On importance sampling with mixtures for random walks with heavy tails

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

Value at Risk (VaR) is an important tool for estimating the risk of a financial portfolio under significant loss. Although Monte Carlo simulation is a powerful tool for estimating VaR, it is quite inefficient since the event of significant loss is usually rare. Previous studies suggest that the performance of the Monte Carlo simulation can be improved by impor-tance sampling if the market returns follow the normality or the distributions. The first contribution of our paper is to extend the importance sampling method for dealing with jump-diffusion market returns, which can more precisely model the phenomenon of high peaks, heavy tails, and jumps of market returns mentioned in numerous empirical study papers. This paper also points out that for portfolios of which the huge loss is triggered by significantly distinct events, naively applying importance sampling method can result in poor performance. The second contribution of our paper is to develop the hybrid importance sampling method for the aforementioned problem. Our method decomposes a Monte Carlo simulation into sub simulations, and each sub simulation focuses only on one huge loss event. Thus the perform-ance for each sub simulation is improved by importance sampling method, and overall performance is optimized by determining the allotment of samples to each sub simulation by Lagrange’s multiplier. Numerical experiments are given to verify the superiority of our method.

Estimating the expectations of functionals applied to sums of random variables (RVs) is a well-known problem encountered in many challenging applications. Generally, closed-form expressions of these quantities are out of reach. A naive Monte Carlo simulation is an alternative approach. However, this method requires numerous samples for rare event problems. Therefore, it is paramount to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), known for its efficiency in requiring fewer computations to achieve the same accuracy requirements. We propose a state-dependent IS scheme based on a stochastic optimal control formulation, where the control is dependent on state and time. We aim to calculate rare event quantities that could be written as an expectation of a functional of the sums of independent RVs. The proposed algorithm is generic and can be applied without restrictions on the univariate distributions of RVs or the functional applied to the sum. We apply this approach to the log-normal distribution to compute the left tail and cumulative distribution of the ratio of independent RVs. For each case, we numerically demonstrate that the proposed state-dependent IS algorithm compares favorably to most well-known estimators dealing with similar problems.

We consider the problem of efficient estimation via simulation of first passage time probabilities for a multidimensional random walk with heavy-tailed increments. In addition to being a natural generalization to the problem of computing ruin probabilities in insurance - in which the focus is the maximum of a one-dimensional random walk with negative drift - this problem captures important features of large deviations for multidimensional heavy-tailed processes (such as the role played by the mean of the process in connection to the location of the target set). We develop a state-dependent importance sampling estimator for this class of multidimensional problems. Then, using techniques based on Lyapunov inequalities, we argue that our estimator is strongly efficient in the sense that the relative mean squared error of our estimator can be made arbitrarily small by increasing the number of replications, uniformly as the probability of interest approaches 0.

Fully modified IV, GIVE and GMM estimation with possibly non-stationary regressors and instruments

The Bayesian estimation of dynamic stochastic general equilibrium (DSGE) models implies usage of Monte Carlo Markov Chain (MCMC) algorithms. The analysis of MCMC algorithms is made for densities with unfavorable properties common in DSGE models (restricted density support, heavy tails, sharp peaks, non-convex log-density, and non-convex density). Three groups of algorithms are investigated: random walk (RW), MALA and suggested LTG (local truncated Gauss). Three versions of MALA and LTG are investigated: the version using local hessian and gradient of log-density, the version using only local gradient and the version that uses only information about the mode. Performance of MALA and LTG are close to each other. There is some advantage of LTG in average and in test with DSGE model. RW performance is worse than MALA or LTG (especially for small-scale cases). The computational costs are almost the similar for RW and approximation based versions of MALA or LTG. Existence of heavy tails leads to decrease of advantage of MALA and LTG algorithms. Acceptance rate (corresponding to the lowest inefficiency of sample) can be quite different from conventional values.

The total energy of tetrahedral CH4 at the equilibrium bond length of 2.05 bohr is calculated by the fixed-node random walk method with importance sampling. The best result is obtained with a single-determinant trial function of double-zeta STO-basis-set quality. This yields an upper bound to the total energy of −40.506 hartree which is 0.048 hartree below the expectation value of the energy from the best variational calculations and only 0.008 hartree above the experimental nonrelativistic energy. Extended calculations with array processors and extrapolation to zero time step were used to obtain a statistical precision of 0.002 hartree (1.4 kcal/mol) for the random walk result.

SimRank is a similarity measure between vertices in a graph, which has become a fundamental technique in graph analytics. Recently, many algorithms have been proposed for efficient evaluation of SimRank similarities. However, the existing SimRank computation algorithms either overlook uncertainty in graph structures or is based on an unreasonable assumption (Du et al). In this paper, we study SimRank similarities on uncertain graphs based on the possible world model of uncertain graphs. Following the random-walk-based formulation of SimRank on deterministic graphs and the possible worlds model of uncertain graphs, we define random walks on uncertain graphs for the first time and show that our definition of random walks satisfies Markov's property. We formulate the SimRank measure based on random walks on uncertain graphs. We discover a critical difference between random walks on uncertain graphs and random walks on deterministic graphs, which makes all existing SimRank computation algorithms on deterministic graphs inapplicable to uncertain graphs. To efficiently compute SimRank similarities, we propose three algorithms, namely the baseline algorithm with high accuracy, the sampling algorithm with high efficiency, and the two-phase algorithm with comparable efficiency as the sampling algorithm and about an order of magnitude smaller relative error than the sampling algorithm. The extensive experiments and case studies verify the effectiveness of our SimRank measure and the efficiency of our SimRank computation algorithms.

In this paper we construct Galois towers with good asymptotic properties over any non-prime finite field Fℓ; i.e., we construct sequences of function fields N=(N1⊂N2⊂⋯) over Fℓ of increasing genus, such that all the extensions Ni/N1 are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with these properties are important for applications in various fields including coding theory and cryptography.

Recurrent events data are commonly seen in longitudinal follow-up studies. Dependent censoring often occurs due to death or exclusion from the study related to the disease process. In this article, we assume flexible marginal regression models on the recurrence process and the dependent censoring time without specifying their dependence structure. The proposed model generalizes the approach by Ghosh and Lin (2003, Biometrics 59, 877-885). The technique of artificial censoring provides a way to maintain the homogeneity of the hypothetical error variables under dependent censoring. Here we propose to apply this technique to two Gehan-type statistics. One considers only order information for pairs whereas the other utilizes additional information of observed censoring times available for recurrence data. A model-checking procedure is also proposed to assess the adequacy of the fitted model. The proposed estimators have good asymptotic properties. Their finite-sample performances are examined via simulations. Finally, the proposed methods are applied to analyze the AIDS linked to the intravenous experiences cohort data.

The objective of this paper is to introduce the break-preserving local linear (BPLL) estimator for the estimation of unstable volatility functions for independent and asymptotically independent processes. Breaks in the structure of the conditional mean and/or the volatility functions are common in Finance. Nonparametric estimators are well suited for these events due to the flexibility of their functional form and their good asymptotic properties. However, the local polynomial kernel estimators are not consistent at points where the volatility function has a break. The estimator presented in this paper generalises the classical local linear (LL). The BPLL estimator maintains the desirable properties of the LL estimator with regard to the bias and the boundary estimation while it estimates the breaks consistently. An extensive Monte Carlo study is shown as well as detailed proofs of the estimator asymptotic behaviour.

Although the periodogram maximizer has good asymptotic properties in the case of a single sinusoid in colored noise, in low SNR conditions and/or in the case of small sample size, the data need to be prewhitened in order for the periodogram maximizer be likely to occur near the true frequency. In this paper we derive an efficient least squares algorithm for estimating the frequencies, amplitudes, phases and autoregressive parameters for a sum of sinusoids in complex autoregressive noise. The procedure is seen to be a generalization of the Levinson-Durbin algorithm, an efficient technique for estimating the parameters in an autoregression.

We propose a class of strongly efficient rare-event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on a recently established heavy-tailed sample-path large deviations result. The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are “universal” in the sense that a single importance sampling scheme applies to a very general class of rare events that arise in heavy-tailed systems. In particular, our estimators can deal with rare events that are caused by multiple big jumps (therefore, beyond the usual principle of a single big jump) as well as multidimensional processes such as the buffer content process of a queueing network. We illustrate the versatility of our approach with several applications that arise in the context of mathematical finance, actuarial science, and queueing theory.

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

In this paper we present a new scheduler, the ?-scheduler, which performs better on heavy tailed traffic than the Foreground-Background (FB) scheduler which is known to be optimal in scheduling traffic which has unknown characteristics. The ?-scheduler is able to provide a closer approximation to the Shortest Remaining Processing Time (SRPT) scheduler which is known to provide optimal scheduling in the case when the length of the packets to be scheduled is known. We are able to improve our ?-scheduler SRPT approximation by basing our expected remaining processing time on an estimate of the shape parameter, ?, from the standard heavy tailed Pareto complementary probability distribution function, P[X > t]=ct??. We review various methods of estimating ? based on linear regression methods on the standard Log-Log Complement Distribution plot first proposed by Crovella in [5] and studied in more detail in [4]. We show that even using the standard sliding window least mean square estimator our scheduler exhibits improved performance over the FB scheduler. The particular scheduling problem we investigate in detail concerns the servicing of a number of ingress flows being fed by heavy tailed distributions. The egress channel is a bottleneck link, for example a shared wireless TDMA domain. Each flow is characterized as having a distinct shape parameter which may change (slowly) over time. While the scheduling is real time, the {?i} estimation does not need to be.