This paper considers how students might be taught statistical consulting. Desirable characteristics of a good consultant are described. While skill in statistical techniques is considered, it is just one of several areas of expertise that are discussed. Suggestions are made as to what should be included in a consulting course to equip a student with these characteristics.

The advantages and limitations of frequency domain and time domain methods for estimating the interannual variability arising from day-to-day weather events are summarized. A modification of the time domain method is developed and its application in examining a precondition for the frequency domain method is demonstrated. A combined estimation procedure is proposed: it takes advantage of the strengths of both methods. The estimation procedures are tested with sets of synthetic data and are applied to long time series of three meteorological parameters. The impacts of the different methods on tests of potential long-range predictability for seasonal means are also discussed.

The dynamics of host-macroparasite infections pose considerable challenges for stochastic modelling because of the need to take into account a large number of relevant factors and many nonlinear interactions between them. This paper focuses attention on the infection transmission process and the effects of specific modelling assumptions about the mechanisms involved. Some dramatically simplified linear models are considered; they are based on multidimensional linear birth and death processes, and are designed to illuminate qualitative effects of interest. Both single and compound infections are allowed. It is shown that such simple models can generate and increase dispersion of parasite counts, even among homogeneous hosts.

The paper considers one of the standard processes for modeling returns in finance, the stochastic volatility process with regularly varying innovations. The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions. Unlike other non-linear models used in finance, such as GARCH and bilinear models, sample autocorrelations of a stochastic volatility process have attractive asymptotic properties. Specifically, in the infinite variance case, the sample autocorrelation function converges to zero in probability at a rate that is faster the heavier the tails of the marginal distribution. This behavior is analogous to the asymptotic behavior of the sample autocorrelations of independent identically distributed random variables.

We provide a set of verifiable sufficient conditions for proving in a number of practical examples the equivalence of the martingale and the PDE approaches to the valuation of derivatives. The key idea is to use a combination of analytic and probabilistic assumptions that covers typical models in finance falling outside the range of standard results from the literature. Applications include Heston's stochastic volatility model and the Black-Karasinski term structure model.

This short note shows that the Lundberg exponential upper bound in the ruin problem of non-life insurance with compound Poisson claims is also valid for the Poisson shot noise delayed-claims model, and that the optimal exponent depends only on the distribution of the total claim per accident, not on the time it takes to honour the claim. This result holds under Cramer's condition.

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L 1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

The bifurcating autoregressive model has been used previously to model cell lineage data. A feature of this model is that each line of descendants from an initial cell follows an AR(1) model, and that the environmental effects on sisters are correlated. However, this model concentrates on modelling the correlations between mother and daughter cells and between sister cells, and does not explain the large correlations between more distant relatives observed by some authors. Here the model is extended, firstly by allowing lines of descent to follow an ARMA(p,q) model rather than an AR(1) model, and secondly by allowing correlations between the environmental effects of relatives more distant than sisters. The models are applied to several data sets consisting of independent cell lineage trees.

An open hierarchical (manpower) system divided into a totally ordered set of k grades is discussed. The transitions occur only from one grade to the next or to an additional (k+1)th grade representing the external environment of the system. The model used to describe the dynamics of the system is a continuous-time homogeneous Markov chain with k+1 states and infinitesimal generator R = (r ij ) satisfying r ij = 0 if i > j or i + 1 < j ≤ k (i, j = 1,…,k+1), the transition matrix P between times 0 and 1 being P = expR . In this paper, two-wave panel data about the hierarchical system are considered and the resulting fact that, in general, the maximum-likelihood estimated transition matrix cannot be written as an exponential of an infinitesimal generator R having the form described above. The purpose of this paper is to investigate when this can be ascribed to the effect of sampling variability.

We analyse a class of diffusion models that (i) allows an explicit expression for the likelihood function of discrete time observation, (ii) allows the possibility of heavy-tailed observations, and (iii) allows an analysis of the tails of the increments. The class simply consists of transformed Ornstein–Uhlenbeck processes and is of relevance for heavy-tailed time series. We also treat the question of the existence of an equivalent martingale measure for the class of models considered.

We consider a sequence matching problem involving the optimal alignment score for contiguous sequences; rewarding matches and penalizing for deletions and mismatches. Arratia and Waterman conjectured in [1] that the score constant a(μ, δ) is a strictly monotone function (i) in δ for all positive δ and (ii) in μ if 0 ≤ μ ≤ 2δ. Here we prove that (i) is true for all δ and (ii) is true for some μ.

We study the present value Z ∞ = ∫0 ∞ e-X t- dY t where (X,Y) is an integrable Lévy process. This random variable appears in various applications, and several examples are known where the distribution of Z ∞ is calculated explicitly. Here sufficient conditions for Z ∞ to exist are given, and the possibility of finding the distribution of Z ∞ by Markov chain Monte Carlo simulation is investigated in detail. Then the same ideas are applied to the present value Z - ∞ = ∫0 ∞ exp{-∫0 t R s ds}dY t where Y is an integrable Lévy process and R is an ergodic strong Markov process. Numerical examples are given in both cases to show the efficiency of the Monte Carlo methods.

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p 1,p 2,…, p n is often approximated by a Poisson distribution with parameter λ = p 1 + p 2 + p n . Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p 1,p 2,…,p n .

Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.

This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ2 < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law B p v n *N nbef which depends on the total population size N n bef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

We formulate stochastic indicator parameters that characterize pollution levels in geographical regions with heterogeneous contaminant distributions. The indicator parameters are expressed in terms of the random fields representing the contaminant distributions and the critical threshold level specified by health and environmental standards. Certain theoretical results are proven regarding univariate and bivariate indicator parameters. The analytical expressions obtained are general and can be used in practice for various types of contaminant distributions. A test of ergodicity-breaking is suggested for scientific and engineering applications in terms of the indicator parameters. Fractal characteristics of the indicator parameters are discussed. The effects of modelling and observation scale on exceedance contamination analysis are examined. Indicator random field parameters are studied on both continuum and lattice domains using analytical means and numerical simulations.

This paper discusses the distribution of tumor size at detection derived within the framework of a new stochastic model of carcinogenesis. This distribution assumes a simple limiting form, with age at detection tending to infinity which is found to be a generalization of the distribution that arises in the length-biased sampling. Two versions of the model are considered with reference to spontaneous and induced carcinogenesis; both of them show similar asymptotic behavior. When the limiting distribution is applied to real data analysis its adequacy can be tested through testing the conditional independence of the size, V, and the age, A, at detection given A > t*, where the value of t* is to be estimated from the given sample. This is illustrated with an application to data on premenopausal breast cancer. The proposed distribution offers the prospect of the estimation of some biologically meaningful parameters descriptive of the temporal organization of tumor latency. An estimate of the model stability to the prior distribution of tumor size and some other stability results for the Bayes formula are given.

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

The theory of majorization is applied to investigate the effects of heterogeneity in a class of epidemics. In particular, the heterogeneity of the components of the contact vector of a given susceptible is studied. It is shown that if the heterogeneity can be partially ordered through majorization, then the probability functions of escaping infection can be similarly ordered. The results can be applied to AIDS research on the spread of the HIV virus.

Let ?(u) be the probability of eventual ruin in the classical Sparre Andersen model of risk theory if the initial risk reserve is u. For a large class of such models ?(u) behaves asymptotically like a multiple of exp (–Ru) where R is the adjustment coefficient; R depends on the premium income rate, the claim size distribution and the distribution of the time between claim arrivals. Estimation of R has been considered by many authors. In the present paper we deal with confidence bounds for R. A variety of methods is used, including jackknife estimation of asymptotic variances and the bootstrap. We show that, under certain assumptions, these procedures result in interval estimates that have asymptotically the correct coverage probabilities. We also give the results of a simulation study that compares the different techniques in some particular cases.

The multihit–one target model induces a stochastic ordering of cell survival with respect to the cell sensitivity characteristics. This property can be used for a description of cell killing effects in heterogeneous populations of cells on the basis of randomized versions of the model. In such versions, either the critical number of lesions or the mean number of hits per unit dose (sensitivity), or both, are assumed to be random. We give some new results specifying conditions under which the randomized multihit models are identifiable, with a focus on the following cases: (1) the critical number of radiation-induced lesions, m, is random; (2) the sensitivity parameter, x, is random given m is known or otherwise; (3) x and m form a pair of independent random variables.