Approximate Log-likelihood Function Research Articles

Probabilistic properties and parametric inference of small variance nonlinear self-stabilizing stochastic differential equations

We consider a process (Xt) solution of a one-dimensional nonlinear self-stabilizing stochastic differential equation, with classical drift term V(α,x) depending on an unknown parameter α, self-stabilizing term Φ(β,x) depending on another unknown parameter β and small noise amplitude ɛ. Self-Stabilization is the effect of including a mean-field interaction in addition to the state-dependent drift. Adding this term leads to a nonlinear or McKean-Vlasov Markov process with transitions depending on the distribution of Xt. We study the probabilistic properties of (Xt) as ɛ tends to 0 and exhibit a Gaussian approximating process for (Xt). Next, we study the estimation of (α,β) from a continuous observation of (Xt,t∈[0,T]). We build explicit estimators using an approximate log-likelihood function obtained from the exact log-likelihood function of a proxi-model. We prove that, for fixed T, as ɛ tends to 0, α can be consistently estimated with rate ɛ−1but notβ. Then, considering ni.i.d. sample paths (Xti,i=1,…,n), we build consistent and asymptotically Gaussian estimators of (α,β) with rates nɛ−1 for α and n for β. Finally, we prove that the statistical experiments generated by (Xt) and the proxi-model are asymptotically equivalent in the sense of the Le Cam Δ-distance both for the continuous observation of one path and for ni.i.d. paths under the condition nɛ→0, which justifies our statistical method.

Prognostics of fractional degradation processes with state-dependent delay

In modern industrial processes, the remaining useful life (RUL) of core manufacturing equipments is regarded as an important indicator for assessing the continuous serving ability by considering safety and reliability. Accurate RUL predictions contribute to saving maintenance costs, and can be applied to the life extension technologies. Being subjected to complicated noise environments, the fractional characteristic usually exists in the stochastic heterogeneous diffusions. Traditional methods mostly utilize the fractional Brownian motion (FBM) to describe a simple class of memory effect in the time domain, but lose sight of potential time-varying state-dependencies from historical information. In view of uncertain lagging levels, the state-dependent delay (SDD) is introduced to construct a novel nonlinear fractional degradation model in this paper. Based on a specific discretization scheme, the unknown parameters are estimated by optimizing an approximate log-likelihood function. The RUL distribution is then derived under a Markovian statistical transformation. Finally, a case study on certain hearth wall degradation processes is provided to validate the proposed prognostic method in production practice.

Estimating a class of diffusions from discrete observations via approximate maximum likelihood method

ABSTRACTAn approximate maximum likelihood method of estimation of diffusion parameters based on discrete observations of a diffusion X along fixed time-interval and Euler approximation of integrals is analysed. We assume that X satisfies a stochastic differential equation (SDE) of form , with non-random initial condition. SDE is nonlinear in generally. Based on assumption that maximum likelihood estimator of the drift parameter based on continuous observation of a path over exists we prove that measurable estimator of the parameters obtained from discrete observations of X along by maximization of the approximate log-likelihood function exists, being consistent and asymptotically normal, and tends to zero with rate in probability when tends to zero with T fixed. The same holds in case of an ergodic diffusion when T goes to infinity in a way that goes to zero with equidistant sampling, and we applied these to show consistency and asymptotical normality of , and asymptotic efficiency of in this case.

A new estimator of skewness parameter in stable distribution

Power-law tail behavior and the summation scheme of stable laws are the basis for their frequent use as models for data when fat tails and asymmetric feature above a Gaussian distribution are observed. Considering the skewness parameter to be another important index to gauge the tail behavior of stable laws besides characteristic exponent, the logistic function is employed to link the skewness parameter to a linear predictor induced by covariates, which constitutes a tail regression model. Then an approximate log-likelihood function is used to obtain skewness parameter regression estimators, and the asymptotic normality of these estimators is proved. Finally, A real example is presented to illustrate that the estimators provided by the paper have some abilities to explain economic meaning and forecast performances as expected too.

Maximum empirical likelihood estimation of time delay in independently and identically distributed noise

The authors apply the maximum empirical likelihood method to the problem of estimating the time delay of a measured digital signal when the signal can be seen as an instance of a stationary random process with additive independently and identically distributed (i.i.d.) noise. It is shown that, under these assumptions, an approximate log-likelihood function can be estimated from the measured data itself, and therefore a maximum likelihood estimate can be obtained without the prior knowledge of the formula for the signal likelihood. The Cramer–Rao lower bounds (CRLB) for two additive noise models (mixed-Gaussian and generalised normal distribution) are derived. The authors also show that the error produced by the maximum log-likelihood estimates (when the likelihood function is estimated from the measured data) better approximates the CRLB than other estimators for noise models other than Gaussian or Laplacian (special case of the generalised normal).

Penalized Covariance Matrix Estimation Using a Matrix-Logarithm Transformation

For statistical inferences that involve covariance matrices, it is desirable to obtain an accurate covariance matrix estimate with a well-structured eigen-system. We propose to estimate the covariance matrix through its matrix logarithm based on an approximate log-likelihood function. We develop a generalization of the Leonard and Hsu log-likelihood approximation that no longer requires a nonsingular sample covariance matrix. The matrix log-transformation provides the ability to impose a convex penalty on the transformed likelihood such that the largest and smallest eigenvalues of the covariance matrix estimate can be regularized simultaneously. The proposed method transforms the problem of estimating the covariance matrix into the problem of estimating a symmetric matrix, which can be solved efficiently by an iterative quadratic programming algorithm. The merits of the proposed method are illustrated by a simulation study and two real applications in classification and portfolio optimization. Supplementary materials for this article are available online.

TOA-Based Robust Wireless Geolocation and Cramér-Rao Lower Bound Analysis in Harsh LOS/NLOS Environments

We consider time-of-arrival based robust geolocation in harsh line-of-sight/non-line-of-sight environments. Herein, we assume the probability density function (PDF) of the measurement error to be completely unknown and develop an iterative algorithm for robust position estimation. The iterative algorithm alternates between a PDF estimation step, which approximates the exact measurement error PDF (albeit unknown) under the current parameter estimate via adaptive kernel density estimation, and a parameter estimation step, which resolves a position estimate from the approximate log-likelihood function via a quasi-Newton method. Unless the convergence condition is satisfied, the resolved position estimate is then used to refine the PDF estimation in the next iteration. We also present the best achievable geolocation accuracy in terms of the Cramér-Rao lower bound. Various simulations have been conducted in both real-world and simulated scenarios. When the number of received range measurements is large, the new proposed position estimator attains the performance of the maximum likelihood estimator (MLE). When the number of range measurements is small, it deviates from the MLE, but still outperforms several salient robust estimators in terms of geolocation accuracy, which comes at the cost of higher computational complexity.

Uniform Approximate Estimation for Nonlinear Nonhomogenous Stochastic System with Unknown Parameter

The error bound in probability between the approximate maximum likelihood estimator (AMLE) and the continuous maximum likelihood estimator (MLE) is investigated for nonlinear nonhomogenous stochastic system with unknown parameter. The rates of convergence of the approximations for Itô and ordinary integral are introduced under some regular assumptions. Based on these results, the in probability rate of convergence of the approximate log‐likelihood function to the true continuous log‐likelihood function is studied for the nonlinear nonhomogenous stochastic system involving unknown parameter. Finally, the main result which gives the error bound in probability between the ALME and the continuous MLE is established.

Parameter Estimation of the Extended Vasiček Model

In this paper, an estimate of the drift and diffusion parameters of the extended Vasicek model is presented. The estimate is based on the method of maximum likelihood. We derive a closed-form expansion for the transition (probability) density of the extended Vasicek process and use the expansion to construct an approximate log-likelihood function of a discretely sampled data of the process. Approximate maximum likelihood estimators (AMLEs) of the parameters are obtained by maximizing the approximate log-likelihood function. The convergence of the AMLEs to the true maximum likelihood estimators is obtained by increasing the number of terms in the expansions with a small time step size.

Tail Index Regression

In extreme value statistics, the tail index is an important measure to gauge the heavy-tailed behavior of a distribution. Under Pareto-type distributions, we employ the logarithmic function to link the tail index to the linear predictor induced by covariates, which constitutes the tail index regression model. We then propose an approximate log-likelihood function to obtain regression parameter estimators, and subsequently show the asymptotic normality of those estimators. Numerical studies are presented to illustrate theoretical findings.

Tail Index Regression

In extreme value statistics, the tail index is an important measure to gauge the heavy-tailed behavior of a distribution. Under Pareto-type distributions, we employ the logarithmic function to link the tail index to the linear predictor induced by covariates, which constitutes the tail index regression model. We then propose an approximate log-likelihood function to obtain regression parameter estimators, and subsequently show the asymptotic normality of those estimators. Numerical studies are presented to illustrate theoretical findings.

A shared random effect parameter approach for longitudinal dementia data with non-ignorable missing data.

A significant source of missing data in longitudinal epidemiologic studies on elderly individuals is death. It is generally believed that these missing data by death are non-ignorable to likelihood based inference. Inference based on data only from surviving participants in the study may lead to biased results. In this paper we model both the probability of disease and the probability of death using shared random effect parameters. We also propose to use the Laplace approximation for obtaining an approximate likelihood function so that high dimensional integration over the distributions of the random effect parameters is not necessary. Parameter estimates can be obtained by maximizing the approximate log-likelihood function. Data from a longitudinal dementia study will be used to illustrate the approach. A small simulation is conducted to compare parameter estimates from the proposed method to the 'naive' method where missing data is considered at random.

Maximum likelihood estimation for stationary point processes

In this paper we derive the log likelihood function for point processes in terms of their stochastic intensities by using the martingale approach. For practical purposes we work with an approximate log likelihood function that is shown to possess the usual asymptotic properties of a log likelihood function. The resulting estimates are strongly consistent and asymptotically normal (under some regularity conditions). As a by-product, a strong law of large numbers and a central limit theorem for martingales in continuous times are derived.

On the residuals of autoregressive processes and polynomial regression

The residual processes of a stationary AR( p) process and of polynomial regression are considered. The residuals are obtained from ordinary least squares fitting. In the AR case, the partial sums converge to Brownian motion. In the polynomial case, they converge to generalized Brownian bridges. Other uses of the residuals are considered. Parameter estimation based on approximate log likelihood function of the residuals is considered.

Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure

A homogeneous spatial point pattern is regarded as one of thermal equilibrium configurations whose points interact on each other through a certain pairwise potential. Parameterizing the potential function, the likelihood is then defined by the Gibbs canonical ensemble. A Monte Carlo simulation method is reviewed to obtain equilibrium point patterns which correspond to a given potential function. An approximate log likelihood function for gas-like patterns is derived in order to compute the maximum likelihood estimates efficiently. Some parametric potential functions are suggested, and the Akaike Information Criterion is used for model selection. The feasibility of our procedure is demonstrated by some computer experiments. Using the procedure, some real data are investigated.