Showing papers in "Journal of the royal statistical society series b-methodological in 1962"

Further Results on Tests of Separate Families of Hypotheses

Abstract: SUMMARY It is required to test a composite null hypothesis. High power is desired against a composite alternative hypothesis that is not in the same parametric family as the null hypothesis. In an earlier paper a modification of the Neyman-Pearson maximum-likelihood ratio test was suggested for this problem. The present paper gives some general comments on the formulation of the problem, a general large-sample form for the test, and, finally, a number of examples.

A Waiting Line with Interrupted Service, Including Priorities

Abstract: SUMMARY A single-server system with stationary compound Poisson input and general independent service times, the latter being subject to random interruptions of independently but otherwise arbitrarily distributed durations, is studied. For a variety of service-interruption interactions (including the preemptiverepeat) the distributions of busy period duration, of queue length, and of waiting time are characterized by transforms and by moments. Applications are made to priority scheduling problems. MANY situations in which waiting lines develop are characterized by the occurrence of interruptions in customer service. Such interruptions may be caused by breakdowns of a machine that provides service, for example, an electronic computer. Also, if certain customers are assigned priority, then the appearance of one of these may bring about an interruption in the servicing of low-priority customers. In practice it would not be surprising to find systems that experience interruptions of both sorts. In this paper we consider the effect of service interruptions upon a waiting-line process of the following kind: customers appear in accordance with a stationary compound Poisson process (i.e. bunches of customers arrive randomly), and are served in turn by a single facility. The basic customer servicing times are independently, identically, but otherwise arbitrarily, distributed. Interruptions appear at random, in the sense that, if the system is currently free of interruption, the time until the next interruption occurs is exponentially distributed. Interruption durations are identically, independently and arbitrarily distributed. Without interruptions the process described has been discussed by Gaver (1959); the present paper is an adaptation of the approach of the latter paper to the needs of the interruption problem. Previous treatments of similar problems, emphasizing priorities, have been given by Cobham (1954), Stephan (1956), Kesten and Runnenberg (1957), White and Christie (1958), Morse (1958, Chapter 9) and Miller (1960). The influence of service interruptions upon waiting-line behaviour cannot be investigated without specifying in detail the interaction between the interruption process and the service process. Throughout the present paper it will be assumed that all interruptions occurring during a particular customer's service period must take effect during, or immediately after, that period. Thus interruptions may be preemptive, summarily breaking in upon a service in progress, or postponable to the end of that period, but not beyond. If interruptions are preemptive it may be possible to resume service from the point at which interruption took place when the interruption is cleared; such an interaction (between service and interruption) is called

On Queues in Heavy Traffic

Abstract: SUMMARY We say that a single server queue is in "heavy traffic" when the traffic intensity p is less than, but very near, unity. Then the equilibrium waiting time w will be large, and it is proved that, under certain conditions, the It is therefore of some interest to ask if there are any results which hold under somewhat weaker conditions. It is not to be expected that these will be results valid in general, but rather that they will hold in some limiting case. The limiting case to be considered here is that usually known as "heavy traffic". A common way of formulating queueing problems with a single server is in terms of a traffic intensity p defined as the average number of customers arriving during a service period. For p > 1 the queue will be stable, while for p > 1 its length will increase indefinitely as time goes on. In particular, when p < 1 an equilibrium waiting-time distribution will exist, and as p approaches 1 from below, this waiting time will become long. This situation, when p is less than, but near, unity, we describe as "heavy traffic". The main conclusion of the present paper may roughly be stated as follows: If w is the equilibrium waiting time, then the distribution of (1- p)w is asymptotically negative exponential as p tends to 1. The parameter of this exponential distribution is given in terms of the means and variances of the input and service processes. In the particular case of the queue GI/G/l the result has been proved by the author (1961). We shall first indicate heuristically why this result is to be expected, and what conditions it is necessary to impose. A rigorous proof is then given for a wide class of queues, and this is shown to include a number of cases of interest. The correspond- ing analysis for the queue length and the busy period is also briefly discussed. All the work in this paper may be re-interpreted in the usual way in terms of random walks or semi-infinite dams.

On a simple procedure of unequal probability sampling without replacement

Abstract: GIVEN is a finite population of N units with characteristics Yt (t = 1,2, ..., N) whose total Y = Yi +Y2 + . .. +YN is to be estimated. If a sample of size n is to be drawn from such a population, it is often advantageous to select the units with unequal probability. For example, such a procedure may be useful when measures of sizes xt are known for all N units in the population which are positively correlated with the characteristics yt. In such cases, one may utilize the knowledge of the xt by selecting units with probabilities proportional to sizes xt, although this is, of course, not the only way of using the known xt. Of the literature on sampling with unequal probabilities and without replacement we mention papers by Horvitz and Thompson (1952), Narain (1951), Yates and Grundy (1953), Des Raj (1956) and Hartley and Rao (1962). There are some limitations, of varying importance, attached to all these methods. Briefly speaking, the method of Horvitz and Thompson (1952) is applicable only under severe restrictions on the prescribed probabilities, the unbiased procedures of Narain (1951), Yates and Grundy (1953) and Des Raj (1956) require a cumbersome evaluation of working probabilities, and Hartley and Rao (1962) give only asymptotic variance formulae for the estimates of Y for large and moderate size populations N. The present method is an attempt to avoid all these disadvantages at the expense of a slight loss in efficiency. It has the following properties: (i) It permits the computation of an estimator of the population total which has always a smaller variance than the standard estimator in sampling with unequal probabilities and with replacement. (ii) Unlike the unbiased procedures of Narain (1951), Yates and Grundy (1953) and Des Raj (1956), the present method does not entail heavy computations, even for sample size n > 2, for drawing the sample or computation of the estimator and its variance estimate.

Confidence Sets for the Mean of a Multivariate Normal Distribution

Abstract: An attempt is made to determine confidence sets for the mean of a multivariate normal distribution with known covariance matrix that take advantage of the fact that the sample mean is not the best estimate when the loss is a nonsingular quadratic function of the error vector. Only the case of high dimension is considered. The geometrical size and shape of the confidence sets, the probability of covering false values, and the relation to posterior probabilities are studied, unfortunately somewhat incompletely.

A Note on a Generalized Inverse of a Matrix with Applications to Problems in Mathematical Statistics

Abstract: Some years ago the author defined a pseudo inverse of a singular matrix and used it in representing a solution of normal equations and for obtaining variances and covariances of estimates in the theory of least squares (Rao, 1955). This provided a unified approach to least squares theory, including the case when the normal equations become singular. This note attempts to collect a few mathematical results, some of which are known in literature, associated with the inversion of singular and rectangular matrices, and to indicate briefly their use in problems of mathematical statistics.

Efficient Estimates and Optimum Inference Procedures in Large Samples

Abstract: The concept of efficiency in estimation is linked with closeness of approximation to the derivative of log likelihood, which plays an important role in statistical inference in large samples. Various orders of efficiency are defined depending on degrees of closeness, and properties of estimates satisfying these criteria are studied. Such measures of efficiency appear to be more appropriate than the one related to asymptotic variance of an estimate for judging the performance of an estimate, when used as a substitute for the whole sample in drawing inference about unknown parameters. It is found that, under some conditions, the maximum likelihood estimate has some optimum properties which distinguish it from all other large sample estimates.

On the Lengths of Intervals in a Stationary Point Process

Abstract: SUMMARY The paper is concerned with the lengths of intervals in a stationary point process. Relations are given between the various probability functions, and moments are considered. Two different random variables are introduced for the lengths of intervals, according to whether the measurement is made from an arbitrary event or beginning at an arbitrary time, and their properties are compared. In particular, new properties are derived for the correlation coefficients between the lengths of successive intervals. Examples are given. A theorem is proved, giving conditions under which two independent stationary point processes with independent intervals may be superposed, giving a new point process which also has independent intervals. Mention is made of the application to the theory of binary random processes and to the zeros of a Gaussian process.

Some Statistical Aspects of Adaptive Optimization and Control

Abstract: It is often necessary to adjust some variable X such as the concentration of consecutive batches of a product, to keep X close to a specified target value. A second more complicated problem occurs when the independent variables X in a response function η(X) are to be adjusted so that the derivatives ∂η/∂Xare kept close to a target value zero, thus maximizing or minimizing the response. These are shown to be problems of prediction, essentially, and the paper is devoted mainly to the estimation from past data of the “best” adjustments to be applied in the first problem.

On Multivariate Sign Tests

Abstract: SUMMARY Certain properties of the multivariate normal integral are used to develop sign tests for the equality of means in two correlated multivariate populations with p < 4 characters each. A numerical example from the anthropometric data of Frets illustrates the particular case p = 2. CERTAIN properties of the multivariate normal integral provide a statistical test for the equality of means in two correlated multivariate populations which is analogous to the familiar sign test for samples from a single bivariate distribution (Dixon and Mood, 1946). Hodges (1955) also has proposed a general bivariate sign test.

On the Distribution of Certain Quadratic Forms in Normal Variates

Abstract: SUMMARY This paper is concerned with the characterization of the variance covariance matrix for a normal family in which certain quadratic forms are independent and follow cx2 distributions. It is also proved that the necessary and sufficient condition for any quadratic or linear form to be distributed independently of a sum of a finite number of non-negative definite quadratic forms is that it is distributed independently of each one of them.

Problems of Selection with Restrictions

Abstract: The paper considers a practically important generalization of the theory of regression. A linear function of a set of variables x 1 ,..., x p , called predictor variables, is constructed so as to maximize its correlation with a criterion variable y 1 , subject to the condition that its correlations with other criterion variables y 2 ,..., y q are non-negative. It is suggested that a linear function so determined is useful when selection of individuals is done on the basis of x 1 ,..., x p to achieve the maximum possible progress in the mean of y 1 , while ensuring that no deterioration takes place in the mean values of y 2 ,..., y a in the selected group, compared with the original group of individuals from which selection is made.

A Simple Randomization Procedure

Abstract: The paper describes a randomization procedure consisting in distributing a deck of cards into 10 decks using random decimal digits and repeating this step with each deck consisting of three or more cards. One random digit is used for randomizing a deck of two cards. This procedure, which is essentially a particular case of a general procedure described by Rao (1961), is called the multistage randomization procedure, or MRP. Some applications are described. A recursive formula is given for the expected number of random digits required by MRP for the randomization of n symbols. A measure of the efficiency of a randomization procedure is presented. The efficiency of MRP is compared with the efficiencies of two other randomization procedures, and it is proved that MRP has an asymptotic efficiency of lOOper cent.