Uniform Asymptotic Normality Research Articles

On the consistency of the maximum likelihood estimator for the three parameter lognormal distribution

The three parameter log-normal distribution is a popular non-regular model, but surprisingly, whether the local maximum likelihood estimator (MLE) for parameter estimation is consistent or not has been speculated about since the 1960s. This note gives a rigorous proof for the existence of a consistent MLE for the three parameter log-normal distribution, which solves a problem that has been recognized and unsolved for 50 years. Our results also imply a uniform local asymptotic normality condition for the three parameter log-normal distribution. In addition, we give results on the asymptotic normality and the uniqueness of the local MLE.

Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions

We show that spline and wavelet series regression estimators for weakly dependent regressors attain the optimal uniform (i.e. sup-norm) convergence rate (n/logn)−p/(2p+d) of Stone (1982), where d is the number of regressors and p is the smoothness of the regression function. The optimal rate is achieved even for heavy-tailed martingale difference errors with finite (2+(d/p))th absolute moment for d/p<2. We also establish the asymptotic normality of t statistics for possibly nonlinear, irregular functionals of the conditional mean function under weak conditions. The results are proved by deriving a new exponential inequality for sums of weakly dependent random matrices, which is of independent interest.

Modeling and forecasting of stock index volatility with APARCH models under ordered restriction

This article examines volatility models for modeling and forecasting the Standard &amp; Poor 500 (S&amp;P 500) daily stock index returns, including the autoregressive moving average, the Taylor and Schwert generalized autoregressive conditional heteroscedasticity (GARCH), the Glosten, Jagannathan and Runkle GARCH and asymmetric power ARCH (APARCH) with the following conditional distributions: normal, Student's t and skewed Student's t‐distributions. In addition, we undertake unit root (augmented Dickey–Fuller and Phillip–Perron) tests, co‐integration test and error correction model. We study the stationary APARCH (p) model with parameters, and the uniform convergence, strong consistency and asymptotic normality are prove under simple ordered restriction. In fitting these models to S&amp;P 500 daily stock index return data over the period 1 January 2002 to 31 December 2012, we found that the APARCH model using a skewed Student's t‐distribution is the most effective and successful for modeling and forecasting the daily stock index returns series. The results of this study would be of great value to policy makers and investors in managing risk in stock markets trading.

Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems

We develop uniformly valid confidence regions for regression coefficients in a high-dimensional sparse median regression model with homoscedastic errors. Our methods are based on a moment equation that is immunized against nonregular estimation of the nuisance part of the median regression function by using Neyman’s orthogonalization. We establish that the resulting instrumental median regression estimator of a target regression coefficient is asymptotically normally distributed uniformly with respect to the underlying sparse model and is semiparametrically efficient. We also generalize our method to a general nonsmooth Z-estimation framework where the number of target parameters is possibly much larger than the sample size. We extend Huber's results on asymptotic normality to this setting, demonstrating uniform asymptotic normality of the proposed estimators over rectangles, constructing simultaneous confidence bands on all of the target parameters, and establishing asymptotic validity of the bands uniformly over underlying approximately sparse models.

A New Non-Parametric Stationarity Test of Time Series in the Time Domain

Summary We propose a new double-order selection test for checking second-order stationarity of a time series. To develop the test, a sequence of systematic samples is defined via Walsh functions. Then the deviations of the autocovariances based on these systematic samples from the corresponding autocovariances of the whole time series are calculated and the uniform asymptotic joint normality of these deviations over different systematic samples is obtained. With a double-order selection scheme, our test statistic is constructed by combining the deviations at different lags in the systematic samples. The null asymptotic distribution of the statistic proposed is derived and the consistency of the test is shown under fixed and local alternatives. Simulation studies demonstrate well-behaved finite sample properties of the method proposed. Comparisons with some existing tests in terms of power are given both analytically and empirically. In addition, the method proposed is applied to check the stationarity assumption of a chemical process viscosity readings data set.

Strong uniform consistency rates and asymptotic normality of conditional density estimator in the single functional index modeling for time series data

In this paper, we investigate a nonparametric estimation of the conditional density of a scalar response variable given a random variable taking values in separable Hilbert space. We establish under general conditions the uniform almost complete convergence rates and the asymptotic normality of the conditional density kernel estimator, when the variables satisfy the strong mixing dependency, based on the single-index structure. The asymptotic \((1-\zeta )\) confidence intervals of conditional density function are given, for \(0 < \zeta < 1\). We further demonstrate the impact of this functional parameter to the conditional mode estimate. Simulation study is also presented. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is discussed, but not tackled.

Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions

We show that spline and wavelet series regression estimators for weakly dependent regressors attain the optimal uniform (i.e., sup-norm) convergence rate (n/log n)^{-p/(2p d)} of Stone (1982), where d is the number of regressors and p is the smoothness of the regression function. The optimal rate is achieved even for heavy-tailed martingale difference errors with finite (2 (d/p))th absolute moment for d/p

Hermite Series Estimation in Nonlinear Cointegrating Models

This paper discusses nonparametric series estimation of integrable cointegration models using Hermite functions. We establish the uniform consistency and asymptotic normality of the series estimator. The Monte Carlo simulation results show that the performance of the estimator is numerically satisfactory. We then apply the estimator to estimate the stock return predictive function. The out-of-sample evaluation results suggest that dividend yield has nonlinear predictive power for stock returns while book-to-market ratio and earning-price ratio have little predictive power.

Nonparametric estimation of mean and covariance structures for longitudinal data

AbstractIn this article we propose a novel nonparametric regression method to model the mean and covariance structures for longitudinal data. A modification of local linear smoothing estimation techniques is used to estimate the parameters and unknown functions in the model. Theoretical properties including uniform consistency and asymptotic normality are studied under certain mild conditions. Simulation studies are carried out to evaluate the efficacy of the proposed method, and real data analysis is provided for illustration. The Canadian Journal of Statistics 41: 557–574; 2013 © 2013 Statistical Society of Canada

Estimation of the conditional tail index using a smoothed local Hill estimator

For heavy-tailed distributions, the so-called tail index is an important parameter that controls the behavior of the tail distribution and is thus of primary interest to estimate extreme quantiles. In this paper, the estimation of the tail index is considered in the presence of a finite-dimensional random covariate. Uniform weak consistency and asymptotic normality of the proposed estimator are established and some illustrations on simulations are provided.

Nonparametric dynamic panel data models: Kernel estimation and specification testing

Motivated by the first-differencing method for linear panel data models, we propose a class of iterative local polynomial estimators for nonparametric dynamic panel data models with or without exogenous regressors. The estimators utilize the additive structure of the first-differenced model—the fact that the two additive components have the same functional form, and the unknown function of interest is implicitly defined as a solution of a Fredholm integral equation of the second kind. We establish the uniform consistency and asymptotic normality of the estimators. We also propose a consistent test for the correct specification of linearity in typical dynamic panel data models based on the L2 distance of our nonparametric estimates and the parametric estimates under the linear restriction. We derive the asymptotic distributions of the test statistic under the null hypothesis and a sequence of Pitman local alternatives, and prove its consistency against global alternatives. Simulations suggest that the proposed estimators and tests perform well for finite samples. We apply our new method to study the relationships among economic growth, the initial economic condition and capital accumulation, and find a significant nonlinear relation between economic growth and the initial economic condition.

SOME ASYMPTOTIC RESULTS OF KERNEL DENSITY ESTIMATOR IN LENGTH-BIASED SAMPLING

In this paper, we prove the strong uniform consistency and asymptotic normality of the kernel density estimator proposed by Jones [12] for length-biased data.The approach is based on the invariance principle for the empirical processes proved by Horvath [10]. All simulations are drawn for different cases to demonstrate both, consistency and asymptotic normality and the method is illustrated by real automobile brake pads data.

Estimating asymptotic dependence functionals in multivariate regularly varying models

This paper deals with semiparametric estimation of the asymptotic portfolio risk factor γ ξ introduced in [G. Mainik and L. Rüschendorf, On optimal portfolio diversification with respect to extreme risks, Finance Stoch., 14:593–623, 2010] for multivariate regularly varying random vectors in $ \mathbb{R}_{+}^d $ . The functional γ ξ depends on the spectral measure Ψ, the tail index α, and the vector ξ of portfolio weights. The representation of γ ξ is extended to characterize the portfolio loss asymptotics for random vectors in ℝ d . The earlier results on uniform strong consistency and uniform asymptotic normality of the estimates of γ ξ are extended to the general setting, and the regularity assumptions are significantly weakened. Uniform consistency and asymptotic normality are also proved for the estimators of the functional $ \gamma_\xi^{{{1} \left/ {\alpha } \right.}} $ that characterizes the asymptotic behavior of the portfolio loss quantiles. The techniques developed here can also be applied to other dependence functionals.

Strong uniform consistency and asymptotic normality of a kernel based error density estimator in functional autoregressive models

Estimating the innovation probability density is an important issue in any regression analysis. This paper focuses on functional autoregressive models. A residual-based kernel estimator is proposed for the innovation density. Asymptotic properties of this estimator depend on the average prediction error of the functional autoregressive function. Sufficient conditions are studied to provide strong uniform consistency and asymptotic normality of the kernel density estimator.

Uniform asymptotic normality of the matrix-variate Beta-distribution

With the upper bound of Kullback-Leibler distance between a matrix variate Beta-distribution and a normal distribution, this paper gives the conditions under which a matrix-variate Beta-distribution will approach uniformly and asymptotically a normal distribution.

Lens data depth and median

We define the lens depth (LD) function LD(t; F) of a vector t∈R d with respect to a distribution function F to be the probability that t is contained in a random hyper-lens formed by the intersection of two closed balls centred at two i.i.d observations from F. We show that LD is a statistical depth function and explore its properties, including affine invariance, symmetry, maximality at the centre and monotonicity. We define the sample LD and investigate its uniform consistency, asymptotic normality and computational complexity in high-dimensional settings. We define the lens median (LM), a multivariate analogue of the univariate median, as the point where the LD is maximised. The sample LM is the vector that is covered by the most number of hyper-lenses formed between any two sample observations. We state its asymptotic consistency and normality and examine its breakdown point and relative efficiency. The sample LM is robust and efficient for estimating the centre of a unimodal distribution. A comparison of LD and LM to existing data depth functions and medians in terms of computational complexity, robustness, efficiency and breakdown point is presented.

Asymptotic Properties of Conditional Quantile Estimator Under Left-Truncated and α-Mixing Conditions

In this article, we establish strong uniform convergence and asymptotic normality of estimators of conditional quantile and conditional distribution function for a left truncated model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary α-mixing sequence. The results of Lemdani et al. (2009) are relaxed from the i.i.d. assumption to α-mixing setting. Finite sample behavior of the estimators is investigated via simulations as well.

Generalised kernel smoothing for non-negative stationary ergodic processes

In this paper, we consider a generalised kernel smoothing estimator of the regression function with non-negative support, using gamma probability densities as kernels, which are non-negative and have naturally varying shapes. It is based on a generalisation of Hille's lemma and a perturbation idea that allows us to deal with the problem at the boundary. Its uniform consistency and asymptotic normality are obtained at interior and boundary points, under a stationary ergodic process assumption, without using traditional mixing conditions. The asymptotic mean squared error of the estimator is derived and the optimal value of smoothing parameter is also discussed. Graphical illustrations of the proposed estimator are provided for simulated as well as for real data. A simulation study is also carried out to compare our method with the competing local linear method.

Uniform weak convergence of the time-dependent poverty measures for continuous longitudinal data

The poverty analysis may require the observation of the same set of households over the time in order to explain the evolution of the poverty situation and to try to explain their behavior. In this case, the poverty measures have to be determined continuously in some interval [0, T] and the sample poverty index becomes time-dependent. In this paper, we settle the global problem of the weak convergence of the time-dependent poverty measures in the functional space of continuous functions defined on [0, T]. We entirely describe the uniform asymptotic normality of the class of nonweighted poverty indices including the Foster–Greer–Thorbecke and Chakravarty ones, which both have the special property of satisfying all the needed axioms for a poverty index.

Bias reduction in kernel density estimation via Lipschitz condition

In this paper we propose a new nonparametric kernel-based estimator for a density function f which achieves bias reduction relative to the classical Rosenblatt–Parzen estimator. Contrary to some existing estimators that provide for bias reduction, our estimator has a full asymptotic characterisation including uniform consistency and asymptotic normality. In addition, we show that bias reduction can be achieved without the disadvantage of potential negativity of the estimated density – a deficiency that results from using higher order kernels. Our results are based on imposing global Lipschitz conditions on f and defining a novel corresponding kernel. A Monte Carlo study is provided to illustrate the estimator's finite sample performance.