133 research outputs found
A sequential Monte Carlo approach to computing tail probabilities in stochastic models
Sequential Monte Carlo methods which involve sequential importance sampling
and resampling are shown to provide a versatile approach to computing
probabilities of rare events. By making use of martingale representations of
the sequential Monte Carlo estimators, we show how resampling weights can be
chosen to yield logarithmically efficient Monte Carlo estimates of large
deviation probabilities for multidimensional Markov random walks.Comment: Published in at http://dx.doi.org/10.1214/10-AAP758 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Efficient importance sampling for Monte Carlo evaluation of exceedance probabilities
Large deviation theory has provided important clues for the choice of
importance sampling measures for Monte Carlo evaluation of exceedance
probabilities. However, Glasserman and Wang [Ann. Appl. Probab. 7 (1997)
731--746] have given examples in which importance sampling measures that are
consistent with large deviations can perform much worse than direct Monte
Carlo. We address this problem by using certain mixtures of exponentially
twisted measures for importance sampling. Their asymptotic optimality is
established by using a new class of likelihood ratio martingales and renewal
theory.Comment: Published at http://dx.doi.org/10.1214/105051606000000664 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices
Several classical results on boundary crossing probabilities of Brownian
motion and random walks are extended to asymptotically Gaussian random fields,
which include sums of i.i.d. random variables with multidimensional indices,
multivariate empirical processes, and scan statistics in change-point and
signal detection as special cases. Some key ingredients in these extensions are
moderate deviation approximations to marginal tail probabilities and weak
convergence of the conditional distributions of certain ``clumps'' around
high-level crossings. We also discuss how these results are related to the
Poisson clumping heuristic and tube formulas of Gaussian random fields, and
describe their applications to laws of the iterated logarithm in the form of
the Kolmogorov--Erd\H{o}s--Feller integral tests.Comment: Published at http://dx.doi.org/10.1214/009117905000000378 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Combining domain knowledge and statistical models in time series analysis
This paper describes a new approach to time series modeling that combines
subject-matter knowledge of the system dynamics with statistical techniques in
time series analysis and regression. Applications to American option pricing
and the Canadian lynx data are given to illustrate this approach.Comment: Published at http://dx.doi.org/10.1214/074921706000001049 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bias correction and confidence intervals following sequential tests
An important statistical inference problem in sequential analysis is the
construction of confidence intervals following sequential tests, to which
Michael Woodroofe has made fundamental contributions. This paper reviews
Woodroofe's method and other approaches in the literature. In particular it
shows how a bias-corrected pivot originally introduced by Woodroofe can be used
as an improved root for sequential bootstrap confidence intervals.Comment: Published at http://dx.doi.org/10.1214/074921706000000590 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Evaluating probability forecasts
Probability forecasts of events are routinely used in climate predictions, in
forecasting default probabilities on bank loans or in estimating the
probability of a patient's positive response to treatment. Scoring rules have
long been used to assess the efficacy of the forecast probabilities after
observing the occurrence, or nonoccurrence, of the predicted events. We develop
herein a statistical theory for scoring rules and propose an alternative
approach to the evaluation of probability forecasts. This approach uses loss
functions relating the predicted to the actual probabilities of the events and
applies martingale theory to exploit the temporal structure between the
forecast and the subsequent occurrence or nonoccurrence of the event.Comment: Published in at http://dx.doi.org/10.1214/11-AOS902 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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