Showing papers on "Asymptotic distribution published in 2023"

A New Class of Generalized Probability-Weighted Moment Estimators for the Pareto Distribution

Abstract: Estimation based on probability-weighted moments is a well-established method and an excellent alternative to the classic method of moments or the maximum likelihood method, especially for small sample sizes. In this research, we developed a new class of estimators for the parameters of the Pareto type I distribution. A generalization of the probability-weighted moments approach is the foundation for this new class of estimators. It has the advantage of being valid in the entire parameter space of the Pareto distribution. We established the asymptotic normality of the new estimators and applied them to simulated and real datasets in order to illustrate their finite sample behavior. The results of comparisons with the most used estimation methods were also analyzed.

Two-stage conditional density estimation based on Bernstein polynomials

Abstract: In this thesis, we propose a new nonparametric approach based on Bernstein polynomials to estimate the conditional density function. The proposed estimators have desired properties at the boundaries, and can outperform the kernel and local linear estimators in terms of Integrated Mean Square Error for an appropriate choice of the polynomials’ order. The idea is constructing a two-stage conditional probability density function estimator based on Bernstein polynomials. Specifically, the Nadaraya-Watson (NW) and local linear (LL) conditional distribution function estimators were smoothed using Bernstein polynomials in the first stage. Secondly, the proposed estimators are obtained by differentiating the smoothed Bernstein NW and LL estimators. Further, the asymptotic properties of these estimators are established such as asymptotic bias, variance and normality under mild regularity conditions. Finally, a simulation study is carried out to assess the relative advantage of our estimators compared to other estimators. Also, the well known Old Faithful Geyser data were analyzed using the proposed estimators.

Epidemic change-point detection in general integer-valued time series

Abstract: In this paper, we consider the structural change in a class of discrete valued time series, where the true conditional distribution of the observations is assumed to be unknown. The conditional mean of the process depends on a parameter θ∗ which may change over time. We provide sufficient conditions for the consistency and the asymptotic normality of the Poisson quasi-maximum likelihood estimator (QMLE) of the model. We consider an epidemic change-point detection and propose a test statistic based on the QMLE of the parameter. Under the null hypothesis of a constant parameter (no change), the test statistic converges to a distribution obtained from increments of a Browninan bridge. The test statistic diverges to infinity under the epidemic alternative, which establishes that the proposed procedure is consistent in power. The effectiveness of the proposed procedure is illustrated by simulated and real data examples.

Statistical inference on stationary shot noise random fields

Abstract: We study the asymptotic behaviour of a stationnary shot noise random field. We use the notion of association to prove the asymptotic normality of the moments and a multidimensional version for the correlation functions. The variance of the moment estimates is detailed as well as their correlation. When the field is isotropic, the estimators are improved by reducing the variance. These results will be applied to the estimation of the model parameters in the case of a Gaussian kernel, with a focus on the correlation parameter. The asymptotic normality is proved and a simulation study is carried out.

A robust spline approach in partially linear additive models

Abstract: Partially linear additive models generalize linear regression models by assuming that the relationship between the response and a set of explanatory variables is linear on some of the covariates, while the other ones enter into the model through unknown univariate smooth functions. The harmful effect of outliers either in the residuals or in the covariates involved in the linear component has been described in the situation of partially linear models, that is, when only one nonparametric component is involved. When dealing with additive components, the problem of providing reliable estimators when atypical data arise is of practical importance motivating the need of robust procedures. Based on this fact, a family of robust estimators for partially linear additive models that combines B-splines with robust linear MM-regression estimators is proposed. Under mild assumptions, consistency results and rates of convergence for the proposed estimators are derived. Furthermore, the asymptotic normality for the linear regression estimators is obtained. A Monte Carlo study is carried out to compare, under different models and contamination schemes, the performance of the robust MM-proposal based on B-splines with its classical counterpart and also with a quantile approach. The obtained results show the benefits of using the robust MM-approach. The analysis of a real data set illustrates the usefulness of the proposed method.

Local asymptotic normality for ergodic jump-diffusion processes via transition density approximation

Abstract: We show local asymptotic normality (LAN) for a statistical model of discretely observed ergodic jump-diffusion processes, where the drift coefficient, diffusion coefficient, and jump structure are parametrized. Under the LAN property, we can discuss the asymptotic efficiency of regular estimators, and the quasi-maximum-likelihood and Bayes-type estimators proposed in Shimizu and Yoshida (Stat. Inference Stoch. Process. 9 (2006) 227–277) and Ogihara and Yoshida (Stat. Inference Stoch. Process. 14 (2011) 189–229) are shown to be asymptotically efficient in this model. Moreover, we can construct asymptotically uniformly most powerful tests for the parameters. Unlike with a model for diffusion processes, Aronson-type estimates of the transition density functions do not hold, which makes it difficult to prove LAN. Therefore, instead of Aronson-type estimates, we employ the idea of Theorem 1 in Jeganathan (Sankhyā Ser. A 44 (1982) 173–212) and use the L2 regularity condition. Moreover, we show that local asymptotic mixed normality of a statistical model is implied from that for a model generated by approximated transition density functions under suitable conditions. Together with density approximation by means of thresholding techniques, the LAN property for the jump-diffusion processes is proved.

Fitting additive risk models using auxiliary information

Abstract: There has been a growing interest in incorporating auxiliary summary information from external studies into the analysis of internal individual‐level data. In this paper, we propose an adaptive estimation procedure for an additive risk model to integrate auxiliary subgroup survival information via a penalized method of moments technique. Our approach can accommodate information from heterogeneous data. Parameters to quantify the magnitude of potential incomparability between internal data and external auxiliary information are introduced in our framework while nonzero components of these parameters suggest a violation of the homogeneity assumption. We further develop an efficient computational algorithm to solve the numerical optimization problem by profiling out the nuisance parameters. In an asymptotic sense, our method can be as efficient as if all the incomparable auxiliary information is accurately acknowledged and has been automatically excluded from consideration. The asymptotic normality of the proposed estimator of the regression coefficients is established, with an explicit formula for the asymptotic variance‐covariance matrix that can be consistently estimated from the data. Simulation studies show that the proposed method yields a substantial gain in statistical efficiency over the conventional method using the internal data only, and reduces estimation biases when the given auxiliary survival information is incomparable. We illustrate the proposed method with a lung cancer survival study.

A Time-Varying Coefficient Double Threshold GARCH Model with Explanatory Variables

Abstract: In this article, we consider the nonparametric inference for the time-varying coefficient double-threshold generalized autoregressive conditional heteroscedastic models. The quasi-maximum exponential likelihood estimators (QMELEs) of the model’s parameters and the asymptotic properties of the estimators are obtained. The simulation study implies that the distribution of the estimators is asymptotically normal. A real data application to stock returns is given. Both the simulations and real data example imply that the model and the QMELE are proper, compatible and accurately fit the financial time series data of the Nikkei 225.

On Consistency and Asymptotic Normality of Least Absolute Deviation Estimators for 2-dimensional Sinusoidal Model

Abstract: Estimation of the parameters of a 2-dimensional sinusoidal model is a fundamental problem in digital signal processing and time series analysis. In this paper, we propose a robust least absolute deviation (LAD) estimators for parameter estimation. The proposed methodology provides a robust alternative to non-robust estimation techniques like the least squares estimators, in situations where outliers are present in the data or in the presence of heavy tailed noise. We study important asymptotic properties of the LAD estimators and establish the strong consistency and asymptotic normality of the LAD estimators of the signal parameters of a 2-dimensional sinusoidal model. We further illustrate the advantage of using LAD estimators over least squares estimators through extensive simulation studies. Data analysis of a 2-dimensional texture data indicates practical applicability of the proposed LAD approach.

Optimal subsampling for the Cox proportional hazards model with massive survival data

Abstract: The use of massive survival data has become common in survival analysis. In this study, a subsampling algorithm is proposed for the Cox proportional hazards model with time-dependent covariates when the sample is extraordinarily large but computing resources are relatively limited. A subsample estimator is developed by maximizing the weighted partial likelihood; it is shown to have consistency and asymptotic normality. By minimizing the asymptotic mean squared error of the subsample estimator, the optimal subsampling probabilities are formulated with explicit expressions. Simulation studies show that the proposed method can satisfactorily approximate the estimator of the full dataset. The proposed method is then applied to corporate loan and breast cancer datasets, with different censoring rates, and the outcomes confirm its practical advantages.

An Analog of the Bickel–Rosenblatt Test for Error Density in the Linear Regression Model

Abstract: This paper addresses the problem of testing the goodness-of-fit hypothesis pertaining to error density in multiple linear regression models with non-random and random predictors. The proposed tests are based on the integrated square difference between a nonparametric density estimator based on the residuals and its expected value under the null hypothesis when all regression parameters are known. We derive the asymptotic distributions of this sequence of test statistics under the null hypothesis and under certain global alternatives. The asymptotic null distribution of a suitably standardized test statistic based on the residuals is the same as in the case of known underlying regression parameters. Under the global $$L_2$$ alternatives of [2], the asymptotic distribution of this sequence of statistics is affected by not knowing the parameters and, in general, is different from the one obtained in [2] for the zero intercept linear autoregressive time series context.

Limit Distributions for the Estimates of the Digamma Distribution Parameters Constructed from a Random Size Sample

Abstract: In this paper, we study a new type of distribution that generalizes distributions from the gamma and beta classes that are widely used in applications. The estimators for the parameters of the digamma distribution obtained by the method of logarithmic cumulants are considered. Based on the previously proved asymptotic normality of the estimators for the characteristic index and the shape and scale parameters of the digamma distribution constructed from a fixed-size sample, we obtain a statement about the convergence of these estimators to the scale mixtures of the normal law in the case of a random sample size. Using this result, asymptotic confidence intervals for the estimated parameters are constructed. A number of examples of the limit laws for sample sizes with special forms of negative binomial distributions are given. The results of this paper can be widely used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support.

On local likelihood asymptotics for Gaussian mixed-effects model with system noise

Abstract: The Gaussian mixed-effects model driven by a stationary integrated Ornstein-Uhlenbeck process has been used for analyzing longitudinal data having an explicit and simple serial-correlation structure in each individual. However, the theoretical aspect of its asymptotic inference is yet to be elucidated. We prove the local asymptotics for the associated log-likelihood function, which in particular guarantees the asymptotic optimality of the suitably chosen maximum-likelihood estimator. We illustrate the obtained asymptotic normality result through some simulations for both balanced and unbalanced datasets.

Optimal subsampling algorithm for composite quantile regression with distributed data

Abstract: For massive data stored at multiple machines, we propose a distributed subsampling procedure for the composite quantile regression. By establishing the consistency and asymptotic normality of the composite quantile regression estimator from a general subsampling algorithm, we derive the optimal subsampling probabilities and the optimal allocation sizes under the L-optimality criteria. A two-step algorithm to approximate the optimal subsampling procedure is developed. The proposed methods are illustrated through numerical experiments on simulated and real datasets.

Estimation and asymptotics for vector autoregressive models with unit roots and Markov switching trends

Abstract: We provide a formal definition of an M-state multivariate Markov switching (MS) trend, describe its asymptotic distribution, and consider vector autoregressive processes with MS trends which contain either unit roots or a stationary part. Then, we estimate the coefficients of such models via ordinary least squares (OLS), and determine the asymptotic distributions of OLS estimators in terms of functionals on a multivariate Brownian motion.

An Evaluation of Non-Iterative Estimators in Confirmatory Factor Analysis

Abstract: In confirmatory factor analysis (CFA), model parameters are usually estimated by iteratively minimizing the Maximum Likelihood (ML) fit function. In optimal circumstances, the ML estimator yields the desirable statistical properties of asymptotic unbiasedness, efficiency, normality, and consistency. In practice, however, real-life data tend to be far from optimal, making the algorithm prone to convergence failure, inadmissible solutions, and bias. In this study, we revisited some old, yet largely neglected non-iterative alternatives and compared their performance to more recently proposed procedures in an extensive simulation study. We conclude that closed-form expressions may serve as viable alternatives for ML, with the Multiple Group Method – the oldest method under consideration – showing favorable results across all settings.

On uniform consistency of Neyman’s type nonparametric tests

Abstract: The goodness-of-fit problem is explored, when the test statistic is a linear combination of squared Fourier coefficients’ estimates coming from the Fourier decomposition of a probability density. Common examples of such statistics include Neyman’s test statistics and test statistics, generated by L2-norms of kernel estimators. We prove the asymptotic normality of the test statistic for both the null and alternative hypothesis. Using these results we deduce conditions of uniform consistency for nonparametric sets of alternatives, which are defined in terms of distribution or density functions. Results on uniform consistency, related to the distribution functions, can be seen as a statement showing to what extent the distance method, based on a given test statistic, makes the hypothesis and alternatives distinguishable. In this case, the deduced conditions of uniform consistency are close to necessary. For sequences of alternatives - defined in terms of density functions - approaching the hypothesis in L2-metric, we find necessary and sufficient conditions for their consistency. This result is obtained in terms of the concept of maxisets, the description of which for given test statistics is found in this publication.

The asymptotic distribution of a truncated sample mean for the extremely heavy-tailed distributions

Abstract: This article deals with the asymptotic distribution for the extremely heavy-tailed distributions with infinite mean or variance by using a truncated sample mean. We obtain three necessary and sufficient conditions under which the asymptotic distribution of the truncated test statistics converges to normal, neither normal nor stable or converges to −∞ or the combination of stable distributions, respectively. The numerical simulation illustrates an application of the theoretical results above in the hypothesis testing.

Smoothed empirical likelihood estimation and automatic variable selection for an expectile high-dimensional model with possibly missing response variable

Abstract: We consider a linear model which can have a large number of explanatory variables, the errors with an asymmetric distribution or some values of the explained variable are missing at random. In order to take in account these several situations, we consider the non parametric empirical likelihood (EL) estimation method. Because a constraint in EL contains an indicator function then a smoothed function instead of the indicator will be considered. Two smoothed expectile maximum EL methods are proposed, one of which will automatically select the explanatory variables. For each of the methods we obtain the convergence rate of the estimators and their asymptotic normality. The smoothed expectile empirical log-likelihood ratio process follow asymptotically a chi-square distribution and moreover the adaptive LASSO smoothed expectile maximum EL estimator satisfies the sparsity property which guarantees the automatic selection of zero model coefficients. In order to implement these methods, we propose four algorithms.

Estimating Generalized Additive Conditional Quantiles for Absolutely Regular Processes

Abstract: We propose a nonparametric method for estimating the conditional quantile function that admits a generalized additive specification with an unknown link function. This model nests single-index, additive, and multiplicative quantile regression models. Based on a full local linear polynomial expansion, we first obtain the asymptotic representation for the proposed quantile estimator for each additive component. Then, the link function is estimated by noting that it corresponds to the conditional quantile function of a response variable given the sum of all additive components. The observations are supposed to be a sample from a strictly stationary and absolutely regular process. We provide results on (uniform) consistency rates, second order asymptotic expansions and point wise asymptotic normality of each proposed estimator.

Semiparametrically Optimal Cointegration Test

Abstract: This paper aims to address the issue of semiparametric efficiency for cointegration rank testing in finite-order vector autoregressive models, where the innovation distribution is considered an infinite-dimensional nuisance parameter. Our asymptotic analysis relies on Le Cam's theory of limit experiment, which in this context takes the form of Locally Asymptotically Brownian Functional (LABF). By leveraging the structural version of LABF, an Ornstein-Uhlenbeck experiment, we develop the asymptotic power envelopes of asymptotically invariant tests for both cases with and without a time trend. We propose feasible tests based on a nonparametrically estimated density and demonstrate that their power can achieve the semiparametric power envelopes, making them semiparametrically optimal. We validate the theoretical results through large-sample simulations and illustrate satisfactory size control and excellent power performance of our tests under small samples. In both cases with and without time trend, we show that a remarkable amount of additional power can be obtained from non-Gaussian distributions.

Inference in Conditional Vector Error Correction Models With a Small Signal-to-Noise Ratio*

Abstract: It is widely documented that while contemporaneous spot and forward financial prices trace each other extremely closely, their difference is often highly persistent and the conventional cointegration tests may suggest lack of cointegration. This chapter studies the possibility of having cointegrated errors that are characterized simultaneously by high persistence (near-unit root behavior) and very small (near zero) variance. The proposed dual parameterization induces the cointegration error process to be stochastically bounded which prevents the variables in the cointegrating system from drifting apart over a reasonably long horizon. More specifically, this chapter develops the appropriate asymptotic theory (rate of convergence and asymptotic distribution) for the estimators in unconditional and conditional vector error correction models (VECM) when the error correction term is parameterized as a dampened near-unit root process (local-to-unity process with local-to-zero variance). The important differences in the limiting behavior of the estimators and their implications for empirical analysis are discussed. Simulation results and an empirical analysis of the forward premium regressions are also provided.

Parameter estimation in mixed fractional stochastic heat equation

Abstract: The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H<3/4$. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.

Semiparametric regression analysis of length‐biased and partly interval‐censored data with application to an AIDS cohort study

Abstract: Length‐biased data occur often in many scientific fields, including clinical trials, epidemiology surveys and genome‐wide association studies, and many methods have been proposed for their analysis under various situations. In this article, we consider the situation where one faces length‐biased and partly interval‐censored failure time data under the proportional hazards model, for which it does not seem to exist an established method. For the estimation, we propose an efficient nonparametric maximum likelihood method by incorporating the distribution information of the observed truncation times. For the implementation of the method, a flexible and stable EM algorithm via two‐stage data augmentation is developed. By employing the empirical process theory, we establish the asymptotic properties of the resulting estimators. A simulation study conducted to assess the finite‐sample performance of the proposed method suggests that it works well and is more efficient than the conditional likelihood approach. An application to an AIDS cohort study is also provided.