CONVERSATIONS WITH EDWIN J.G. PITMAN

Edwin Pitman was born on 29th October, 1897, in Melbourne, Victoria. He was Professor of Mathematics at the University of Tasmania from 1926 until his retirement in 1962. His publications on distribution-free methods in Statistics established a sound basis for the development of these methods. His theoretical papers gave a sound mathematical treatment of the concepts of intrinsic accuracy and sufficiency, and a definitive treatment of estimation problems for location and scale parameters. The concept of asymptotic relative efficiency is due to him. He was elected a Fellow of the Australian Academy of Science (FAA) in 1954, a Fellow of the Institute of Mathematical Statistics in 1948, President of the Australian Mathematical Society in 1958 and 1959, and Vice-President of the International Statistical Institute in 1960. He was elected an Honorary Life Member of the Statistical Society of Australia in 1966, an Honorary Fellow of the Royal Statistical Society in 1965, and an Honorary Life Member of the Australian Mathematical Society in 1968.
In 1977, in honour of his eightieth birthday, the University of Tasmania conferred on him an Honorary Doctorate of Science, and the Statistical Society of Australia instituted the Pitman Medal (1979), to be awarded to a member of the Society for high distinction in Statistics.
The conversations recorded here took place in May, 1991, at Pitman's home in Hobart, Tasmania.     

Generalized Additive Modelling of Mixed Distribution Markov Models with Application to Melbourne's Rainfall

The paper considers the modelling of time series using a generalized additive model with first-order Markov structure and mixed transition density having a discrete component at zero and a continuous component with positive sample space. Such models have application, for example, in modelling daily occurrence and intensity of rainfall, and in modelling numbers and sizes of insurance claims. The paper shows how these methods extend the usual sinusoidal seasonal assumption in standard chain-dependent models by assuming a general smooth pattern of occurrence and intensity over time. These models can be fitted using standard statistical software. The methods of Grunwald & Jones (2000) can be used to combine these separate occurrence and intensity models into a single model for amount. The models are used to investigate the relationship between the Southern Oscillation Index and Melbourne's rainfall, illustrated with 36 years of rainfall data from Melbourne, Australia.     

Pitman Medal Awarded to P. A. P. Moran

In recognition of his unique and distinguished contributions to statistics, Professor P. A. P. Moran, FAA, FRS, Foundation Professor of Statistics, Institute of Advanced Studies, Australian National University, has been awarded the Pitman Medal of the Statistical Society of Australia. The Medal was presented to Professor Moran on 26th August, 1982, at the Sixth Australian Statistical Conference in Melbourne.     

A SINGLE CHANNEL ON/OFF MODEL WITH TCP-LIKE CONTROL

We model behavior of a TCP-like source transmitting over a single channel to a server that processes work at a constant rate τ. Transmission by the source follows an on/off mechanism. When the overall load in the system is below a critical constant γ, transmission rates increase linearly but when the load exceeds γ, then transmission rates decrease geometrically fast. We study the system by means of an embedded Markov chain, which gives the buffer content at the start of transmissions. Attention is paid to the time necessary to transmit a file of size L and both the tail behavior and expectation of the distribution of file transmission time are considered.     

STABILITY IN QUEUES WITH IMPATIENT CUSTOMERS

This paper deals with GI/G/1 queueing systems with impatient customers that have individual deadlines until their beginning of service. The impatience law depends on the number of waiting customers and on the elapsed service time of the customer in service. An exhaustive analysis of the asymptotic behavior of the model, combining ideas of stochastic dominance of well–known processes and some properties of models with finite capacity, is provided. We prove that the model is ergodic, null recurrent or transient if the corresponding traffic parameter in a simple associated model is respectively lower than, equal to, or greater than one.     

BAYESIAN SMOOTHING WITH OCCASIONAL JUMPS

Consider a sequence of independent random variables X ₁, X ₂,…,X _n observed at n equally spaced time points where X _i has a probability distribution which is known apart from the values of a parameter θ _i ∈ R which may change from observation to observation. We consider the problem of estimating θ = (θ₁, θ₂,…,θ _n ) given the observed values of X ₁, X ₂,…,X _n . The paper proposes a prior distribution for the parameters θ for which sets of parameter values exhibiting no change, or no change apart from a few sudden large changes, or lots of small changes, all have positive prior probability. Markov chain sampling may be used to calculate Bayes estimates of the parameters. We report the results of a Monte Carlo study based on Poisson distributed data which compares the Bayes estimator with estimators obtained using cubic splines and with estimators derived from the Schwarz criterion. We conclude that the Bayes method is preferable in a minimax sense since it never produces the disastrously large errors of the other methods and pays only a modest price for this degree of safety. All three methods are used to smooth mortality rates for oesophageal cancer in Irish males aged 65–69 over the period 1955 through 1994.     

CONTROLLED SAMPLING WITH VARYING PROBABILITIES WITH AND WITHOUT REPLACEMENT

Goodman and Kish (1950) introduced the problem of controlled selection in the sense of decreasing the selection probability of nonpreferred combinations, such as, for example, combinations which present organisational difficulties involving additional cost, etc. The authors (Avadhani and Sukhatme (1965), Sukhatme and Avadhani (1965)) have evolved certain techniques of controlled selection which eliminate altogether some non-preferred combinations and reduce the probability of selection of the remaining combinations, if any, to the minimum possible extent without deviating from the fundamental principles of random sampling. However, it is often felt convenient in practice to draw sampling units one after another from the population rather than combinations of units. But, at present no technique by which one can select the units one after another and at the same time reduce the probability of selection of non-preferred units is available in literature.
In this paper we have suggested a solution to this problem which not only minimizes the selection probability of non-preferred units (and of samples containing predominantly large numbers of nonpreferred units) but also provides more efficient estimators than the the usual probability proportional to size (P.P.S) sampling scheme.     

ACCELERATED FAILURE TIME MODELS WITH NONLINEAR COVARIATES EFFECTS

As a flexible alternative to the Cox model, the accelerated failure time (AFT) model assumes that the event time of interest depends on the covariates through a regression function. The AFT model with non‐parametric covariate effects is investigated, when variable selection is desired along with estimation. Formulated in the framework of the smoothing spline analysis of variance model, the proposed method based on the Stute estimate ( Stute, 1993 [Consistent estimation under random censorship when covariables are present, J. Multivariate Anal. 45 , 89–103]) can achieve a sparse representation of the functional decomposition, by utilizing a reproducing kernel Hilbert norm penalty. Computational algorithms and theoretical properties of the proposed method are investigated. The finite sample size performance of the proposed approach is assessed via simulation studies. The primary biliary cirrhosis data is analyzed for demonstration.     

POISSON REGRESSION WITH A PERIODIC FUNCTION

ABSTRACT

Let {y_t } be a Poisson-like process with the mean μ _t which is a periodic function of time t. We discuss how to fit this type of data set using quasi-likelihood method. Our method provides a new avenue to fit a time series data when the usual assumption of stationarity and homogeneous residual variances are invalid. We show that the estimators obtained are strongly consistent and also asymptotically normal.     

SOJOURN TIMES IN NON-HOMOGENEOUS QBD PROCESSES WITH PROCESSOR SHARING

We study sojourn times of customers in a processor sharing queue with a service rate that varies over time, depending on the number of customers and on the state of a random environment. An explicit expression is derived for the Laplace–Stieltjes transform of the sojourn time conditional on the state upon arrival and the amount of work brought into the system. Particular attention is paid to the conditional mean sojourn time of a customer as a function of his required amount of work, and we establish the existence of an asymptote as the amount of work tends to infinity. The method of random time change is then extended to include the possibility of a varying service rate. By means of this method, we explain the well-established proportionality between the conditional mean sojourn time and required amount of work in processor sharing queues without random environment. Based on numerical experiments, we propose an approximation for the conditional mean sojourn time. Although first presented for exponentially distributed service requirements, the analysis is shown to extend to phase-type services. The service discipline of discriminatory processor sharing is also shown to fall within the framework.     

BOOK REVIEWS

Book reviewed in this article:
Hannan, E. J. Multiple Time Series.
Bancroft, T. A., and Brown S. A. Statistical Papers in Honour of George W. Snedecor.
Friedman, A. Differential Games
Henderson, Ronald F., Harcourt, Alison, Harper, R. J. A., and Shaver, Sheila. The Melbourne Poverty Survey: Further Notes on Methods and Results .
Lukacs, Eugene. Characteristic Functions
Searle, S. R. Linear Models     

THE BOTTOMLESS DAM WITH SEASONAL INPUTS

The stationary distribution of the depletion (from the maximum content) of a bottomless dam is derived for the case where the inputs and releases are seasonal. The method used is an extension of the method of Wishart for a problem in queueing theory (without involving seasonality). A numerical example is given illustrating the method for the case of two seasons.
In some recent work on dams (see Phatarfod, 1979; Pakes & Phatarfod, 1978), it was shown that it was more appropriate to consider the bottomless (or the infinitely deep) dam model rather than the topless (or the infinitely high) dam model as an approximation to the finite dam. The main consideration is that in real-life reservoir situations, the planned release or draft is almost always less than the mean inflow. It is obvious that with this condition the content of a topless dam will not have a stationary distribution, whereas if we consider the depletion of a dam from its top, this will have a stationary distribution. An additional reason for considering the bottomless dam model is that the model is mathematically more tractable; this property was noticed before for continuous time models (see e.g. Hasofer, 1966)     

A STOCHASTIC LOGISTIC MODEL WITH TIME DELAY

The stochastic version of the logistic model for population growth is generalized to take account of continuously distributed time delay with an exponentially decaying kernel. The theory of diffusion processes is used to analyse the probability density function of the population size. The explicit expression for the stationary distribution is worked out and the effect of time delay on various statistics is discussed.     

Q-LEARNING WITH CENSORED DATA

We develop methodology for a multistage-decision problem with flexible number of stages in which the rewards are survival times that are subject to censoring. We present a novel Q-learning algorithm that is adjusted for censored data and allows a flexible number of stages. We provide finite sample bounds on the generalization error of the policy learned by the algorithm, and show that when the optimal Q-function belongs to the approximation space, the expected survival time for policies obtained by the algorithm converges to that of the optimal policy. We simulate a multistage clinical trial with flexible number of stages and apply the proposed censored-Q-learning algorithm to find individualized treatment regimens. The methodology presented in this paper has implications in the design of personalized medicine trials in cancer and in other life-threatening diseases.     

FAMILY BUILDING DIFFERENCES BETWEEN SAME SEX AND MIXED SEX FAMILIES IN AUSTRALIA

From this investigation it seems that an aversion among some couples to a same sex family has some relevance in family building. However, although there are significant differences in the proportions of same sex and mixed sex families having an additional child, the overall effect on average family size of the (apparent) attempt to achieve a child of each sex appears to be slight—an increase of less than 3% in the case of the Melbourne data. Despite the evidence that a significantly greater proportion of women with same sex families have another child, relatively few women admit that the sex structure of the children was the main factor in such a decision. Another insight into the relationship between sex structure and family building was that the desire for a child of each sex may have a negative effect on fertility in that couples who have already achieved such a configuration may decide not to have an additional child that they had originally planned to have. Obviously, it is difficult to separate attitudes to the importance of having both sons and daughters from rationalisations associated with the sex structure of the respondent's own family. Nevertheless the desire for children of each sex seems to be related to traditional attitudes to woman's role, Southern European background, an earlier year of birth, and a lower level of education. However, at the same time women in these categories, particularly the last, seem to be relatively less likely to exhibit behaviour indicative of a controlled response to the sex structure of the family; thus among such groups the probability of same sex families having another child is found to be relatively similar to the proportion of mixed sex families having another child. In other words, they are less likely than other women, who care less about having both sons and daughters, to stop at two (or three) children when at least one son and one daughter have been attained. What of the future relationship between proportions of same sex and mixed sex families having another child? Although it would seem that the combined effect of better educational opportunities and less traditional attitudes of each new generation would further reduce the importance attached to having both sons and daughters, at the same time one would expect an increase in a couple's ability to stop family building when two or three children of the desired sex structure had been achieved. Another factor is that possibly an increasing preference for smaller families will outweigh the wish to have a third (or fourth) child to achieve one child of each sex or for any other reason.     

EDITING A LARGE-SCALE DEMOGRAPHIC SURVEY–THE EXPERIENCE OF THE MELBOURNE FAMILY SURVEY1

This paper discusses the editing procedures for a survey on family formation conducted in Melbourne during 1971. Examples are given of the incidence and types of inconsistencies revealed during the editing stage in the mainly quantitative variables concerning childbearing, numbers of children, and the demographic characteristics of the children. The principal point emphasized is the need for automatic detection and manual correction of coding deficiencies in a survey of this type, i.e. an intensive, “once-off” survey, with a sample size of a few thousand, from which detailed and complex analyses are planned.     

COMMON FEATURES IN TIME SERIES WITH BOTH DETERMINISTIC AND STOCHASTIC SEASONALITY

This paper extends the notions of common cycles and common seasonal features to time series having deterministic and stochastic seasonality at different frequencies. The conditions under which quarterly time series with these characteristics have common features are investigated, various representations are presented and statistical inference is discussed. Finally, the analysis is applied to study comovements between different components of consumption and income using UK data.     

OPTIMAL EXPERIMENTAL DESIGNS FOR MULTILEVEL MODELS WITH COVARIATES

In this paper optimal experimental designs for multilevel models with covariates and two levels of nesting are considered. Multilevel models are used to describe the relationship between an outcome variable and a treatment condition and covariate. It is assumed that the outcome variable is measured on a continuous scale. As optimality criteria D-optimality, and L-optimality are chosen. It is shown that pre-stratification on the covariate leads to a more efficient design and that the person level is the optimal level of randomization. Furthermore, optimal sample sizes are given and it is shown that these do not depend on the optimality criterion when randomization is done at the group level.     

A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE-PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS

A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.     

HITTING LINES and CIRCLES WITH DIFFUSION PROCESSES

For (x(t),y(t)), a diffusion process starting from (x(0),y(0)) = (x,y), the problem of computing the moment generating function of the first passage time T(x, y) to a given subset D of IR²is considered. A particular case of the method of similarity solutions is used. The problems that can be solved explicitly are those for which D is either a straight line or a circle.