \\ \ ~~ ~ ~\~ !!\\ e: &'\ !\ ~ 1\\ - \' \\ For r-l < XJ. ~ r, "ijle line 3XJ.+x2 I: m can be crossed at points ~, ll, and.Q.. Let the number of paths to these points be Po(3,m), P1 (3,m) and P2(3 ,m) respectively. The number of paths to both points 11 and .Q. is the same as to the lattice point just to their left. The result then follows. • Similarly it may be shown that for general k P(k,m) l!!l P(k,m-l) + P(k,m-k) • Assuming then that P(k,m) can be written as a linear combination of the roth power of k values \ , ••• , \:, i •e • k m P(k,m) ....~aiti ' i~ we then find that \ , ••• , \: are the k roots of tk_t\c-l :0: 1. This is easily shown, for we have ~' a.1 to1 m = ~ a oto m-l + ~ a. t m-k 111 i , or so that the to are roots of an equation of the form given above. 1 Without loss of generality, we may assume that t. 1 and we know that ~ t i !!il t. for i J will be the dominating term is the only root greater than unity. This can be shown by considering the variation of the function tk_tk-l_l with t. monotonic increasing function for t values of t. ~ 1 and is positive for large > 1. Thus P(k,m) is approximately a geometric progression for increasing integral values of m, P(k,m) ~ This is a For "l:Fl, it is negative, showing that one and only one root exists for t where <j 1. In the expression for P(k,m), ~ \ since \ ~ is as yet undetermined. rV al\m , 43 To obtain al' consider the system of equations TA=,2 where 1 T = ~ ~2 ~ Ja. 1 t 2 0 • 00 1 • \: t 2 2 k-l t2 k-l AU .,. (~ a :e,t 2 0= o (1 2 \:2 k-l \: •• 0 ~) ••• k ) is the vector whose elements are the number of paths for m'" 0, 1, 2, •• 0' k-l respectively. To find A, we must invert T; however, since' only ~ is needed, we need find the cofactors of the first column only, i.e., the elements of 1 1· the firs t row of T-. Call these c J. Then T is called the matrix of Vandermonde, and its determinantal value is k(k-l) 7T (t.-t.) i(j \ Ie I t I I. J ~. a (-1) 2 7T (t.-t j ). i<j ~ .. 44 Using this value and a theorem of elementary symmetric functions,* we find C lp := ~p 1 7T (t -t.) j>l 1 , p <k J and Applying the theorem mentioned above, we find 7T \k + k-l (\-t.) j>l lIB J. ~t-1 Therefore * thm If ~, x 2 ' •• c, .x are roots of the equation n .& k k-l Hence for the equation t -t -1 -= 0, so that ~ t i "" 1, the roots are zero. 1Tti lIB (_l)k-l and all other sums of products of 45 t k-2(t k + k-l) 1 1 so that in order to find al' only one root, namely t , need be obtained. l a can be further simplified by multiplying numerator and denomil nator by (1 - 1 2 r) Many of the terms in the numerator cancel out and 0 1 we obtain t k ~ = k(\~l)+l '.An asymptotic expression for t l • can be fotmd, which is valid for ~ large k, so that the equation need not be solved explicitly. If a closer approx:i.ma.tion is desired, successive approximations can be obtained satisfying the equation to the required degree of accuracy. Use of high speed computing is well suited to this need. Now if \ is the root greater than unity and if we let t l =1 + then the equation becomes taking logari thIns k-l Therefore Substituting for r:l ,..J t' ~ lIll _ lQg S log(l+~) /V _ log ~ @ -log f3 k-l on the right, and ignoring log log terms, we have ~, 46 On keeping the log log terms in the approximation, we find similarly t:L rV k . l/k (log k~ , hence for large k, we will have It is interesting to note that the bounds on t for (io g k) -11k are not very wide, l approaches unity quite rapidly. ~ The following table gives the values of and 4, ~ and ~ for k == 2, 3 the actual number of paths, P(k,m) and the estimated number of "'- paths, P(k,m}, using the values of t l and aI' for various values of m. TIl .40 Hyperbolic. Stop Rule It is often desirable to sample until a specified accuracy is obtained. This accuracy might be given in terms of the standard error of the statistic used; for example, we might want the standard error of the difference between two means to be a given small number, sa:v:rt. Assuming independence and a common variance, 'P'o expression is or ci, this e e e kl!O!.~ k 1.61803 t1 III ~ ... 1.17082 m P(2,m) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 ·2 3 5 8 13 21 34 55 89 144 2.33 .377 610 987 1597· 2584 4181 6765 1\ ~ liII '3 k-ij ... 1.46557 ~ 8.1 • 1.31342 ... 1.38028 a1 - 1'.43970 ,1\ ;'\ P(2,m) P(3,m) P(3,m) P(4,m) P(4,m) 1.17 1.89 3.07 4.96 8.02 12.98 21.01 33.99 55.00 89.00 l44.0 233.0 377.0 610.0 987.0 1597. 2584. 4181. 6765. 1 2 '3 4 6 9 13 19 28 1.31 1.92 2.82 4.13 6.06 8.88 13.01 19.07 27.95 40.97 60.04 88.00 129.0 189.0 . 277.0 406.0 595.0 872.0 1278. 1 2 3 4 5 7 10 14 19 26 36 50 69 95 131 181 250 345 476 1.44 1.99 2.74 3.79 5.23 7.21 9.96 13.74 18.97 26.18 36.14 49.88 68.85 95.03 131.16 181.0 249.9 345.0 476.1 hI 60 88 129 189 . 277 406 595 872 1278 -F -.:J ,- 48 wheren l is the number of o·bserva.tions from population I and n population 2 from :a. The stop rule then becomesg selec·t observations until the above inequality is satisfied. A graph of the boundary appears in Figure Figure - -. 4. 4 Hyperbolic Stop Rule x2 (0 ,c) . -_ _- L --=~~_ &..-_------(c,O) ~ To be consistent with the previous sections, replace n. by x., ~ ~ since the n, can assume only integral values; then (2) is a rec- - - ~ taIlgular hyperbola in the ~,x2""Plane. We are interested only in the positive branch of the hyperbola., which has asympotes at:x.t .. c and x 2 :e c, since (2) is equivalent to ... c 2 The slope of this curve is c 2 , 2 (~-c) =0 -k , using the notation of III.2 for the slope. F'or a gi van va1ue 0 f ~, cr r say, x 2 ... H ' an(i the slope a._ t the point is 49 2 = --:c~_ (r_c)2 The equation of a straight line passing through 0 the point (r,cL) with this slope is r=c We may now consider this as a special case of the straight line of Section II102 for in the neighborhood of reg~ded as linear departures from XI == :~ , the curve may be As in the case of the straight line, we expect the 0 XJ. "" :~ to be 0 ( {M) 0 For the straight line of slope -k we had co z2(q+kp) 2pq dz In· iths case, where m ~ 2 2 and k (r-c) cr = c 2 (r-c) that The distribution of z becomes = ~dz ~ _. ~ e ...L 22 p dz, 2np .- and the distribution of the sample size n, where 2 cr r n""r+-lOllr-c r-c ' is In the case when p m q, the modal line intersects the hyperbola 2 ' 50 a t the point where the slope is 1, and the variance vanishes. The fact is that now the variance does not go to infinity with c, but tends to a finite limit. In practice these results may not be useful because the variance is usually unknown. For this case the variance may be sequentially esti- mated, as each observation is drawn, by the method siJuilar to the' one due to Anscombe referred to in Chapter II o Since common variance is assumed, the estimate can be based on the pooled sums of squares from the two populations. After a sample is taken, make the transformation i f the observation is from population I, where w. is the value of the J random variable on the jth observation from this population, or Vi ~ id+1) {i "1+1 i f the 0 ~"d 2 bservation is from population II and wj is the value of the - - random variable on the jth observation from population II. Then where there are nl and n observations taken from populations I and II 2 respectively. In this case c 1\2 l!!l co(J , which is a function ofn, so that the curve changes as the sample size increases. 51 Rea.rran ging the inequa.1i ty, we see that the criterion for stopping d becomes~ continue sampling until the following inequality is satisfied A short hypothetical numerical example is given for 11.1ustrative purposes. Suppose the variance of the mea.n difference is desired to be less than or equal to 1/2, Le., c o = 2, and suppose the observations were taken in the following order, with the values of the random variable given in parenthesesg Then the analysis can be set up as follows: w!~ Vi ~Vi ~Ui+2Vi W(nl'n 2) .50 2 2.00 2.00 017 067 4 0.00 2.00 2.67 4.75 4.75 4.80 1.80(3,2) 2.67(4,2) 4.29(4,3) 5.62(5,3) 3 2008 2 75 4 5 005 2080 5 4 ~Ui i wi U i 1 3 050 2 4 3 0 Hence sampling stops when n • 8, n A2 (J ·e . ... 4·6 80 5' I.. !Ol 0 • s.. "i:' Jlli~larly p ... t l 8!Il 5, n 2 '" and q = III 8'~ . 30 (J2 is estimated by ... Chapter IV EXTENSION OF RESULTS IV 010 General Function in Two Dimensions The first generalization that will be considered is the na.tural ~xtension to a general negatively sloping function in two dimensions defining the stop rule 0 Let this function be an.ci let Pl and P2 be the proportions of populations I and II in the mixture. a f(:x:t,x2) fl go @ ~ \ a f(3.'X2) f2 = @~ where both f l and f 2 are of the same sign. I :x:t = API x 2 ... AP2 It is reasonable to expect that Xi will be proportional to Pi' i.e. equal to APi' so that f(A. Pl,A.P 2) =0 will give the value of A. at the curve, and hence define where the modal line crosses the curve. Assuming that this curve will almost certainly be crossed in the ne:ighborhood of this intersection, the results for the straight line may be used in the appro:x::i.mation • As before, Ie t 53 e- n 2- nP 2 Y2"" rn Then the transformation of Watson is equivalent to where Again the final term on the right is to allow for discontinuity effec ts at the boundary. Now so n "" A+O({1) , rn vt+ocJ5 , m . and we may write , then E(n) "" 1.+0(1) V(nl "" ~~2V(Yl)+0(1) , "" A.PlP2~ 2+0(1) To the desired degree of approximation, the bounded terms may be ignored 0 Again this argument is heuristic and the mathematically -- rigorous proof will be given for the more general function in the next section. An application to the hyperbolic stop rule will indicate how the resul t holds for non-linear stop rules. 54 The basic equation is 1-+ 1-.1 . x2 ~ c 0 Replacing x. by A.p., we have 1. 1. Hence Now x. dPV '1.p. .1.1. so that f i ;-J - 1 2 2 ' A. p. 1. .. and A.(f2~fl)2PlP2 2!lll , (fl PI+f 2P2) A.(P2-Pl)2 PlP2 2 2 PI P2 c(P2-PI) ~ . 2 2 PI P2 2 = yen) , which are the values found in Chapter III for this specia.l case. IV 02 0 Functions in s Dimensions For the general function in more than two dimensions, a generalization of the heuristic method and results of the preceding. section is first used. Let the func tion be represen ted by •• 0, x ) s s and let the proportion of popula.tion i in the mixture be Pi· fPi!lll L 55 Define s quantities y.]. by n.-np. ]. ]. Yi rn == where CIj. == I-Pi' and s partial derivatives by @f(:x:t, "', x ) f. ]. which all have the same sign s = ---=-----C> xi Xj == 'A.Pj j == 1, 2, ..• , s 0 Then will define the modal line to the surface and give the value of 'A. at the surface. The s dimensional generalization of the transformation then becomes ~ _ . :2 f ]..y.]. + 0 (1) 2'fi Pi E(n) == V(n) == 'A.+O (1) 'A. I"':' f. P .) ~ ]. 2 V(2 f .Yi)+O(l) ]. I ]. Now the problem reduces to one of find in g the variance of .L. ~ f ].. y ]..• V~f.y.) ~ ].]. == 2 ""f. p.q. - .L.]. ~f.f.p.p. .L.].].]. J ]. J 56 The generalization of the valid proof of Chapter III considers where s/J is the moment generating function of the multinomial distribution. Setting t.~. equal to tf.~ and using the restriction ~ ~ Ln. = c, ~ ~ the identity becomes e -tc as before, where K ... log ,;. Soiving for t in terms of K and taking the terms in the expansion . 2 up to the term in K as was done in Chapter III, the results of the heuristic proof then follow. In attempting to apply these results to the hyperbolic stop rule when more than two populations are concerned, we consider the between population component of variance, which is the variance among population means and is defined by 2 O'b - 13=1 where we have s populations and n i th population. , observations are taken from the i For sg)2, the term in parentheses is 57 and i f set equal to a constant gives the hyperbolic stop rule. Hence making the standard error of the difference between two means a given popul~tion small n1.Ul)-ber is equivalent to making the ance Q" component of vari- constant. Then for the 13 'populations, the direct generalization 2ninj C =0 2;ni or equivalen tJ.y defining a rectangular hyperboloid in tionship involved in the stop rule. the sample size n ... :2 ni dimensions, becomes the rela- 13 To find the mean and variance of to attain the boundary, we use the results of the general case. The mean A. is found by setting ~A.PiA.Pj , ~A.Pi >= c go c , then A. ~p.P. ~ J. J so that A. := _ ....c"'-_ 2 Pi P j To derive the var;i.ance, we replace n i by A.P , then i 58 ~Pifi Hence becomes 2PiPj and f:i..-fj = PfP i ' Chapter V This dissertation was concerned with finding the asymptotic distribution of the sample size required to reach a certain fixed boundary when the sampling is of a sequential nature. For the most general case consider s populations mixed together in the proportions PI' P2' • 0., ps. Observations are drawn one by one and then can be correctly classified by populations. Let n, be the number of observations taken from the J. 1 -t'J.on; then J..th popuua 1'1 =: :2 ni is the total sample size 0 A sample path is defined by a sequence of points in s dimensions, each point having integral coordinates, the coordinate Xi at any given time being the number of observations n i thus far in the sample. Sampling ceases when the sample path crosses a fixed boundary. If this boundary is of the form (A) where f has derivatives with respect to ~, x 2' 000, X and has negas tive slope everywhere, or equivalently if all the derivatives are of the same sign, then it is shown that the sample size,n is asymptotically ~orrnally distributed with mean 'X. and variance , where X ' == 'X.p, J J j == 1, 2, ••. , s 60 and '1. is defined by Special cases of this general result are given for s=2, where the - boundaries are a rectangular hyperbola I - +I - = I - :lCJ. ~ c and a negatively sloping straight line Results for a rectangular hyperboloid in s dimensions are also given. One case where the boundary is not of the form (A) is also considered • This boundary is :lCJ. = 111. and x2 = m2 and the mean and vari- ance of the sample size are given in integral form. If the boundary is a straight line of the form kXI + x 2 = m, k a positive integer, and if the probability of any path finally crossing the line is the same for all paths, then where P(k,m,r) is the number of paths crossing the line for (r-l <~ and P(k,m} is the total number of paths to the line. An approximation to P(k,m) is considered and it is shown for - increasing integral values of m, that the sequence so formed is approximately a geometric progression. P(k,m} I'V That is atm , and the values of a and t can readily be obtained. ~ r) 61 BIBLIOGRAPHY Anscombe, F. J. 1949. Large Sample Theory of Sequential Estimation. Biometrika ,J2g455=458. Anscombe, F. J~ 1952. Large Sample Theory of Sequential Estimation. Proceedings of the Camb:i'idge Philosophical Society 1l!ig 600=607 • Anscombe, F.J .1953. Sequential Estima.tion. (Series B) 12.g1=21. Jour. RoYal Stat. Soc. Barnard, G. A~ 1946. Sequential Tests of Industrial Statistics. . Royal Stat. Soc. (Supplement) .§.gl=21. 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