extremes of moving averages of random variables with finite endpoint where zj are iid and nonnegative random variables and c 0 are constants satisfying summability conditions at least sufficient to make the random series above converge we suppose that the distribution of zj is regularly varying near 0 and discuss lower tail behavior of finite and infinite linear combinations the behavior is quite different in the two cases for finite linear combinations the lower tail is again regularly varying but for infinite moving averages the lower tail is f varying i e it is in the domain of attraction of a type i extreme value distribution in the sense of minima convergence of point processes based on the moving averages is shown to hold in both the finite and infinite order cases and suitable conclusions are drawn from such convergences a useful analytic tool is asymptotic normality of the esscher transform of the common distribution of the z s the extreme value results of this paper are in terms of minima of the moving average processes but results can be adapted to study maxima of moving averages of random variables in the domain of attraction of the type iii extreme value distribution for maxima

fractional langevin model of memory in financial markets the separation of the microscopic and macroscopic time scales is necessary for the validity of ordinary statistical physics and the dynamical description embodied in the langevin equation when the microscopic time scale diverges the differential equations on the macroscopic level are no longer valid and must be replaced with fractional differential equations of motion in particular we obtain a fractional differential stochastic equation of motion after decades of statistical analysis of financial time series certain stylized facts have emerged including the statistics of stock price fluctuations having fat tails and their linear correlations in time being exceedingly short lived on the other hand the magnitude of these fluctuations and other such measures of market volatility possess temporal correlations that decay as an inverse power law one explanation of this long term memory is that it is a consequence of the time scale separation between microscopic and macroscopic economic variables we propose a fractional langevin equation as a dynamical model of the observed memory in financial time series

asymptotic properties of moment estimators for overdispersed counts and proportions summary consistency and asymptotic normality of moment estimates of regression parameters and an overdispersion parameter are shown for data with extra binomial and extrapoisson variation the asymptotic covariance of the regression parameters is found to be unaffected by estimation of the overdispersion parameter the moment estimates may be obtained using iterated weighted least squares

robust estimation of the variance in moment methods for extra binomial and extra poisson variation when faced with data in the form of overdispersed counts or proportions moment methods allow consistent parameter estimation when only the form of the mean and variance is specified if the variance form is misspecified these methods still yield consistent parameter estimates though with lower efficiency and the variances of the estimates will be inconsistent a variance correction is available that yields consistent variance estimates in these circumstances the asymptotic and small sample efficiencies of this correction are calculated and its performance under variance misspecification is studied a group randomized breast self examination prevention study that is now underway serves as a focal point for the study of these properties the use of the variance correction in modelling is illustrated on a teratology data set

the estimation of age period and cohort effects for vital rates in models for vital rates which include effects due to age period and cohort there is aliasing due to a linear dependence among these three factors this dependence arises both when age and period intervals are equal and when they are not one solution to the dependence is to set an arbitrary constraint on the parameters estimable functions of the parameters are invariant to the particular constraint applied for evenly spaced intervals deviations from linearity are estimable but only a linear function of the three slopes is estimable when age and period intervals have different widths further aliasing occurs it is assumed that the number of deaths in the numerator of the rate equation has a poisson distribution the calculations are illustrated with data on mortality from prostate cancer among nonwhites in the u s

statistical inference for poisson and multinomial models for capture recapture experiments summary the classical multinomial model used for estimating the size of a closed population is compared to the highly flexible poisson models introduced by cormack 1981 the multinomial model and generalizations of it which allow for dependence between samples may be obtained from that of cormack by conditioning on the population size the maximum likelihood estimators for n the population size and 0 the vector of parameters describing the capture process are the same in both models completely general formulae for the asymptotic variances of the maximum likelihood estimates of n for both models are given the substantial differences between the variances under the two models are discussed hypotheses concerning 0 may be tested using the log likelihood ratio the procedures which result from both models are asymptotically equivalent under the null hypothesis but differ in power under the alternative

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

diffusively coupled allee effect on heterogeneous and homogeneous graphs abstract we study how the allee effect with diffusion changes depending on the graph topology we present the metapopulation dynamic model on heterogeneous and homogeneous graphs for the allee effect of mobile individuals we consider the small graphs with three and four nodes and the star and complete graphs with n nodes a subpopulation patch is represented by a node on a graph a link represents a migration path between subpopulations mobile individuals move by random walk through a link between nodes individuals grow or die out according to allee population dynamics within each subpopulation the population dynamics in the metapopulation model are presented by reaction diffusion equations to evaluate the population size of each subpopulation node we obtain the solutions of the reaction diffusion equations numerically for the small graphs star graphs and complete graphs the equilibrium points of the allee metapopulation model depend highly on the network topology the dependence of the critical allee parameter on the number of nodes is derived for star and complete graphs

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

a simple em algorithm for capture recapture data with categorical covariates a simple em algorithm is proposed for obtaining maximum likelihood estimates when fitting a loglinear model to data from k capture recapture samples with categorical covariates the method is used to analyze data on screening for the early detection of breast cancer

tests of hypotheses in overdispersed poisson regression and other quasi likelihood models abstract test statistics for evaluating the significance of added variables in a regression equation are developed for mixed poisson models where the structural parameter that determines the mean variance relationship var 2 is estimated by the method of moments and the regression coefficients are estimated by quasi likelihood the formulas presented for test statistics and related estimating equations are applicable generally to quasi likelihood models specified by an arbitrary mean value function x together with a variance function v that contains one or more unknown parameters two versions of the wald and score tests are investigated one calculated from the usual model based covariance matrix whose validity depends on correct specification of the variance function and another using an empirical covariance matrix that has a more general asymptotic justification monte carlo simulations demonstrate that the quasi likelihood method of moments ql m procedures yield ap

estimating the size of a population from a single sample summary methods for estimating the size of a population of individuals usually require multiple samples from the group we consider a population composed of an unknown number n of individuals on one or more of k 1 ordered lists a single sample of individuals from the population those on list k together with the identification of the list on which they last appeared prior to list k is obtained under relatively weak assumptions on the probability model an unbiased maximum likelihood estimator of n is obtained an expression is derived for the bias of the estimator and its consequence on the true probability of coverage of the confidence interval when the model s assumptions do not hold applications of this method are discussed and an illustrative example is presented estimation of the size of a closed population is frequently based on a capture recapture cr model the data may be viewed as k 1 samples or lists of individuals who are tagged or otherwise identified so that the lists on which they appear are known the capture history of each individual in the population corresponds to a k dimensional vector whose ith element is 1 if the individual is on the ith list or sampling occasion and 0 otherwise an unobserved individual is represented by a vector all of whose elements are 0 in the cr setting the size of the population the parameter to be estimated is the observed number of distinct individuals appearing on at least one of the lists plus the unobserved number who do not appear on any of the lists viewing the observations as falling into a 2k multidimensional cross classification bishop fienberg and holland 1975 reviewed the use of log linear models for estimating the size of the population recently wolter 1986 proposed various two dimensional cross tabulation coverage models for census data the literature appears to be silent on the problem of estimating the number of individuals n on k lists when only one of the lists say the last one is observable the individuals on this list in the cr setting are those caught in the last capture the problem arises in many areas for example suppose an unknown number of patients are treated in a clinic during k 52 weeks in the last week of the 52 week period a survey of those who are treated is to be performed can information on the surveyed individuals be obtained that would permit estimation of the distinct number of patients treated in the year neither the standard capture recapture models of seber 1982 nor the contingency table approaches

patterns in reaction diffusion systems generated by global alternation of dynamics abstract we recently proposed a mechanism for inducing a turing instability by alternation of reaction diffusion dynamics each of which is pattern free herein we shed light on the question of which particular switching schemes produce that instability and which do not we also consider a particular reaction diffusion model to illustrate the mechanism

stratified capture recapture estimation of the size of a closed population this paper considers the estimation of the size of a closed population in a stratified capturerecapture experiment using the stratified petersen lincoln model introduced by darroch 1961 biometrika 48 241 260 a generalization of darroch s model where the tagging or the recovery probabilities depend on explanatory variables possibly related to the capture mechanism is proposed a simple scoring algorithm for the estimation of the nuisance parameters i e the vectors of tagging and of recovery probabilities and the matrix for the distribution of the units in the capturerecapture area is derived this algorithm is based on a new closed form expression for the inverse of the fisher information matrix of the conditional likelihood for the estimation of the nuisance parameters large sample tests for the tagging and the recovery probabilities are obtained pooling of adjacent recovery strata with equal capture probabilities is proposed as a method for getting accurate estimators of the size of the population conditional and unconditional estimators of the size of the population are proposed analyses of data on smolt migration illustrate the methodology presented in this paper

analysis of two way layout of count data involving multiple counts in each cell abstract multiple counts may occur in each cell of an a b two way layout balanced or unbalanced of two fixed factors a and b standard log linear model analysis based on a poisson distribution assumption of the cell counts is not applicable here because of the unbalanced nature of the table or because the poisson distribution assumption is not valid we develop c tests for interaction and main effects assuming data to be poisson distributed and also assuming that data within the cells have extra over under dispersion beyond that explained by a poisson distribution for this we consider an extended negative binominal distribution and a semiparametric model using the quasi likelihood we show that in all situations the c tests for interaction are of very simple forms for c tests for the main effect in presence of no interaction such simplification is possible only under certain conditions a score test for detecting extra dispersion in presence of interaction is also obtained and is of sim

inference for poisson and multinomial models for capture recapture experiments summary capture recapture models have been formulated both as poisson and as multinomial distributions maximum likelihood estimates of parameters under the two models are compared for parameters which do not involve the population size the asymptotic covariances are shown to be the same for certain classes of capture recapture experiments both poisson and multinomial models can be used in the poisson models cormack 1979 the numbers of observed individuals with different total capture histories w are regarded as independent poisson variates with means npf o n being some measure of the unknown size of the population and p 0 being specified functions of parameters 0 representing the demography of the population and the behavioural interaction of individuals with the sampling process in the multinomial models darroch 1958 n individuals are regarded as being multinomially distributed into a number of capture histories the observable ones with probabilities ph and the unobservable category with probability l p where p xp s 1 sandland cormack 1984 explored the relationships between the poisson and multinomial models and showed that the asymptotic variance of the maximum likelihood estimate of n under the poisson model is related to that obtained fienberg 1972 under the multinomial model by

pattern formation by super diffusion in fitzhugh nagumo model abstract the aim of this paper is to examine the turing instability and pattern formation in the fitzhugh nagumo model with super diffusion in two dimensional numerical simulation we also studied the effects of the super diffusive exponent on pattern formation concluding that with the presence of super diffusion the stable homogenous steady state becomes unstable by using the stability analysis of local equilibrium point we procure the conditions which ensure that the turing and hopf bifurcations occur for pattern selection the weak nonlinear multi scale analysis is used to derive the amplitude equations of the stationary patterns we then apply amplitude equations and observe that this model has very rich dynamical behaviors such as stripes spots and hexagon patterns the complexity of the dynamics in this system is theoretically discussed and graphically displayed in numerical simulation the simulation helps us to show the effectiveness of theoretical analysis and patterns which appear numerically

spatially explicit maximum likelihood methods for capture recapture studies live trapping capture recapture studies of animal populations with fixed trap locations inevitably have a spatial component animals close to traps are more likely to be caught than those far away this is not addressed in conventional closed population estimates of abundance and without the spatial component rigorous estimates of density cannot be obtained we propose new flexible capture recapture models that use the capture locations to estimate animal locations and spatially referenced capture probability the models are likelihood based and hence allow use of akaike s information criterion or other likelihood based methods of model selection density is an explicit parameter and the evaluation of its dependence on spatial or temporal covariates is therefore straightforward additional nonspatial variation in capture probability may be modeled as in conventional capture recapture the method is tested by simulation using a model in which capture probability depends only on location relative to traps point estimators are found to be unbiased and standard error estimators almost unbiased the method is used to estimate the density of red eyed vireos vireo olivaceus from mist netting data from the patuxent research refuge maryland u s a estimates agree well with those from an existing spatially explicit method based on inverse prediction a variety of additional spatially explicit models are fitted these include models with temporal stratification behavioral response and heterogeneous animal home ranges

spatial coherence resonance in excitable media we study the phenomenon of spatial coherence resonance in a two dimensional model of excitable media with fitzhugh nagumo local dynamics in particular we show that there exists an optimal level of additive noise for which an inherent spatial scale of the excitable media is best pronounced we argue that the observed phenomenon occurs due to the existence of a noise robust excursion time that is characteristic for the local dynamics whereby the diffusion constant representing the rate of diffusive spread determines the actual resonant spatial frequency additionally biological implications of presented results in the field of neuroscience are outlined

testing covariance structure in multivariate models application to family disease data abstract recent interest in modeling multivariate responses for members of groups has emphasized the need for testing goodness of fit here we describe a way to test the covariance structure of a multivariate distribution parameterized by a vector the idea is to extend this distribution the null distribution to a more general distribution that depends on an additional scalar and a specific quadratic function of the response vector chosen to capture features of an alternative covariance structure when 0 the more general distribution reduces to the null one standard likelihood theory yields a score test for 0 that is a test of fit of the null distribution the score statistic is the standardized difference between observed and expected values of the quadratic function where the expectation is taken with respect to the null distribution with replaced by its maximum likelihood estimate applying the methods to case control data on familial cancers of the ovary and breast we illu

econometric models for count data with an application to the patents r d relationship this paper focuses on developing and adapting statistical models of counts nonnegative integers in the context of panel data and using them to analyze the relationship between patents and r d expenditures since a variety of other economic data come in the form of repeated counts of some individual actions or events the methodology should have wide applications the statistical models we develop are applications and generalizations of the poisson distribution two important issues are i given the panel nature of our data how can we allow for separate persistent individual fixed or random effects ii how does one introduce the equivalent of disturbances in the equation into the analysis of poisson and other discrete probability functions the first problem is solved by conditioning on the total sum of outcomes over the observed years while the second problem is solved by introducing an additional source of randomness allowing the poisson parameter to be itself randomly distributed and compounding the two distributions lastly we develop a test statistic for the presence of serial correlation when fixed effects estimators are used in nonlinear conditional models

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

a semiparametric model for the analysis of recurrent event panel data summary in many longitudinal studies interest focuses on the occurrence rate of some phenomenon for the subjects in the study when the phenomenon is nonterminating and possibly recurring the result is a recurrent event data set examples include epileptic seizures and recurrent cancers when the recurring event is detectable only by an expensive or invasive examination only the number of events occurring between follow up times may be available this article presents a semiparametric model for such data based on a multiplicative intensity model paired with a fully flexible nonparametric baseline intensity function a random subject specific effect is included in the intensity model to account for the overdispersion frequently displayed in count data estimators are determined from quasi likelihood estimating functions because only first and second moment assumptions are required for quasi likelihood the method is more robust than those based on the specification of a full parametric likelihood consistency of the estimators depends only on the assumption of the proportional intensity model the semiparametric estimators are shown to be highly efficient compared with the usual parametric estimators as with semiparametric methods in survival analysis the method provides useful diagnostics for specific parametric models including a quasi score statistic for testing specific baseline intensity functions the techniques are used to analyze cancer recurrences and a pheromone based mating disruption experiment in moths a simulation study confirms that for many practical situations the estimators possess appropriate small sample characteristics

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of data

statistics of heteroscedastic extremes we extend classical extreme value theory to non identically distributed observations when the tails of the distribution are proportional much of extreme value statistics remains valid the proportionality function for the tails can be estimated non parametrically along with the common extreme value index for a positive extreme value index joint asymptotic normality of both estimators is shown they are asymptotically independent we also establish asymptotic normality of a forecasted high quantile and develop tests for the proportionality function and for the validity of the model we show through simulations the good performance of the procedures and also present an application to stock market returns a main tool is the weak convergence of a weighted sequential tail empirical process

a variance components approach to categorical data models with heterogeneous cell populations analysis of spatial gradients in lung cancer mortality rates in north carolina counties a mixed categorical continuous variable model is proposed for the analysis of mortality rates this model differs from other available models such as weighted least squares and loglinear models in that the within cell populations are assumed to be heterogeneous in their levels of mortality risk heterogeneity implies that in addition to the sampling variance considered in other available models there will be a second component of variance due solely to within cell heterogeneity maximum likelihood procedures are presented for the estimation of the model parameters these procedures are based on the assumption that the distribution function for each cell death count is the negative binomial probability function this assumption is shown to be equivalent to assuming a mixture of poisson processes with the differential risk levels among individuals within each cell being governed by a two parameter gamma distribution the model is applied to data on lung cancer mortality for 1970 1975 for the 100 counties of north carolina the analysis shows that though a gradient in lung cancer mortality rates exist in space the gradient is restricted to specific demographic categories identified by race age and sex

diagnostics for overdispersion abstract diagnostic tools are proposed for assessing the dependence of extrabinomial or extra poisson variation on explanatory variables and for comparing several common models for overdispersion these tools are based on tests for regression terms in the dispersion parameter of a generalized linear model using double exponential family and pseudolikelihood formulations score tests do not require the full fitting of models for variation and lead to easy graphical and numerical procedures based on squared residuals deviance or pearson robust modifications of these are motivated by levene s test in linear models the diagnostic tools are intended primarily to ensure prudent modeling of the variance to make correct inferences about parameters in the mean

biometrika centenary sample surveys summary despite karl pearson s stated aim that the evolutionist has to become in the widest sense of the words a registrar general for all forms of life the papers published in biometrika on sample surveys have been mainly theoretical and do not start to appear until the 1940s there has been little work on sampling in the life sciences with the exception of capture recapture methods the major areas have been in variable probability designs the foundations of inference for finite populations analytic surveys and the role of selection mechanisms in inference in recent years the work on survey analysis has been the outcome of an interaction between sample selection mechanisms and mainstream theory with interesting consequences for both areas

applications and extensions of chao s moment estimator for the size of a closed population summary this article revisits chao s 1989 biometrics 45 427 438 lower bound estimator for the size of a closed population in a mark recapture experiment where the capture probabilities vary between animals model mh first an extension of the lower bound to models featuring a time effect and heterogeneity in capture probabilities mth is proposed the biases of these lower bounds are shown to be a function of the heterogeneity parameter for several loglinear models for mth small sample bias reduction techniques for chao s lower bound estimator are also derived the application of the loglinear model underlying chao s estimator when heterogeneity has been detected in the primary periods of a robust design is then investigated a test for the null hypothesis that chao s loglinear model provides unbiased abundance estimators is provided the strategy of systematically using chao s loglinear model in the primary periods of a robust design where heterogeneity has been detected is investigated in a monte carlo experiment its impact on the estimation of the population sizes and of the survival rates is evaluated in a monte carlo experiment

the fisher information matrix for log linear models arguing conditionally on observed explanatory variable summary for poisson or multinomial contingency table data the conditional distribution is product multinomial when conditioning on observed values of explanatory variables birch 1963 showed that under the restriction formed by keeping the marginal totals of one margin fixed at their observed values the poisson multinomial and product multinomial likelihoods are proportional and give the same estimates for common parameters in the log linear model here the inverses of the fisher information matrices are shown to be identical over common parameters so that the asymptotic covariance matrices of the estimates correspond

mark recapture models with parameters constant in time the jolly seber method which allows for both death and immigration is easy to apply but often requires a larger number of parameters to be estimated tha would otherwise be necessary if i survival rate phi or ii probability of capture p or iii both phi and p can be assumed constant over the experimental period models with a reduced number of parameters are desirable in the present paper maximum likelihood ml solutions for these three situations are derived from the general ml equations of jolly 1979 in sampling biological populations r m cormack g p patil and d s robson eds 277 282 a test is proposed for heterogeneity arising from a breakdown of assumptions in the general jolly seber model tests for constancy of phi and p are provided an example is given in which these models are fitted to data from a local butterfly population

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

turing instability induced by random network in fitzhugh nagumo model abstract although there is general agreement that the network plays an essential role in the biological system how the connection probability of network affects the natural model especially neural network is poorly understood in this paper we show the impact of the network on turing instability in the fitzhugh nagumo fn model then we obtain the condition of how the turing bifurcation saddle node bifurcation and turing instability occur by using the gershgorin circle theorem we investigate the relationship between degree and eigenvalue of the network matrix and obtain the approximate range of eigenvalue of the network matrix also we derive the instability condition about diffusion and the connection probability in the network organized system and then we obtain the estimated range of connection probability meanwhile we apply these results to explaining the spiking of neuron and find this system has rich dynamics behavior finally the numerical simulation verifies our analytical results

stability of turing patterns in the brusselator model the selection and competition of turing patterns in the brusselator model are reviewed the stability of stripes and hexagons towards spatial perturbations is studied using the amplitude equation formalism for hexagonal patterns these equations include both linear and nonpotential spatial terms enabling distorted solutions the latter modify substantially the stability diagrams and select patterns with wave numbers quite different from the critical value the analytical results from the amplitude formalism agree with direct simulations of the model moreover we show that slightly squeezed hexagons are locally stable in a full range of distortion angles the stability regions resulting from the phase equation are similar to those obtained numerically by other authors and to those observed in experiments

capture recapture models allowing for age dependent survival and capture rates a k sample capture recapture model for an open population of animals which allows for different identifiable age categories to have different survival and capture probabilities is developed explicit maximum likelihood estimators of population size and survival rates together with their asymptotic variances and covariances are given a test of whether survival and capture rates are independent of age is shown to be of simple hypergeometric form an illustrative example based on resighting data of neck collared giant canada geese branta canadergsis maxima is presented the relationship of this model to other capture recapture and band recovery models is discussed

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

turing pattern dynamics in an activator inhibitor system with superdiffusion the fractional operator is introduced to an activator inhibitor system to describe species anomalous superdiffusion the effects of the superdiffusive exponent on pattern formation and pattern selection are studied our linear stability analysis shows that the wave number of the turing pattern increases with the superdiffusive exponent a weakly nonlinear analysis yields a system of amplitude equations and the analysis of these amplitude equations predicts parameter regimes where hexagons stripes and their coexistence are expected numerical simulations of the activator inhibitor model near the stability boundaries confirm our analytical results since diffusion rate manifests in both diffusion constant and diffusion exponent we numerically explore their interactions on the emergence of turing patterns when the activator and inhibitor have different superdiffusive exponents we find that the critical ratio of the diffusion rate of the inhibitor to the activator required for the formation of the turing pattern increases monotonically with the superdiffusive exponent we conclude that small ratio than unity of anomalous diffusion exponent between the inhibitor and activator is more likely to promote the emergence of the turing pattern relative to the normal diffusion

derivation of amplitude equations for nonlinear oscillators subject to arbitrary forcing by using a generalization of the multiple scales technique we develop a method to derive amplitude equations for zero dimensional forced systems the method allows to consider either additive or multiplicative forcing terms and can be straightforwardly applied to the case that the forcing is white noise we give examples of the use of this method to the case of the van der pol duffing oscillator the writing of the amplitude equations in terms of a lyapunov potential allow us to obtain an analytical expression for the probability distribution function which reproduces reasonably well the numerical simulation results

a general methodology for the analysis of capture recapture experiments in open populations we trace the development of a likelihood function representation for the open population capture recapture jolly seber experiment we find that the modelling of the birth process in the general model is not consistent with the reduced death only model and that all formulations to date lead to difficulties in imposing constraints upon the parameters of the birth process we propose a generalisation to the usual jolly seber representation that models births using a multinomial distribution from a super population we show how this leads to simplifications in the numerical optimization of the likelihood and how constraints upon the parameters of the model can now be easily imposed we show how covariate models using auxiliary variables such as sampling effort or weather conditions to explain capture or survival rates can also be easily added we also show how this model can be generalised to more than one group of animals finally a numerical example is provided which fits a class of models where the capture probabilities survival probabilities and birth probabilities can each vary over time or among groups or both this permits sequential model fitting within a comprehensive model framework an approach akin to that of lebreton et al ecological monographs 62 67 118

log linear models for capture recapture summary log linear models are developed for capture recapture experiments and their advantages and disadvantages discussed ways in which they can be extended sometimes with only partial success to open populations subpopulations trap dependence and long chains of recapture periods are presented the use of residual patterns and analysis of subsets of data to identify behavioural patterns and acceptable models is emphasised and illustrated with two examples 1 capture recapture the biological problem with which we are concerned is that of estimating the parameters of a free living population of unknown size from a sequence of s samples in each of which any individual either is or is not observed the models in this paper require that each animal be marked naturally or artificially in such a way that we can observe the numbers of animals which have any particular pattern of being or not being observed through the sequence of samples there are t 2s 1 different observable patterns which we shall refer to as capture histories although there is no requirement for physical capture and denote each pattern by a sequence of l s and 2 s representing respectively seen or not seen in a particular sample a general pattern of capture history will be denoted a with ra the number of individuals observed to have this pattern thus for example with s 3 samples and a 121 r 2 is the number of animals seen in the first and third samples but not in the second the pattern with all 2 s is by definition unobservable in general the population will be subject to demographic change by birth or immigration and by death or emigration the simplest assumption for the response of an animal to the observer one which underlies the classical petersen schnabel or jolly seber estimators is that every animal alive in the population at a sampling time has the same probability of being observed this probability may vary from one sample to another or be constant over all or a number of samples if this assumption fails it may do so for either or both of two reasons the first is that of trap dependence the probability of an individual being observed in a sample depending on its past history of capture the second is that of heterogeneity between individuals who because of differences in their behaviour or in where they live inherently have different probabilities of being observed in any sample the basic idea of heterogeneity refers to properties of individuals which remain constant over time but individual animals may also have time varying behaviour the aspect most commonly considered being temporary departure of an individual from the population various excellent texts describe theoretical and practical aspects of capture recapture notably the comprehensive one by seber 1982 the introduction by begon 1979 the

fitting time series models to nonstationary processes a general minimum distance estimation procedure is presented for nonstationary time series models that have an evolutionary spectral representation the asymptotic properties of the estimate are derived under the assumption of possible model misspecification for autoregressive processes with time varying coefficients the estimate is compared to the least squares estimate furthermore the behavior of estimates is explained when a stationary model is fitted to a nonstationary process

efficiency of martingale methods in recapture studies summary the asymptotic variances of various estimators of the population size from a recapture study are calculated the estimation methods compared are maximum likelihood and a class of natural martingale estimators with both time variation and behavioural response it is found that i the martingale methods appropriate for time variation or behavioural response are fully efficient ii there is no loss of efficiency at all in allowing for time variation iii there is large loss of efficiency in allowing for behavioural response iv there is a natural martingale method that allows for both time variation and behavioural response

obtaining confidence limits on parameters of the jolly seber model for capture recapture data on propose quelques transformations de la methode pour les estimateurs des tailles de la population et des probabilites de survie

pattern formation in the fitzhugh nagumo model in this paper we investigate the effect of diffusion on pattern formation in fitzhugh nagumo model through the linear stability analysis of local equilibrium we obtain the condition how the turing bifurcation hopf bifurcation and the oscillatory instability boundaries arise by using the method of the weak nonlinear multiple scales analysis and taylor series expansion we derive the amplitude equations of the stationary patterns the analysis of amplitude equations shows the occurrence of different complex phenomena including turing instability eckhaus instability and zigzag instability in addition we apply this analysis to fitzhugh nagumo model and find that this model has very rich dynamical behaviors such as spotted stripe and hexagon patterns finally the numerical simulation shows that the analytical results agree with numerical simulation

comparison of methods for the analysis of longitudinal interval count data longitudinal studies are often concerned with estimating the recurrence rate of a non fatal event in many cases only the total number of events occurring during successive time intervals is known we compared a mixed poisson gamma regression method proposed by thall and a quasi likelihood method proposed by zeger and liang for the analysis of such data in the case where the mean was correctly specified using simulation techniques with large samples both methods produced similar standard errors in most situations except in the case of time dependent covariates with non poisson gamma data where they were seriously underestimated by the thall method a simple method for discriminating between the variance forms of the two methods is described the findings are applied to the analyses of clinical trials of non melanoma skin cancer and familial polyposis this study extends the findings of breslow concerning variance misspecification in overdispersed poisson and quasi likelihood models to the longitudinal setting

variances of mark recapture estimates when mark recapture data are analysed by log linear models as described by cormack 1989 biometrics 45 395 413 the analysis is in terms of parameters that are related nonlinearly to the parameters of biological interest for several models formulae are given expressing the variance of the interesting parameter estimates in terms of those provided by a computer log linear analysis a device is presented that allows such expressions to be computed simply

should annual maximum temperatures follow a generalized extreme value distribution summary motivated by the study of annual temperature extremes two new results on the limiting distribution of block maxima of random variables with varying upper bounds are obtained one gives a generalized extreme value distribution as the limit but with a different shape parameter from that obtained when the bound on the random variables does not vary the other gives a limiting distribution that is only a generalized extreme value in certain cases both results consider triangular arrays of random variables in order to mimic the property of an upper bound that changes slowly with the day of the year as seems to occur for temperature data at many locations an analysis of 140 years of daily temperatures in new york city shows mixed results in terms of the ability of the theory presented here to provide new insights into the behaviour of extreme temperatures at this location

robust estimation in heteroscedastic linear models we consider a heteroscedastic linear model in which the variances are given by a parametric function of the mean responses and a parameter theta we propose robust estimates for the regression parameter beta and show that as long as a reasonable starting estimate of theta is available our estimates of beta are asymptotically equivalent to the natural estimate obtained with known variances a particular method for estimating theta is proposed and shown by monte carlo to work quite well especially in power and exponential models for the variances we also briefly discuss a feedback estimate of beta

interval estimation for mark recapture studies of closed populations textbooks continue to recommend the use of an asymptotic normal distribution to provide an interval estimate for the unknown size n of a closed population studied by a mark recapture experiment or multiple record system a likelihood interval approach is proposed and its implementation demonstrated for a range of models for such studies including all main effect and interaction models for incomplete contingency tables

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

plant capture estimation of the size of a homogeneous population we consider maximum likelihood estimation of the size of a target population to which has been added a known number of planted individuals the standard equal catchability model used in mark recapture is assumed to be applicable to the augmented population after proving the unimodality of the profile likelihood for the target population size we obtain both the maximum likelihood estimator of this size and interval estimators based on its asymptotic distribution

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures