Defective Galton–Watson processes

The Galton–Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to k offspring with probability p_k, k ? 0. In this paper, we consider defective Galton–Watson processes having defective reproduction laws, so that ∑_{k ? 0}p_k = 1 ? ? for some ? ∈ (0, 1). In this setting, each particle may send the process to a graveyard state Δ with probability ?. Such a Markov chain, having an enhanced state space {0, 1, …}∪{Δ}, gets eventually absorbed either at 0 or at Δ. Assuming that the process has avoided absorption until the observation time t, we are interested in its trajectories as t → ∞ and ? → 0.     

Sequential experimental design and response optimisation

We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy _k . This may correspond to the case of a regression model, where one observesy _k =f(θ,x _k )+ε _k , with ε _k some random error, or to the Bernoulli case wherey _k ∈{0, 1}, with Pr[y _k =1|θ,x _k |=f(θ,x _k ). Special attention is given to sequences given by , with an estimated value of θ obtained from (x₁, y₁),...,(x _k ,y _k ) andd _k (x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon, where one wants to maximize ∑ _i=1 ^N w _i f(θ, x _i ) with {w _i } a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method for a binary response problem.     

Planning step-stress test under Type-I censoring for the exponential case

We consider in this work a k-level step-stress accelerated life-test (ALT) experiment with unequal duration steps τ=(τ₁, …, τ_k). Censoring is allowed only at the change-stress point in the final stage. An exponential failure time distribution with mean life that is a log-linear function of stress, along with a cumulative exposure model, is considered as the working model. The problem of choosing the optimal τ is addressed using the variance-optimality criterion. Under this setting, we then show that the optimal k-level step-stress ALT model with unequal duration steps reduces just to a 2-level step-stress ALT model.     

Selection procedures for sparse data

The problem of selecting the best of k populations is studied for data which are incomplete as some of the values have been deleted randomly. This situation is met in extreme value analysis where only data exceeding a threshold are observable. For increasing sample size we study the case where the probability that a value is observed tends to zero, but the sparse condition is satisfied, so that the mean number of observable values in each population is bounded away from zero and infinity as the sample size tends to infinity. The incomplete data are described by thinned point processes which are approximated by Poisson point processes. Under weak assumptions and after suitable transformations these processes converge to a Poisson point process. Optimal selection rules for the limit model are used to construct asymptotically optimal selection rules for the original sequence of models. The results are applied to extreme value data for high thresholds data.     

Exchangeability,predictive sufficiency and Bayesian bootstrap

Summary Let {X _n } be a sequence of random variables conditionally independent and identically distributed given the random variable Θ. The aim of this paper is to show that in many interesting situations the conditional distribution of Θ, given (X ₁,…,X _n ), can be approximated by means of the bootstrap procedure proposed by Efron and applied to a statisticT _n (X ₁,…,X _n ) sufficient for predictive purposes. It will also be shown that, from the predictive point of view, this is consistent with the results obtained following a common Bayesian approach.     

Planning step-stress test plans under Type-I censoring for the log-location-scale case

In this paper, we consider a k-level step-stress accelerated life-testing (ALT) experiment with unequal duration steps τ=(τ₁, …, τ _k ). Censoring is allowed only at the change-stress point in the final stage. A general log-location-scale lifetime distribution with mean life which is a linear function of stress, along with a cumulative exposure model, is considered as the working model. Under this model, the determination of the optimal choice of τ for both Weibull and lognormal distributions are addressed using the variance–optimality criterion. Numerical results show that for a general log-location-scale distributions, the optimal k-step-stress ALT model with unequal duration steps reduces just to a 2-level step-stress ALT model.     

Partial and inverse autocorrelations in portmanteau-type tests for time series

We present a decomposition of the correlation coefficient between x_t and x_t?k into three terms that include the partial and inverse autocorrelations. The first term accounts for the portion of the autocorrelation that is explained by the inner variables {x_t?1 , x_t?2 , …, x _{t? k+1}}, the second one measures the portion explained by the outer variables {x _t+1, x _t+2, } ∪ {x _t?k?1, x _t?k?2,…} and the third term measures the correlation between x _t and x_t?k given all other variables. These terms, squared and summed, can form the basis of three portmanteau-type tests that are able to detect both deviation from white noise and lack of fit of an entertained model. Quantiles of their asymptotic sample distributions are complicated to derive at an adequate level of accuracy, so they are approximated using the Monte Carlo method. A simulation experiment is carried out to investigate significance levels and power of each test, and compare them to the portmanteau test.     

Estimation of system reliability

In this paper, we estimate the reliability of a system with k components. The system functions when at least s (1≤s≤k) components survive a common random stress. We assume that the strengths of these k components are subjected to a common stress which is independent of the strengths of these k components. If (X ₁,X ₂,…,X _k ) are strengths of k components subjected to a common stress (Y), then the reliability of the system or system reliability is given byR=P[Y<X _(k−s+1)] whereX _(k−s+1) is (k−s+1)-th order statistic of (X ₁,…,X _k ). We estimate R when (X ₁,…,X _k ) follow an absolutely continuous multivariate exponential (ACMVE) distribution of Hanagal (1993) which is the submodel of Block (1975) and Y follows an independent exponential distribution. We also obtain the asymptotic normal (AN) distribution of the proposed estimator.     

On the α-risks for shewhart control charts

The performance of the usual Shewhart control charts for monitoring process means and variation can be greatly affected by nonnormal data or subgroups that are correlated. Define the α_k-risk for a Shewhart chart to be the probability that at least one “out-of-control” subgroup occurs in k subgroups when the control limits are calculated from the k subgroups. Simulation results show that the α_k-risks can be quite large even for a process with normally distributed, independent subgroups. When the data are nonnormal, it is shown that the α_k-risk increases dramatically. A method is also developed for simulating an “in-control” process with correlated subgroups from an autoregressive model. Simulations with this model indicate marked changes in the α_k-risks for the Shewhart charts utilizing this type of correlated process data. Therefore, in practice a process should be investigated thoroughly regarding whether or not it is generating normal, independent data before out-of-control points on the control charts are interpreted to be due to some real assignable cause.     

Testing concerning the homogeneity of some predator–prey populations

In this paper, we study an inference problem for a stochastic model where k deterministic Lotka–Volterra systems of ordinary differential equations (ODEs) are perturbed with k pairs of random errors. The k deterministic systems describe the ecological interaction between k predator–prey populations. These k deterministic systems depend on unknown parameters. We consider the testing problem concerning the homogeneity between k pairs of the interaction parameters of the ODEs. We assume that the k pairs of random errors are independent and that, each pair follows correlated Ornstein–Uhlenbeck processes. Thus, we extend the stochastic model suggested in Froda and Colavita [2005. Estimating predator–prey systems via ordinary differential equations with closed orbits. Aust. N.Z. J. Stat. 2, 235–254] as well as in Froda and Nkurunziza [2007. Prediction of predator–prey populations modeled by perturbed ODE. J. Math. Biol. 54, 407–451] where k=1. Under this statistical model, we propose a likelihood ratio test and study the asymptotic properties of this test. Finally, we highlight the performance of our method through some simulations studies.     

Hidden Second‐order Stationary Spatial Point Processes

In the existing statistical literature, the almost default choice for inference on inhomogeneous point processes is the most well‐known model class for inhomogeneous point processes: reweighted second‐order stationary processes. In particular, the K‐function related to this type of inhomogeneity is presented as the inhomogeneous K‐function. In the present paper, we put a number of inhomogeneous model classes (including the class of reweighted second‐order stationary processes) into the common general framework of hidden second‐order stationary processes, allowing for a transfer of statistical inference procedures for second‐order stationary processes based on summary statistics to each of these model classes for inhomogeneous point processes. In particular, a general method to test the hypothesis that a given point pattern can be ascribed to a specific inhomogeneous model class is developed. Using the new theoretical framework, we reanalyse three inhomogeneous point patterns that have earlier been analysed in the statistical literature and show that the conclusions concerning an appropriate model class must be revised for some of the point patterns.     

Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions

For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  _θ P _θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.     

Improved estimators for the selected location parameters

_i , i = 1, 2, ..., k be k independent exponential populations with different unknown location parameters θ _i , i = 1, 2, ..., k and common known scale parameter σ. Let Y _i denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k population is selected using the subset selection procedure according to which the population π _i is selected iff Y _i ≥Y ₍₁₎−d, where Y ₍₁₎ is the largest of the Y _i 's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained. The improved estimators are consistent and their risks are shown to be O(kn ⁻²). As a special case, we obtain the coresponding results for the estimation of θ₍₁₎, the parameter associated with Y ₍₁₎. Received: January 6, 1998; revised version: July 11, 2000     

Testing stability in a spatial unilateral autoregressive model

ABSTRACT

Least squares estimator of the stability parameter ? ? |α| + |β| for a spatial unilateral autoregressive process X_{k, ?} = αX_{k ? 1, ?} + βX_{k, ? ? 1} + ?_{k, ?} is investigated and asymptotic normality with a scaling factor n^5/4 is shown in the unstable case ? = 1. The result is in contrast to the unit root case of the AR(p) model X_k = α₁X_{k ? 1} + ??? + α_pX_{k ? p} + ?_k, where the limiting distribution of the least squares estimator of the unit root parameter ? ? α₁ + ??? + α_p is not normal.     

Consistency of stochastic approximation algorithm with quasi-associated random errors

ABSTRACT

Many mathematical and physical problems are led to find a root of a real function f. This kind of equation is an inverse problem and it is difficult to solve it. Especially in engineering sciences, the analytical expression of the function f is unknown to the experimenter, but it can be measured at each point x_k with M(x_k) as expected value and induced error ξ_k. The aim is to approximate the unique root θ under some assumptions on the function f and errors ξ_k. We use a stochastic approximation algorithm that constructs a sequence (x_k)_{k ? 1}. We establish the almost complete convergence of the sequence (x_k)_k to the exact root θ by considering the errors (ξ_k)_k quasi-associated and we illustrate the method by numerical examples to show its efficiency.     

A functional marked point process model for lupus data

Modelling for marked point processes is an important problem, but has received remarkably little attention in the statistical literature. The authors developed a marked point process model that incorporates the use of functional data analysis in a joint estimation of the frequency function of the point process and the intensity of the mark, with application to data from 22 lupus patients consisting of times of flares in symptom severity combined with a quantitative assessment of the severity. The data indicate that a rapid decrease in drug dose is significantly associated with a decrease in flare frequency. Experiments with simulated data designed to model the actual data further support this conclusion. The Canadian Journal of Statistics 40: 517–529; 2012 © 2012 Statistical Society of Canada     

DISTRIBUTION FREE MULTIPLE COMPARISONS FOR MONOTONICALLY ORDERED TREATMENT EFFECTS

A method of calculating simultaneous one-sided confidence intervals for all ordered pairwise differences of the treatment effects_j–_i, 1 i < j k, in a one-way model without any distributional assumptions is discussed. When it is known a priori that the treatment effects satisfy the simple ordering_1k, these simultaneous confidence intervals offer the experimenter a simple way of determining which treatment effects may be declared to be unequal, and is more powerful than the usual two-sided Steel-Dwass procedure. Some exact critical points required by the confidence intervals are presented for k= 3 and small sample sizes, and other methods of critical point determination such as asymptotic approximation and simulation are discussed.     

Limit results for ordered uniform spacings

Let Δ _k:n  = X _k,n  − X _k-1,n (k = 1, 2, . . . , n + 1) be the spacings based on uniform order statistics, provided X _0,n  = 0 and X _n+1,n  = 1. Obtained from uniform spacings, ordered uniform spacings 0 = Δ_0,n  < Δ_1,n  < . . . < Δ _n+1,n , are discussed in the present paper. Distributional and limit results for them are in the focus of our attention.     

Quadratic subspaces and construction of Bayes invariant quadratic estimators of variance components in mixed linear models

Gnot et al. (J Statist Plann Inference 30(1):223–236, 1992) have presented the formulae for computing Bayes invariant quadratic estimators of variance components in normal mixed linear models of the form where the matrices V _i , 1 ≤ i ≤ k − 1, are symmetric and nonnegative definite and V _k is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV ₁ M,...,MV _k-1 M,M, where M is an orthogonal projector on the null space of X′. In the paper we discuss methods of construction of such a basis. We survey Malley’s algorithms for finding the smallest quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms. We also consider a class of matrices sharing some of the symmetries common to MV ₁ M,...,MV _k-1 M,M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving the efficiency of Malley’s algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices MV ₁ M,...,MV _k-1 M,M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods. Dedicated to the memory of Professor Stanisław Gnot.