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We build a unifying convex analysis framework characterizing the statistical properties of a large class of penalized estimators, both under a regular and an irregular design. Our framework interprets penalized estimators as proximal estimators, defined by a proximal operator applied to a corresponding initial estimator. We characterize the asymptotic properties of proximal estimators, showing that their asymptotic distribution follows a closed-form formula depending only on (i) the asymptotic distribution of the initial estimator, (ii) the estimator's limit penalty subgradient and (iii) the inner product defining the associated proximal operator. In parallel, we characterize the Oracle features of proximal estimators from the properties of their penalty's subgradients. We exploit our approach to systematically cover linear regression settings with a regular or irregular design. For these settings, we build new $\sqrt{n}-$consistent, asymptotically normal Ridgeless-type proximal estimators, which feature the Oracle property and are shown to perform satisfactorily in practically relevant Monte Carlo settings.

Dyadic data, where outcomes reflecting pairwise interaction among sampled units are of primary interest, arise frequently in social science research. Regression analyses with such data feature prominently in many research literatures (e.g., gravity models of trade). The dependence structure associated with dyadic data raises special estimation and, especially, inference issues. This chapter reviews currently available methods for (parametric) dyadic regression analysis and presents guidelines for empirical researchers., Comment: 20 pages, 1 table

In recent years, shrinkage priors have received much attention in high-dimensional data analysis from a Bayesian perspective. Compared with widely used spike-and-slab priors, shrinkage priors have better computational efficiency. But the theoretical properties, especially posterior contraction rate, which is important in uncertainty quantification, are not established in many cases. In this paper, we apply global-local shrinkage priors to high-dimensional multivariate linear regression with unknown covariance matrix. We show that when the prior is highly concentrated near zero and has heavy tail, the posterior contraction rates for both coefficients matrix and covariance matrix are nearly optimal. Our results hold when number of features p grows much faster than the sample size n, which is of great interest in modern data analysis. We show that a class of readily implementable scale mixture of normal priors satisfies the conditions of the main theorem., Comment: 22 pages, 1 figure

On the one hand, we investigate the Bahadur representation for sample quantiles under $\varphi$-mixing sequence with $\varphi(n)=O(n^{-3})$ and obtain a rate as $O(n^{-\frac{3}{4}}\log n)$, $a.s.$. On the other hand, by relaxing the condition of mixing coefficients to $\sum\nolimits_{n=1}^\infty\varphi^{1/2}(n)<\infty$, a rate $O(n^{-1/2}(\log n)^{1/2})$, $a.s.$, is also obtained.

We present a key example from sequential analysis, which illustrates that conditional bias reduction can cause infinite mean absolute error., Comment: 4 pages

In this dissertation, I derive a new method to estimate the Vapnik-Chervonenkis Dimension (VCD) for the class of linear functions. This method is inspired by the technique developed by Vapnik et al. Vapnik et al. (1994). My contribution rests on the approximation of the expected maximum difference between two empirical Losses (EMDBTEL). In fact, I use a cross-validated form of the error to compute the EMDBTEL, and I make the bound on the EMDBTEL tighter by minimizing a constant in of its right upper bound. I also derive two bounds for the true unknown risk using the additive (ERM1) and the multiplicative (ERM2) Chernoff bounds. These bounds depend on the estimated VCD and the empirical risk. These bounds can be used to perform model selection and to declare with high probability, the chosen model will perform better without making strong assumptions about the data generating process (DG). I measure the accuracy of my technique on simulated datasets and also on three real datasets. The model selection provided by VCD was always as good as if not better than the other methods under reasonable conditions., Comment: 131 pages, 34 figures, and 44 tables

Many statistical estimands can expressed as continuous linear functionals of a conditional expectation function. This includes the average treatment effect under unconfoundedness and generalizations for continuous-valued and personalized treatments. In this paper, we discuss a general approach to estimating such quantities: we begin with a simple plug-in estimator based on an estimate of the conditional expectation function, and then correct the plug-in estimator by subtracting a minimax linear estimate of its error. We show that our method is semiparametrically efficient under weak conditions and observe promising performance on both real and simulated data., Comment: 67 pages, 3 figures

Markov regime switching models have been widely used in numerous empirical applications in economics and finance. However, the asymptotic distribution of the maximum likelihood estimator (MLE) has not been proven for some empirically popular Markov regime switching models. In particular, the asymptotic distribution of the MLE has been unknown for models in which some elements of the transition probability matrix have the value of zero, as is commonly assumed in empirical applications with models with more than two regimes. This also includes models in which the regime-specific density depends on both the current and the lagged regimes such as the seminal model of Hamilton (1989) and switching ARCH model of Hamilton and Susmel (1994). This paper shows the asymptotic normality of the MLE and consistency of the asymptotic covariance matrix estimate of these models.

The Latent Block Model (LBM) is a model-based method to cluster simultaneously the $d$ columns and $n$ rows of a data matrix. Parameter estimation in LBM is a difficult and multifaceted problem. Although various estimation strategies have been proposed and are now well understood empirically, theoretical guarantees about their asymptotic behavior is rather sparse and most results are limited to the binary setting. We prove here theoretical guarantees in the valued settings. We show that under some mild conditions on the parameter space, and in an asymptotic regime where $\log(d)/n$ and $\log(n)/d$ tend to $0$ when $n$ and $d$ tend to infinity, (1) the maximum-likelihood estimate of the complete model (with known labels) is consistent and (2) the log-likelihood ratios are equivalent under the complete and observed (with unknown labels) models. This equivalence allows us to transfer the asymptotic consistency, and under mild conditions, asymptotic normality, to the maximum likelihood estimate under the observed model. Moreover, the variational estimator is also consistent and, under the same conditions, asymptotically normal., Comment: 36 pages, 2 figure

The Dantzig selector for a special parametric model of diffusion processes is studied in this paper. In our model, the diffusion coefficient is given as the exponential of the linear combination of other processes which are regarded as covariates. We propose an estimation procedure which is an adaptation of the Dantzig selector for linear regression models and prove the $l_q$ consistency of the estimator for all $q \in [1,\infty]$., Comment: 15 pages

The large-sample properties of likelihood-based statistical inference under mixture models have received much attention from statisticians. Although the consistency of the nonparametric MLE is regarded as a standard conclusion, many researchers ignore the precise conditions required on the mixture model. An incorrect claim of consistency can lead to false conclusions even if the mixture model under investigation seems well behaved. Under a finite normal mixture model, for instance, the consistency of the plain MLE is often erroneously assumed in spite of recent research breakthroughs. This paper streamlines the consistency results for the nonparametric MLE in general, and in particular for the penalized MLE under finite normal mixture models.

Aggregate shocks affect most households' and firms' decisions. Using three stylized models we show that inference based on cross-sectional data alone generally fails to correctly account for decision making of rational agents facing aggregate uncertainty. We propose an econometric framework that overcomes these problems by explicitly parameterizing the agents' inference problem relative to aggregate shocks. Our framework and examples illustrate that the cross-sectional and time-series aspects of the model are often interdependent. Therefore, estimation of model parameters in the presence of aggregate shocks requires the combined use of cross-sectional and time series data. We provide easy-to-use formulas for test statistics and confidence intervals that account for the interaction between the cross-sectional and time-series variation. Lastly, we perform Monte Carlo simulations that highlight the properties of the proposed method and the risks of not properly accounting for the presence of aggregate shocks.

We study theoretical properties of regularized robust M-estimators, applicable when data are drawn from a sparse high-dimensional linear model and contaminated by heavy-tailed distributions and/or outliers in the additive errors and covariates. We first establish a form of local statistical consistency for the penalized regression estimators under fairly mild conditions on the error distribution: When the derivative of the loss function is bounded and satisfies a local restricted curvature condition, all stationary points within a constant radius of the true regression vector converge at the minimax rate enjoyed by the Lasso with sub-Gaussian errors. When an appropriate nonconvex regularizer is used in place of an l_1-penalty, we show that such stationary points are in fact unique and equal to the local oracle solution with the correct support---hence, results on asymptotic normality in the low-dimensional case carry over immediately to the high-dimensional setting. This has important implications for the efficiency of regularized nonconvex M-estimators when the errors are heavy-tailed. Our analysis of the local curvature of the loss function also has useful consequences for optimization when the robust regression function and/or regularizer is nonconvex and the objective function possesses stationary points outside the local region. We show that as long as a composite gradient descent algorithm is initialized within a constant radius of the true regression vector, successive iterates will converge at a linear rate to a stationary point within the local region. Furthermore, the global optimum of a convex regularized robust regression function may be used to obtain a suitable initialization. The result is a novel two-step procedure that uses a convex M-estimator to achieve consistency and a nonconvex M-estimator to increase efficiency., Comment: 56 pages, 8 figures

We demonstrate that the primal-dual witness proof method may be used to establish variable selection consistency and $\ell_\infty$-bounds for sparse regression problems, even when the loss function and/or regularizer are nonconvex. Using this method, we derive two theorems concerning support recovery and $\ell_\infty$-guarantees for the regression estimator in a general setting. Our results provide rigorous theoretical justification for the use of nonconvex regularization: For certain nonconvex regularizers with vanishing derivative away from the origin, support recovery consistency may be guaranteed without requiring the typical incoherence conditions present in $\ell_1$-based methods. We then derive several corollaries that illustrate the wide applicability of our method to analyzing composite objective functions involving losses such as least squares, nonconvex modified least squares for errors-in variables linear regression, the negative log likelihood for generalized linear models, and the graphical Lasso. We conclude with empirical studies to corroborate our theoretical predictions., Comment: 51 pages, 13 figures

This paper proposes a local representation for Empirical Likelihood (EL). EL admits the classical local linear quadratic representation by its likelihood ratio property. A local estimator is derived by using the new representation. Consistency, local asymptotic normality, and asymptotic optimality results hold for the new estimator. In particular, when the regularity conditions do not include any differentiability assumption, these asymptotic results are still valid for the local estimator. Simulations illustrate that the local method improves the inference accuracy of EL.

We establish a new framework for statistical estimation of directed acyclic graphs (DAGs) when data are generated from a linear, possibly non-Gaussian structural equation model. Our framework consists of two parts: (1) inferring the moralized graph from the support of the inverse covariance matrix; and (2) selecting the best-scoring graph amongst DAGs that are consistent with the moralized graph. We show that when the error variances are known or estimated to close enough precision, the true DAG is the unique minimizer of the score computed using the reweighted squared l_2-loss. Our population-level results have implications for the identifiability of linear SEMs when the error covariances are specified up to a constant multiple. On the statistical side, we establish rigorous conditions for high-dimensional consistency of our two-part algorithm, defined in terms of a "gap" between the true DAG and the next best candidate. Finally, we demonstrate that dynamic programming may be used to select the optimal DAG in linear time when the treewidth of the moralized graph is bounded., Comment: 41 pages, 7 figures

We provide novel theoretical results regarding local optima of regularized $M$-estimators, allowing for nonconvexity in both loss and penalty functions. Under restricted strong convexity on the loss and suitable regularity conditions on the penalty, we prove that \emph{any stationary point} of the composite objective function will lie within statistical precision of the underlying parameter vector. Our theory covers many nonconvex objective functions of interest, including the corrected Lasso for errors-in-variables linear models; regression for generalized linear models with nonconvex penalties such as SCAD, MCP, and capped-$\ell_1$; and high-dimensional graphical model estimation. We quantify statistical accuracy by providing bounds on the $\ell_1$-, $\ell_2$-, and prediction error between stationary points and the population-level optimum. We also propose a simple modification of composite gradient descent that may be used to obtain a near-global optimum within statistical precision $\epsilon$ in $\log(1/\epsilon)$ steps, which is the fastest possible rate of any first-order method. We provide simulation studies illustrating the sharpness of our theoretical results., Comment: 58 pages, 13 figures. To appear in JMLR

We consider a multidimensional diffusion X with drift coefficient b({\alpha},X(t)) and diffusion coefficient {\epsilon}{\sigma}({\beta},X(t)). The diffusion is discretely observed at times t_k=k{\Delta} for k=1..n on a fixed interval [0,T]. We study minimum contrast estimators derived from the Gaussian process approximating X for small {\epsilon}. We obtain consistent and asymptotically normal estimators of {\alpha} for fixed {\Delta} and {\epsilon}\rightarrow0 and of ({\alpha},{\beta}) for {\Delta}\rightarrow0 and {\epsilon}\rightarrow0. We compare the estimators obtained with various methods and for various magnitudes of {\Delta} and {\epsilon} based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework., Comment: 31 pages, 2 figures, 2 tables

A maximum likelihood based model selection of discrete Bayesian networks is considered. The model selection is performed through scoring function $S$, which, for a given network $G$ and $n$-sample $D_n$, is defined to be the maximum log-likelihood $l$ minus a penalization term $\lambda_n h$ proportional to network complexity $h(G)$, $$ S(G|D_n) = l(G|D_n) - \lambda_n h(G). $$ The data is allowed to have missing values at random that has prompted, to improve the efficiency of estimation, a replacement of the standard log-likelihood with the sum of sample average node log-likelihoods. The latter avoids the exclusion of most partially missing data records and allows the comparison of models fitted to different samples. Provided that a discrete Bayesian network is identifiable for a given missing data distribution, we show that if the sequence $\lambda_n$ converges to zero at a slower rate than $n^{-{1/2}}$ then the estimation is consistent. Moreover, we establish that BIC model selection ($\lambda_n=0.5\log(n)/n$) applied to the node-average log-likelihood is in general not consistent. This is in contrast to the complete data case where BIC is known to be consistent. The conclusions are confirmed by numerical examples.

This paper deals with four types of point estimators based on minimization of information-theoretic divergences between hypothetical and empirical distributions. These were introduced (i) by Liese & Vajda (2006) and independently Broniatowski & Keziou (2006), called here power superdivergence estimators, (ii) by Broniatowski & Keziou (2009), called here power subdivergence estimators, (iii) by Basu et al. (1998), called here power pseudodistance estimators, and (iv) by Vajda (2008) called here Renyi pseudodistance estimators. The paper studies and compares general properties of these estimators such as consistency and influence curves, and illustrates these properties by detailed analysis of the applications to the estimation of normal location and scale.

Since its early use in least squares regression problems, the l1-penalization framework for variable selection has been employed in conjunction with a wide range of loss functions encompassing regression, classification and survival analysis. While a well developed theory exists for the l1-penalized least squares estimates, few results concern the behavior of l1-penalized estimates for general loss functions. In this paper, we derive two results concerning penalized estimates for a wide array of penalty and loss functions. Our first result characterizes the asymptotic distribution of penalized parametric M-estimators under mild conditions on the loss and penalty functions in the classical setting (fixed-p-large-n). Our second result explicits necessary and sufficient generalized irrepresentability (GI) conditions for l1-penalized parametric M-estimates to consistently select the components of a model (sparsistency) as well as their sign (sign consistency). In general, the GI conditions depend on the Hessian of the risk function at the true value of the unknown parameter. Under Gaussian predictors, we obtain a set of conditions under which the GI conditions can be re-expressed solely in terms of the second moment of the predictors. We apply our theory to contrast l1-penalized SVM and logistic regression classifiers and find conditions under which they have the same behavior in terms of their model selection consistency (sparsistency and sign consistency). Finally, we provide simulation evidence for the theory based on these classification examples., Comment: 55 pages, 4 figures, also available as a technical report from the Statistics Department at Indiana University

A parameter estimation problem is considered for a linear stochastic hyperbolic equation driven by additive space-time Gaussian white noise. The damping/amplification operator is allowed to be unbounded. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed.

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 recursive maximum likelihood algorithm is demonstrated, and relatively tight bounds on the convergence rate are derived. As opposed to the existing result on the asymptotic behavior of maximum likelihood estimation in hidden Markov models, the results of this paper are obtained without assuming that the log-likelihood function has an isolated maximum at which the Hessian is strictly negative definite.

In this paper the integer-valued autoregressive model of order one, contaminated with additive or innovational outliers is studied in some detail. Moreover, parameter estimation is also addressed. Supposing that the time points of the outliers are known but their sizes are unknown, we prove that the Conditional Least Squares (CLS) estimators of the offspring and innovation means are strongly consistent. In contrast, however, the CLS estimators of the outliers' sizes are not strongly consistent, although they converge to a random limit with probability 1. This random limit depends on the values of the process at the outliers' time points and on the values at the preceding time points and in case of additive outliers also on the values at the following time points. We also prove that the joint CLS estimator of the offspring and innovation means is asymptotically normal. Conditionally on the above described values of the process, the joint CLS estimator of the sizes of the outliers is also asymptotically normal., Comment: 106 pages; the proofs of the existence of CLS estimators are corrected

We focus on estimating the integrated covariance of log-price processes in the presence of market microstructure noise. We construct an efficient unbiased estimator for the quadratic covariation of two It\^{o} processes in the case where high-frequency asynchronous discrete returns under market microstructure noise are observed. This estimator is based on synchronization and multi-scale methods and attains the optimal rate of convergence. A Monte Carlo study analyzes the finite sample size characteristics of our estimator., Comment: 29 pages, including 4 pictures

A parameter estimation problem is considered for a diagonaliazable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter $H\geq 1/2$. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation.

A nearly unstable sequence of stationary spatial autoregressive processes is investigated, when the sum of the absolute values of the autoregressive coefficients tends to one. It is shown that after an appropriate norming the least squares estimator for these coefficients has a normal limit distribution. If none of the parameters equals zero than the typical rate of convergence is n., Comment: 26 pages To appear in: J. Multivariate Anal

In finite mixtures of location-scale distributions, if there is no constraint or penalty on the parameters, then the maximum likelihood estimator does not exist because the likelihood is unbounded. To avoid this problem, we consider a penalized likelihood, where the penalty is a function of the minimum of the ratios of the scale parameters and the sample size. It is shown that the penalized maximum likelihood estimator is strongly consistent. We also analyze the consistency of a penalized maximum likelihood estimator where the penalty is imposed on the scale parameters themselves., Comment: 29 pages, 2 figures

The difference equations $\xi_{k}=af(\xi_{k-1})+\epsilon_{k}$, where $(\epsilon_k)$ is a square integrable difference martingale, and the differential equation ${\rm d}\xi=-af(\xi){\rm d}t+{\rm d}\eta$, where $\eta$ is a square integrable martingale, are considered. A family of estimators depending, besides the sample size $n$ (or the observation period, if time is continuous) on some random Lipschitz functions is constructed. Asymptotic optimality of this estimators is investigated., Comment: 10 pages

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., Comment: Warning: due to a flaw in the publishing process, some of the references in the published version of the article are confused

We point out some pitfalls related to the concept of an oracle property as used in Fan and Li (2001, 2002, 2004) which are reminiscent of the well-known pitfalls related to Hodges' estimator. The oracle property is often a consequence of sparsity of an estimator. We show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function; in particular, the maximal risk diverges to infinity whenever the loss function is unbounded. For ease of presentation the result is set in the framework of a linear regression model, but generalizes far beyond that setting. In a Monte Carlo study we also assess the extent of the problem in finite samples for the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li (2001). We find that this estimator can perform rather poorly in finite samples and that its worst-case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity., Comment: 18 pages, 5 figures

In finite mixtures of location-scale distributions, if there is no constraint on the parameters then the maximum likelihood estimate does not exist. But when the ratios of the scale parameters are restricted appropriately, the maximum likelihood estimate exists. We prove that the maximum likelihood estimator (MLE) is strongly consistent, if the ratios of the scale parameters are restricted from below by $\exp(-n^{d}), 0 < d < 1 $, where $n$ is the sample size., Comment: 11 pages, 2 figures. This paper is superseded by arXiv:0710.2183