First Order Asymptotic Approximation Research Articles

Sequential Anomaly Detection Under Sampling Constraints

The problem of sequential anomaly detection is considered, where multiple data sources are monitored in real time and the goal is to identify the “anomalous” ones among them, when it is not possible to sample all sources at all times. A detection scheme in this context requires specifying not only when to stop sampling and which sources to identify as anomalous upon stopping, but also which sources to sample at each time instance until stopping. A novel formulation for this problem is proposed, in which the number of anomalous sources is not necessarily known in advance and the number of sampled sources per time instance is not necessarily fixed. Instead, an arbitrary lower bound and an arbitrary upper bound are assumed on the number of anomalous sources, and the fraction of the expected number of samples over the expected time until stopping is required to not exceed an arbitrary, user-specified level. In addition to this sampling constraint, the probabilities of at least one false alarm and at least one missed detection are controlled below user-specified tolerance levels. A general criterion is established for a policy to achieve the minimum expected time until stopping to a first-order asymptotic approximation as the two familywise error rates go to zero. Moreover, the asymptotic optimality is established of a family of policies that sample each source at each time instance with a probability that depends on past observations only through the current estimate of the subset of anomalous sources. This family includes, in particular, a novel policy that requires minimal computation under any setup of the problem.

Probabilistic modeling of threshold stress intensity factor for fatigue endurance reliability prediction

This study develops a probabilistic model for the threshold stress intensity factor range, which is a critical parameter in infinite fatigue life design under material flaws. The model is based on the proposed concept of probability of propagation in the probabilistic framework, allowing for deriving the probability density function of the threshold intensity factor range. The uncertainty in fatigue crack growth can naturally be incorporated into the resulting distribution. By further introducing the derived distribution into the Kitagawa–Takahashi diagram, the fatigue endurance reliability model can be established in a rational manner. With the first-order asymptotic approximation, the analytical form of fatigue endurance reliability index is obtained. The usefulness of the overall method is demonstrated using realistic engineering application examples.

Sequential multiple testing with generalized error control: An asymptotic optimality theory

The sequential multiple testing problem is considered under two generalized error metrics. Under the first one, the probability of at least $k$ mistakes, of any kind, is controlled. Under the second, the probabilities of at least $k_1$ false positives and at least $k_2$ false negatives are simultaneously controlled. For each formulation, the optimal expected sample size is characterized, to a first-order asymptotic approximation as the error probabilities go to 0, and a novel multiple testing procedure is proposed and shown to be asymptotically efficient under every signal configuration. These results are established when the data streams for the various hypotheses are independent and each local log-likelihood ratio statistic satisfies a certain Strong Law of Large Numbers. In the special case of i.i.d. observations in each stream, the gains of the proposed sequential procedures over fixed-sample size schemes are quantified.

Asymptotic Bayesian Theory of Quickest Change Detection for Hidden Markov Models

In the 1960s, Shiryaev developed a Bayesian theory of change-point detection in the i.i.d. case, which was generalized in the early 2000s by Tartakovsky and Veeravalli and recently by Tartakovsky (2017) for general stochastic models assuming a certain stability of the log-likelihood ratio process. Hidden Markov models represent a wide class of stochastic processes in a variety of applications. In this paper, we investigate the performance of the Bayesian Shiryaev change-point detection rule for hidden Markov models. We propose a set of regularity conditions under which the Shiryaev procedure is first-order asymptotically optimal in a Bayesian context, minimizing moments of the detection delay up to certain order asymptotically as the probability of false alarm goes to zero. The developed theory for hidden Markov models is based on Markov chain representation for the likelihood ratio and r -quick convergence for Markov random walks. In addition, applying Markov nonlinear renewal theory, we present a high-order asymptotic approximation for the expected delay to detection and a first-order asymptotic approximation for the probability of false alarm of the Shiryaev detection rule. We also study asymptotic properties of another popular change detection rule, the Shiryaev–Roberts rule, and provide some interesting examples.

Numerical Studies on Asymptotics of European Option Under Multiscale Stochastic Volatility

Multiscale stochastic volatilities models relax the constant volatility assumption from Black-Scholes option pricing model. Such models can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. Christoffersen et al. Manag Sci 55(2):1914–1932 (2009) presented a model where the underlying price is governed by two volatility components, one changing fast and another changing slowly. Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013) transformed Christoffersen’s model and computed an approximate formula for pricing American options. They used Duhamel’s principle to derive an integral form solution of the boundary value problem associated to the option price. Using method of characteristics, Fourier and Laplace transforms, they obtained with good accuracy the American option prices. In a previous research of the authors (Canhanga et al. 2014), a particular case of Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013) model is used for pricing of European options. The novelty of this earlier work is to present an asymptotic expansion for the option price. The present paper provides experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices will be compared to the approximation obtained by Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013).

Comments on: Reflections on the Probability Space Induced by Moment Conditions with Implications for Bayesian Inference

The traditional Bayesian approach to inference is based on the combination of a fully specified density for the data conditional on the model parameters (the likelihood) with prior views on those parameters. Aside from other methodological considerations, the advantage of using prior information may be particularly important in low-frequency macro/finance time series contexts in which the number of observations is insufficient to precisely pin down the values of the unknown model parameters. Nevertheless, a potential drawback of the traditional Bayesian approach is that it is a full information procedure, which requires the correct specification of features of the distribution of the observed variables that the researcher might not be particularly interested in. In fact, many researchers prefer to use limited-information frequentist procedures, often with semi-parametric components, because under certain regularity conditions they reduce the potential for inconsistent estimation resulting from incorrect distributional assumptions. Whether those regularity conditions hold in any particular application (see Sims 2007 ), or whether the finite-sample performance of the limited-information, semi-parametric procedures agrees with the usual first-order asymptotic approximations even when they hold (see e.g., Cattaneo and Jansson 2014 ), is a different matter.

Low-frequency dispersion characteristics of a multilayered coaxial cable

This paper provides an exact asymptotic analysis regarding the low-frequency dispersion characteristics of a multilayered coaxial cable. A layer-recursive description of the dispersion function is derived that is well suited for asymptotic analysis. The recursion is based on two well-behaved (meromorphic) subdeterminants defined by a perfectly electrically conducting (PEC) and a perfectly magnetically conducting termination, respectively. For an open waveguide structure, the dispersion function is a combination of two such functions, and there is only one branch point that is related to the exterior domain. It is shown that if there is one isolating layer and a PEC outer shield, then the classical Weierstrass preparation theorem can be used to prove that the low-frequency behavior of the propagation constant is governed by the square root of the complex frequency, and an exact analytical expression for the dominating term of the asymptotic expansion is derived. It is furthermore shown that the same asymptotic expansion is valid to its lowest order even if the outer shield has finite conductivity and there is an infinite exterior region with finite nonzero conductivity. As a practical application of the theory, a high-voltage direct current (HVDC) power cable is analyzed and a numerical solution to the dispersion relation is validated by comparisons with the asymptotic analysis. The comparison reveals that the low-frequency dispersion characteristics of the power cable is very complicated and a first-order asymptotic approximation is valid only at extremely low frequencies (below 1 Hz). It is noted that the only way to come to this conclusion is to actually perform the asymptotic analysis. Hence, for practical modeling purposes, such as with fault localization, an accurate numerical solution to the dispersion relation is necessary and the asymptotic analysis is useful as a validation tool.

Parametric bootstrap under model mis-specification

Under model correctness, highly accurate inference on a scalar interest parameter in the presence of a nuisance parameter can be achieved by several routes, among them considering the bootstrap distribution of the signed root likelihood ratio statistic. The context of model mis-specification is considered and inference based on a robust form of the signed root statistic is discussed in detail. Stability of the distribution of the statistic allows accurate inference, outperforming that based on first-order asymptotic approximation, by considering the bootstrap distribution of the statistic under the incorrectly assumed distribution. Comparisons of this simple approach with alternative analytic and non-parametric inference schemes are discussed.

An analytic model for the water blast response of one-dimensional marine structure coated with elastic foam

The water blast response of one-dimensional marine structures coated with elastic foam is theoretically investigated. The foam coating is modeled by a group of concentrated masses separated by parallel massless nonlinear spring and damper, simulating the micro inertia, stiffness, and rate-dependent effects exhibited by common cellular materials. An analytic model for the wet face response, inclusive of fluid-structure interaction, is developed. The model addresses cavitation and takes into account the momentum of reconstituted water attached to the wet face. The first-order double asymptotic approximation method is used to compute the pressure during the spring-back and succeeding phase. The fidelity of the analytic model has been assessed by the finite element analysis.

Unconditional efficient one‐sided confidence limits for the odds ratio based on conditional likelihood

We compare various one-sided confidence limits for the odds ratio in a 2 x 2 table. The first group of limits relies on first-order asymptotic approximations and includes limits based on the (signed) likelihood ratio, score and Wald statistics. The second group of limits is based on the conditional tilted hypergeometric distribution, with and without mid-P correction. All these limits have poor unconditional coverage properties and so we apply the general transformation of Buehler (J. Am. Statist. Assoc. 1957; 52:482-493) to obtain limits which are unconditionally exact. The performance of these competing exact limits is assessed across a range of sample sizes and parameter values by looking at their mean size. The results indicate that Buehler limits generated from the conditional likelihood have the best performance, with a slight preference for the mid-P version. This confidence limit has not been proposed before and is recommended for general use, especially when the underlying probabilities are not extreme.

Asymptotic analysis of powers of matrices

We analyze the representation ofAnas a linear combination ofAj,0≤j≤k−1, whereAis ak×kmatrix. We obtain a first-order asymptotic approximation ofAnasn→∞, without imposing any special conditions onA. We give some examples showing the application of our results.

Bootstrap inference in systems of single equation error correction models

We analyze OLS-based tests of long-run relationships, weak exogeneity and short-run dynamics in conditional error correction models. Unweighted sums of single equation test statistics are used for hypothesis testing in pooled systems. When model errors are (conditionally) heteroskedastic tests of weak exogeneity and short run dynamics are affected by nuisance parameters. Similarly, on the pooled level the advocated test statistics are no longer pivotal in presence of cross-sectional error correlation. We prove that the wild bootstrap provides asymptotically valid critical values under both conditional heteroskedasticity and cross-sectional error correlation. A Monte-Carlo study reveals that in small samples the bootstrap outperforms first-order asymptotic approximations in terms of the empirical size even if the asymptotic distribution of the test statistic does not depend on nuisance parameters. Opposite to feasible GLS methods the approach does not require any estimate of cross-sectional correlation and copes with time-varying patterns of contemporaneous error correlation.

A mixed model formulation for designing cluster randomized trials with binary outcomes

Cluster randomized trials (CRTs) are unlike traditional individually randomized trials because observations within the same cluster are positively correlated and the sample size (number of clusters) is relatively small. Although formulae for sample size and power estimates of CRT designs do exist, these formulae rely upon first-order asymptotic approximations for the distribution of the average intervention effect and are inaccurate for CRTs that have a small number of clusters. These formulae also assume that the intracluster correlation (ICC) is the same for each cluster in the CRT. However, for CRTs in which the clusters are classrooms or medical practices, the degree of ICC is often a factor of how many students are in each classroom or how many patients are in each practice. Specifically, smaller clusters are expected to have larger ICC than larger clusters. A weighted sum of the cluster means, D, is the statistic often used to estimate the average intervention effect in a CRT. Therefore, we propose that a saddlepoint approximation is a natural choice to approximate the distributions of the cluster means more precisely than a standard large-sample approximation. We parameterize the ICC for each cluster as a random effect with a predefined prior distribution that is dependent upon the size of each cluster. After integrating over the range of the random effect, we use Monte Carlo methods to generate sample cluster means, which are in turn used to approximate the distribution of D with saddlepoint methods. Through numerical examples and an actual application, we show that our method has accuracy that is equal to or better than that of existing methods. Futhermore, our method accommodates CRTs in which the correlation within cluster is expected to diminish with the cluster size.

Bootstrap Methods for Markov Processes

The block bootstrap is the best known bootstrap method for time-series data when the analyst does not have a parametric model that reduces the data generation process to simple random sampling. However, the errors made by the block bootstrap converge to zero only slightly faster than those made by first-order asymptotic approximations. This paper describes a bootstrap procedure for data that are generated by a Markov process or a process that can be approximated by a Markov process with sufficient accuracy. The procedure is based on estimating the Markov transition density nonparametrically. Bootstrap samples are obtained by sampling the process implied by the estimated transition density. Conditions are given under which the errors made by the Markov bootstrap converge to zero more rapidly than those made by the block bootstrap.

Bootstrap inference in single equation error correction models

In the sequel of its seminal application in Davidson, Hendry, Srba and Yeo (1978) the single equation error correction model has been widely used in empirical practice. Providing a clear distinction between short- and long-run dynamics this model allows OLS-methods to be as efficient as (multivariate) full information maximum likelihood methods under a few assumptions on weak exogeneity and cointegration. We consider OLS-based tests on long-run relationships, weak exogeneity and short-run dynamics. For the latter issues it is known that common test-statistics are no longer pivotal if model errors exhibit conditional heteroskedasticity. We show that the wild bootstrap provides convenient critical values for the considered OLS-based statistics under both homoskedastic and conditionally heteroskedastic model errors. The wild bootstrap is easy to implement and turns out to improve considerably the empirical size of common test statistics compared to first order asymptotic approximations. We prove further that the wild bootstrap retains its validity for inference within a system of pooled equations exhibiting cross sectional correlation. Opposite to feasible GLS methods our approach does not require any parametric specification of cross sectional correlation, and copes with time varying patterns of contemporaneous error correlation.

A Martingale-based bootstrap inference with censored data

In this paper, me shall investigate a bootstrap method hasd on a martingale representation of the relevant statistic for inference to a class of functionals of the survival distribution. The method is similar in spirit to Efron's (1981) bootstrap, and thus in the present paper will be referred to as “martingale-based bootstrap” The method was derived from Lin,Wei and Ying (1993), who appiied the method in checking the Cox model with cumulative sums of martingale-based residuals. It is shown that this martingale-based bootstrap gives a correct first-order asymptotic approximation to the distribution function of the corresponding functional of the Kaplan-Meier estimator. As a consequence, confidence intervals constructed by the martingale-based bootstrap have asymptotially correct coverage probability. Our simulation study indicats that the martingale-based bootst strap method for a small and moderate sample sizes can be uniformly better than the usual bootstrap method in estimating the sampling distribution for a mean function and a point probability in survival analysis.

Nonparametric bootstrap analysis with applications to demographic effects in demand functions

A new bootstrap proposal, labeled smooth conditional moment (SCM) bootstrap, is introduced for independent but not necessarily identically distributed data, where the classical bootstrap procedure fails. The procedure is shown to encompass the iid and wild bootstrap procedures as special cases. The SCM bootstrap is applied to the construction of confidence intervals for nonparametric equivalence scales estimates, and of critical values for consistent nonparametric test statistics. In both cases, the bootstrap approximations to the small sample properties are usually far superior to those provided by the first-order asymptotic approximations.

Amplitude modulated chaos in two degree-of-freedom systems with quadratic nonlinearities

Two degree-of-freedom systems with weak quadratic nonlinearities are studied under weak external and parametric excitations respectively. All six possible cases, that arise in the presence of 1∶2 internal resonance, are investigated. The method of averaging is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order asymptotic approximation to the response. An analytical technique, based on Melnikov's method is used to predict the parameter range for which chaotic dynamics exists in the undamped averaged system. Numerical studies show that such chaotic responses are quite common in these quadratic systems, and they seem to persist even in the presence of damping.

Bootstrap Methods for Median Regression Models

The least-absolute-deviations (LAD) estimator for a median-regression model does not satisfy the standard conditions for obtaining asymptotic refinements through use of the bootstrap because the LAD objective function is not smooth. This paper overcomes this problem by smoothing the objective function. The smoothed estimator is asymptotically equivalent to the standard LAD estimator. With bootstrap critical values, the rejection probabilities of symmetrical t and X 2 tests based on the smoothed estimator are correct through O(n -γ ) under the null hypothesis, where γ<1 but can be arbitrarily close to 1. In contrast, first-order asymptotic approximations make errors of size O(n -γ ). These results also hold for symmetrical t and X 2 tests for censored median regression models.

Forced solitary waves and hydraulic falls in two-layer flows

Long nonlinear waves in the two-layer flow of an inviscid, incompressible fluid are considered. Both the free surface and the interface of the two fluids are unknown free boundaries. The flow is forced by an obstruction on the bottom and/or an external pressure (usually called the wind stress) on the free surface. Away from the critical depth ratio between the surface layer and the internal layer of the fluids, the weak nonlinearity is of second order. Then there exists a balance between the dispersion and the second-order nonlinearity. The first-order asymptotic approximations of both free-surface elevation and interface elevation satisfy forced Korteweg-de Vries equations (fKdV). Because of the existence of two modes, the total number of types of solutions is double that for the single-layer flow. There are two hydraulic falls and four solitary waves for positive forcing. The first hydraulic fall and the first two solitary waves correspond to the fast mode. The remaining ones correspond to the slow mode. The first hydraulic fall was numerically found by Forbes (1989) by directly integrating the Laplace equation with nonlinear boundary conditions. There are two types of forcing according to the length of the base of the obstruction. One type is called local and the length of its base has the same scale as the height of the forcing. The expression of the forcing in the fKdV is given by the Dirac delta function. Based upon our long-wave scale the horizontal laboratory lengthscale is shrunk by ε½, while the vertical laboratory lengthscale is unchanged. Hence the semicircular bumps in the papers by Vanden-Broeck (1987), and Forbes (1988, 1989) are all considered to be local in the present paper. For the locally forced cases, stationary problems have been solved analytically. We have derived analytical expressions for the upstream speeds UL and Uc, at which the hydraulic falls can occur and solitary waves respectively cease to exist. The formulae are reduced to those due to Miles (1986) in the case of a single-layer flow and very small forcing. A full comparison among the asymptotic, computational and experimental results is provided. The comparison shows that the difference is less than 10% for UL and Uc in most of the parameter range where the asymptotic method is valid, the computational scheme converges and experiments were conducted (see figure 10). The second type of forcing is called non-local and the length of the base of the obstruction has the same scale as the wavelength. The existence and behaviour of the stationary solutions of the non-locally forced fKdV are described. Surprisingly, there can be more than four solitary waves sustained on the site of some negative forcing.