Don McLeish

A general method for debiasing a Monte Carlo estimator

Consider a process, stochastic or deterministic, obtained by using a numerical integration scheme, or from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation. We will assume that we can sample from the distribution of the process from time 0 to finite time n. We propose a scheme for unbiased estimation of the limiting value of the process, together with estimates of standard error and apply this to examples including numerical integrals, root-finding and option pricing in a Heston Stochastic Volatility model. This results in unbiased estimators in place of biased ones i nmany potential applications.

Investigation of phase noise of ring oscillators with time-varying current and noise sources by time-scaling thermal noise

This paper presents a new methodology of analyzing phase noise in a ring oscillator by time-scaling the thermal noise. Close-form solutions that relate the probability distribution and power-spectral density of the phase noise to circuit parameters have been obtained. These close-form solutions characterize the behavior of phase noise even when the circuit is varying with time in a nonlinear fashion. Specifically, the theory predicts that for a given oscillation frequency, phase noise roughly decreases as the cube of the delay cell charging current value at threshold crossing; thus, it provides new design insights. Simulations were run and verified this dependency.

Conditioning for variance reduction in estimating the sensitivity of simulations

AbstractWe consider first a discrete event static system that is to be simulated at values of a parameter or vector of parametersθ. The system is assumed driven by an inputX, where typicallyX is a vector of variables whose densityfθ(x) depends on the parameterθ. For the purpose of optimizing, finding roots, or graphing the expected performanceEθL(X) for performance measureL, it is useful to estimate not only the expected value but also its gradient. An unbiased estimator for the latter is the score function estimator

Bounded Relative Error Importance Sampling and Rare Event Simulation

L(X)S(\theta ) = L(X)\frac{\partial }{{\partial \theta }}\ln f_\theta (x).

Simulating random variables using moment-generating functions and the saddlepoint approximation

This estimator and likelihood ratio analogues typically require variance reduction, and we consider conditioning on the value of the score function for this purpose. The efficiency gains due to performing the Monte Carlo conditionally can be very large. Extension to discrete event dynamic systems such as theM/G/1 queue and other more complicated systems is considered.

Nearly Exact Option Price Simulation Using Characteristic Functions

This paper presents new theoretical results on the phase noise of a class of unsaturated ring oscillators. These new results focus on cycle-to-cycle correlation as a source of timing jitter and highlight its importance in contributing to the phase noise of these unsaturated ring oscillators. Because the outputs of saturated ring oscillators always reach the power supply, such cycle-to-cycle correlation is not present there. This paper presents an in-depth treatment of the two-stage implementation of these unsaturated ring oscillators. It is shown that the timing jitter of each quarter period of this implementation follows an autoregressive process. The results also include a closed-form solution for the power spectral density of the phase noise in terms of circuit parameters such as transistor sizing and bias currents, thus providing new design insights. Using this solution, this paper shows that, under a proper choice of circuit parameters, the phase noise can improve by up to 20 dBc/Hz. Simulations which verify the dependence of timing jitter on cycle-to-cycle correlation and also the 20-dBc/Hz improvement are performed.

ON THE MARTINGALE PROPERTY IN STOCHASTIC VOLATILITY MODELS BASED ON TIME-HOMOGENEOUS DIFFUSIONS: MARTINGALE PROPERTY IN STOCHASTIC VOLATILITY MODELS

We consider estimating tail events using exponential families of importance sampling distributions. When the cannonical sufficient statistic for the exponential family mimics the tail behaviour of the underlying cumulative distribution function, we can achieve bounded relative error for estimating tail probabilities. Examples of rare event simulation from various distributions including Tukeys g&h distribution are provided.

A particular diffusion model for incomplete longitudinal data: application to the multicenter AIDS cohort study

cial to find algorithms for enumerating evidence sources in particular domains, but argue that it is just as important to have a theory of what the actual posterior probabilities are given the evidence. But most puzzles in commonsense reasoning are solved when one hypothesis becomes overwhelmingly more likely than its competitors, because overwhelming evidence has popped up. If the two best hypotheses have probabilities of 0.45 and 0.35, it’s not at all obvious that one should commit oneself one way or the other. But if the only issue is, Is one piece of evidence in favor of hypothesis H much stronger than all the evidence in favor of the other hypotheses?, then an intricate calculus of number combinations may be pointless. It seems to me that there are lots of ways besides Bayes of thinking about plausible reasoning. For instance, one way to evaluate the likelihood that an agent would adopt a certain plan in a certain circumstance is to mn your own planning algorithm (making adjustments for the agent’s idiosyncracies) and see how highly it rates that plan versus others. The resulting numbers are in no sense probabilities. For one thing, plan ratings might not be single numbers; bat more fundamentally, it’s absurd to suppose that the purpose of my own planning algorithm is to judge the chances that I’ll adopt a certain plan. The adaptation of plan-rating numbers to serve as weights on hypotheses about the plans of other agents is just one way in which numbers might get into the hypothesis-ranking system. There are probably lots of other diverse and arbitrary sources of hypothesis weights. It’s possible that there is no elegant unifying theory for all these numbers. Obviously, we should resist that conclusion as long as there is the slightest chance that a unifying theory can be found. But to simply seize on probability theory as the only candidate there will ever be betrays a lack of imagination.