An autoregressive process with Markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable Markov chain. In this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite. Consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden Markov chain conditional on the observations.
We consider maximum likelihood estimation for both causal and noncausal autoregressive time series processes with non-Gaussian $α$-stable noise. A nondegenerate limiting distribution is given for maximum likelihood estimators of the parameters of the autoregressive model equation and the parameters of the stable noise distribution. The estimators for the autoregressive parameters are $n^{1/α}$-consistent and converge in distribution to the maximizer of a random function. The form of this limiting distribution is intractable, but the shape of the distribution for these estimators can be examine…
The first purpose of this article is to obtain a.s. asymptotic properties of the maximum likelihood estimator in the autoregressive process driven by a stationary Gaussian noise. The second purpose is to show the local asymptotic normality property of the likelihoods ratio in order to get a notion of asymptotic efficiency and to build an asymptotically uniformly invariant most powerful procedure for testing the significance of the autoregressive parameter.
This paper considers the asymptotic properties of the recursive maximum likelihood estimation in hidden Markov models. The paper is focused on the asymptotic behavior of the log-likelihood function and on the point-convergence and convergence rate of the recursive maximum likelihood estimator. Using the principle of analytical continuation, the analyticity of the asymptotic log-likelihood function is shown for analytically parameterized hidden Markov models. Relying on this fact and some results from differential geometry (Lojasiewicz inequality), the almost sure point-convergence of the recur…
In this paper we are interested in the Maximum Likelihood Estimator (MLE) of the vector parameter of an autoregressive process of order $p$ with regular stationary Gaussian noise. We exhibit the large sample asymptotical properties of the MLE under very mild conditions. Simulations are done for fractional Gaussian noise (fGn), autoregressive noise (AR(1)) and moving average noise (MA(1)).
In this note, we propose a new approach for the proof of the consistency and normality of the maximum likelihood estimator for nonlinear AR processes with markov-switching under the assumptions of uniform exponential forgetting of the prediction filter and $α$-mixing property. We show that in the linear and Gaussian case our assumptions are fully satisfied.
In many complex statistical models maximum likelihood estimators cannot be calculated. In the paper we solve this problem using Markov chain Monte Carlo approximation of the true likelihood. In the main result we prove asymptotic normality of the estimator, when both sample sizes (the initial and Monte Carlo one) tend to infinity. Our result can be applied to models with intractable norming constants and missing data models.
A hidden Markov model with trends is a hidden Markov model whose emission distributions are translated by a trend that depends on the current hidden state and on the current time. Contrary to standard hidden Markov models, such processes are not homogeneous and cannot be made homogeneous by a simple de-trending step. We show that when the trends are polynomial, the maximum likelihood estimator is able to recover the trends together with the other parameters and is strongly consistent. More precisely, the supremum norm of the difference between the true trends and the estimated ones tends to ze…
Nonparametric identification and maximum likelihood estimation for finite-state hidden Markov models are investigated. We obtain identification of the parameters as well as the order of the Markov chain if the transition probability matrices have full-rank and are ergodic, and if the state-dependent distributions are all distinct, but not necessarily linearly independent. Based on this identification result, we develop nonparametric maximum likelihood estimation theory. First, we show that the asymptotic contrast, the Kullback--Leibler divergence of the hidden Markov model, identifies the true…
This paper deals with a parametrized family of partially observed bivariate Markov chains. We establish that, under very mild assumptions, the limit of the normalized log-likelihood function is maximized when the parameters belong to the equivalence class of the true parameter, which is a key feature for obtaining the consistency of the maximum likelihood estimators (MLEs) in well-specified models. This result is obtained in the general framework of partially dominated models. We examine two specific cases of interest, namely, hidden Markov models (HMMs) and observation-driven time series mode…
We study some regularity properties in locally stationary Markov models which are fundamental for controlling the bias of nonparametric kernel estimators. In particular, we provide an alternative to the standard notion of derivative process developed in the literature and that can be used for studying a wide class of Markov processes. To this end, for some families of V-geometrically ergodic Markov kernels indexed by a real parameter u, we give conditions under which the invariant probability distribution is differentiable with respect to u, in the sense of signed measures. Our results also co…
The AMP Markov property is a recently proposed alternative Markov property for chain graphs. In the case of continuous variables with a joint multivariate Gaussian distribution, it is the AMP rather than the earlier introduced LWF Markov property that is coherent with data-generation by natural block-recursive regressions. In this paper, we show that maximum likelihood estimates in Gaussian AMP chain graph models can be obtained by combining generalized least squares and iterative proportional fitting to an iterative algorithm. In an appendix, we give useful convergence results for iterative p…
In this paper we consider the problem of detecting a change in the parameters of an autoregressive process, where the moments of the innovation process do not necessarily exist. An empirical likelihood ratio test for the existence of a change point is proposed and its asymptotic properties are studied. In contrast to other work on change point tests using empirical likelihood, we do not assume knowledge of the location of the change point. In particular, we prove that the maximizer of the empirical likelihood is a consistent estimator for the parameters of the autoregressive model in the case …
In this paper, we study a notion of local stationarity for discrete time Markov chains which is useful for applications in statistics. In the spirit of some locally stationary processes introduced in the literature, we consider triangular arrays of time-inhomogeneous Markov chains, defined by some families of contracting Markov kernels. Using the Dobrushin's contraction coefficients for various metrics, we show that the distribution of such Markov chains can be approximated locally with the distribution of ergodic Markov chains and we also study some mixing properties. From our approximation r…
We introduce a simple modification to the standard maximum likelihood estimation (MLE) framework. Rather than maximizing a single unconditional likelihood of the data under the model, we maximize a family of \textit{noise conditional} likelihoods consisting of the data perturbed by a continuum of noise levels. We find that models trained this way are more robust to noise, obtain higher test likelihoods, and generate higher quality images. They can also be sampled from via a novel score-based sampling scheme which combats the classical \textit{covariate shift} problem that occurs during sample …
It is an important task in the literature to check whether a fitted autoregressive moving average (ARMA) model is adequate, while the currently used tests may suffer from the size distortion problem when the underlying autoregressive models have low persistence. To fill this gap, this paper proposes two empirical likelihood-based portmanteau tests. The first one is naive but can serve as a benchmark, and the second is for the case with infinite variance innovations. The asymptotic distributions under the null hypothesis are derived under mild moment conditions, and their usefulness is demonstr…
Let $(Y_k)_{k\in \mathbb{Z}}$ be a stationary sequence on a probability space $(Ω,\mathcal{A},\mathbb{P})$ taking values in a standard Borel space $\mathsf{Y}$. Consider the associated maximum likelihood estimator with respect to a parametrized family of hidden Markov models such that the law of the observations $(Y_k)_{k\in \mathbb{Z}}$ is not assumed to be described by any of the hidden Markov models of this family. In this paper we investigate the consistency of this estimator in such misspecified models under mild assumptions.
Change point analysis has become an important research topic in many fields of applications. Several research work has been carried out to detect changes and its locations in time series data. In this paper, a nonparametric method based on the empirical likelihood is proposed to detect the structural changes of the parameters in autoregressive (AR) models . Under certain conditions, the asymptotic null distribution of the empirical likelihood ratio test statistic is proved to be the extreme value distribution. Further, the consistency of the test statistic has been proved. Simulations have bee…
Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete separable metric space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state space models, as well as general results on linear Gaussian state space models and finite state models. A novel aspect of our approach is an information-theoretic technique for proving identifiability, whi…
We consider parameter estimation in finite hidden state space Markov models with time-dependent inhomogeneous noise, where the inhomogeneity vanishes sufficiently fast. Based on the concept of asymptotic mean stationary processes we prove that the maximum likelihood and a quasi-maximum likelihood estimator (QMLE) are strongly consistent. The computation of the QMLE ignores the inhomogeneity, hence, is much simpler and robust. The theory is motivated by an example from biophysics and applied to a Poisson- and linear Gaussian model.
The paper studies large sample asymptotic properties of the Maximum Likelihood Estimator (MLE) for the parameter of a continuous time Markov chain, observed in white noise. Using the method of weak convergence of likelihoods due to I.Ibragimov and R.Khasminskii, consistency, asymptotic normality and convergence of moments are established for MLE under certain strong ergodicity conditions of the chain.
This paper considers maximum likelihood (ML) estimation in a large class of models with hidden Markov regimes. We investigate consistency of the ML estimator and local asymptotic normality for the models under general conditions which allow for autoregressive dynamics in the observable process, Markov regime sequences with covariate-dependent transition matrices, and possible model misspecification. A Monte Carlo study examines the finite-sample properties of the ML estimator in correctly specified and misspecified models. An empirical application is also discussed.
We propose a sparse coefficient estimation and automated model selection procedure for autoregressive (AR) processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator (PCMLE) satisfies a strong consistency, $O_P(N^{-1/2})$ consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator (LASSO…
This paper presents a model of asymmetric bifurcating autoregressive process with random coefficients. We couple this model with a Galton Watson tree to take into account possibly missing observations. We propose least-squares estimators for the various parameters of the model and prove their consistency with a convergence rate, and their asymptotic normality. We use both the bifurcating Markov chain and martingale approaches and derive new important general results in both these frameworks.
We propose a method for obtaining maximum likelihood estimates (MLEs) of a Markov-Modulated Jump-Diffusion Model (MMJDM) when the data is a discrete time sample of the diffusion process, the jumps follow a Laplace distribution, and the parameters of the diffusion are controlled by a Markov Jump Process (MJP). The data can be viewed as incomplete observation of a model with a tractable likelihood function. Therefore we use the EM-algorithm to obtain MLEs of the parameters. We validate our method with simulated data. The motivation for obtaining estimates of this model is that stock prices have …
NTT Communication Science Laboratories（NTT通信科学实验室）；NTT Secure Platform Laboratories（NTT安全平台实验室）
We propose a supervised anomaly detection method based on neural density estimators, where the negative log likelihood is used for the anomaly score. Density estimators have been widely used for unsupervised anomaly detection. By the recent advance of deep learning, the density estimation performance has been greatly improved. However, the neural density estimators cannot exploit anomaly label information, which would be valuable for improving the anomaly detection performance. The proposed method effectively utilizes the anomaly label information by training the neural density estimator so th…
In this paper we present a novel method for estimating the parameters of a parametric diffusion processes. Our approach is based on a closed-form Maximum Likelihood estimator for an approximating Continuous Time Markov Chain (CTMC) of the diffusion process. Unlike typical time discretization approaches, such as psuedo-likelihood approximations with Shoji-Ozaki or Kessler's method, the CTMC approximation introduces no time-discretization error during parameter estimation, and is thus well-suited for typical econometric situations with infrequently sampled data. Due to the structure of the CTMC,…
Two-dimensional (2-D) autoregressive moving average (ARMA) models are commonly applied to describe real-world image data, usually assuming Gaussian or symmetric noise. However, real-world data often present non-Gaussian signals, with asymmetrical distributions and strictly positive values. In particular, SAR images are known to be well characterized by the Rayleigh distribution. In this context, the ARMA model tailored for 2-D Rayleigh-distributed data is introduced -- the 2-D RARMA model. The 2-D RARMA model is derived and conditional likelihood inferences are discussed. The proposed model wa…
In this work we introduce the class of beta autoregressive fractionally integrated moving average models for continuous random variables taking values in the continuous unit interval $(0,1)$. The proposed model accommodates a set of regressors and a long-range dependent time series structure. We derive the partial likelihood estimator for the parameters of the proposed model, obtain the associated score vector and Fisher information matrix. We also prove the consistency and asymptotic normality of the estimator under mild conditions. Hypotheses testing, diagnostic tools and forecasting are als…
We show that large-scale typicality of Markov sample paths implies that the likelihood ratio statistic satisfies a law of iterated logarithm uniformly to the same scale. As a consequence, the penalized likelihood Markov order estimator is strongly consistent for penalties growing as slowly as log log n when an upper bound is imposed on the order which may grow as rapidly as log n. Our method of proof, using techniques from empirical process theory, does not rely on the explicit expression for the maximum likelihood estimator in the Markov case and could therefore be applicable in other setting…
Deep autoregressive models compute point likelihood estimates of individual data points. However, many applications (i.e., database cardinality estimation) require estimating range densities, a capability that is under-explored by current neural density estimation literature. In these applications, fast and accurate range density estimates over high-dimensional data directly impact user-perceived performance. In this paper, we explore a technique, variable skipping, for accelerating range density estimation over deep autoregressive models. This technique exploits the sparse structure of range …
In this paper we derive the consistency of the penalized likelihood method for the number state of the hidden Markov chain in autoregressive models with Markov regimen. Using a SAEM type algorithm to estimate the models parameters. We test the null hypothesis of hidden Markov Model against an autoregressive process with Markov regime.
The Markovian arrival process (MAP) has proven a versatile model for fitting dependent and non-exponential interarrival times, with a number of applications to queueing, teletraffic, reliability or finance. Despite theoretical properties of MAPs and models involving MAPs are well studied, their estimation remains less explored. This paper examines maximum likelihood estimation of the second-order MAP using a recently obtained parameterization of the two-state MAPs.
Due to the intractable partition function, the exact likelihood function for a Markov random field (MRF), in many situations, can only be approximated. Major approximation approaches include pseudolikelihood and Laplace approximation. In this paper, we propose a novel way of approximating the likelihood function through first approximating the marginal likelihood functions of individual parameters and then reconstructing the joint likelihood function from these marginal likelihood functions. For approximating the marginal likelihood functions, we derive a particular likelihood function from a …
ExxonMobil Research and Engineering Company（埃克森美孚研究与工程公司）
We report an exact likelihood computation for Linear Gaussian Markov processes that is more scalable than existing algorithms for complex models and sparsely sampled signals. Better scaling is achieved through elimination of repeated computations in the Kalman likelihood, and by using the diagonalized form of the state transition equation. Using this efficient computation, we study the accuracy of kernel learning using maximum likelihood and the posterior mean in a simulation experiment. The posterior mean with a reference prior is more accurate for complex models and sparse sampling. Because …
This report introduces a parsimonious structure for mixture of autoregressive models, where the weighting coefficients are determined through latent random variables as functions of all past observations. These variables follow a hidden Markov model. We modify EM and Baum-Welch algorithms to estimate the parameters of the model.
Jilin University（吉林大学）；Zhuhai College of Jilin University（吉林大学珠海学院）
Flow-based generative models are a family of exact log-likelihood models with tractable sampling and latent-variable inference, hence conceptually attractive for modeling complex distributions. However, flow-based models are limited by density estimation performance issues as compared to state-of-the-art autoregressive models. Autoregressive models, which also belong to the family of likelihood-based methods, however suffer from limited parallelizability. In this paper, we propose Dynamic Linear Flow (DLF), a new family of invertible transformations with partially autoregressive structure. Our…
We propose an inferential approach for maximum likelihood estimation of the hidden Markov models for continuous responses. We extend to the case of longitudinal observations the finite mixture model of multivariate Gaussian distributions with Missing At Random (MAR) outcomes, also accounting for possible dropout. The resulting hidden Markov model accounts for different types of missing pattern: (i) partially missing outcomes at a given time occasion; (ii) completely missing outcomes at a given time occasion (intermittent pattern); (iii) dropout before the end of the period of observation (mono…
Time-homogeneous Markov chains are often used as disease progression models in studies of cost-effectiveness and optimal decision-making. Maximum likelihood estimation of these models can be challenging when data are collected at a time interval longer than the model's transition cycle length. For example, it may be necessary to estimate a monthly transition model from data collected annually. The likelihood for a time-homogeneous Markov chain with transition matrix $\mathbf{P}$ and data observed at intervals of $T$ cycles is a function of $\mathbf{P}^T.$ The maximum likelihood estimate of $\m…
We develop a new statistical model to analyse time-varying ranking data. The model can be used with a large number of ranked items, accommodates exogenous time-varying covariates and partial rankings, and is estimated via the maximum likelihood in a straightforward manner. Rankings are modelled using the Plackett-Luce distribution with time-varying worth parameters that follow a mean-reverting time series process. To capture the dependence of the worth parameters on past rankings, we utilise the conditional score in the fashion of the generalised autoregressive score (GAS) models. Simulation e…
We consider high-dimensional distribution estimation through autoregressive networks. By combining the concepts of sparsity, mixtures and parameter sharing we obtain a simple model which is fast to train and which achieves state-of-the-art or better results on several standard benchmark datasets. Specifically, we use an L1-penalty to regularize the conditional distributions and introduce a procedure for automatic parameter sharing between mixture components. Moreover, we propose a simple distributed representation which permits exact likelihood evaluations since the latent variables are interl…
Neural density estimators are flexible families of parametric models which have seen widespread use in unsupervised machine learning in recent years. Maximum-likelihood training typically dictates that these models be constrained to specify an explicit density. However, this limitation can be overcome by instead using a neural network to specify an energy function, or unnormalized density, which can subsequently be normalized to obtain a valid distribution. The challenge with this approach lies in accurately estimating the normalizing constant of the high-dimensional energy function. We propos…
We consider the issue of performing accurate small sample inference in beta autoregressive moving average model, which is useful for modeling and forecasting continuous variables that assumes values in the interval $(0,1)$. The inferences based on conditional maximum likelihood estimation have good asymptotic properties, but their performances in small samples may be poor. This way, we propose bootstrap bias corrections of the point estimators and different bootstrap strategies for confidence interval improvements. Our Monte Carlo simulations show that finite sample inference based on bootstra…
In this paper, we introduce a variant of hidden Markov models in which the transition probabilities between the states, as well as the emission distributions, are not constant in time but vary in a periodic manner. This class of models, that we will call seasonal hidden Markov models (SHMM) is particularly useful in practice, as many applications involve a seasonal behaviour. However, up to now, there is no theoretical result regarding this kind of model. We show that under mild assumptions, SHMM are identifiable: we can identify the transition matrices and the emission distributions from the …
This paper introduces a new parsimonious structure for mixture of autoregressive models. the weighting coefficients are determined through latent random variables, following a hidden Markov model. We propose a dynamic programming algorithm for the application of forecasting. We also derive the limiting behavior of unconditional first moment of the process and an appropriate upper bound for the limiting value of the variance. This can be considered as long run behavior of the process. Finally we show convergence and stability of the second moment. Further, we illustrate the efficacy of the prop…
This work concerns estimation of linear autoregressive models with Markov-switching using expectation maximisation (E.M.) algorithm. Our method generalise the method introduced by Elliot for general hidden Markov models and avoid to use backward recursion.
We introduce Hyper-Conditioned Neural Autoregressive Flow (HCNAF); a powerful universal distribution approximator designed to model arbitrarily complex conditional probability density functions. HCNAF consists of a neural-net based conditional autoregressive flow (AF) and a hyper-network that can take large conditions in non-autoregressive fashion and outputs the network parameters of the AF. Like other flow models, HCNAF performs exact likelihood inference. We conduct a number of density estimation tasks on toy experiments and MNIST to demonstrate the effectiveness and attributes of HCNAF, in…
New York University（纽约大学）；Google AI（谷歌人工智能）；Facebook AI Research（脸书人工智能研究院）
Many sequence-to-sequence generation tasks, including machine translation and text-to-speech, can be posed as estimating the density of the output y given the input x: p(y|x). Given this interpretation, it is natural to evaluate sequence-to-sequence models using conditional log-likelihood on a test set. However, the goal of sequence-to-sequence generation (or structured prediction) is to find the best output y^ given an input x, and each task has its own downstream metric R that scores a model output by comparing against a set of references y*: R(y^, y* | x). While we hope that a model that ex…
It is quite common that the structure of a time series changes abruptly. Identifying these change points and describing the model structure in the segments between these change points is of interest. In this paper, time series data is modelled assuming each segment is an autoregressive time series with possibly different autoregressive parameters. This is achieved using two main steps. The first step is to use a likelihood ratio scan based estimation technique to identify these potential change points to segment the time series. Once these potential change points are identified, modified param…
