A Quasi-Likelihood Approach to Parameter Estimation for Simulatable Statistical Models

Abstract

This paper introduces a parameter estimation method for a general class of statistical models. The method exclusively relies on the possibility to conduct simulations for the construction of interpolation-based metamodels of informative empirical characteristics and some subjectively chosen correlation structure of the underlying spatial random process. In the absence of likelihood functions for such statistical models, which is often the case in stochastic geometric modelling, the idea is to follow a quasi-likelihood (QL) approach to construct an optimal estimating function surrogate based on a set of interpolated summary statistics. Solving these estimating equations one can account for both the random errors due to simulations and the uncertainty about the meta-models. Thus, putting the QL approach to parameter estimation into a stochastic simulation setting the proposed method essentially consists of finding roots to a sequence of approximating quasiscore functions. As a simple demonstrating example, the proposed method is applied to a special parameter estimation problem of a planar Boolean model with discs. Here, the quasi-score function has a half-analytical, numerically tractable representation and allows for the comparison of the model parameter estimates found by the simulation-based method and obtained from solving the exact quasi-score equations.

The Principal Component Analysis of Tribological Tests of Surface Layers Modified with IF-WS₂ Nanoparticles

“…This procedure may be evaluated by a covariance matrix or a correlation matrix and different results are obtained leading to different inference. Some articles related to such problems of objectiveness and stability were published [13], especially about data driven methods [14,15].…”

The Principal Component Analysis of Tribological Tests of Surface Layers Modified with IF-WS₂ Nanoparticles

“…This procedure may be evaluated by a covariance matrix or a correlation matrix and different results are obtained leading to different inference. Some articles related to such problems of objectiveness and stability were published [13], especially about data driven methods [14,15].…”

On the estimation of parameters of a spheroid distribution from planar sections

“…where t (k) is some step length parameter (Osborne, 1992;Baaske et al, 2014), in case the quasiscore Q and the quasi-information matrix I T are available as a closed form expression which can be evaluated at least numerically. In particular, this would include to know expectations and variances of the employed summary statistics T w.r.t.…”

“…The results of QLE are given in Table 6, together with the asymptotic standard errors of these estimates as the entries of diag(I(θ QL ) −1 ), cf. Section 4 and Baaske et al (2014).…”

“…For more involved simulations, however, a Monte Carlo error cannot be avoided, making a direct application of (9) unreasonable. In Baaske et al (2014) an idea is presented how the quasilikelihood estimation approach can be applied even in that case. A respective routine has been developed within the contributed R package qle (Baaske, 2018).…”

“…A general problem which holds both for finding a minimum or a root is to explore efficiently the space of possible parameters in case the objective function can be determined only by simulations. For quasi-likelihood estimation an approach is given in Baaske et al (2014).…”

The Principal Component Analysis of Tribological Tests of Surface Layers Modified with IF-WS₂ Nanoparticles

“…This procedure may be evaluated by a covariance matrix or a correlation matrix and different results are obtained leading to different inference. Some articles related to such problems of objectiveness and stability were published [13], especially about data driven methods [14,15].…”

On the estimation of parameters of a spheroid distribution from planar sections

“…where t (k) is some step length parameter (Osborne, 1992;Baaske et al, 2014), in case the quasiscore Q and the quasi-information matrix I T are available as a closed form expression which can be evaluated at least numerically. In particular, this would include to know expectations and variances of the employed summary statistics T w.r.t.…”

“…The results of QLE are given in Table 6, together with the asymptotic standard errors of these estimates as the entries of diag(I(θ QL ) −1 ), cf. Section 4 and Baaske et al (2014).…”

“…For more involved simulations, however, a Monte Carlo error cannot be avoided, making a direct application of (9) unreasonable. In Baaske et al (2014) an idea is presented how the quasilikelihood estimation approach can be applied even in that case. A respective routine has been developed within the contributed R package qle (Baaske, 2018).…”

“…A general problem which holds both for finding a minimum or a root is to explore efficiently the space of possible parameters in case the objective function can be determined only by simulations. For quasi-likelihood estimation an approach is given in Baaske et al (2014).…”

The Principal Component Analysis of Tribological Tests of Surface Layers Modified with IF-WS₂ Nanoparticles

“…This procedure may be evaluated by a covariance matrix or a correlation matrix and different results are obtained leading to different inference. Some articles related to such problems of objectiveness and stability were published [13], especially about data driven methods [14,15].…”

On the estimation of parameters of a spheroid distribution from planar sections

“…where t (k) is some step length parameter (Osborne, 1992;Baaske et al, 2014), in case the quasiscore Q and the quasi-information matrix I T are available as a closed form expression which can be evaluated at least numerically. In particular, this would include to know expectations and variances of the employed summary statistics T w.r.t.…”

“…The results of QLE are given in Table 6, together with the asymptotic standard errors of these estimates as the entries of diag(I(θ QL ) −1 ), cf. Section 4 and Baaske et al (2014).…”

“…For more involved simulations, however, a Monte Carlo error cannot be avoided, making a direct application of (9) unreasonable. In Baaske et al (2014) an idea is presented how the quasilikelihood estimation approach can be applied even in that case. A respective routine has been developed within the contributed R package qle (Baaske, 2018).…”

“…A general problem which holds both for finding a minimum or a root is to explore efficiently the space of possible parameters in case the objective function can be determined only by simulations. For quasi-likelihood estimation an approach is given in Baaske et al (2014).…”