Minimum Contrast Estimators Research Articles

Statistical inference in discretely observed fractional Ornstein–Uhlenbeck processes

In this paper, we develop a two-stage method for estimating all the unknown parameters in the fractional Ornstein–Uhlenbeck model from discrete observations. The estimation procedure is built upon the marriage of the power variation and the minimum contrast estimation. In the first stage, the Hurst coefficient is estimated by the increment Bernoulli statistic and the volatility parameter is estimated by power variations. The asymptotic theory of the proposed estimators in the diffusion term are established under an in-fill asymptotic scheme. In the second stage, two drift parameters are estimated based on the minimum contrast estimation. Their asymptotic theory are analyzed via use of a double asymptotic scheme. The results from Monte Carlo studies illustrate that the proposed estimators have reasonable finite sample properties. An empirical illustration based on realized volatility indicates that the realized volatility is rough.

Efficient nonparametric estimation of generalised autocovariances

This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalised autocovariance function of a stationary random process. The generalised autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. We consider two cases: the fixed bandwidth design and the adaptive bandwidth design. The general result on the asymptotic efficiency, established for linear processes, is then applied to the class of stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule–Walker system of equations in the generalised autocovariance estimator.

Parameter estimation for linear parabolic SPDEs in two space dimensions based on high frequency data

AbstractWe consider parameter estimation for a linear parabolic second‐order stochastic partial differential equation (SPDE) in two space dimensions driven by two types of ‐Wiener processes based on high frequency data in time and space. We first estimate the parameters which appear in the eigenfunctions of the differential operator of the SPDE using the minimum contrast estimator based on the thinned data with respect to space, and then construct an approximate coordinate process of the SPDE. Furthermore, we propose estimators of the coefficient parameters of the SPDE utilizing the approximate coordinate process based on the thinned data with respect to time. We also give some simulation results.

Rate of convergence in the Kolmogorov distance for the minimum contrast estimator in the Heston model

Rate of convergence in the Kolmogorov distance for the minimum contrast estimator in the Heston model

Berry–Esseen inequalities for the fractional Black–Karasinski model of term structure of interest rates

Abstract The Black–Karasinski model is a one-factor non-affine interest rate model as it describes interest rate movements driven by a single source of randomness and the drift function is a nonlinear function of the interest rate. The drift parameters represent the level and the speed of mean reversion of the interest rate. It belongs to the class of no-arbitrage models. The paper introduces some new approximate minimum contrast estimators of the mean reversion speed parameter in the model based on discretely sampled data which are efficient and studies their asymptotic distributional properties with precise rates of convergence.

Adaptive estimator for a parabolic linear SPDE with a small noise

We deal with parametric estimation for a parabolic linear second-order stochastic partial differential equation (SPDE) with a small dispersion parameter based on high-frequency data which are observed in time and space. Using the thinned data with respect to space obtained from the high-frequency data, the minimum contrast estimators of two coefficient parameters of the SPDE are proposed. With these estimators and the thinned data with respect to time obtained from the high-frequency data, we construct an approximation of the coordinate process of the SPDE. Using the approximate coordinate process, we obtain the adaptive estimator of a coefficient parameter of the SPDE. Moreover, we give simulation results of the proposed estimators of the SPDE.

Parameter Estimation for Stochastic Wave Equation Based on Observation Window

Statistical inference for a linear stochastic hyperbolic equation with two unknown parameters is studied. Based on observation of coordinates of the solution or their linear combinations (i.e., on the observation window in the space domain), minimum contrast estimators are introduced. Strong consistency and asymptotic normality are proved. The results are applied to stochastic wave equation perturbed by Brownian noise and they are illustrated by a numerical simulation.

Efficient Nonparametric Estimation of Generalized Autocovariances

This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalized autocovariance function of a stationary random process. The generalized autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. The general result on the asymptotic efficiency is then applied to the class of Gaussian stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule-Walker system of equations in the generalized autocovariance estimator.

Assessment of Leaf Chlorophyll Content Models for Winter Wheat Using Landsat-8 Multispectral Remote Sensing Data

The leaf chlorophyll content (LCC) is a critical index to characterize crop growth conditions, photosynthetic capacity, and physiological status. Its dynamic change characteristics are of great significance for monitoring crop growth conditions and understanding the process of material and energy exchange between crops and the environment. Extensive research has focused on LCC retrieval with hyperspectral data onboard various sensor platforms. Nevertheless, limited attention has been paid to LCC inversion from multispectral data, such as the data from Landsat-8, and the potentials and capabilities of the data for crop LCC estimation have not been fully explored. The present study made use of Landsat-8 Operational Land Imager (OLI) imagery and the corresponding field experimental data to evaluate their capabilities and potentials for LCC modeling using four different retrieval methods: vegetation indices (VIs), machine learning regression algorithms (MLRAs), lookup-table (LUT)-based inversion, and hybrid regression approaches. The results showed that the modified triangular vegetation index (MTVI2) exhibited the best estimate accuracy for LCC retrieval with a root mean square error (RMSE) of 5.99 μg/cm2 and a relative RMSE (RRMSE) of 10.49%. Several other vegetation indices that were established from red and near-infrared (NIR) bands also exhibited good accuracy. Models established from Gaussian process regression (GPR) achieved the highest accuracy for LCC retrieval (RMSE = 5.50 μg/cm2, RRMSE = 9.62%) compared with other MLRAs. Moreover, red and NIR bands outweighed other bands in terms of GPR modelling. LUT-based inversion methods with the “K(x) = −log (x) + x” cost function that belongs to the “minimum contrast estimates” family showed the best estimation results (RMSE = 8.08 μg/cm2, RRMSE = 14.14%), and the addition of multiple solution regularization strategies effectively improved the inversion accuracy. For hybrid regression methods, the use of active learning (AL) techniques together with GPR for LCC modelling significantly increased the estimation accuracy, and the combination of entropy query by bagging (EQB) AL and GPR had the best accuracy for LCC estimation (RMSE = 12.43 μg/cm2, RRMSE = 21.77%). Overall, our study suggest that Landsat-8 OLI data are suitable for crop LCC retrieval and could provide a basis for LCC estimation with similar multispectral datasets.

A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations

A new modification of the minimum-contrast estimator (the weighted MCE) of drift parameter in a linear stochastic evolution equation with additive fractional noise is introduced in the setting of the spectral approach (Fourier coordinates of the solution are observed). The reweighing technique, which utilizes the self-similarity property, achieves strong consistency and asymptotic normality of the estimator as number of coordinates increases and time horizon is fixed (the space consistency). In this respect, this modification outperforms the standard (non-weighted) MCE. Compared to other drift estimators studied within spectral approach (e.g., maximum likelihood, trajectory fitting), the weighted MCE is rather universal. It covers discrete time as well as continuous-time observations and it is applicable to processes with any value of Hurst index [Formula: see text]. To the author’s best knowledge, this is so far the first space-consistent estimator studied for [Formula: see text].

Central limit theorems and minimum-contrast estimators for linear stochastic evolution equations

ABSTRACTCentral limit theorems and asymptotic properties of the minimum-contrast estimators of the drift parameter in linear stochastic evolution equations driven by fractional Brownian motion are studied. Both singular ( and regular ( types of fractional Brownian motion are considered. Strong consistency is achieved by ergodicity of the stationary solution. The fundamental tool for the limit theorems and asymptotic normality (shown for Hurst parameter ) is the so-called 4th moment theorem considered on the second Wiener chaos. This technique provides also the Berry-Esseen-type bounds for the speed of the convergence. The general results are illustrated for parabolic equations with distributed and pointwise fractional noises.

Retrieval of Maize Leaf Area Index Using Hyperspectral and Multispectral Data

Field spectra acquired from a handheld spectroradiometer and Sentinel-2 images spectra were used to investigate the applicability of hyperspectral and multispectral data in retrieving the maize leaf area index in low-input crop systems, with high spatial and intra-annual variability, and low yield, in southern Mozambique, during three years. Seventeen vegetation indices, comprising two and three band indices, and nine machine learning regression algorithms (MLRA) were tested for the statistical approach while five cost functions were tested in the look-up-table (LUT) inversion approach. The three band vegetation indices were selected, specifically the modified difference index (mDId: 725; 715; 565) for the hyperspectral dataset and the modified simple ratio (mSRc: 740; 705; 865) for the multispectral dataset of field spectra and the three band spectral index (TBSIb: 665; 865; 783) for the Sentinel-2 dataset. The relevant vector machine was the selected MLRA for the two datasets of field spectra (multispectral and hyperspectral) while the support vector machine was selected for the Sentinel-2 data. When using the LUT inversion technique, the minimum contrast estimation and the Bhattacharyya divergence cost functions were the best performing. The vegetation indices outperformed the other two approaches, with the TBSIb as the most accurate index (RMSE = 0.35). At the field scale, spectral data from Sentinel-2 can accurately retrieve the maize leaf area index in the study area.

Parameter Estimation for Stochastic Partial Differential Equations of Second Order

Stochastic partial differential equations of second order with two unknown parameters are studied. Based on ergodicity, two suitable families of minimum contrast estimators are introduced. Strong consistency and asymptotic normality of estimators are proved. The results are applied to hyperbolic equations perturbed by Brownian noise.

Discrete-Time Statistical Inference for Multiscale Diffusions

We study statistical inference for small-noise-perturbed multiscale dynamical systems under the assumption that we observe a single time series from the slow process only. We construct estimators for both averaging and homogenization regimes, based on an appropriate misspecified model motivated by a second-order stochastic Taylor expansion of the slow process with respect to a function of the time-scale separation parameter. In the case of a fixed number of observations, we establish consistency, asymptotic normality, and asymptotic statistical efficiency of a minimum contrast estimator (MCE), the limiting variance having been identified explicitly; we furthermore establish consistency and asymptotic normality of a simplified MCE, which is, however, not, in general, efficient. These results are then extended to the case of high-frequency observations under a condition restricting the rate at which the number of observations may grow vis-a-vis the separation of scales. Numerical simulations illustrate the ...

Network Quantile Autoregression

It is a challenging task to understand the complex dependency structures in an ultra-high dimensional network, especially when one concentrates on the tail dependency. To tackle this problem, we consider a network quantile autoregres- sion model (NQAR) to characterize the dynamic quantile behavior in a complex system. In particular, we relate responses to its connected nodes and node spe- ci c characteristics in a quantile autoregression process. A minimum contrast estimation approach for the NQAR model is introduced, and the asymptotic properties are studied. Finally, we demonstrate the usage of our model by in- vestigating the nancial contagions in the Chinese stock market accounting for shared ownership of companies.

Robust parameter estimation for stationary processes by an exotic disparity from prediction problem

A new class of disparities from the point of view of prediction problem is proposed for minimum contrast estimation of spectral densities of stationary processes. We investigate asymptotic properties of the minimum contrast estimators based on the new disparities for stationary processes with both finite and infinite variance innovations. The relative efficiency and the robustness against randomly missing observations are shown in our numerical simulations.

Asymptotic properties of parameter estimates for random fields with tapered data

In this paper we present novel results on the asymptotic behavior of the so-called Ibragimov minimum contrast estimates. The case of tapered data for various models of Gaussian random fields is investigated. The CLT for quadratic forms with tapered data is presented.

Contrast Estimation for Parametric Stationary Determinantal Point Processes

AbstractWe study minimum contrast estimation for parametric stationary determinantal point processes. These processes form a useful class of models for repulsive (or regular, or inhibitive) point patterns and are already applied in numerous statistical applications. Our main focus is on minimum contrast methods based on the Ripley's K‐function or on the pair correlation function. Strong consistency and asymptotic normality of theses procedures are proved under general conditions that only concern the existence of the process and its regularity with respect to the parameters. A key ingredient of the proofs is the recently established Brillinger mixing property of stationary determinantal point processes. This work may be viewed as a complement to the study of Y. Guan and M. Sherman who establish the same kind of asymptotic properties for a large class of Cox processes, which in turn are models for clustering (or aggregation).

Asymptotic properties of the minimum contrast estimators for projections of inhomogeneous space-time shot-noise Cox processes

We consider a flexible class of space-time point process models—inhomogeneous shot-noise Cox point processes. They are suitable for modelling clustering phenomena, e.g. in epidemiology, seismology, etc. The particular structure of the model enables the use of projections to the spatial and temporal domain. They are used to formulate a stepwise estimation method to estimate different parts of the model separately. In the first step, the Poisson likelihood approach is used to estimate the inhomogeneity parameters. In the second and third steps, the minimum contrast estimation based on K-functions of the projected processes is used to estimate the interaction parameters. We study the asymptotic properties of the resulting estimators and formulate a set of conditions sufficient for establishing consistency and asymptotic normality of the estimators under the increasing domain asymptotics.

Two-step estimation procedures for inhomogeneous shot-noise Cox processes

In the present paper, we discuss and compare several two-step estimation procedures for inhomogeneous shot-noise Cox processes. The intensity function is parametrized by the inhomogeneity parameters while the pair-correlation function is parametrized by the interaction parameters. The suggested procedures are based on a combination of Poisson likelihood estimation of the inhomogeneity parameters in the first step and an adaptation of a method from the homogeneous case for estimation of the interaction parameters in the second step. The adapted methods, based on minimum contrast estimation, composite likelihood and Palm likelihood, are compared both theoretically and by means of a simulation study. The general conclusion from the simulation study is that the three estimation methods have similar performance. Two-step estimation with Palm likelihood has not been considered before and is motivated by the superior performance of the Palm likelihood in the stationary case for estimation of certain parameters of interest. Asymptotic normality of the two-step estimator with Palm likelihood is proved.