33,094 research outputs found
Best estimation of functional linear models
Observations which are realizations from some continuous process are frequent
in sciences, engineering, economics, and other fields. We consider linear
models, with possible random effects, where the responses are random functions
in a suitable Sobolev space. The processes cannot be observed directly. With
smoothing procedures from the original data, both the response curves and their
derivatives can be reconstructed, even separately. From both these samples of
functions, just one sample of representatives is obtained to estimate the
vector of functional parameters. A simulation study shows the benefits of this
approach over the common method of using information either on curves or
derivatives. The main theoretical result is a strong functional version of the
Gauss-Markov theorem. This ensures that the proposed functional estimator is
more efficient than the best linear unbiased estimator based only on curves or
derivatives.Comment: the best information from the two samples of functions and
derivatives: a strong version of the Gauss-Markov theorem. Relaxed an hidden
hypothesis on linear independence of the Riesz representation of the
Karhunen-Loeve bas
Techniques for the Fast Simulation of Models of Highly dependable Systems
With the ever-increasing complexity and requirements of highly dependable systems, their evaluation during design and operation is becoming more crucial. Realistic models of such systems are often not amenable to analysis using conventional analytic or numerical methods. Therefore, analysts and designers turn to simulation to evaluate these models. However, accurate estimation of dependability measures of these models requires that the simulation frequently observes system failures, which are rare events in highly dependable systems. This renders ordinary Simulation impractical for evaluating such systems. To overcome this problem, simulation techniques based on importance sampling have been developed, and are very effective in certain settings. When importance sampling works well, simulation run lengths can be reduced by several orders of magnitude when estimating transient as well as steady-state dependability measures. This paper reviews some of the importance-sampling techniques that have been developed in recent years to estimate dependability measures efficiently in Markov and nonMarkov models of highly dependable system
Markov Functional Market Model nd Standard Market Model
The introduction of so called Market Models (BGM) in 1990s has developed
the world of interest rate modelling into a fresh period. The obvious
advantages of the market model have generated a vast amount of research
on the market model and recently a new model, called Markov functional
market model, has been developed and is becoming increasingly popular.
To be clearer between them, the former is called standard market model
in this paper.
Both standard market models and Markov functional market models are
practically popular and the aim here is to explain theoretically how each
of them works in practice. Particularly, implementation of the standard
market model has to rely on advanced numerical techniques since Monte
Carlo simulation does not work well on path-dependent derivatives. This
is where the strength of the Longstaff-Schwartz algorithm comes in. The
successful application of the Longstaff-Schwartz algorithm with the standard
market model, more or less, adds another weight to the fact that the
Longstaff-Schwartz algorithm is extensively applied in practice
Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification
Gaussian processes are a natural way of defining prior distributions over
functions of one or more input variables. In a simple nonparametric regression
problem, where such a function gives the mean of a Gaussian distribution for an
observed response, a Gaussian process model can easily be implemented using
matrix computations that are feasible for datasets of up to about a thousand
cases. Hyperparameters that define the covariance function of the Gaussian
process can be sampled using Markov chain methods. Regression models where the
noise has a t distribution and logistic or probit models for classification
applications can be implemented by sampling as well for latent values
underlying the observations. Software is now available that implements these
methods using covariance functions with hierarchical parameterizations. Models
defined in this way can discover high-level properties of the data, such as
which inputs are relevant to predicting the response
A shared-parameter continuous-time hidden Markov and survival model for longitudinal data with informative dropout
A shared-parameter approach for jointly modeling longitudinal and survival data is proposed. With respect to available approaches, it allows for time-varying random effects that affect both the longitudinal and the survival processes. The distribution of these random effects is modeled according to a continuous-time hidden Markov chain so that transitions may occur at any time point. For maximum likelihood estimation, we propose an algorithm based on a discretization of time until censoring in an arbitrary number of time windows. The observed information matrix is used to obtain standard errors. We illustrate the approach by simulation, even with respect to the effect of the number of time windows on the precision of the estimates, and by an application to data about patients suffering from mildly dilated cardiomyopathy
Space-time duality for fractional diffusion
Zolotarev proved a duality result that relates stable densities with
different indices. In this paper, we show how Zolotarev duality leads to some
interesting results on fractional diffusion. Fractional diffusion equations
employ fractional derivatives in place of the usual integer order derivatives.
They govern scaling limits of random walk models, with power law jumps leading
to fractional derivatives in space, and power law waiting times between the
jumps leading to fractional derivatives in time. The limit process is a stable
L\'evy motion that models the jumps, subordinated to an inverse stable process
that models the waiting times. Using duality, we relate the density of a
spectrally negative stable process with index to the density of
the hitting time of a stable subordinator with index , and thereby
unify some recent results in the literature. These results also provide a
concrete interpretation of Zolotarev duality in terms of the fractional
diffusion model.Comment: 16 page
Exact simulation pricing with Gamma processes and their extensions
Exact path simulation of the underlying state variable is of great practical
importance in simulating prices of financial derivatives or their sensitivities
when there are no analytical solutions for their pricing formulas. However, in
general, the complex dependence structure inherent in most nontrivial
stochastic volatility (SV) models makes exact simulation difficult. In this
paper, we present a nontrivial SV model that parallels the notable Heston SV
model in the sense of admitting exact path simulation as studied by Broadie and
Kaya. The instantaneous volatility process of the proposed model is driven by a
Gamma process. Extensions to the model including superposition of independent
instantaneous volatility processes are studied. Numerical results show that the
proposed model outperforms the Heston model and two other L\'evy driven SV
models in terms of model fit to the real option data. The ability to exactly
simulate some of the path-dependent derivative prices is emphasized. Moreover,
this is the first instance where an infinite-activity volatility process can be
applied exactly in such pricing contexts.Comment: Forthcoming The Journal of Computational Financ
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