Eeden, C. (Constance) van

English

CWI's Institutional Repository

20 A SEQUENTIAL TEST WITH THREE POSSIBLE DECISIONS FOR COMPARING TWO UNKNOWN PROBABILITIES, BASED ON GROUPS OF OBSERVATIONS 1 by Constance van Eeden Mathematical Centre, Amsterdam 1. INTRODUCTION Sequential tests for comparing two unknown probabilities p and p' with two pos-sible decisions, i.e. tests resulting in one of the decisions p <p' orp > p' (1.1) have been developed by A. Wald [8] for pairs of observations and by the Statistical Research Group of the Columbia University [6] for groups of observations. Such sequential tests with two possible decisions have the drawback that they al-ways result in one of the decisions p < p' or p > p', even if p = p'; therefore it is very useful for practical purposes to develop tests with three possible decisions, i.e. tests resulting in one of the decisions p <p',p > p' orp:;:: p'. (1.2) A generalization in this direction of the above-mentioned test of Wald has been given by J. de Boer [2]. Whereas his test applies to pairs of observations - a case which will not be considered in this paper - the next pages generalize the sequential test de-scribed in [6], applying to groups of observations, to a test with the three possible decisions (1.2). For this generalization an approximation is used, which makes it un-suitable for small groups. In order to explain the test clearly and to give a picture of its relation with the other tests mentioned, a short description of these is given in the following section. 2. SHORT DESCRIPTION OF RELATED TESTS a. Two-decision-test for probabilities based on groups of observations [6]. Consider two independent series of independent trials, e.g. two processes A and A', each trial resulting in a success or a failure with probabilities p, 1 - p and p', 1 - p' for the two processes respectively. Suppose the group of trials constituting the i-th stage of the test, consists of n; trials for process A and n; trials for process A'. If the numbers of success are a; and a; respectively,2 a; and a; both possess a binomial probability distribution with parameters n;, p and n;, p' respectively. The following transformation of a binomial probability distribution is then used: if, in general, m possesses a binomial probability distribution with parameters M (num-bers of trials) and p (probability of a success), then the random variable 1 Report SP 34 of the Statistical Department of the Mathematical Centre, Amsterdam. 2 Random variables will be distinguished from numbers (e.g. from the value they take in an ex-periment) by printing them in bold type. 21 ) y = 2 arc sin ym/M for O < m < M y = y2/ M form= 0 y = ,__y2/M form= M is, for large M, approximately normally distributed with mean µ = 2 arc sinyp and variance cr2 = (n + 1)/n2• 3 (2.1) (2.2) (2.3) In this way the variable m is transformed in a variable y with a mean which is prac-tically independent of Mand a variance which is practically independent of p. The transformation (2.1) is applied to ai and to a;; after this transformation these random variables will be denoted by ui and u; respectively and the sequential test of Wald [8] with two possible decisions for the mean of a normal distribution with known variance is then applied to def x1 "'= u1- u;. This test will be generalized by applying to the xi the sequential test of Sobel and Wald with three possible decisions for the mean of a normal distribution with known variance. [7] -b. Two-decision-test for the mean of a normal variate with known variance [8]. For the case that the successive observations x1 , x2 , ••• are independent observa-tions of one random variable x, possessing a normal probability distribution with mean µ and known variance cr2, Wald's sequential test with two possible decisions forµ has been described in [8, p. 117-124]. This test will be described here for the case that the variance is not constant [6]. This results in a small change in Wald's test; the proof of the validity of this test follows at once from Wald's own proofs. For the test a value µ.0 of µ. must be chosen, the two possible decisions being: µ. < µ0 and µ > µ0 , where we may substitute ~ resp. ~ for < resp. >. Furthermore two values µ1 and µ2 must be chosen with such that the decision µ > µ is considered an incorrect decision if µ S µ1 and the 0 -decision µ < µ0 is considered incorrect ifµ ~ µ2; for values ofµ between µ1 and µ2 it is not important which decision is taken. The concepts "correct" and "incorrect decision" are thus defined as follows. 3 Tables of y (in radians) and a2 are given in [6] for M = 10 (1) 50 and OS m SM (cf. also section 3). - -22 TABLE 1. CORRECT AND INCORRECT DECISIONS correct incorrect value ofµ decision < µ = µ1 µ < fLo µ > fLo > µ = µ2 µ > fLo µ < fLo l µ < fLo µ1 < [L < µ2 and -µ > fLo The interval (µ1, µ2) is called the indifference region. If: oc = the probability of acceptance of µ > µ0 if µ = µ1, ~ = the probability of acceptance ofµ < µ0 ifµ= µ2, and if ix and ~ are chosen both < ½ the probability of an incorrect decision is ~ ix forµ 2: µ1 and~~ forµ~ µ2. The value µ0 is of no further importance for the performance of the test. The test is carried out as follows (er; is the known variance of xi). Additional obser-vations are taken as long as: where A = (1 - ~)/ix > 1 B = ~ I (1 - ix) < 1. (2.4) The test is terminated as soon as (2.4) does not hold and the decision µ > µ0 is then taken if If the random variables X; all have the same variance cr2 the test may be carried out graphically, as indicated by Wald (8, p. 118-121]. c. Three-decision-test for the mean of a normal variate with known variance [7]. The sequential test with three possible decisions for the mean µ of a normal distri-bution with known variance, developed by Sobel and Wald has been described in [7] for the case that the variance of xi is a constant. This restriction is again dropped here. For the test two values µ0 andµ~ and four values µ1, µ2 , µ3 and µ4 must be chosen such that the three possible decisions being: (1). µ < µo (2). µ > µ~ 23 (3). µo ~ µ ~ µ~ . The intervals (µ1, µ2) and (µ3, µ4) are the indifference regions. The concepts "correct" and "incorrect decision" are defined as follows. TABLE 2. CORRECT AND INCORRECT DECISIONS correct I incorrect value ofµ decision < µ=µ1 (1) (3) and (2) µ1 < µ < µ2 (1) and (3) (2) < < µ2 =µ=µ3 (3) (2) and (1) µ3 < µ < µ4 (3) and (2) (1) > µ=µ4 (2) (1) and (3) The values µ0 and µ~ are of no further importance for the performance of the test. Suppose Tis the sequential test of section 2b for testing µ = µ1 against µ = µ2, then this test leads to a decision as soon as -n µ1 + µ2 / 2 1n B/(µ2 - µ1) < i"l:, (xi- -2-) cri < ln A/(µ2 - µ1) does not hold, where oc = the probability of accepting µ ~ µ0 according to T ifµ = µ1, {, = the probability of accepting µ < µ0 according to T ifµ = µ2. (2.5) Suppose furthermore that T' is the analogous sequential test for testing µ = µ3 againstµ = µ4 then T' leads to a decision as soon as n µ3 + µ / ln B' /(µ4 - µ3) < i"l:l (xi- - 2-~ crf < ln A' /(µ4 -µ3) does not hold, where oc' = the probability of accepting µ > µ~ according to T' ifµ = µ3, (,' = the probability of acceptingµ~µ~ according to T' ifµ = µ4. We introduce the following notation (2.6) (2.7) 24 then according to T a decision is taken as soon as b < Yn < a does not hold and according to T' as soon as b' <y' < a' n does not hold. From (2.7) it follows that a and a' are positive, band b' negative and If now the inequalities Yn > Y~· ~ b~b' ( a~a' (2.5a) (2.6a) (2.8) {2.9) are fulfilled, it follows from (2.8) and (2.9) that T' cannot lead to the decision µ > µ~ before the decision µ ~ µ0 has been found according to T and that T cannot give the decisionµ < µ0 before the decisionµ~µ~ has been given by T'. In that case only the following decisions according to T and T' are possible: I. T' gives the decisionµ~µ~ and T gives (at the same or a later step) the decision µ < µ0 or the decision µ ~ µ0 • 2i T gives the decisionµ~ µ0 arid T' gives (at the same or a later step) the decision µ ~ µ~ or the decision µ > µ~. The sequential test with three possible decisions is then defined as follows: Additional observations are taken as long as not both tests T and T' have given a decision. As soon as both tests are terminated a decision is taken according to the following rules: l. µ < µ0 if T' has given the decision µ ~ µ~ and T the decision µ < µ0 , 2. µ > µ~ if T has given the decision µ ~ µ0 and T' the decision µ. > µ~, 3. µ.0 < µ. ~ µ.~ if T has given the decision µ. ~ µ0 and T' the decision µ. ~ µ.~. If (2.9) does not hold there exists the possibility of accepting µ > µ~ according to T' and of afterwards accepting µ. < µ. { < µ.~) according to T. This kind of contradictory result is excluded by (2.9). 0 If the random variables X; all have the same variance cr2 the test may be carried out graphically [7]. 3. SEQUENTAL TEST WITH THREE POSSIBLE DECISIONS FOR THE COMPARISON OF TWO PROBABILITIES On the basis of the test of section 2 c a sequential test with three possible decisions for comparing two unknown probabilities p and p' may be developed as follows. To the variables a; and a'. (cf. section 1) one of the following transformations is • l applied ,, y = 2 arcsin -ym/M, 4 l y' = 2 arcsin ym/M y' = 2 arcsin V¼M y' = 1t-2 arcsin V¼M l y" = 2 arcsin ym/M y" =,Y2/M y" = 1t- ,Y2/M 25 for 0 < m < M, 5 form= 0, form= M, for0 < m < M, form= 0, form= M, (3.1) (3.2) (3.3) where m possesses a binomial probability distribution with parameters M and p. The transformation (3.1) is introduced by Fisher [3], the transformation (3.2) by Bartlett [1] and (3.3) is given in [6]. For further information about the transformations we refer to [5, p. 395-416]. Denoting the variables ai and a;, after their transformation, by ui and u; the sequen-tial test of section 2c is applied to the random variables i = 1, 2, ... , which possess, for large n; and n;, approximately a normal probability distribution with mean µ = 2 arcsin-yp-2 arcsin W =2 arcsin(-ypq'--yp'q) (q= 1-p; q' = 1 - p') (3.4) and variance a~= (n1• + 1)/n~ + (n'. + 1)/n'. 2• l l l l (3.5) Two values µ0 andµ~ and four values µ1, µ2, µ3 and µ4 ofµ must be chosen, with: fL1 < !Lo < !Lz < !L3 < µ~ < !L4 and four values cc, cc', 13 and 13' (all < ½) with (cf. (2.9)) llnB/(µ2 -µ 1) 2: lnB'/(µ4 -µ3), In A/(µ2 - µ) 2: In A' /(µ4 - µ3), where ~A=(l-13)/cc, B=13/(l-cc), (A'= (1-(3')/cc' , B' = 13'/(l-cc'). (3.6) (3.7) Having chosen these values the abovementioned test may be applied, leading to one of the decisions : ) 1. µ < !Lo• 2. µ > µ~, 3. !Lo 2: µ ~ µ~. We shall translate these decisions in terms of p and p'. Let ypq'-,Yp'q = 3, (3.8) (3.9) 4 Tables ofy =2 arcsinyx are given in [4,p. 70-7l]for x =0,000(0,001) 1,000, with yin radians. s Tables of y = 1 arcsiny¼n and y = 1t - 1 arcsiny¼n are given in [5, p. 406] for n = 10(1)50, ~ 26 then µ = 2 arcsin a (3.10) The functional relationship (3.9) between p and p' for given 82 consists (fig. 1) of the arcs P Q and RS of the ellipse: p2 + p'2- 2 pp' (1 - 2 82) - 2 82 (p + p') + 84 = o. (3.11) Choosing two values 80 and 8~ and four values 81, 82, 83 and 84 of 8 with 81 < 80 < 82 < 83 < 8~ < 84, (3.12) the decisions (3.8) are equivalent with the following decisions for 8 (3.10): ) 1. 8 < 80 , 2. 8 > 8~, 3. 80 ;£ 8 ;£ 8~ (3.13) and hence with the following decisions for p and p': I p 0 l 1. the point (p, p') lies above the arc RS of figure 1 with 8 = 80 , 2. the point (p, p') lies below the arc P Q with 8 = 8~, (3.14) 3. the point (p, p') lies on or between the arcs P Q and RS with 8 = 8~ and 8 = 80 respectively. (1-b:i.1) ' s / '- / ' '-/ '- / (1.1-bl) '- / Q '-'- / '- / / / '-/ '- / '- / '- / '- / '- ✓ / / '-'-/ "' / '-/ '-. / '-/ '-'-/ / '-. / '-/ '-. / '-. / "-/ "-p (~0) p FIG. 1. Functional relationship between p and p' for given value of 82• 27 The values 31, ~2, 33 and 34 may be chosen by means of fig. 2, where the arcs P Q and RS are given for several values of 3. One may also choose these values as follows: 1. Wald [8] uses the ratio u = pq' /p'q. On the line p + p' = 1, 3 can be expressed in terms of u: u = (I + 3)2 / (1 - 3)2, which is equivalent to ~ 3=0 if U=l ( 3 = (u + 1 - 2 yu) / (u - 1) if u -:i 1. Choosing four values for u with U1 < U2 < 1 < U3 < U4 p' FIG. 2, Functional relationship between p and p' for several values of 3. (3.15) (3.16) (3.17) 28 one finds four values for 8 such that 81 < 82 < 0 < 83 < 84. 2. On the line p + p' = l the equality 8 =p-p' holds. Choosing four values for p - p' one finds four values for 8. (3.18) (3.19) The four values of 8 (or u respectively) must furthermore be chosen such that (3.7) holds. Usually one will choose these values symmetrically, i.e. such that and to If (3.20) holds, (3.7) is equivalent to ~ B~B' ( A~A'. Remark It is not necessary that p and p' are constants. We only need a constant 8. REFERENCES (3.20) (3.21) (3.22) (3.23) [l] Bartlett, M. S. Some examples of statistical methods of research in agriculture and applied biology. "Suppl. to the JI. Royal Stat. Soc.", 4 : 2, p. 137-183, 1937. [2] Boer, J. de. Sequential test with three possible decisions for testing an unknown probability. "Appl. Sci. Res.", 3, p. 249-259, 1953. [3] Fisher, R. A. On the dominance ratio. "Proc. Royal Soc. of Edinburgh", 42, 1921/'22. [4] Hald, A. Statistical tables and formulas. New York, J. Wiley and Sons, 1952. 97 p. [5] Selected techniques of statistical analysis for scientific and industrial research and production and management engineering; ed. by C. Eisenhart, M. W. Hastay, W. A. Wallis a.o. New York, Mc Graw Hill, 1947. XIV + 473 p. [6] Sequential analysis of statistical data; applications: Section 3. Statistical Research Group of the Columbia University. New York, 1945. [7] Sobel, M., and A. Wald. A sequential decision procedure for choosing one of three hypotheses concerning the unknown mean of a normal distribution. "Ann. Mathern. Stat.", 20: 4, p. 502-522, 1949. [8] Wald, A. Sequential analysis. New York, J. Wiley and Sons, 1947. XII + 212 p. Resume: Le "Statistical Research Group" de l'Universite de Columbia a developpe un test sequentiel avec deux decisions possibles pour examiner l'egalite de deux probabilites inconnues, base sur la com-paraison successive de groupes d'observations. Dans le present article !'auteur decrit un test analogue avec trois decisions possibles - ce qui est plus utile en pratique - en employant la methode de MM. M. Sobel et A. Wald. Elle donne aussi un expose des methodes mentionnees par d'autres auteurs. 

A sequential test with three possible decisions for comparing two unknown probabilities, based on groups of observations

Abstract

Similar works

Full text

Available Versions

CWI's Institutional Repository

NARCIS