Journal of the Royal Statistical Society Series B (Statistical Methodology)

Vere‐Jones, D.

doi: 10.1111/j.2517-6161.1970.tb00814.xpmid: N/A

This paper attempts to survey some of the stochastic models which have been proposed for the sequence of energies and origin times of earthquakes from a given region, and to describe some examples of their application. to a good approximation in a regional study, each earthquake may be regarded as a point event, and consequently the main emphasis is on stochastic point processes. The theory of such processes is developed in a form suitable for this context, with particular emphasis being given to the clustering models of Neymann–Scott and Bartlett–Lewis. The use of these models is illustrated with reference to earthquake data from New Zealand. A final Section is concerned with stochastic models for aftershock sequences—the trains of smaller shocks which frequently follow the occurrence of large shallow earthquakes.

doi: 10.1111/j.2517-6161.1970.tb00815.xpmid: N/A

Altham, Patricia M. E.

doi: 10.1111/j.2517-6161.1970.tb00816.xpmid: N/A

The generalization of Edwards's argument for the measure of association of the rows and columns of 2 times 2 table, to that of an r x s table whose rows and columns are assumed unordered, shows, not surprisingly, that association ought to be measured by some function of the (r — 1)(s — 1) cross‐ratios. Such a function is suggested by the introduction of a metric on certain equivalence classes. The properties of such metrics are examined, and in particular comparisons are made with Good's suggestion of the use of the algebraic rank of the contingency table, and with Lindley's significance test for association in the r x s table.

Mardia, K. V.

doi: 10.1111/j.2517-6161.1970.tb00817.xpmid: N/A

A non‐parametric test for the bivariate c‐sample problem is proposed. The test is an extension of a bivariate two‐sample test given by Mardia (1967) and possesses various desirable properties. We derive the asymptotic distribution of this test and obtain the Pitman efficiency of the test relative to Wilks's criterion A for the translation types of alternatives. We give the critical values of the distribution and some approximations. The test is applied to a numerical example.

Gnanadesikan, R.; Wilk, M. B.

doi: 10.1111/j.2517-6161.1970.tb00818.xpmid: N/A

This paper describes a generalization of probability plotting to supplement general analysis of variance procedures. The mean squares in a general orthogonal analysis of variance are ordered and plotted against the corresponding expected values of standardized ordered mean squares. Since the mean squares may have differing degrees of freedom, alternate conceptions are possible with respect to association of the ordered mean squares with their parent distributions.

Kingman, J. F. C.

doi: 10.1111/j.2517-6161.1970.tb00819.xpmid: N/A

Some simple inequalities are established for the stationary waiting time distribution in the queue GI/G/k, both for the single‐server case k = 1 and for general values of k.

Mazumdar, Sati

doi: 10.1111/j.2517-6161.1970.tb00820.xpmid: N/A

For single‐server, first‐come‐first‐served queues, specified completely by the sequences of interarrival times and service times, if these sequences are stationary and if (in some sense) there is not too much dependence within or between them, it has been shown by Kingman (1962) that the waiting‐time distribution is, in heavy traffic, approximately negative exponential. This result has been extended in the present paper when there are two types of customer, one type of customer having priority of service over that of the other.

Stephens, M. A.

doi: 10.1111/j.2517-6161.1970.tb00821.xpmid: N/A

This paper gives modifications of eleven statistics, usually used for goodness of fit, so as to dispense with the usual tables of percentage points. Some test situations are illustrated, and formulae given for calculating significance levels.

Gulati, B. R.; Kounias, E. G.

doi: 10.1111/j.2517-6161.1970.tb00822.xpmid: N/A

Consider a finite n‐dimensional projective space PG(n, s) over a Galois field of order s = ph (where p, h are positive integers and p is a prime characteristic of the field). A set of k distinct points in PG(n, s), no four coplanar, is said to be complete if there exists no other set with kx points with k1 > k. The number of points in a maximal complete set is denoted by m4(n+ 1, s). The exact value of m4(n + 1, 2) is known for n ≤ 7. When n ≥ 8, the best upper bound on m4(n+1, 2) is due to Seiden (1964). It is the purpose of this paper to show that m4(4, s) = s+1 for s > 4 and to obtain bounds for m4(n + 1, s), n > 3.

Loynes, R. M.

doi: 10.1111/j.2517-6161.1970.tb00823.xpmid: N/A

If, by analogy with means and medians, one computes that value such that the sum of the absolute pth powers of the sample deviations about that point takes on its minimum, one obtains an estimate of the central value of a symmetric distribution. Such estimates, for p ≥ 1 and under reasonable conditions, are known to be asymptotically normal as the sample size becomes large, and here the asymptotic relative efficiency of any pair is obtained, both for the general case and when the distribution sampled is also supposed unimodal.