12 research outputs found
Maximum likelihood estimation by monte carlo simulation:Toward data-driven stochastic modeling
We propose a gradient-based simulated maximum likelihood estimation to estimate unknown parameters in a stochastic model without assuming that the likelihood function of the observations is available in closed form. A key element is to develop Monte Carlo-based estimators for the density and its derivatives for the output process, using only knowledge about the dynamics of the model. We present the theory of these estimators and demonstrate how our approach can handle various types of model structures. We also support our findings and illustrate the merits of our approach with numerical results
Differential Sensitivity in Discontinuous Models
Differential sensitivity measures provide valuable tools for interpreting
complex computational models used in applications ranging from simulation to
algorithmic prediction. Taking the derivative of the model output in direction
of a model parameter can reveal input-output relations and the relative
importance of model parameters and input variables. Nonetheless, it is unclear
how such derivatives should be taken when the model function has
discontinuities and/or input variables are discrete. We present a general
framework for addressing such problems, considering derivatives of
quantile-based output risk measures, with respect to distortions to random
input variables (risk factors), which impact the model output through
step-functions. We prove that, subject to weak technical conditions, the
derivatives are well-defined and derive the corresponding formulas. We apply
our results to the sensitivity analysis of compound risk models and to a
numerical study of reinsurance credit risk in a multi-line insurance portfolio
Quantile Optimization via Multiple Timescale Local Search for Black-box Functions
We consider quantile optimization of black-box functions that are estimated
with noise. We propose two new iterative three-timescale local search
algorithms. The first algorithm uses an appropriately modified
finite-difference-based gradient estimator that requires + 1 samples of
the black-box function per iteration of the algorithm, where is the number
of decision variables (dimension of the input vector). For higher-dimensional
problems, this algorithm may not be practical if the black-box function
estimates are expensive. The second algorithm employs a
simultaneous-perturbation-based gradient estimator that uses only three samples
for each iteration regardless of problem dimension. Under appropriate
conditions, we show the almost sure convergence of both algorithms. In
addition, for the class of strongly convex functions, we further establish
their (finite-time) convergence rate through a novel fixed-point argument.
Simulation experiments indicate that the algorithms work well on a variety of
test problems and compare well with recently proposed alternative methods
Monte Carlo and Quasi-Monte Carlo Density Estimation via Conditioning
Estimating the unknown density from which a given independent sample originates is more difficult than estimating the mean, in the sense that for the best popular non-parametric density estimators, the mean integrated square error converges more slowly than at the canonical rate of . When the sample is generated from a simulation model and we have control over how this is done, we can do better. We examine an approach in which conditional Monte Carlo yields, under certain conditions, a random conditional density which is an unbiased estimator of the true density at any point. By averaging independent replications, we obtain a density estimator that converges at a faster rate than the usual ones. Moreover, combining this new type of estimator with randomized quasi-Monte Carlo to generate the samples typically brings a larger improvement on the error and convergence rate than for the usual estimators, because the new estimator is smoother as a function of the underlying uniform random numbers.IVADO Research Grant, NSERC-Canada Discorvery Grant, Canada Research Chair, Inria International Chair, ERDF, ESF, EXP. 2019/0043