Multiple root estimation problems in statistical inference arise in many contexts in the literature. In the context of maximum likelihood estimation, the existence of multiple roots causes uncertainty in the computation of maximum likelihood estimators using hill-climbing algorithms, and consequent difficulties in the resulting statistical inference.
In this paper, we study the multiple roots phenomenon in maximum likelihood estimation for factor analysis. We prove that the corresponding likelihood equations have uncountably many feasible solutions even in the simplest cases. For the case in which the observed data are two-dimensional and the unobserved factor scores are one-dimensional, we prove that the solutions to the likelihood equations form a one-dimensional real curve.
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 exact maximum likelihood estimate (MLE) provides a test statistic for the unit root test that is more powerful \citep[p. 577]{Fuller96} than the usual least squares approach. In this paper a new derivation is given for the asymptotic distribution of this test statistic that is simpler and more direct than the previous method. The response surface regression method is used to obtain a fast algorithm that computes accurate finite-sample critical values. This algorithm is available in the R package {\tt mleur} that is available on CRAN. The empirical power of the new test is shown to be much …
The asymptotic theory of various estimators based on Gaussian likelihood has been developed for the unit root and near unit root cases of a first-order moving average model. Previous studies of the MA(1) unit root problem rely on the special autocovariance structure of the MA(1) process, in which case, the eigenvalues and eigenvectors of the covariance matrix of the data vector have known analytical forms. In this paper, we take a different approach to first consider the joint likelihood by including an augmented initial value as a parameter and then recover the exact likelihood by integrating…
We examine the normal approximation of the modified likelihood root, an inferential tool from higher-order asymptotic theory, for the linear exponential and location-scale family. We show that the $r^\star$ statistic can be thought of as a location and scale adjustment to the likelihood root $r$ up to $O_p(n^{-3/2})$, and more generally $r^\star$ can be expressed as a polynomial in $r$. We also show the linearity of the modified likelihood root in the likelihood root for these two families.
This paper gives a new approach for the maximum likelihood estimation of the joint of the location and scale of the Cauchy distribution. We regard the joint as a single complex parameter and derive a new form of the likelihood equation of a complex variable. Based on the equation, we provide a new iterative scheme approximating the maximum likelihood estimate. We also handle the equation in an algebraic manner and derive a polynomial containing the maximum likelihood estimate as a root. This algebraic approach provides another scheme approximating the maximum likelihood estimate by root-findin…
The first-order moving average model or MA(1) is given by $X_t=Z_t-θ_0Z_{t-1}$, with independent and identically distributed $\{Z_t\}$. This is arguably the simplest time series model that one can write down. The MA(1) with unit root ($θ_0=1$) arises naturally in a variety of time series applications. For example, if an underlying time series consists of a linear trend plus white noise errors, then the differenced series is an MA(1) with unit root. In such cases, testing for a unit root of the differenced series is equivalent to testing the adequacy of the trend plus noise model. The unit root…
There is widespread confusion about the role of projectivity in likelihood-based inference for random graph models. The confusion is rooted in claims that projectivity, a form of marginalizability, may be necessary for likelihood-based inference and consistency of maximum likelihood estimators. We show that likelihood-based superpopulation inference is not affected by lack of projectivity and that projectivity is not a necessary condition for consistency of maximum likelihood estimators.
Motivated by studying asymptotic properties of the maximum likelihood estimator (MLE) in stochastic volatility (SV) models, in this paper we investigate likelihood estimation in state space models. We first prove, under some regularity conditions, there is a consistent sequence of roots of the likelihood equation that is asymptotically normal with the inverse of the Fisher information as its variance. With an extra assumption that the likelihood equation has a unique root for each $n$, then there is a consistent sequence of estimators of the unknown parameters. If, in addition, the supremum of…
Inference about a scalar parameter of interest typically relies on the asymptotic normality of common likelihood pivots, such as the signed likelihood root, the score and Wald statistics. Nevertheless, the resulting inferential procedures are known to perform poorly when the dimension of the nuisance parameter is large relative to the sample size and when the information about the parameters is limited. In many such cases, the use of asymptotic normality of analytical modifications of the signed likelihood root is known to recover inferential performance. It is proved here that parametric boot…
Wilk's theorem, which offers universal chi-squared approximations for likelihood ratio tests, is widely used in many scientific hypothesis testing problems. For modern datasets with increasing dimension, researchers have found that the conventional Wilk's phenomenon of the likelihood ratio test statistic often fails. Although new approximations have been proposed in high dimensional settings, there still lacks a clear statistical guideline regarding how to choose between the conventional and newly proposed approximations, especially for moderate-dimensional data. To address this issue, we deve…
In this paper, we study the nonparametric maximum likelihood estimator for an event time distribution function at a point in the current status model with observation times supported on a grid of potentially unknown sparsity and with multiple subjects sharing the same observation time. This is of interest since observation time ties occur frequently with current status data. The grid resolution is specified as $cn^{-γ}$ with $c>0$ being a scaling constant and $γ>0$ regulating the sparsity of the grid relative to $n$, the number of subjects. The asymptotic behavior falls into three cases depend…
For the basic maximum likelihood estimating function of the two parameters Weibull distribution, a simple proof on its global monotonicity is given to ensure the existence and uniqueness of its solution. The boundary of the function's first-order derivative is defined based on its scale-free property. With a bounded derivative, the possible range of the root of this function can be determined. A novel root-finding algorithm employing these established results is proposed accordingly, its convergence is proved analytically as well. Compared with other typical algorithms for this problem, the ef…
This paper establishes the almost sure convergence and asymptotic normality of levels and differenced quasi maximum-likelihood (QML) estimators of dynamic panel data models. The QML estimators are robust with respect to initial conditions, conditional and time-series heteroskedasticity, and misspecification of the log-likelihood. The paper also provides an ECME algorithm for calculating levels QML estimates. Finally, it uses Monte Carlo experiments to compare the finite sample performance of levels and differenced QML estimators, the differenced GMM estimator, and the system GMM estimator. In …
In the application of autoregressive models the order of the model is often estimated using either a sequence of likelihood ratio tests, a likelihood based information criterion, or a residual based test. The properties of such procedures has been discussed extensively under the assumption that the characteristic roots of the autoregression are stationary. While non-stationary situations have also been considered the results in the literature depend on conditions to the characteristic roots. It is here shown that these methods for lag length determination can be used regardless of the assumpti…
The generalized varying coefficient partially linear model with growing number of predictors arises in many contemporary scientific endeavor. In this paper we set foot on both theoretical and practical sides of profile likelihood estimation and inference. When the number of parameters grows with sample size, the existence and asymptotic normality of the profile likelihood estimator are established under some regularity conditions. Profile likelihood ratio inference for the growing number of parameters is proposed and Wilk's phenomenon is demonstrated. A new algorithm, called the accelerated pr…
Motivated by recent works on the high-dimensional logistic regression, we establish that the existence of the maximum likelihood estimate exhibits a phase transition for a wide range of generalized linear models with binary outcome and elliptical covariates. This extends a previous result of Candès and Sur who proved the phase transition for the logistic regression with Gaussian covariates. Our result reveals a rich structure in the phase transition phenomenon, which is simply overlooked by Gaussianity. The main tools for deriving the result are data separation, convex geometry and stochastic …
Asymptotic distribution for the proportional covariance model under multivariate normal distributions is derived. To this end, the parametrization of the common covariance matrix by its Cholesky root is adopted. The derivations are made in three steps. First, the asymptotic distribution of the maximum likelihood estimators of the proportionality coefficients and the Cholesky inverse root of the common covariance matrix is derived by finding the information matrix and its inverse. Next, the asymptotic distributions for the case of the Cholesky root of the common covariance matrix and finally fo…
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…
The extremes of a stationary time series typically occur in clusters. A primary measure for this phenomenon is the extremal index, representing the reciprocal of the expected cluster size. Both a disjoint and a sliding blocks estimator for the extremal index are analyzed in detail. In contrast to many competitors, the estimators only depend on the choice of one parameter sequence. We derive an asymptotic expansion, prove asymptotic normality and show consistency of an estimator for the asymptotic variance. Explicit calculations in certain models and a finite-sample Monte Carlo simulation study…
Explicit functional forms for the generator derivatives of well-known one-parameter Archimedean copulas are derived. These derivatives are essential for likelihood inference as they appear in the copula density, conditional distribution functions, or the Kendall distribution function. They are also required for several asymmetric extensions of Archimedean copulas such as Khoudraji-transformed Archimedean copulas. Access to the generator derivatives makes maximum-likelihood estimation for Archimedean copulas feasible in terms of both precision and run time, even in large dimensions. It is shown…
The mixed fractional Vasicek model, which is an extended model of the traditional Vasicek model, has been widely used in modelling volatility, interest rate and exchange rate. Obviously, if some phenomenon are modeled by the mixed fractional Vasicek model, statistical inference for this process is of great interest. Based on continuous time observations, this paper considers the problem of estimating the drift parameters in the mixed fractional Vasicek model. We will propose the maximum likelihood estimators of the drift parameters in the mixed fractional Vasicek model with the Radon-Nikodym d…
Bayesian properties of the signed root likelihood ratio statistic are analysed. Conditions for first-order probability matching are derived by the examination of the Bayesian posterior and frequentist means of this statistic. Second-order matching conditions are shown to arise from matching of the Bayesian posterior and frequentist variances of a mean-adjusted version of the signed root statistic. Conditions for conditional probability matching in ancillary statistic models are derived and discussed.
Generalized likelihood ratio statistics have been proposed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193] as a generally applicable method for testing nonparametric hypotheses about nonparametric functions. The likelihood ratio statistics are constructed based on the assumption that the distributions of stochastic errors are in a certain parametric family. We extend their work to the case where the error distribution is completely unspecified via newly proposed sieve empirical likelihood ratio (SELR) tests. The approach is also applied to test conditional estimating equations on the…
We consider the problem of constructing confidence intervals for nonparametric functional data analysis using empirical likelihood. In this doubly infinite-dimensional context, we demonstrate the Wilks's phenomenon and propose a bias-corrected construction that requires neither undersmoothing nor direct bias estimation. We also extend our results to partially linear regression involving functional data. Our numerical results demonstrated the improved performance of empirical likelihood over approximation based on asymptotic normality.
University of Iowa（爱荷华大学）；Wuhan University（武汉大学）；Duke-NUS Medical School（杜克-新加坡国立大学医学院）
Feature selection is important for modeling high-dimensional data, where the number of variables can be much larger than the sample size. In this paper, we develop a support detection and root finding procedure to learn the high dimensional sparse generalized linear models and denote this method by GSDAR. Based on the KKT condition for $\ell_0$-penalized maximum likelihood estimations, GSDAR generates a sequence of estimators iteratively. Under some restricted invertibility conditions on the maximum likelihood function and sparsity assumption on the target coefficients, the errors of the propo…
We study the distributions of the LASSO, SCAD, and thresholding estimators, in finite samples and in the large-sample limit. The asymptotic distributions are derived for both the case where the estimators are tuned to perform consistent model selection and for the case where the estimators are tuned to perform conservative model selection. Our findings complement those of Knight and Fu (2000) and Fan and Li (2001). We show that the distributions are typically highly nonnormal regardless of how the estimator is tuned, and that this property persists in large samples. The uniform convergence rat…
In this paper, we propose methods for the estimation of parameters for the three-parameter Reflected Weibull distribution. The Moment estimator , Maximum likelihood estimator and Location and Scale Parameters free maximum likelihood estimator. The Location and Scale Parameters free maximum likelihood estimator is based on a data transformation, which avoids the problem of unbounded likelihood estimator. Through Mont Carlo simulations, we further show that the Location and Scale Parameters free maximum likelihood estimator performs better than methods moment and maximum likelihood estimator in …
We have observed an interesting, yet unexplained, phenomenon: Semidefinite programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend to be tight in recovery problems with noisy data, even when MLE cannot exactly recover the ground truth. Several results establish tightness of SDP based relaxations in the regime where exact recovery from MLE is possible. However, to the best of our knowledge, their tightness is not understood beyond this regime. As an illustrative example, we focus on the generalized Procrustes problem.
We provide evidence that a root-mean-square test of goodness-of-fit can be significantly more powerful than state-of-the-art exact tests in detecting deviations from Hardy-Weinberg equilibrium. Unlike Pearson's chi-square test, the log--likelihood-ratio test, and Fisher's exact test, which are sensitive to relative discrepancies between genotypic frequencies, the root-mean-square test is sensitive to absolute discrepancies. This can increase statistical power, as we demonstrate using benchmark datasets and through asymptotic analysis. With the aid of computers, exact P-values for the root-mean…
A linear Gaussian state-space smoothing algorithm is presented for estimation of derivatives from a sequence of noisy measurements. The algorithm uses numerically stable square-root formulas, can handle simultaneous independent measurements and non-equally spaced abscissas, and can compute state estimates at points between the data abscissas. The state space model's parameters, including driving noise intensity, measurement variance, and initial state, are determined from the given data sequence using maximum likelihood estimation computed using a expectation maximisation iteration. In tests w…
Full likelihood-based inference for high-dimensional multivariate extreme value distributions, or max-stable processes, is feasible when incorporating occurrence times of the maxima; without this information, $d$-dimensional likelihood inference is usually precluded due to the large number of terms in the likelihood. However, some studies have noted bias when performing high-dimensional inference that incorporates such event information, particularly when dependence is weak. We elucidate this phenomenon, showing that for unbiased inference in moderate dimensions, dimension $d$ should be of a m…
We propose a new testing procedure about the tail weight parameter of multivariate Student $t$ distributions by having recourse to the Le Cam methodology. Our test is asymptotically as efficient as the classical likelihood ratio test, but outperforms the latter by its flexibility and simplicity: indeed, our approach allows to estimate the location and scatter nuisance parameters by any root-$n$ consistent estimators, hereby avoiding numerically complex maximum likelihood estimation. The finite-sample properties of our test are analyzed in a Monte Carlo simulation study, and we apply our method…
This paper extends some prominent statistical results including \emph{Fisher Theorem and Wilks phenomenon} to the penalized maximum likelihood estimation with a quadratic penalization. It appears that sharp expansions for the penalized MLE \(\tilde{\thetav}_{G} \) and for the penalized maximum likelihood can be obtained without involving any asymptotic arguments, the results only rely on smoothness and regularity properties of the of the considered log-likelihood function. The error of estimation is specified in terms of the effective dimension \(p_G \) of the parameter set which can be much s…
It is well known that the empirical likelihood ratio confidence region suffers finite sample under-coverage issue, and this severely hampers its application in statistical inferences.} The root cause of this under-coverage is an upper limit imposed by the convex hull of the estimating functions that is used in the construction of the profile empirical likelihood. For i.i.d data, various methods have been proposed to solve this issue by modifying the convex hull, but it is not clear how well these methods perform when the data are no longer independent. In this paper, we propose an adjusted blo…
We study tail risk dynamics in high-frequency financial markets and their connection with trading activity and market uncertainty. We introduce a dynamic extreme value regression model accommodating both stationary and local unit-root predictors to appropriately capture the time-varying behaviour of the distribution of high-frequency extreme losses. To characterize trading activity and market uncertainty, we consider several volatility and liquidity predictors, and propose a two-step adaptive $L_1$-regularized maximum likelihood estimator to select the most appropriate ones. We establish the o…
We propose a new methodology for denoising, variance-stabilizing and normalizing signals whose both mean and variance are parameterized by a single unknown varying parameter, such as Poisson or scaled chi-squared. Key to our methodology is the observation that the signed and square-rooted generalized log-likelihood ratio test for the equality of the local means is approximately and asymptotically distributed as standard normal under the null. We use these test statistics within the Haar wavelet transform at each scale and location, referring to them as the likelihood ratio Haar (LRH) coefficie…
Penalized likelihood inference for beta-binomial meta-analysis of proportions of rare events
The Graduate Institute for Advanced Studies, The Graduate University for Advanced Studies（高级研究研究生院，高级研究院）；Department of Interdisciplinary Statistical Mathematics, The Institute of Statistical Mathematics（统计数学系，统计数学研究所）
Nonparametric generalized likelihood ratio test is popularly used for model checking for regressions. However, there are two issues that may be the barriers for its powerfulness. First, the bias term in its liming null distribution causes the test not to well control type I error and thus Monte Carlo approximation for critical value determination is required. Second, it severely suffers from the curse of dimensionality due to the use of multivariate nonparametric function estimation. The purpose of this paper is thus two-fold: a bias-correction is suggested to this test and a dimension reducti…
Accurate inference on a scalar interest parameter in the presence of a nuisance parameter may be obtained using an adjusted version of the signed root likelihood ratio statistic, in particular Barndorff-Nielsen's $R^*$ statistic. The adjustment made by this statistic may be decomposed into a sum of two terms, interpreted as correcting respectively for the possible effect of nuisance parameters and the deviation from standard normality of the signed root likelihood ratio statistic itself. We show that the adjustment terms are determined to second-order in the sample size by their means. Explici…
Does the asymptotic variance of the maximum composite likelihood estimator of a parameter of interest always decrease when the nuisance parameters are known? Will a composite likelihood necessarily become more efficient by incorporating addi- tional independent component likelihoods, or by using component likelihoods with higher dimension? In this note we show through illustrative examples that the an- swer to both questions is no, and indeed the opposite direction might be observed. The role of information bias is highlighted to understand the occurrence of these paradoxical phenomenon.
The discrepant posterior phenomenon (DPP) is a counter-intuitive phenomenon that can frequently occur in a Bayesian analysis of multivariate parameters. It refers to the phenomenon that a parameter estimate based on a posterior is more extreme than both of those inferred based on either the prior or the likelihood alone. Inferential claims that exhibit DPP defy the common intuition that the posterior is a prior-data compromise, and the phenomenon can be surprisingly ubiquitous in well-behaved Bayesian models. In this paper we revisit this phenomenon and, using point estimation as an example, d…
Parameters defined via General Estimating Equations (GEE) can be estimated by maximizing the Empirical Likelihood (EL). Newey and Smith (2004) have recently shown that this EL estimator exhibits desirable higher-order asymptotic properties, namely, that its O(n^-1) bias is small and that bias-corrected EL is higher-order efficient. Although EL possesses these properties when the model is correctly specified, this paper shows that, in the presence of model misspecification, EL may cease to be root n convergent when the functions defining the moment conditions are unbounded (even when their expe…
Parameters defined via general estimating equations (GEE) can be estimated by maximizing the empirical likelihood (EL). Newey and Smith [Econometrica 72 (2004) 219--255] have recently shown that this EL estimator exhibits desirable higher-order asymptotic properties, namely, that its $O(n^{-1})$ bias is small and that bias-corrected EL is higher-order efficient. Although EL possesses these properties when the model is correctly specified, this paper shows that, in the presence of model misspecification, EL may cease to be root n convergent when the functions defining the moment conditions are …
We consider a joint asymptotic framework for studying semi-nonparametric regression models where (finite-dimensional) Euclidean parameters and (infinite-dimensional) functional parameters are both of interest. The class of models in consideration share a partially linear structure and are estimated in two general contexts: (i) quasi-likelihood and (ii) true likelihood. We first show that the Euclidean estimator and (pointwise) functional estimator, which are re-scaled at different rates, jointly converge to a zero-mean Gaussian vector. This weak convergence result reveals a surprising joint as…
Gaussian quasi-likelihood estimation of the parameter $θ$ in the square-root diffusion process is studied under high frequency sampling. Different from the previous study of Overbeck and Rydén(1998) under low-frequency sampling, high-frequency of data provides very simple form of the asymptotic covariance matrix. Through easy-to-compute preliminary contrast functions, a practical two-stage manner without numerical optimization is formulated in order to conduct not only an asymptotically efficient estimation of the drift parameters, but also high-precision estimator of the diffusion parameter. …
We investigate the estimation of parameters in the random coefficient autoregressive model. We consider a nonstationary RCA process and show that the innovation variance parameter cannot be estimated by the quasi-maximum likelihood method. The asymptotic normality of the quasi-maximum likelihood estimator for the remaining model parameters is proven so the unit root problem does not exist in the random coefficient autoregressive model.
