Showing papers in "Journal of the royal statistical society series b-methodological in 1961"

The Moment Generating Function of the Truncated Multi-Normal Distribution

Abstract: SUMMARY In this paper the moment generating function (m.g.f.) of the truncated n-dimensional normal distribution is obtained. From the m.g.f., formulae for E(Xi) and E(Xi Xj) are derived, and are used to investigate certain special cases. Some applications of these results to statistical genetics are also discussed.

Prediction by Exponentially Weighted Moving Averages and Related Methods

Abstract: SUMMARY The mean square error of prediction is calculated for an exponentially weighted moving average (e.w~.m.a.), when the series predicted is a Markov series, or a Markov series with superimposed error. The best choice of damping constant is given; the choice is not critical. There is a value of the Markov correlation po below which it is impossible to predict, with an e.w.m.a., the local variations of the series. The mean square error of an e.w.m.a. is compared with the minimum possible value, namely that for the best linear predictor (Wiener). A modified e.w.m.a. is constructed having a mean square error approaching that of the Wiener predictor. This modification will be of value if the Markov correlation parameter is negative, and possibly also when the Markov parameter is near po.

A Simple Method of Trend Construction

Abstract: SUMMARY The principle adopted here in the construction of a trend for a time series consists in minimizing a linear combination of two sums of squares, of which one refers to the second differences of the trend values, the other to the deviations of the observations from the trend values. Properties of the general solution are deduced, and the solution is explicitly obtained for up to 7 observations. In the special case in which the sum of the two sums of squares is minimized, the exact solution is derived for up to 15 observations. An approximation formula, suitable for practical use when there are 8 or more observations, is also given. The method is illustrated by examples, in which it is applied to an artificially constructed and to an actual time series.

A Note on Maximum-Likelihood in the Case of Dependent Random Variables

Abstract: SUMMARY The validity of certain statistical procedures depends on the "classical" properties of the method of maximum-likelihood, viz., its consistency and the fact that it leads to an estimator whose asymptotic variance can be derived easily from the likelihood function. These properties are well established for the case where successive observations are independent. Here it is shown that their extension to the dependent case involves suitable behaviour of two semi-martingales and a martingale, and general conditions are given which ensure this behaviour. While interest is centred on the theoretical rather than on the practical side of the problem, these conditions may have some practical value.

Some Stochastic Processes with Absorbing Barriers

Abstract: SUMMARY Models for growth distributions of biological organisms are investigated in this paper. They are specializations of the birth-death equations with two absorbing barriers. The solutions we seek are the distributions of the time to grow from an initial dose to an infective dose. Equations for these functions are simply obtained from the birth-death equations. Two main cases of interest are then examined. In the first case the birth and death rates are proportional to the number present and in the second case they are constant. Graphs are developed for the solution of the first case. Useful applications depend on this result.

Closed Queueing Systems: A Generalization of the Machine Interference Model

Abstract: SUMMARY A closed loop of queues in series is considered in which the time taken by any customer to travel between consecutive service gates is a random variable. The case of a single server at each gate is analysed, where in addition to the customers from the closed loop, each server may also have customers arriving from sources outside the system and, after completion of service, departing to destinations again outside the system. Limiting cases are examined, firstly where the number of customers in the closed loop becomes large, and secondly where, in addition to this condition, the average transit time between one pair of consecutive service gates also tends to infinity.