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Characteristic values of spatially varying material properties in existing structures
Published in Airong Chen, Xin Ruan, Dan M. Frangopol, Life-Cycle Civil Engineering: Innovation, Theory and Practice, 2021
S. Geyer, I. Papaioannou, D. Straub
In many cases, the solution of Equation (4) needs to be found numerically as the proportionality constant cannot be evaluated directly. This can be avoided by the use of conjugate priors, which allow direct evaluation of the posterior distribution Raiffa_1961,Gelman_2013. Conjugate priors are characterized by the fact that the posterior distribution is of the same distribution type as the prior distribution, which can be achieved by specific choices of the prior distribution and the likelihood function. In such case, the prior distribution is termed a conjugate prior for the likelihood function and the Bayesian updating reduces to a parameter update of the chosen parametric family Gelman_2013. The presented model in this contribution makes use of the conjugacy of the chosen prior distribution and likelihood function.
Moments and Wavelets in Signal Estimation
Published in Donald B. Owen, Subir Ghosh, William R. Schucany, William B. Smith, Statistics of Quality, 2020
Edward J. Wegman, Hung T. Le, Wendy L. Poston, Jeffrey L. Solka
Consider a general function, f(x), to be estimated based on some sampled data, say, x1, x2,…, xn. This is, in fact, the most elementary estimation problem in statistical inference. Often the function f in question is the probability distribution function or the probability density function, and most frequently the approach taken is to place the function within a parametric family indexed by some parameter, say, θ. Rather than estimate f directly, the parameter θ is estimated, with fθ then being estimated by f^θ=f^θ^. Under a variety of circumstances, it is much more desirable to take a nonparametric approach so as to avoid problems associated with misspecification of parametric family. This is particularly the case when data are relatively plentiful and the information captured by the parametric model is not needed for statistical efficiency.
The Bootstrap
Published in Tucker S. McElroy, Dimitris N. Politis, Time Series, 2019
Tucker S. McElroy, Dimitris N. Politis
The nonlinear function H might belong to a parametric family, in which case estimating those parameters (say, by nonlinear Least Squares) yields an estimator H^; else, H might be estimated nonparametrically. In any event, we can compute residuals via () et=Xt−H^(Xt−1,…,Xt−p)
Generalized Computer Model Calibration for Radiation Transport Simulation
Published in Technometrics, 2021
Michael Grosskopf, Derek Bingham, Marvin L. Adams, W. Daryl Hawkins, Delia Perez-Nunez
For our application, the response variable comes in the form of neutron counts. For these data, as well as other noncontinuous responses, we propose to replace the Gaussian observation likelihood with a distribution from a parametric family with density f(x), with the expected value equal to the latent mean :with observation likelihood parameters (in Equation (4) is the residual variance). The simulator at some value of its calibration parameters is treated as an observation of the underlying mean function of the process generating the field data with some systematic discrepancy.
On the connection between a skew product IFS and the ergodic optimization for a finite family of potentials
Published in Dynamical Systems, 2019
On one hand we have a geometrical property which is the existence of an invariant subset Λ of the cylinder having a random SRB measure, which we are going to define later on. On the other hand, we establish a subtle and non-trivial relation between the boundary of this set and the analogous of the Aubry-Mather theory. More specifically, as in the Lagrangian case, the solution of the Bellman equation with a discounting parameter plays the role of the viscosity solution of the Hamilton-Jacobi equation with a viscosity parameter , allowing us to employ the Fenchel-Rockafellar duality theorem to relate the Bellman equation solution and the problem of maximizing the integral of a ‘Lagrangian function’ which in our case is a parametric family of potential functions. The key element here is the set of measures over which the maximization is performed. Thanks to the analogy with the Lagrangian case, we are able to define the concept of holonomic measure in our setting, which turns out to be the right choice to reproduce the expected results such as the characterization of the critical value and the support of the maximizing measures.
Statistical modelling for cancer mortality
Published in Letters in Biomathematics, 2019
Generally, a separate hazard rate is used for each piece of the time scale. The piecewise exponential approach is a natural one for life-table analysis where the period of follow-up is divided into intervals, since a common assumption is that the hazard function is approximately constant within interval (Holford, 1976). Perhaps the most appealing as well as popular feature (in survival analysis) of the hazard function is that it allows a needful way for specifying the effect of covariates on survival. The ‘proportional hazards model’ introduced by Cox (1972) is as follows: where is the underlying hazard function which is chosen from any parametric family (such as exponential, Weibull, etc.) or it may be left unspecified and β is a column vector of unknown parameters specifying the effect of covariates (Cox, 1972). The ratio of hazard functions for any two individuals with covariate vectors and is which is independent of t. It is noted that ‘the ratio of hazard functions’ does not depend on t provides a convenient way of summarizing the effect of a covariate on survival. ‘Non-proportional hazards’ models can be constructed by allowing to depend on X, whereas time varying covariates allow X to depend on t.