In Bayesian hypothesis testing and model selection, prior distributions must be chosen carefully. For example, setting arbitrarily large prior scales for location parameters, which is common practice in estimation problems, can lead to undesirable behavior in testing (Lindley's paradox). We study the properties of some restricted type II maximum likelihood (type II ML) priors on regression coefficients. In type II ML, hyperparameters are "estimated" by maximizing the marginal likelihood of a model. In this article, we define priors by estimating their variances or covariance matrices, adding restrictions which ensure that the resulting priors are at least as vague as conventional proper priors for model uncertainty. We find that these type II ML priors typically yield results that are close to answers obtained with the Bayesian Information Criterion (BIC).
Bayesian inference provides a flexible way of combining data with prior information. However, quantile regression is not equipped with a parametric likelihood, and therefore, Bayesian inference for quantile regression demands careful investigation. This paper considers the Bayesian empirical likelihood approach to quantile regression. Taking the empirical likelihood into a Bayesian framework, we show that the resultant posterior from any fixed prior is asymptotically normal; its mean shrinks toward the true parameter values, and its variance approaches that of the maximum empirical likelihood …
The asymmetric Laplace density (ALD) is used as a working likelihood for Bayesian quantile regression. Sriram et al.(2013) derived posterior consistency for Bayesian linear quantile regression based on the misspecified ALD. While their paper also argued for $\sqrt{n}-$consistency, Sriram and Ramamoorthi (2017) highlighted that the argument was only valid for a rate less than $\sqrt{n}$. So, the question of $\sqrt{n}-$rate has remained unaddressed, but is necessary to carry out meaningful Bayesian inference based on the ALD. In this paper, we derive results to obtain posterior consistency rates…
We introduce a point process regression model that is applicable to price models and limit order book models. Hawkes type autoregression in the intensity process is generalized to a stochastic regression to covariate processes. We establish the so-called quasi likelihood analysis, which gives a polynomial type large deviation estimate for the statistical random field. We derive large sample properties of the maximum likelihood type estimator and the Bayesian type estimator when the intensity processes become large under a finite time horizon. There appears non-ergodic statistics. A classical a…
Bayesian posterior distributions arising in modern applications, including inverse problems in partial differential equation models in tomography and subsurface flow, are often computationally intractable due to the large computational cost of evaluating the data likelihood. To alleviate this problem, we consider using Gaussian process regression to build a surrogate model for the likelihood, resulting in an approximate posterior distribution that is amenable to computations in practice. This work serves as an introduction to Gaussian process regression, in particular in the context of buildin…
In this paper, we consider the multicollinearity problem in the gamma regression model when model parameters are linearly restricted. The linear restrictions are available from prior information to ensure the validity of scientific theories or structural consistency based on physical phenomena. In order to make relevant statistical inference for a model any available knowledge and prior information on the model parameters should be taken into account. This paper proposes therefore an algorithm to acquire Bayesian estimator for the parameters of a gamma regression model subjected to some linear…
Challenging research in various fields has driven a wide range of methodological advances in variable selection for regression models with high-dimensional predictors. In comparison, selection of nonlinear functions in models with additive predictors has been considered only more recently. Several competing suggestions have been developed at about the same time and often do not refer to each other. This article provides a state-of-the-art review on function selection, focusing on penalized likelihood and Bayesian concepts, relating various approaches to each other in a unified framework. In an…
In this work we consider Bayesian inference problems with intractable likelihood functions. We present a method to compute an approximate of the posterior with a limited number of model simulations. The method features an inverse Gaussian Process regression (IGPR), i.e., one from the output of a simulation model to the input of it. Within the method, we provide an adaptive algorithm with a tempering procedure to construct the approximations of the marginal posterior distributions. With examples we demonstrate that IGPR has a competitive performance compared to some commonly used algorithms, es…
Approximate Bayesian computation (ABC) methods, which are applicable when the likelihood is difficult or impossible to calculate, are an active topic of current research. Most current ABC algorithms directly approximate the posterior distribution, but an alternative, less common strategy is to approximate the likelihood function. This has several advantages. First, in some problems, it is easier to approximate the likelihood than to approximate the posterior. Second, an approximation to the likelihood allows reference analyses to be constructed based solely on the likelihood. Third, it is stra…
Imperial College London（帝国理工学院）；University of Cambridge（剑桥大学）
We propose a lower bound on the log marginal likelihood of Gaussian process regression models that can be computed without matrix factorisation of the full kernel matrix. We show that approximate maximum likelihood learning of model parameters by maximising our lower bound retains many of the sparse variational approach benefits while reducing the bias introduced into parameter learning. The basis of our bound is a more careful analysis of the log-determinant term appearing in the log marginal likelihood, as well as using the method of conjugate gradients to derive tight lower bounds on the te…
Gaussian process regression in its most simplified form assumes normal homoscedastic noise and utilizes analytically tractable mean and covariance functions of predictive posterior distribution using Gaussian conditioning. Its hyperparameters are estimated by maximizing the evidence, commonly known as type II maximum likelihood estimation. Unfortunately, Bayesian inference based on Gaussian likelihood is not robust to outliers, which are often present in the observational training data sets. To overcome this problem, we propose a robust process model in the Gaussian process framework with the …
The paper addresses joint sparsity selection in the regression coefficient matrix and the error precision (inverse covariance) matrix for high-dimensional multivariate regression models in the Bayesian paradigm. The selected sparsity patterns are crucial to help understand the network of relationships between the predictor and response variables, as well as the conditional relationships among the latter. While Bayesian methods have the advantage of providing natural uncertainty quantification through posterior inclusion probabilities and credible intervals, current Bayesian approaches either r…
Logistic regression models for binomial responses are routinely used in statistical practice. However, the maximum likelihood estimate may not exist due to data separability. We address this issue by considering a conjugate prior penalty which always produces finite estimates. Such a specification has a clear Bayesian interpretation and enjoys several invariance properties, making it an appealing prior choice. We show that the proposed method leads to an accurate approximation of the reduced-bias approach of Firth (1993), resulting in estimators with smaller asymptotic bias than the maximum-li…
One major drawback of state-of-the-art artificial intelligence is its lack of explainability. One approach to solve the problem is taking causality into account. Causal mechanisms can be described by structural causal models. In this work, we propose a method for estimating bivariate structural causal models using a combination of normalising flows applied to density estimation and variational Gaussian process regression for post-nonlinear models. It facilitates causal discovery, i.e. distinguishing cause and effect, by either the independence of cause and residual or a likelihood ratio test. …
A local projection model is defined by a set of linear regressions that account for the associations between exogenous variables and an endogenous variable observed at different time points. While it is standard practice to separately estimate individual regressions using the ordinary least squares estimator, some recent studies treat a local projection model as a multivariate regression with correlated errors, i.e., seemingly unrelated regressions, and propose Bayesian and non-Bayesian methods to improve the estimation accuracy. However, it is not clear how and when this way of treatment of l…
Unlike the ordinary least-squares (OLS) estimator for the linear model, a ridge regression linear model provides coefficient estimates via shrinkage, usually with improved mean-square and prediction error. This is true especially when the observed design matrix is ill-conditioned or singular, either as a result of highly-correlated covariates or the number of covariates exceeding the sample size. This paper introduces novel and fast marginal maximum likelihood (MML) algorithms for estimating the shrinkage parameter(s) for the Bayesian ridge and power ridge regression models, and an automatic p…
In this letter we generalise Ensemble Kalman inversion techniques to general Bayesian models where previously they were restricted to additive Gaussian likelihoods - all in the difficult setting where the likelihood can be sampled from, but its density not necessarily evaluated.
Bayesian likelihood-free methods implement Bayesian inference using simulation of data from the model to substitute for intractable likelihood evaluations. Most likelihood-free inference methods replace the full data set with a summary statistic before performing Bayesian inference, and the choice of this statistic is often difficult. The summary statistic should be low-dimensional for computational reasons, while retaining as much information as possible about the parameter. Using a recent idea from the interpretable machine learning literature, we develop some regression-based diagnostic met…
Like mean, quantile and variance, mode is also an important measure of central tendency and data summary. Many practical questions often focus on "Which element (gene or file or signal) occurs most often or is the most typical among all elements in a network?". In such cases mode regression provides a convenient summary of how the regressors affect the conditional mode and is totally different from other regression models based on conditional mean or conditional quantile or conditional variance. Some inference methods have been used for mode regression but none of them from the Bayesian perspe…
Florida State University（佛罗里达州立大学）；University of Dayton Research Institute（代顿大学研究院）；Air Force Research Laboratory（空军研究实验室）；Luna Innovations Incorporated（露娜创新公司）
This paper presents a new approach to a robust Gaussian process (GP) regression. Most existing approaches replace an outlier-prone Gaussian likelihood with a non-Gaussian likelihood induced from a heavy tail distribution, such as the Laplace distribution and Student-t distribution. However, the use of a non-Gaussian likelihood would incur the need for a computationally expensive Bayesian approximate computation in the posterior inferences. The proposed approach models an outlier as a noisy and biased observation of an unknown regression function, and accordingly, the likelihood contains bias t…
University of Cambridge（剑桥大学）；Imperial College London（帝国理工学院）
Recent work in scalable approximate Gaussian process regression has discussed a bias-variance-computation trade-off when estimating the log marginal likelihood. We suggest a method that adaptively selects the amount of computation to use when estimating the log marginal likelihood so that the bias of the objective function is guaranteed to be small. While simple in principle, our current implementation of the method is not competitive computationally with existing approximations.
We propose information criteria that measure the prediction risk of a predictive density based on the Bayesian marginal likelihood from a frequentist point of view. We derive criteria for selecting variables in linear regression models, assuming a prior distribution of the regression coefficients. Then, we discuss the relationship between the proposed criteria and related criteria. There are three advantages of our method. First, this is a compromise between the frequentist and Bayesian standpoints because it evaluates the frequentist's risk of the Bayesian model. Thus, it is less influenced b…
The logistic specification has been used extensively in non-Bayesian statistics to model the dependence of discrete outcomes on the values of specified covariates. Because the likelihood function is globally weakly concave estimation by maximum likelihood is generally straightforward even in commonly arising applications with scores or hundreds of parameters. In contrast Bayesian inference has proven awkward, requiring normal approximations to the likelihood or specialized adaptations of existing Markov chain Monte Carlo and data augmentation methods. This paper approaches Bayesian inference i…
We consider nonsynchronous sampling of parameterized stochastic regression models, which contain stochastic differential equations. Constructing a quasi-likelihood function, we prove that the quasi-maximum likelihood estimator and the Bayes type estimator are consistent and asymptotically mixed normal when the sampling frequency of the nonsynchronous data becomes large.
Seemingly unrelated regressions are statistical regression models based on the Gaussian distribution. They are popular in econometrics but also arise in graphical modeling of multivariate dependencies. In maximum likelihood estimation, the parameters of the model are estimated by maximizing the likelihood function, which maps the parameters to the likelihood of observing the given data. By transforming this optimization problem into a polynomial optimization problem, it was recently shown that the likelihood function of a simple bivariate seemingly unrelated regressions model may have several …
The restricted maximum likelihood (REML) estimator of the dispersion matrix for random coefficient models is rewritten in terms of the sufficient statistics of the individual regressions.
Likelihood ratio tests are widely used to test statistical hypotheses about parametric families of probability distributions. If interest is restricted to a subfamily of distributions, then it is natural to inquire if the restricted LRT is superior to the unrestricted LRT. Marden's general LRT conjecture posits that any restriction placed on the alternative hypothesis will increase power. The only published counterexample to this conjecture is rather technical and involves a restriction that maintains the dimension of the alternative. We formulate the dimension-restricted LRT conjecture, which…
In the random-effects model of meta-analysis a canonical representation of the restricted likelihood function is obtained. This representation relates the mean effect and the heterogeneity variance estimation problems. An explicit form of the variance of weighted means statistics determined by means of a quadratic form is found. The behavior of the mean squared error for large heterogeneity variance is elucidated. It is noted that the sample mean is not admissible nor minimax under a natural risk function for the number of studies exceeding three.
To address the problem of multicollinearity in the logistic regression model, in this paper we propose a new estimator called Stochastic restricted almost unbiased logistic Liu-estimator (SRAULLE) when the prior information is available in the form of stochastic linear restrictions. A Monte Carlo simulation study was carried out to compare the performance of the proposed estimator with some existing estimators in the scalar mean squared error (SMSE) sense. Finally, a real data example was given to appraise the performance of the estimators.
We deal with a general class of extreme-value regression models introduced by Barreto- Souza and Vasconcellos (2011). Our goal is to derive an adjusted likelihood ratio statistic that is approximately distributed as \c{hi}2 with a high degree of accuracy. Although the adjusted statistic requires more computational effort than its unadjusted counterpart, it is shown that the adjustment term has a simple compact form that can be easily implemented in standard statistical software. Further, we compare the finite sample performance of the three classical tests (likelihood ratio, Wald, and score), …
Shape restricted regressions, including isotonic regression and concave regression as special cases, are studied using priors on Bernstein polynomials and Markov chain Monte Carlo methods. These priors have large supports, select only smooth functions, can easily incorporate geometric information into the prior, and can be generated without computational difficulty. Algorithms generating priors and posteriors are proposed, and simulation studies are conducted to illustrate the performance of this approach. Comparisons with the density-regression method of Dette et al. (2006) are included.
The main Theorem of Jain et al.[Jain, K., Singh, S., and Sharma, S. (2011), Re- stricted estimation in multivariate measurement error regression model; JMVA, 102, 2, 264-280] is established in its full generality. Namely, we derive the joint asymp- totic normality of the unrestricted estimator (UE) and the restricted estimators of the matrix of the regression coefficients. The derived result holds under the hypothesized restriction as well as under the sequence of alternative restrictions. In addition, we establish Asymptotic Distributional Risk for the estimators and compare their relative pe…
We wish to estimate conditional density using Gaussian Mixture Regression model with logistic weights and means depending on the covariate. We aim at selecting the number of components of this model as well as the other parameters by a penalized maximum likelihood approach. We provide a lower bound on penalty, proportional up to a logarithmic term to the dimension of each model, that ensures an oracle inequality for our estimator. Our theoretical analysis is supported by some numerical experiments.
In this paper, we develop modified versions of the likelihood ratio test for multivariate heteroskedastic errors-in-variables regression models. The error terms are allowed to follow a multivariate distribution in the elliptical class of distributions, which has the normal distribution as a special case. We derive the Skovgaard adjusted likelihood ratio statistics, which follow a chi-squared distribution with a high degree of accuracy. We conduct a simulation study and show that the proposed tests display superior finite sample behavior as compared to the standard likelihood ratio test. We ill…
It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, in the context of biased shrinkage ridge estimation, Chang (2015) introduced an almost unbiased Liu estimator in the logistic regression model. Making use of his approach, when some prior knowledge in the form of linear restrictions are also available, we introduce a restricted almost unbiased Liu estimator in the logistic regression model. Statistical properties of this newly defined estimator are derived and some comparison result are also pr…
The stratified proportional intensity model generalizes Cox's proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also…
We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression, and constrained single index model. We review some of the theoretical properties of the least squares estimator (LSE) in these problems, emphasizing on the adaptive nature of the LSE. In particular, we study the behavior of the risk of the LSE, and its pointwise limiting distribution theory, with special emphasis to isotonic regression. We survey various methods for const…
In this article, the analysis of misspecification was extended to the recently introduced stochastic restricted biased estimators when multicollinearity exists among the explanatory variables. The Stochastic Restricted Ridge Estimator (SRRE), Stochastic Restricted Almost Unbiased Ridge Estimator (SRAURE), Stochastic Restricted Liu Estimator (SRLE), Stochastic Restricted Almost Unbiased Liu Estimator (SRAULE), Stochastic Restricted Principal Component Regression Estimator (SRPCR), Stochastic Restricted r-k class estimator (SRrk) and Stochastic Restricted r-d class estimator (SRrd) were examined…
Quantile regression is often used when a comprehensive relationship between a response variable and one or more explanatory variables is desired. The traditional frequentists' approach to quantile regression has been well developed around asymptotic theories and efficient algorithms. However, not much work has been published under the Bayesian framework. One challenging problem for Bayesian quantile regression is that the full likelihood has no parametric forms. In this paper, we propose a Bayesian quantile regression method, the linearly interpolated density (LID) method, which uses a linear …
This paper outlines a new nonparametric estimation procedure for unobserved phi-mixing processes. It is assumed that the only information on the stationary hidden states (Xk) is given by the process (Yk), where Yk is a noisy observation of f(Xk). The paper introduces a maximum pseudo-likelihood procedure to estimate the function f and the distribution of the hidden states using blocks of observations of length b. The identifiability of the model is studied in the particular cases b=1 and b=2. The consistency of the estimators of f and of the distribution of the hidden states as the number of o…
The performance of Least Squares (LS) estimators is studied in isotonic, unimodal and convex regression. Our results have the form of sharp oracle inequalities that account for the model misspecification error. In isotonic and unimodal regression, the LS estimator achieves the nonparametric rate $n^{-2/3}$ as well as a parametric rate of order $k/n$ up to logarithmic factors, where $k$ is the number of constant pieces of the true parameter. In univariate convex regression, the LS estimator satisfies an adaptive risk bound of order $q/n$ up to logarithmic factors, where $q$ is the number of aff…
A polynomial that is nonnegative over a given interval is called a positive polynomial. The set of such positive polynomials forms a closed convex cone $K$. In this paper, we consider the likelihood ratio test for the hypothesis of positivity that the estimand polynomial regression curve is a positive polynomial. By considering hierarchical hypotheses including the hypothesis of positivity, we define nested likelihood ratio tests, and derive their null distributions as mixtures of chi-square distributions by using the volume-of-tubes method. The mixing probabilities are obtained by utilizing t…
A simple construction of adaptive confidence sets is proposed in isotonic, convex and unimodal regression. In univariate isotonic regression, the proposed confidence set enjoys uniform coverage over all non-decreasing regression functions. Furthermore, the diameter of the proposed confidence set automatically adapts to the unknown number of pieces of the true parameter, in the sense that the diameter is bounded from above by the minimax risk over the class of $k$-piecewise constant functions. The diameter of the confidence set is a simple increasing function of the number of jumps of the isoto…
In this paper we consider two closely related problems : estimation of eigenvalues and eigenfunctions of the covariance kernel of functional data based on (possibly) irregular measurements, and the problem of estimating the eigenvalues and eigenvectors of the covariance matrix for high-dimensional Gaussian vectors. In Peng and Paul (2007), a restricted maximum likelihood (REML) approach has been developed to deal with the first problem. In this paper, we establish consistency and derive rate of convergence of the REML estimator for the functional data case, under appropriate smoothness conditi…
In this article we discuss estimation of the common variance of several normal populations with tree order restricted means. We discuss the asymptotic properties of the maximum likelihood estimator of the variance as the number of populations tends to infinity. We consider several cases of various orders of the sample sizes and show that the maximum likelihood estimator of the variance may or may not be consistent or be asymptotically normal.
We consider component-wise estimation of order restricted location/scale parameters $θ_1$ and $θ_2$ ($θ_1\leq θ_2$) of a general bivariate distribution under the squared error loss function. To find improvements over the best location/scale equivariant estimators (BLEE/BSEE) of $θ_1$ and $θ_2$, we study isotonic regression of suitable location/scale equivariant estimators (LEE/SEE) of $θ_1$ and $θ_2$ with general weights. Let $\mathcal{D}_{1,ν}$ and $\mathcal{D}_{2,β}$ denote suitable classes of isotonic regression estimators of $θ_1$ and $θ_2$, respectively. Under the squared error loss funct…
This article presents methods for constructing an asymptotic hyperbolic band under the multiple logistic regression model when the predictor variables are restricted to a specific region $\mathscr{X}$. Scheffé's method yields unnecessarily wide, and hence conservative, bands if the predictor variables can be restricted to a certain region. Piegorsch and Casella (1988) developed a procedure to build an asymptotic confidence band for the multiple logistic regression model over particular regions. Those regions are shown to be special cases of the region $\mathscr{X}$, which was first investigate…
In this paper, we study the asymptotic normality of the conditional maximum likelihood (ML) estimators for the truncated regression model and the Tobit model. We show that under the general setting assumed in his book, the conjectures made by Hayashi (2000) \footnote{see page 516, and page 520 of Hayashi (2000).} about the asymptotic normality of the conditional ML estimators for both models are true, namely, a sufficient condition is the nonsingularity of $\mathbf{x_tx'_t}$.
In many public health problems, an important goal is to identify the effect of some treatment/intervention on the risk of failure for the whole population. A marginal proportional hazards regression model is often used to analyze such an effect. When dependent censoring is explained by many auxiliary covariates, we utilize two working models to condense high-dimensional covariates to achieve dimension reduction. Then the estimator of the treatment effect is obtained by maximizing a pseudo-likelihood function over a sieve space. Such an estimator is shown to be consistent and asymptotically nor…
Although the log-likelihood is widely used in model selection, the log-likelihood ratio has had few applications in this area. We develop a log-likelihood ratio based method for selecting regression models by focusing on the set of models deemed plausible by the likelihood ratio test. We show that when the sample size is large and the significance level of the test is small, there is a high probability that the smallest model in the set is the true model; thus, we select this smallest model. The significance level of the test serves as a parameter for this method. We consider three levels of t…
Classical least squares estimators are well-known to be robust with respect to moment assumptions concerning the error distribution in a wide variety of finite-dimensional statistical problems; generally only a second moment assumption is required for least squares estimators to maintain the same rate of convergence that they would satisfy if the errors were assumed to be Gaussian. In this paper, we give a geometric characterization of the robustness of shape-restricted least squares estimators (LSEs) to error distributions with an $L_{2,1}$ moment, in terms of the `localized envelopes' of the…
