25,160 research outputs found
Demonstration of Enhanced Monte Carlo Computation of the Fisher Information for Complex Problems
The Fisher information matrix summarizes the amount of information in a set
of data relative to the quantities of interest. There are many applications of
the information matrix in statistical modeling, system identification and
parameter estimation. This short paper reviews a feedback-based method and an
independent perturbation approach for computing the information matrix for
complex problems, where a closed form of the information matrix is not
achievable. We show through numerical examples how these methods improve the
accuracy of the estimate of the information matrix compared to the basic
resampling-based approach. Some relevant theory is summarized
Information matrix for hidden Markov models with covariates
For a general class of hidden Markov models that may include time-varying covariates, we illustrate how to compute the observed information matrix, which may be used to obtain standard errors for the parameter estimates and check model identifiability. The proposed method is based on the Oakes’ identity and, as such, it allows for the exact computation of the information matrix on the basis of the output of the expectation-maximization (EM) algorithm for maximum likelihood estimation. In addition to this output, the method requires the first derivative of the posterior probabilities computed by the forward-backward recursions introduced by Baum and Welch. Alternative methods for computing exactly the observed information matrix require, instead, to differentiate twice the forward recursion used to compute the model likelihood, with a greater additional effort with respect to the EM algorithm. The proposed method is illustrated by a series of simulations and an application based on a longitudinal dataset in Health Economics
Fisher information matrix for single molecules with stochastic trajectories
Tracking of objects in cellular environments has become a vital tool in
molecular cell biology. A particularly important example is single molecule
tracking which enables the study of the motion of a molecule in cellular
environments and provides quantitative information on the behavior of
individual molecules in cellular environments, which were not available before
through bulk studies. Here, we consider a dynamical system where the motion of
an object is modeled by stochastic differential equations (SDEs), and
measurements are the detected photons emitted by the moving fluorescently
labeled object, which occur at discrete time points, corresponding to the
arrival times of a Poisson process, in contrast to uniform time points which
have been commonly used in similar dynamical systems. The measurements are
distributed according to optical diffraction theory, and therefore, they would
be modeled by different distributions, e.g., a Born and Wolf profile for an
out-of-focus molecule. For some special circumstances, Gaussian image models
have been proposed. In this paper, we introduce a stochastic framework in which
we calculate the maximum likelihood estimates of the biophysical parameters of
the molecular interactions, e.g., diffusion and drift coefficients. More
importantly, we develop a general framework to calculate the Cram\'er-Rao lower
bound (CRLB), given by the inverse of the Fisher information matrix, for the
estimation of unknown parameters and use it as a benchmark in the evaluation of
the standard deviation of the estimates. There exists no established method,
even for Gaussian measurements, to systematically calculate the CRLB for the
general motion model that we consider in this paper. We apply the developed
methodology to simulated data of a molecule with linear trajectories and show
that the standard deviation of the estimates matches well with the square root
of the CRLB
Testing the Information Matrix Equality with Robust Estimators
We study the behaviour of the information matrix (IM) test when maximum likelihood estimators are replaced with robust estimators. The latter may unmask outliers and hence improve the power of the test. We investigate in detail the local asymptotic power of the IM test in the normal model, for various estimators and under a range of local alternatives. These local alternatives include contamination neighbourhoods, Student's t (with degrees of freedom approaching infinity), skewness, and a tilted normal. Simulation studies for fixed alternatives confirm that in many cases the use of robust estimators substantially increases the power of the IM test.
- …