risk sensitive and robust escape criteria the problem of controlling a noisy process so as to prevent it from leaving a prescribed set has a number of interesting applications in this paper new approaches to this problem are considered first a risk sensitive criterion for a stochastic diffusion process model is examined and it is shown that the value is a classical solution of a related pde the qualitative properties of this criterion are favorably contrasted with those of existing criteria in the risk averse limit it is proved that in the risk averse limit the value of the risk sensitive criterion converges to a viscosity solution of a first order pde it is then demonstrated that the value function of a deterministic differential game is also a viscosity solution to the pde this game gives a robust control formulation of the escape time problem and is analogous to h infty control in particular the opposing player attempts to push the process out of the prescribed set and suffers an l2 cost for his efforts lower bounds on the escape time as a function of this cost are obtained

a note on n estimators for the binomial distribution abstract consider k success counts from a binomial distribution with unknown n and success probability p we examine the problem of estimating n by integrating the likelihood for n and p over a beta density for p we obtain the beta binomial distribution resulting in stable and reasonably efficient estimators of n which compare favorably with and are often better than the estimates introduced by olkin et al 1981

estimation of population trajectories from count data monitoring changes in animal population size is rarely possible through complete censuses frequently the only feasible means of monitoring changes in population size is to use counts of animals obtained by skilled observers as indices to abundance analysis of changes in population size can be severely biased if factors related to the acquisition of data are not adequately controlled for in particular we identify two types of observer effects these correspond to baseline differences in observer competence and to changes through time in the ability of individual observers we present a family of models for count data in which the first of these observer effects is treated as a nuisance parameter conditioning on totals of negative binomial counts yields a dirichlet compound multinomial vector for each observer quasi likelihood is used to estimate parameters related to population trajectory and other parameters of interest model selection is carried out on the basis of akaike s information criterion an example is presented using data on wood thrush from the north american breeding bird survey

minimizing the probability of lifetime ruin under ambiguity aversion we determine the optimal robust investment strategy of an individual who targets at a given rate of consumption and seeks to minimize the probability of lifetime ruin when she does not have perfect confidence in the drift of the risky asset using stochastic control we characterize the value function as the unique classical solution of an associated hamilton jacobi bellman hjb equation obtain feedback forms for the optimal investment and drift distortion and discuss their dependence on various model parameters in analyzing the hjb equation we establish the existence and uniqueness of a viscosity solution using perron s method and then upgrade regularity by working with an equivalent convex problem obtained via the cole hopf transformation we show the original value function may lose convexity for a class of parameters and the isaacs condition may fail numerical examples are also included to illustrate our results

application of the boundary integral equation method to reissner s plate model numerical solution of reissner s model for the bending of elastic plates by an advanced boundary integral equation method is outlined construction of fundamental solutions and computational strategy is fully described implementation of the resulting one dimensional model is illustrated by two test problems including comparison to analytical solutions

a viscosity solution approach to the asymptotic analysis of queueing systems we consider a system of several interconnected queues with a single class of customers and model the state xt of the system as a jump markov process the problem of interest is to estimate the large deviations behavior of the rescaled system xt ext s corresponding to large time and large excursions of the original unscaled system the techniques employed are those of the theory of viscosity solutions to hamilton jacobi equations from the point of view of large deviation theory the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty since an abrupt change in the jump measure occurs from the point of view of viscosity solutions the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations much of the paper is devoted to proving uniqueness of viscosity solutions for these equations and these sections are of independent interest while our use of test functions in proving the uniqueness is an adaptation of the usual technique the construction of the test functions themselves via the legendre transform is new we obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem which suggests the correct integrand in the large deviation rate functional since it is the treatment of the effects due to the boundaries that is novel we devote the majority of the paper to the detailed development of a simple two dimensional system that exhibits all the essential new features however the arguments may be applied to nueueing systems that are considerably more general and we attempt to indicate this generality as well

risk sensitive asset management and cascading defaults we consider an optimal risk sensitive portfolio allocation problem accounting for the possibility of cascading defaults default events have an impact on the distress state of the surviving stocks in the portfolio we study the recursive system of non lipschitz quasilinear parabolic hjb pdes associated with the value function of the control problem in the different default states of the economy we show the existence of a classical solution to this system via super sub solution techniques and give an explicit characterization of the optimal feedback strategy in terms of the value function we prove a verification theorem establishing the uniqueness of the solution a numerical analysis indicates that the investor significantly accounts for contagion effects when making investment decisions and that his strategy depends nonmonotonically on the aggregate risk level

approximation of escape time for lagrangian systems with fast noise this note is concerned with the large deviations asymptotics estimate of the mean escape time for lagrangian systems subjected to fast gaussian noise the solution to the hamilton jacobi equation of the associated variational problem is derived as a sum of two terms dependent on kinetic and potential energy of the system respectively a closed form solution for classes of linear and nonlinear systems is obtained an application to a controlled pointing problem is discussed as an example

risk sensitive portfolio management with cox ingersoll ross interest rates the hjb equation this paper presents an application of risk sensitive control theory in financial decision making the investor has an infinite horizon objective that can be interpreted as maximizing the portfolio s risk adjusted exponential growth rate there are two assets a stock and a bank account and two underlying brownian motions so this model is incomplete the novel feature here is that the interest rate for the bank account is governed by cox ingersoll ross dynamics this is significant for risk sensitive portfolio management because the factor process unlike in the gaussian and all other cases treated in the literature cannot be negative under appropriate parameterization

augmented levy michell equations for flexural plates abstract a new sixth order isotropic elastic and flexural plate theory is presented based on the fourth order equations of levy 1877 and michell 1900 the lack of surface loading in these levy michell equations is overcome by including the dominant flexural component of dougall s 1904 solution for a point load on the surface of a three dimensional isotropic elastic layer the augmented plate equations are arranged to fit the form of a consistent sixth order system of isotropic plate equations which include the well known work of reissner and static equations of mindlin numerical applications to two three dimensional problems with analytical solutions show good comparative results

risk sensitive control on an infinite time horizon stochastic control problems on an infinite time horizon with exponential cost criteria are considered the donsker varadhan large deviation rate is used as a criterion to be optimized the optimum rate is characterized as the value of an associated stochastic differential game with an ergodic expected average cost per unit time cost criterion if we take a small noise limit a deterministic differential game with average cost per unit time cost criterion is obtained this differential game is related to robust control of nonlinear systems

double exponential families and their use in generalized linear regression abstract in one parameter exponential families such as the binomial and poisson the variance is a function of the mean double exponential families allow the introduction of a second parameter that controls variance independently of the mean double families are used as constituent distributions in generalized linear regressions in which both means and variances are allowed to depend on observed covariates the theory is applied to two examples a logistic regression and a large two way contingency table in such cases the binomial model of variance is often untrustworthy for example because genuine random sampling was infeasible the subjects may have been obtained in clumps so that the statistician should really be using smaller sample sizes clumped sampling is just one of many possible causes of overdispersion a habitual source of concern to users of binomial and poisson models this article concerns a class of regression families that allow the statistician to model overdispersion while carrying

bernstein nagumo conditions and solutions to nonlinear differential inequalities abstract for an open bounded subset of r n with smooth boundary and 1 p we establish w 1 p a priori bounds and prove the compactness of solution sets to differential inequalities of the form div a x u f x u u which are bounded in l the main point in this work is that the nonlinear term f may depend on u and may grow as fast as a power of order p in this variable such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as bernstein nagumo growth conditions in addition we use these results to establish a sub supersolution theorem

the general fundamental solution of the sixth order reissner and mindlin plate bending models revisited abstract the sixth order differential equation system of the reissner and mindlin plate bending models describe mathematically the plate problem where lines normal to the mid plane before deformation remain straight and inextensible in the deformed configuration but not necessarily normal to the reference plane anymore the non normality condition is due to the consideration of transverse shearing strains disregarded in the classical bi harmonic kirchhoff plate model as the plate thickness is reduced and or the transverse shear modulus is increased the deformed configuration is less dependent of the transverse shear strains and in the limit as the plate thickness approaches zero and or the transverse shear modulus approaches the transverse shear strain effects vanish and the problem is exactly that described by the classical plate model in this paper we investigate the fundamental solutions of both the fourth order kirchhoff and the sixth order reissner and mindlin plate models we consider a transversely isotropic material and show that the fundamental solution of the bi harmonic problem can be obtained directly from the general fundamental solution of the sixth order plate problem in the limit as the plate thickness approaches zero and or the transverse shear modulus approaches this solution is in agreement with the analytical solution of an infinite thin clamped circular plate submitted to a unitary concentrated load acting at its center

on the reliability of n mixture models for count data n mixture models describe count data replicated in time and across sites in terms of abundance n and detectability p they are popular because they allow inference about n while controlling for factors that influence p without the need for marking animals using a capture recapture perspective we show that the loss of information that results from not marking animals is critical making reliable statistical modeling of n and p problematic using just count data one cannot reliably fit a model in which the detection probabilities are distinct among repeat visits as this model is overspecified this makes uncontrolled variation in p problematic by counter example we show that even if p is constant after adjusting for covariate effects the constant p assumption scientifically plausible alternative models in which n or its expectation is non identifiable or does not even exist as a parameter lead to data that are practically indistinguishable from data generated under an n mixture model this is particularly the case for sparse data as is commonly seen in applications we conclude that under the constant p assumption reliable inference is only possible for relative abundance in the absence of questionable and or untestable assumptions or with better quality data than seen in typical applications relative abundance models for counts can be readily fitted using poisson regression in standard software such as r and are sufficiently flexible to allow controlling for p through the use covariates while simultaneously modeling variation in relative abundance if users require estimates of absolute abundance they should collect auxiliary data that help with estimation of p

an asymptotic approximation to the n mixture model for the estimation of disease prevalence n mixture models are probability models that estimate abundance using replicate observed counts while accounting for imperfect detection in this article we propose an asymptotic approximation to the n mixture model which efficiently estimates large abundances without the computational limitations of the generalized n mixture model introduced by dail and madsen in 2011 it has been suggested in the literature that n mixture models do not perform well when counts from the same sites show weak patterns of population dynamics our proposed model addresses this issue by using the asymptotic information matrix to diagnose model parameter estimability and to derive parameter standard errors a simulation study show that this model performs almost as well as the dail madsen generalized n mixture model at low abundances and improves on it at higher abundances we illustrate the procedure using two data sets the american robin data from dail and madsen 2011 and counts of chlamydia cases in the state of oregon from 2007 2016 the chlamydia data exhibit very large abundances and demonstrate the potential usefulness of the proposed model for disease surveillance data

a weak dynamic programming principle for zero sum stochastic differential games with unbounded controls we analyze a zero sum stochastic differential game between two competing players who can choose unbounded controls the payoffs of the game are defined through backward stochastic differential equations we prove that each player s priority value satisfies a weak dynamic programming principle and thus solves the associated fully nonlinear partial differential equation in the viscosity sense

minimizing escape probabilities a large deviations approach this paper considers the problem of controlling a possibly degenerate diffusion process so as to minimize the probability of escape over a given time interval it is assumed that the control acts on the process through the drift coefficient and that the noise coefficient is small developing a large deviations type of theory for the controlled diffusion produces several results the limit of the normalized log of the minimum exit probability is identified as the value i of an associated deterministic differential game furthermore we identify a deterministic and varepsilon independent mapping g from the sample values varepsilon w s 0 leqq s leqq t into the control space such that if we define the control used at time t by u t g varepsilon w s 0 leqq s leqq t then the resulting control process is progressively measurable and delta optimal in the sense that the limit of the normalized log of the exit probability is within delta of i

a large deviations analysis of range tracking loops large deviations theory is applied to the analysis of a discrete time range tracking loop it is shown that the resulting asymptotics differ from those of the continuous time diffusion limit

risk sensitive control for the parallel server model a markovian queueing model is considered in which servers of various types work in parallel to process jobs from a number of classes at rates mu ij that depend on the class i and the type j the problem of dynamic resource allocation so as to minimize a risk sensitive criterion is studied in a law of large numbers scaling letting x i t denote the number of class i jobs in the system at time t the cost is given by e exp n int 0 th bar x t dt g bar x t where t 0 h and g are given functions satisfying regularity and growth conditions and bar x bar x n n 1 x n cdot it is well known in an analogous context of controlled diffusion and has been shown for some classes of stochastic networks that the limit behavior as n to infty is governed by a differential game dg in which the state dynamics is given by a fluid equation for the formal limit varphi of bar x while the cost consists of int 0 th varphi t dt g varphi t and an additional term th

an escape time criterion for queueing networks asymptotic risk sensitive control via differential games we consider the problem of risk sensitive control of a stochastic network in controlling such a network an escape time criterion can be useful if one wishes to regulate the occurrence of large buffers and buffer overflow in this paper a risk sensitive escape time criterion is formulated which in comparison to the ordinary escape time criteria penalizes exits that occur on short time intervals more heavily the properties of the risk sensitive problem are studied in the large buffer limit and related to the value of a deterministic differential game with constrained dynamics we prove that the game has value and that the value is the viscosity solution of a pde for a simple network the value is computed demonstrating the applicability of the approach

multinomial n mixture models for removal sampling multinomial n mixture models are commonly used to fit data from a removal sampling protocol if the mixing distribution is negative binomial the distribution of the counts does not appear to have been identified and practitioners approximate the requisite likelihood by placing an upper bound on the embedded infinite sum in this paper the distribution which underpins the multinomial n mixture model with a negative binomial mixing distribution is shown to belong to the broad class of multivariate negative binomial distributions specifically the likelihood can be expressed in closed form as the product of conditional and marginal likelihoods and the information matrix shown to be block diagonal as a consequence the nature of the maximum likelihood estimates of the unknown parameters and their attendant standard errors can be examined and tests of the hypothesis of the poisson against the negative binomial mixing distribution formulated in addition appropriate multinomial n mixture models for data sets which include zero site totals can also be constructed two illustrative examples are provided

computational aspects of n mixture models the n mixture model is widely used to estimate the abundance of a population in the presence of unknown detection probability from only a set of counts subject to spatial and temporal replication royle 2004 biometrics 60 105 115 we explain and exploit the equivalence of n mixture and multivariate poisson and negative binomial models which provides powerful new approaches for fitting these models we show that particularly when detection probability and the number of sampling occasions are small infinite estimates of abundance can arise we propose a sample covariance as a diagnostic for this event and demonstrate its good performance in the poisson case infinite estimates may be missed in practice due to numerical optimization procedures terminating at arbitrarily large values it is shown that the use of a bound k for an infinite summation in the n mixture likelihood can result in underestimation of abundance so that default values of k in computer packages should be avoided instead we propose a simple automatic way to choose k the methods are illustrated by analysis of data on hermann s tortoise testudo hermanni

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

n mixture models for estimating population size from spatially replicated counts spatial replication is a common theme in count surveys of animals such surveys often generate sparse count data from which it is difficult to estimate population size while formally accounting for detection probability in this article i describe a class of models n mixture models which allow for estimation of population size from such data the key idea is to view site specific population sizes n as independent random variables distributed according to some mixing distribution e g poisson prior parameters are estimated from the marginal likelihood of the data having integrated over the prior distribution for n carroll and lombard 1985 journal of american statistical association 80 423 426 proposed a class of estimators based on mixing over a prior distribution for detection probability their estimator can be applied in limited settings but is sensitive to prior parameter values that are fixed a priori spatial replication provides additional information regarding the parameters of the prior distribution on n that is exploited by the n mixture models and which leads to reasonable estimates of abundance from sparse data a simulation study demonstrates superior operating characteristics bias confidence interval coverage of the n mixture estimator compared to the caroll and lombard estimator both estimators are applied to point count data on six species of birds illustrating the sensitivity to choice of prior on p and substantially different estimates of abundance as a consequence

a risk sensitive maximum principle the case of imperfect state observation the risk sensitive maximum principle for optimal stochastic control derived by the author in an earlier work system control letters vol 15 1990 is restated this is an immediate generalization of the classic pontryagin principle to which it reduces in the deterministic case and is expressed immediately in terms of observables it is derived on the assumption that the criterion function is the exponential of an additive cost function and is exact under linear quadratic gaussian assumptions but is otherwise valid as a large deviation approximation the principle is extended to the case of imperfect state observation after preliminary establishment of a certainty equivalence principle the derivation yields as byproduct a large deviation version of the updating equation for nonlinear filtering the development is heuristic it is believed that the mathematical arguments given are the essential ones and provide a self contained treatment at this level