Abstracts of Papers Source: The Annals of Mathematical Statistics, Vol. 36, No. 3 (Jun., 1965), pp. 1073-1087 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2238226 Accessed: 17-12-2015 13:44 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematical Statistics. http://www.jstor.org This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions ABSTRACTS OF PAPERS (Abstractsof papers presentedat the CentralRegional meeting, Lincoln, Nebraska, April 1-3, 1965. Additional abstracts appeared in earlier issues.) 15. On Some c-Sample NonparametricTests for Location in the BehrensFisher Situation. P. V. RAMACHANDRAMURTY, Case Institute of Tech- nology. When the variances of the underlying populations differ, the usual rank tests for location in the c-sample problem (c _ 2) do not have the stated level of significance even asymptotically since the asymptotic variance of the test statistics depends on the unknown variances. An obvious way of overcoming this difficulty is by replacing the asymptotic variance by a consistent nonparametric estimator. The resulting test does not usually have the mini mum property of consistency for all alternatives because of the indistinguishability of the null hypothesis and the alternative. We show here that for the Kruskal-Wallis test the above method can be applied in such a way that the resulting test is consistent for all alternatives for which the underlying populations are symmetric; in fact, for all generalized Behrens-Fisher alternatives. When there are only two populations, the above method gives a test which is asymptotically equivalent to the studentised Wilcoxon test given by Sen. (Ann. Inst. Statist. Math. Tokyo 14. This test is shown to be better than a conservative test proposed by Pothoff (Ann. Math. Statist. 34). (Abstracts of papers presented at the Eastern Regional meeting, Tallahassee, Florida, April 29-May 1, 1965. Additional Abstracts appeared in earlier issues.) 12. Adaptive Statistical Proceduresin Reliabilityand MaintenanceProblems. JOSEPHL. GASTWIRTHand J. H. VENTER,Johns Hopkins University and PotchefstroomUniversity. (Invited address) In the present paper the following problem is considered: A "system" with an exponentially distributed lifetime is to be inspected at times t, < t2 < *-- . If inspection reveals that the system is in-operative, it is repaired, otherwise nothing is done. The problem is to choose the sequence {tJ in an optimal manner. Most of the previous literature is concerned with the choice of the times {tij when X (the failure rate) is known. In the present work we assume that X is unknown and several sequential inspection plans are proposed. These plans use the information as it becomes available from inspection to estimate X and thereby approach the plan that would be optimum if Xwere known. We discuss the problem of finding optimum sequential plans (called adaptive planas)when the objective is either to obtain the maximum limiting average information about the parameter X or to minimize the average expected loss. In this second case our loss function takes into account the cost of an inspection, the cost of repair or replacement and the cost incurred while the system is inoperative. 13. Principles for Ranking IndependentVariablesby Order of Importancein the General Linear Hypothesis Model. KLAus ABT, U. S. Naval Weapons Laboratory. Three principles for the ranking of individual, or groups of, independent variables ("IV's") by order of importance are stated and justified. These principles are applicable 1073 This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1074 ABSTRACTS to the general linear hypothesis model, i.e. to ordinary multiple regression problems as well as to non-orthogonal analysis of variance and covariance. The corresponding computational procedures are suitable only for large scale electronic computers. By reference to an NWL-coded IBM STRETCH program, it is illustrated how the combined application of the three principles can be used to screen IV's and, consequently, to establish a "significant model". The First Principle is that of ranking the IV's according to their prediction power for the dependent variable. As a measure of this prediction power, the tail area under the F-distribution curve to the right of the calculated F-value for testing the corresponding nullhypothesis is recommended. The Second Principle is that of executing the ranking process "reversely", i.e. in deleting the IV's from a postulated model by increasing order of prediction power. The concept of a "Compound" is introduced to describe a specified relation between the dependent variable and a set of IV's. The presence of compounds is shown to necessitate this reverse process. The Third Principle is that of "Admissibility" of IV's for the ranking process. For polynomial terms it is recommended to consider a term of a given order for ranking only after the respective higher order terms have been ranked. For general non-orthogonal ANOVA it is shown that main effects and interaction effects can uniquely be ranked only after the respective higher order effects have been ranked. 14. Small Sample Power for the One-SampleWilcoxonTest for Non-Normal Shift Alternatives. HARVEYJ. ARNOLD,Bucknell University. Klotz [Ann. Math. Statist. 34 (1963) 624] computed the power of the one-sample Wilcoxon rank-sum test for the hypothesis that the Median is zero against various shift alternatives for samples drawn from a normal distribution. This paper reports on computations for samples to size 10 from some non-normal distributions. The power for selected Type I errors a are compared with the power of a best signed-rank procedure obtained by sorting the probabilities in decreasing order for all possible sample configurations for fixed size N and adding up the first 100 a%. The non-normal distributions selected for study are the t distributions with degrees of freedom 2, 1, 2 and 4. The one-sample Wilcoxon rank-sum test, although powerful for normal shift alternatives, deteriorates badly in power for the long-tailed distributions studied as does the one sample t test. However the Wilcoxon test remains more powerful than the t test. The sign test is still more powerful than either Wilcoxon or t. No asymptotic results have been obtained. and 15. ConditionalDistribution of Number of Renewals. D. L. BENTLEY B. J. PROCHASKA,Pomona College and Clemson University. The conditional distribution of number of renewals in the interval (T - a, T- b], 0 < b < a, given a renewal at T = t is derived for a general renewal process. This result is then applied to the case b = 0+ for a type I or non-paralyzable counter with Poisson arrivals and constant dead time. 16. Bayes Sequential Design in the Two Action Case (Preliminiary Report). ROBERTBOHRER,Research Triangle Institute and University of North Carolina. Consideration of Bayes sequential design procedures which depend on observations only through posterior distributions is motivated. For the problem of deciding between two actions, the following design rule x* is proposed for use with given stopping and terminal decision rules. If, for a given posterior distribution, sampling is continued, then x* uses the experiment which would be preferredif one experiment were to be used for all succeeding This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1075 ABSTRACTS trials. Let pn denote the risk of the procedure which follows x* through n trials and subsequently performs the, then, best experiment at all remaining trials; let p* be the risk using x*. Then p >_ pn+1 , and, quite generally, p* = lim pn . For specified stopping and terminal decision rules, x* is proved asymptotically (as cost of experimentation approaches zero) optimum. 17. Effects of Sampling Frequency on Estimates of Regression Parameters when CorrelationAmongErrorsis Ignored (PreliminaryReport). VICTOR CHEW, RCA ServiceCompany. In regression analysis with correlated errors, the covariance matrix of the errors must be completely known, except perhaps for a scalar multiplier, if minimum variance unbiased estimates of the regression parameters are desired. Often this information is not available and it is not possible or feasible, as in real time trajectory computation, to estimate the covariance matrix. Standard least squares method, ignoring the correlation, is then used, sometimes at reduced sampling frequency so as to reduce the effects of the correlation. The problem of sampling frequency has been studied by Proschan (Siam Review3 230-235) for uncorrelated errors. Hoel (Ann. Math. Statist. 32 1042-1047)gives asymptotic results when the errors are generated by a stationary stochastic process. Grenander (Ann. Math. Statist. 25 252-272) has shown that for polynomial regression the standard least squares estimates are asymptotically efficient. The present paper gives small sample results on the effects of sampling frequency on standard least squares estimates when errors follow one of several postulated models. 18. The Varianceof the Numberof Zeros of a StationaryNormalProcess. J. D. Research Triangle Institute. CRYERand M. R. LEADBETTER, Let x (t) be a separable stationary normal stochastic process with zero mean, covariance function r (t) (with r (0) = 1 for convenience), and (integrated) spectrum F (X) having an absolutely continuous component. Let N denote the number of zeros of x (t) for 0 < t < T, and let X2 ff' X2dF(X). It is well known that &N = T(X2)i/r, the best conditions being given by Ylvisaker (Ann. Math. Statist., to appear). The second moment has been given by a number of authors but the sufficient conditions given for the validity of the result include the existence of a sixth derivative for r(t). In this paper FN2 is derived under conditions which are very close to the necessary ones. It is shown that the formula for &N2given for example in Rozanov and Volkonskii (Theor. Probability Appl., 6 (1961)) is valid if X2 < and, for all sufficiently small t, X2 + r"(t) ?< '(t) where '(t) decreases as t decreases to zero and is such that '(t)/t is integrable over [0, T]. 19. K-Sample Analogues of Renyi's Kolmogorov-SmirnovType Theorems Princeton University. MiKL6s CSORGO, Let Xji i = l, ... , n, , j = 1, ... , k, be k mutually independent samples of mutually independent random variables having a common continuous distribution function F(x). Let F., (x), j = 1, *- *, k, be the corresponding empirical distribution functions. We define N = n,/ j ni/ni) and let N oo mean that nj oo, j = 1, * -, k, so that ni/nj -Cj, 2, ... , k, where Cj's are constant for each j. Under these conditions we derive the limit distribution of the random variables Ni SUpa 0. If one lets n -* oowhile holding qi and q2 fixed and then inverts the information matrix, the result is the asymptotic variance-covariance matrix whose elements are the asymptotic variances and covariances of the joint maximum-likelihood estimators of the parameters. Tables are given of the coefficients of (1/n) times powers of the scale parameter 0 in the asymptotic variances and covariances for the cases qi = 0.000(0.005)0.025, q2 = 0.00(0.25)0.75 for both Weibull and gamma populations with shape parameters 1, 2, and 3, omitting cases for which q- = 0 when the shape parameter is 1 or 2. 22. Semifolding sn-k Design. PETERW. M. JOHN, University of California, Davis. Consider a 2n-k fraction such that the main effect A is not a defining contrast but at least one interaction containing A is a defining contrast. Semifolding on A consists of augmenting the design to a 3 (2n-11-1)fraction by repeating at low A the points previously run at high A. If the original design is a form letter plan semifolding will give estimates of the 2 fi containing A. This procedure will not isolate the 2 fi from a three letter plan. 23. A Chi-Square Decision Procedure Using Prior Information.WILLIAME. LEVER,Florida State University. Let U be an observation from the non-central chi square distribution with S degrees of freedom and non-centrality parameter X. Consider the decision procedure, make decision This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1077 ABSTRACTS doif X > A and make decision d1if X > A, given the loss function; L(X,do) = 0 = kowo(X,A) L (X, di ) = kiwi (X, A) =0 if X < A ifX> A; if X _ A ifX>A. If it is assumed that X has an a priori distribution f(X) such that wi (X, A) f(X) is bounded (for (O < X < oo) and i = 0, 1) and that wo(X, A) = Iwi(X, A)1 for O < X < A. Then a decision procedure is developed so that the decision dois taken if U < C and the decision di is taken if U > C, where C is an appropriate constant. 24. Functions of Finite Markov Chains. FREDERICK State University. W. LEYSIEFFER, Florida N - 1, positive initial Let X(t) be a standard Markov chain having states 0, 1, *, probabilities and transition probability matrix P (t) = exp (At). Assume A has distinct eigenvalues. With these properties, X (t) is termed a basic Markov chain. Let f be a function from the state space of X (t) onto 0, 1, --, M - 1 where M < N. Consider the process ... 0,letsbeasequencesi< 1 < i < n. Then (y, s) is termed a sequence pair for Y(t) and the joint distribution function associated with this sequence pair is given by p(y, s) = P[Y(si) = yl, 1 5 i < n]. A characterization for p (y, s) is found leading to the definition of processes of exponential type. Such a process is characterized by joint distribution functions which are linear combinations of exponential terms involving distinct quantities Pk. The order K, of such a process is the minimum number of Pkappearing in the representation for p(y, s) for all sequence pairs (y, s). It is shown, functions of basic Markov chains are of exponential type. Sufficient conditions for a process to be a function of a basic Markov chain with a state space of K points are given. Y(t) = f[X(t)] is a Markov iff K = M. 25. Measures of Dispersion.A. M. MATHAI, McGill University. In this paper an attempt is made to give an axiomatic development of the concept of dispersion in a statistical population. Multivariate analogue and related topics, the concept of distance between two populations and criteria for testing various statistical hypotheses are also examined in the light of the proposed axioms. General principles for estimation and testing hypothesis are also given. Statistical analysis is treated as a study of dispersion in a statistical population. 26. Latticepaths, Kolnogorov-SmirnovStatistics and Compositions.SRIGOPAL MOHANTY, McMasterUniversity. In this paper, minimal lattice paths from the origin to a point (m, n) not crossing two fixed boundaries have been enumerated. With the help of this result, the distributions of Kolmogorov-Smirnov statistics Dm,n and DM,n have been obtained and the number of (n + 1)-compositions of m that are dominated [Narayana and Fulton "A note on the compositions of an integer" Canad. Mdth. Bull. 1 (1958)] by a given (n + l)-composition of m and dominate another given (n + 1)-composition of m has been determined. This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1078 ABSTRACTS 27. A Simple Proof of the ConservativeCharacterof the WilcoxonProcedures in the Discrete Case. GOTTFRIED E. NOETHER, Boston University. A discrete sample may be assumed to have arisen by "projection" from a continuous sample. If the method of "projection" is suitably chosen, it follows immediately that a Wilcoxon confidence interval containing the translation parameter of interest for the continuous sample, does so also for the discrete sample. Thus the confidence coefficient associated with the discrete case is at least as large as in the distribution free continuous case. 28. Effect of Non-Normalityon Two-SidedVariablesSamplingPlans and TwoSided ToleranceLimits which ControlPercentages in Both Tails of the Report). J. N. K. RAOand D. B. Normal Distribution (Prelimninary OWEN,Graduate Research Center of the Southwest. Owen [Control of percentages in both tails of the normal distribution. Technometrics6 (1964) 377-387]has given two-sided variables sampling plans and two-sided tolerance limits which control percentages in both tails of the normal distribution. In this paper a mathematical investigation is made to examine in what way the OC curves of Owen's variables plans and tolerance limits are affected when the characteristic follows a non-normal distribution specified by the third approximation to the law of error (Edgeworth's Form). The approach used here is similar to that of Srivastava [Variables sampling inspection for nonnormal Samples. J. Sci. Engrg. Res. 5 (1961) 146-152]for one-sided variables sampling plans. 29. On Stochastic Approximation Processes. M. T. WASAN,Queen's University, Ontario. The mean square error of a stochastic approximation process of type A3 as defined in (*) has been considered for a finite number of observations by taking a regression function M (x) of the form at - , (x - 0)3. This enables one to make a probability statement about an estimate of 0 by Tchebichev's inequality. For computation, the recursion relation is indicated. For given sequence of positive number {c,}, the criteria are developed for optimum choice on {an} a positive number sequence required for definition of As . It is also shown that the mean square error tends to zero as the number of observations tends to infinity under suitable conditions. One can easily consider similarly the behaviour of the mean square error for other stochastic approximation processes defined in (*) ["On a class of Stochastic Approximation Processes" (1956) Ann. Math. Statist. 27 1044-10591by D. L. Burkholder. 30. A Compound Generalized Extreme Value Distribution. SATYAD. DUBEY, Procterand GambleCo. (By title) A 3-parameter distribution, defined over the whole real line, has been considered in this paper which includes the extreme value distribution of double exponential type as a special case. This distribution is compounded with a gamma distribution. The resulting compound generalized extreme value distribution is shown to include the logistic distribution of Gupta and Shah [Ann. Math. Statist. (1963) 681] as a special case. It is further shown that the compound distribution of this paper yields an intensity function or hazard function which can be recognized as a generalizedlogistic curve. Consequently, this compound distribution may be considered to be a generalizedlogistic distribution. The moment generating function of the generalized logistic distribution has been derived which involves the usual beta function. This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1079 ABSTRACTS 31. The HyperbolicSecantDistribution.W. L. HARKNESSand M. L. HARKNESS, PennsylvaniaState University. (By title) The two-parameterfamily of distributionshaving characteristicfunctions +(t) [sechat]P,a > 0, p > 0, will be calledthe hyperbolicsecant distributions.Thesedistributions have arisenin characterizationsof families of distributionsin which certain types of statistics have quadraticregressionon a secondstatistic [cf. Laha and Lukacs,Biom. 47 (1960)335-343and Bolger and Harkness,Ann. Math.Statist. 36 (1965)].Some of the importantpropertiesof these distributionsare considered,includingmoments,cumulants, and estimationof parameters. 32. Some Necessary Conditionsfor the Existence of PartiallyBalancedArrays. J. N. SRIVASTAVA,University of Nebraska. (By title) Partiallybalancedarrays (PBA) are well known.In Bose and Srivastava[Sankhya26 (1964)],a detailedtheoryof connectedassociationschemesand linearassociationschemes andlinearassociativealgebrasis developed.Let T be a PBAwith s symbols,m constraints and strengtht. For simplicityconsiderfirst the case t = 4, s = 2. Let ui be the number of columnsin any 4-rowedsubmatrixof T, such that in each column,the symbol1 occurs exactly i times, (i = 0, ***, 4). A new methodfor testing the existenceof T with given parameters ,uiis as follows. Consider using T for the estimation of all effects involving 2 or fewernumbersof factorsin a 2" factorialand let the normalequationsbe ML = z, where z dependsupon the observationsand L is the usual vector of estimates. In this paper, firstly the roots of M are explicitly obtained as functions of the ,ui . Since the existence of T implies that M is at least positive semi-definite each root of M is nonnegative. This gives a set of new and rather stringent conditions to be satisfied by the ti , if a PBA with these parameters exists. For, say t < 4, submatrices of M give appropriate new results. The same method, with slight restrictions, is useful for any s > 2. 33. A Group-Testing Problem (Preliminary Report). S. KUMAR,University of Minnesota. The problem is to classify each of the N given units into one of the three disjoint categories by means of group testing. We shall call the three categories good, mediocre and defective. In group testing, a set of x units is tested simultaneously as a group with one of the three possible outcomes: (i) all the x units are good, (ii) at least one of the x units is mediocre and none is defective, (iii) at least one of the x units is defective. It is assumed that the N units can be represented by independent trinomial random variables with q2 of being good, mediocre and defective, respectively. probability qi , q2 and q3 1 - qThe problem is to devise a procedure, for known values of q, , q2 and q3, which minimizes the expected number of tests E(T) required to classify all the N units as good, mediocre or defective. A procedure, which is optimal in a certain class of procedures, is proposed. Furthermore, we prove the following THEOREM: A necessary and sufficient condition that the optimal group test plan among all proceduresis unique and tests one unit at a time is that (1) q, < [(5q22- 6q2 + 5)1 -2 + 1]. If the inequality is reversed in (1), then ET < N for the optimal group test plan and if < holds then ET = N. Some further generalizations are under consideration. This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1080 ABSTRACTS (Abstracts of papers to be presented at the WesternRegional meeting, Berkeley, California, July 19-21, 1965. An additional abstractappeared in an earlier issue.) 2. Analysis of the N-Way ClassificationDesign Model: Non-OrthogonalLayout with One or More ConcomitantVariables (PreliminaryReport). BONITA J. PEURA, U. S. Navy Electronics Laboratory. This paper is concerned with the analysis of main effects and interactions when concomitant variables are present. The regression coefficients of the concomitant variables are assumed to be different for all treatment combinations, thereby giving a more general form of the analysis of covariance. Statistics are constructed by Hotelling's T2, which are then used for testing the various hypotheses. The analysis which is derived is applicable to both orthogonal and non-orthogonal layouts. 3. U-Statistics and Combination of Independent Estimators of Regular Functionals. PRANAB KUMAR SEN, University of California, Berkeley. Suppose we have c independent samples drawn from c different populations with cdf's F1 , * **, FCrespectively. We are interested in estimating an estimable functional 0 = 0 (F). Under the homogeneity of the cdf's F1 , ***, FC, we may either combine the c different U-statistics (which are the MVU estimators) from the c samples into a single statistic having some optimal properties or we may use the pooled sample U-statistic corresponding to the kernel of 0(F) (and this will again be the MVU estimate of 0(F)). Here various properties of these two estimators are studied and contrasted in both the situations, where F1, *.., F, are homogeneous and they are not. Some useful variance inequalities are obtained and their applications considered. 4. On a Test of the Homogeneity of Correlation Coefficients. M. S. SRIVASTAVA, University of Toronto. (By title) In this papera test basedon the maximumof the squaredsamplecorrelationcoefficients .. , Pk of k inde, rk2 is proposedfor testing that the correlationcoefficientsP1, ... pendentbivariatenormalpopulationsare all zero. It has been shownthat the proposed test has the monotonicityproperty;a propertyenjoyedby the likelihoodratio test also. However,it has an added (overthe likelihoodratio test) featureof simplicityin calculation; the ease with whichthe percentagepoints can be obtained. Althoughquestionable,but if one restricts to proceduresbased on r.2 = maxl 2, and c is an integer, c _ 2; (D) (s - 1)2, X2 = 2, where s is V = S2, b = 2s2, nli = 2(s - 1), X1 = 1, r = 2s, k = s, n2 = a prime power, s = pn > 2. (S -1)2, X2 = 2. All AdmissibleLinearEstimates of the Mean Vector (PreliminaryReport). ARTHURCOHEN,Rutgers-TheState University. Let y be an observation on a p X 1 random vector which is distributed according to the multivariate normal distribution with mean vector 0 and covariance matrix I. Consider the problem of estimating 0 when the loss function is the sum of the squared errors in estimating the individuals components of 0. Let G be a p X p matrix. Then it is proved that estimates Gy are admissible if and only if G is symmetric and all its latent roots, gi say, i = 1, 2, *.. , p satisfy, 0 < gi < 1, with at most two of the roots equal to 1. The proof regarding the latent roots uses essentially the ideas of Karlin (1958) and Stein (1955) while the proof of symmetry utilizes a theorem of Sacks (1963). The result stated answers the question raised by this author, Cohen (1965). Generalizations with regard to the underlying distribution and with regard to restricted classes of estimates are also discussed. 3. Linear Combinations of Non-Central Chi-Squared Variates. S. JAMESPRESS, The RAND Corporation. Let U, V be positive semi-definite quadratic forms in normal variates. Then U, V are each distributed as a linear combination (with positive coefficients) of non-central chisquared variates. Let T = U - V. This paper is concerned with the determination of the distribution of T for known values of the parameters. A new mixture representation is presented for the cdf of U in terms of Poisson weight coefficients. The cdf of a positive linear combination of non-central F variates is derived and an exact representation of the density function of T is obtained in terms of confluent hypergeometric functions. Finally, it will be shown how to obtain approximate percentage points of T for the tails of the distribution. (Abstracts of papers not connectedwith any meeting of the Institute.) 1. The Queue GI/M/3 with Service Rate Dependingon the Number of Busy Servers. U. N. BHAT,Universityof WesternAustralia. Let to , t1 , t2, ***be the epochs of arrival in a three server queueing system with a general inter-arrival time distribution and negative exponential service times. Also let the service rate depend on the number of busy servers so that, the service is slowed or accelerated when all the servers are not busy. Let Q(t) be the number of customers in the system at time t and define op (n)(t) = Pr{Q(t) = j, t, < t < t+., Q(t4) > 0 (r = 1, 2, *-- X n) < t,Q(tr) > 0 (r = 1, 2, , n - 1) Q(to) = i, to = 01 andRi(n)(t) = Pr(Q(t.) = 0, tn% Q(to) = i, to = 0}. Transforms of these probabilities are obtained using recurrence relations and they involve the roots of an equation of the type Z2 = wo (0 + 3X - 3Xz) in lzl < 1 where ,6(0) is the Laplace-Stieltjes transform of the inter-arrival time distribution. The probabilities oP(n)(t) and Ri('(t) give the busy period and the busy cycle distributions as well as the transient behaviour of the process Q(t). The method can be easily extended to the s server case whose equilibrium behaviour (of the imbedded Markov chain) has been given by D. G. Kendall [Ann. Math. Statist. 24 (1953) 338-3541. This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1082 ABSTRACTS 2. On a Class of Optimal Stopping Problems. Y. S. CHOW and H. ROBBINS, Purdue University and Columbia University. For fixed n _ 1 and a > 0 let x1, ***, xn be independently and uniformly distributed on [0, 1]. We observe the xi sequentially and must stop with some xi, 1 < i < n; if we stop with xi we lose jaxj . Denote the minimum possible expected loss for all stopping rules by v. It is shown that for 0 5 a < 1, v - 2 (1 - a)n-1, and that for a = 1, v - 2 (log n)-1 as n -4 oo. For a > 1, v tends to a positive limit as n -? oo, which is t for a greater than some 1 < a* < 2. The existence and value of v for untruncated rules is also considered. 3. Sufficiency for Selection Models. D. A. S. FRASER,University of Toronto. Minimal sufficiency is examined for selection models: fixed density function truncated to a carrier set X(0) that depends on the parameter. The minimal sufficient statistic can be given as nlX(0) containing x or as n (x) = {0 x e X (0)}. Regularity conditions are introduced: the boundary is formed by a finite number of conneeted surfaces with a normal vector and differentiability with respect to 0. For any x form the vector of derivatives with respect to the coordinates of 0; let c(x, 0o) be the related vector of unit length, call it a structural vector. The rank of a distribution at 00is the number of different structural vectors if each intersection of differentiable boundary segments is convex and is + co otherwise. THEOREM: If a distribution has finite rank s at Gothenfor a sample of n the local minimal sufficient statistic has dimension at most s, and the dimension s is attained at some point if n _ s. This is used to derive some Koopman-Darmois-Pitman theorems and provides the basis for analyzing models having variable carrier and variable relative density. 4. Bayesian Single Sampling Attribute Plans for Discrete Prior Distributions. ANDERSHALD,University of Copenhagen. The paper gives a rather complete tabulation and discussion of a system of single sampling attribute plans obtained by minimizing average costs under the assumptions that costs are linear in p, the fraction defective, and that distribution of lot quality is a double binomial distribution. The optimum sampling plan (n, c) depends on 6 parameters (N, pr, Ps , pI , P2, W2), where N denotes lot size, (pr, p8) suitably normalized cost parameters, and (pI, P2 , W2) parameters of the prior distribution. A procedure to obtain the exact solution of the problem has been developed in a previous paper and this procedure is used for computing a set of master tables in which pr = P. = 0.01 and 0.10, w2 = 0.05, (plp2) take on suitably chosen values in relation to the value of pr, and 1 _ N < 200,000. The properties of the optimum plans are studied, and simple conversionformulas are derived which makes it possible to find the optimum plan for an arbitrary set of parameters from a plan in the master tables with a "corresponding" set of parameters. The main tool for this investigation is the asymptotic expressionsfor the acceptancenumberand for the sample lnln N +ln X+ 2ln poo)+ o (1),where size, viz. c = npo + a + o (1) and n = (po-1 (ln Npo and qpoare functions of (pI , P2), only, whereas a and X depend on the other parameters also. Efficiency of various other systems of sampling plans is studied in relation to the present model and some general recommendations are made. 5. Rank Tests for Randomized Blocks when the Alternatives Have A Priori Ordering. MYLESHOLLANDER, Stanford University. Let Xij (i = 1, ***, n, j = 1, * , k) be independent random variables with Xij having the continuous distribution Fj(x -bi) where bi is the nuisance parameter corresponding This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1083 ABSTRACTS to "block" i. Also, let Y(i)V= IXiXi,J and R()? = rank of Y(i) in the ranking from = least to greatest of [Y(2]1- . Furthermore, let $1 R(iv)Vi) where 4,(i) = 1 if Xiu < Xi , and 0 otherwise. To test HO:F1= F (unknown), the statistic Y = >A V*. However, it seems likely that V(S) has only countably many distinct values on the space of unbiased S, and that as among the corresponding sets of limits x*, the set for which V(S(x*)) = V* has positive, if not maximal, probability. If this is true, then repeated runs of the k-means would yield a k-partition S with V(S) arbitrarily near V*. Other applications of k-means occur in pattern recognition, in data reduction of large samples of n-dimensional data, in coding problems, and with certain modifications, in the description of categorizing behavior. P(Si), n =o 10. On Some Tests of Homogeneityof Variances.M. L. PURI,CourantInstitute of MathematicalSciences. For testing the equality of c continuous probability distributions on the basis of c independent random samples, various non-parametric tests for dispersions are offered. These tests, include among others, the multi-sample analogues of the two-sample normal-scores test of dispersion and the tests considered by Ansari and Bradley [Ann. Math. Statist. (1960)], Klotz [Ann. Math. Statist. (1962)], Mood [Ann. Math. Statist. (1954)], and Siegel and Tukey [J. Amer. Stat. Assn. (1960)]. Under suitable regularity conditions, it is shown This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1085 ABSTRACTS that these tests have limiting non-central chi-square distributions with c - 1 df. The asymptotic relative efficiencies (in the Pitman sense) of these tests relative to one another and the 3f test (Scheff6, The Analysis of Variance, pp. 83-87) are obtained and are shown to be the same as for the two-sample problem [cf. Ansari-Bradley and Klotz cited above]. 11. Orthonormal Bases of Error Spaces and Their Use for Investigating the Normality and the Variances of Residuals. Institute of Agricultural Research, Rehovot. JOSEPH PUTTER, Volcani If, under the assumptions of a linear model, all the residual errors are independent normals with the same variance, then the same also holds for the elements of any orthonormal basis of the error space of the model. Therefore, these elements can be used for testing the normality and overall variance homogeneity of the residuals. Special orthonormal bases have additional properties, which make them suitable for investigating problems of partial variance homogeneity. The general considerations involved in this approach are explained and illustrated. The case of the unreplicated two-way layout is investigated in some detail, yielding new estimates of the error variances and an exact test of overall variance homogeneity under the assumption of one-way homogeneity. The results are readily extended to general complete multi-factor layouts. Some orthonormal bases are also constructed for the case of linear regression on one variable. 12. On a Stochastic Model of an Epidemic. College of Swansea. C. J. RIDLER-ROWE, University The model is obtained by specifying the birth and death rates for particular states of the population, and it is the irreducible case of a model considered by M. S. Bartlett (Hotelling Festschrift, Stanford Univ. Ptess (1960) 89-96). From a general theorem (see G. E. H. Reuter, Proc. Fourth Berkeley Symp. Math. Statist. Prob. 2 (1961) 421-430) the expectation of the time at which the infectives first die out is the least positive solution of a certain inequality. By finding sufficiently good solutions of this inequality, and by comparing the process with other simpler processes, it is found that the expectation of this extinction time is asymptotically proportional to log(m + n) as m + n -- 00, where n (>O) and m are the initial numbers of infectives and susceptibles respectively. The stochastic process 7r; = considered is non-dissipative, i.e. 1, where-ri is the limit as t -+ co of the transition probability pj (t). By considering the equation Ei 7riqii = 0, where {qi;J is the set of birth and death rates of the process, it is found that Es 7ri converges geometrically on the state space; furthermore, if v' is the birth rate of the susceptibles, then as v -+ 0+, the limits xj are shown to converge to those for the reducible case where v = 0. Es 13. On a Stochastic Model of the Competition Between Two Species. C. J. RMLER-ROWE, University College of Swansea. This model was suggested by G. E. H. Reuter (Proc. Fourth Berkeley Symp. Math. Stat. Prob. 2 (1961) 421-430). Let am, ymn be the bi-rthand death rates respectively of a species of size m, and let ,Bn,amn be the birth and death rates respectively of a species of size n. The functions considered are functions of position which are characterized by being the least positive solutions of certain inequalities. By finding sufficiently good solutions of these inequalities, and by comparing the process with other simpler processes, the following results are obtained. If X (m, n) is the probability that the species of size n will survive the other species, then X (m, n) - 4)1(n-y - ma)/[ms (-y + S)]i} -+0 uniformly as m + n -*oo , exp (-s2/2) ds. A result of G. E. H. Reuter in his above mentioned where4 (t) = (21r)-f2 This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions 1086 ABSTRACTS paper is extended to show that the expectation of the time at which one of the species first becomes extinct is bounded, independently of the initial state of the process. It is also shown that, if E(m, n) is the expectation of the number of births and deaths that occur before one of the species becomes extinct, where the initial numbers of the species are respectively m and n, then E(m, n) - (-y + 6) min (m/-y, n/5) uniformly as miiand n -4 00. 14. On First Hitting Times of Some Recurrent Two-DimensionalRandom Walks. C. J. RIDLER-ROWE,University College of Swansea. A random walk in discrete time and on a two-dimensional lattice is considered under the assumptions that (i) each step has zero first moment and a finite moment of ordergreater than two, and (ii) starting from any given point, each point of the lattice may be visited with positive probability. Let the random variable T(x) be the time at which the random walk first visits the origin, starting from the point x. By finding the generating function of the distribution function of T(x), and then applying a Tauberian argument of a type due to J. Karamata, it is found that, for each a 2 2, Pi T(x) ? jzla' -- 1 - 2a-1 as Jxl - >oo, where lxl denotes the length of the vector x. It is shown that this result may be proved similarly for the random walk in continuous time and on a two-dimensional lattice under certain similar assumptions; and moreover in both cases the result holds if the origin is replaced by any finite set of points. A related property of the asymptotic behaviour of first visits to a fixed line is deduced for a restricted class of random walks in discrete time and on a three-dimensional lattice. 15. Limit Distributions of an Age-Dependent Branching Stochastic Process. ISMAIL N. SHIMI, University of California,Riverside. An age-dependent branching stochastic process is considered. Xg(t) is the number of particles in the population at time t and N is the initial size of the population. If the parameters of the process change, as N -t 0o, in a way similar to the Poisson approximation of a binomial distribution, then if t is fixed (not necessarily large) and N is allowed to tend to infinity a limiting distribution of the stochastic process XN(t) - N is obtained, and shown to be the distribution of a stochastic process with independent increments. This limiting distribution can be used to find an approximation to the distribution of XN(t) when t is fixed (not necessarily large) and N is large. This considerably improves a result previously obtained [Ann. Math. Statist. 35 557-5651where age-dependence was not considered. 16. ProductEstimators. S. K. SRIVASTAVA,LucknowUniversity. The present paper considers the product estimator suggested by Murthy (Sankhya 26 69-74) for the estimation of the population mean of a finite population of a character y using an auxiliary character x. The product estimator takes the form gs/X, where g and x are simple random sample means of y and x and X is the population mean of x. Exact formulae for the bias and mean square error of the estimator have been obtained. The asymptotic distribution of the estimator has been shown to be normal under certain very mild conditions. The results for stratified random sampling and sampling with varying probabilities of selection have also been obtained. University 17. On a Property of the Binomial Distribution. K. SUBRAHMANIAM, of WesternOntario. We have established a uniqueness property of the binomial distribution in reference to the bivariate discrete distributions. We have studied, in particular, (i) the bivariate Poisson This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions ABSTRACTS 1087 with the probability generating function (pgf ) (Zl , Z2) = exp [Xi(zi -1) + X2(Z2 -1) + X12(ZlZ2 - 1)1; (ii) the bivariate binomial distribution with the pgf 4)(zl , Z2) = [1 + X1(zi - 1) )I; (iii) the bivariate negative binomial with the pgf 4 (zI , Z2) = + X2(Z2 - 1) + X12(zizI-1 + X12(z1z2- 1)]-r; (iv) the bivariate logarithmic series distri[1 + XI (zl - 1) + X2(z2-1) ln [Xo+Xl(z - 1) + X2(Z2 - 1) + X12(ZlZ2 - 1)]/ln [Xo]. bution with the pgf 4(zi , Z2) It has been established that the univariate binomial distribution plays a unique role in the conditional distribution of y, given x = X, for all the distributions mentioned above. We have also established the following THEOREM: Let x and y have the bivariate discrete distribution with the pgf 4(z1 , Z2). Also let the conditional distribution of y, given x = X, be theconvolutionof therv xi with therv X2 . Then4 (z1, Z2) = X==0 P (X)410(z2/X)4)2 (Z2/X)ZIx) whereo) and 4)2 are the pgf's of the conditional distributions of XI and X2 given x and P(X) is the probability function of the rv x. 18. Decompositionof a Mixture of Two Poisson Distributions.K. SUBRAHMANIAMand A. K. MD. EHSANESSALEH,University of WesternOntario. This paper deals with the problem of estimating the parameters (Xl , X2and p) in a mixture of two Poisson distributions P(x) = pe-)o (X,/xx) + (1 - p)e-X2(X2z/x) (0 < XI < X2 < oo;0 < p < 1) by the method of factorial moments. The variance-covariance expressions of the estimators are obtained. Also the asymptotic relative efficiencies (ARE) of the estimators.and the joint asymptotic efficiency (JAE) of the estimation relative to maximum likelihood estimation are presented for several combination values of Xl, X2 and p. Massa19. Bias of the One-Sample Cramer-von Mises Test. RORYTHOMPSON, chusetts Institute of Technology. There are one-sided alternatives to any continuous hypothesized population for which the probability of rejection by a test based on a statistic equivalent to E (Fo(Xk) -Ek)2 is less than the size for the Ek0's positive. This content downloaded from 130.194.20.173 on Thu, 17 Dec 2015 13:44:13 UTC All use subject to JSTOR Terms and Conditions