On sufficient conditions for the strong consistency of least-squares estimates

The strong consistency of the least-squares estimates in regression models is obtained when the errors are i.i.d. with absolute moment of order r, 0<r? 2. The assumptions presented for the random error sequence will permit us to obtain improvements of the conditions on the regressors in order to obtain the strong consistency of the least-squares estimates in linear and nonlinear regression models.     

Strong consistency of least squares estimates with i.i.d. errors with mean values not necessarily defined

We establish strong consistency of the least squares estimates in multiple regression models discarding the usual assumption of the errors having null mean value. Thus, we required them to be i.i.d. with absolute moment of order r, 0<r<2, and null mean value when r>1. Only moderately restrictive conditions are imposed on the model matrix. In our treatment, we use an extension of the Marcinkiewicz–Zygmund strong law to overcome the errors mean value not being defined. In this way, we get a unified treatment for the case of i.i.d. errors extending the results of some previous papers.     

Minimum-Distance Estimator for Stable Exponent

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.     

Recursive estimation of the mode of a multivariate density

Let f be an unknown possibly multimodal density on R^d and let X₁, X₂, … be a sequence of independent random vectors with density f. Several recursive estimates of the mode of f are proposed, and sufficient conditions ensuring their weak and strong consistency are established.     

Skewness by Splitting the Scale Parameter

This article describes a recursive nonparametric estimation for the local partial first derivative of an arbitrary function satisfied some regularity conditions and establishes its consistency and asymptotic normality under the assumption of strong mixing sequence. The proposed estimator is a variable window width version of the Watson-Nadaraya type of derivative estimator. The window width varied as more data points become available enables a recursive algorithm that reduce computational complexity from order N ³ normally required by batch methods for kernel regression to order N ². This approach is computationally simple and attractive from practical viewpoint especially when the situation call for frequent updating of first derivative estimates. For example, maintaining a delta-hedged position of a portfolio of equities with index options is one of many applications of such estimation.     

Non-parametric regression for spatially dependent data with wavelets

We study non-parametric regression estimates for random fields. The data satisfies certain strong mixing conditions and is defined on the regular N-dimensional lattice structure. We show consistency and obtain rates of convergence. The rates are optimal modulo a logarithmic factor in some cases. As an application, we estimate the regression function with multidimensional wavelets which are not necessarily isotropic. We simulate random fields on planar graphs with the concept of concliques (cf. [Kaiser MS, Lahiri SN, Nordman DJ. Goodness of fit tests for a class of markov random field models. Ann Statist. 2012;40:104–130]) in numerical examples of the estimation procedure.     

Complete convergence for weighted sums of i.i.d. random variables with applications in regression estimation and EV model

In this paper, we obtain complete convergence results for Stout type weighted sums of i.i.d. random variables. A strong law for weighted sums of i.i.d. random variables is also obtained. As the applications of the strong law, the strong consistency and rate of the nonparametric regression estimations and the rates of the strong consistency of LS estimators for the unknown parameters of the simple linear errors in variables (EV) model are given.     

Estimation of a multiple-threshold AR(p) model

The present paper deals with the multiple-threshold p-order autoregressive model which has been introduced by Tong and Lim [H. Tong, K.S. Lim, Threshold autoregression, limit cycles and cyclical data, J. R. Stat. Soc. Ser. B 42 (1980) 245–292] in nonlinear system modelling. Under some conditions on the coefficients of the model which ensure the stationarity, the existence of moments and the strong mixing property of this process and under other mild assumptions, we establish the asymptotic properties (consistency and asymptotic normality) of the minimum Hellinger distance estimates of the autoregressive coefficients of the model.     

Moment Consistency of the Exchangeably Weighted Bootstrap for Semiparametric M‐estimation

The bootstrap variance estimate is widely used in semiparametric inferences. However, its theoretical validity is a well‐known open problem. In this paper, we provide a first theoretical study on the bootstrap moment estimates in semiparametric models. Specifically, we establish the bootstrap moment consistency of the Euclidean parameter, which immediately implies the consistency of t‐type bootstrap confidence set. It is worth pointing out that the only additional cost to achieve the bootstrap moment consistency in contrast with the distribution consistency is to simply strengthen the L₁ maximal inequality condition required in the latter to the L_p maximal inequality condition for p≥1. The general L_p multiplier inequality developed in this paper is also of independent interest. These general conclusions hold for the bootstrap methods with exchangeable bootstrap weights, for example, non‐parametric bootstrap and Bayesian bootstrap. Our general theory is illustrated in the celebrated Cox regression model.     

Asymptotic properties for LS estimators in EV regression model with dependent errors

In this paper, we establish the strong consistency and asymptotic normality for the least square (LS) estimators in simple linear errors-in-variables (EV) regression models when the errors form a stationary α-mixing sequence of random variables. The quadratic-mean consistency is also considered.     

Model selection consistency of U-statistics with convex loss and weighted lasso penalty

In the paper we consider minimisation of U-statistics with the weighted Lasso penalty and investigate their asymptotic properties in model selection and estimation. We prove that the use of appropriate weights in the penalty leads to the procedure that behaves like the oracle that knows the true model in advance, i.e. it is model selection consistent and estimates nonzero parameters with the standard rate. For the unweighted Lasso penalty, we obtain sufficient and necessary conditions for model selection consistency of estimators. The obtained results strongly based on the convexity of the loss function that is the main assumption of the paper. Our theorems can be applied to the ranking problem as well as generalised regression models. Thus, using U-statistics we can study more complex models (better describing real problems) than usually investigated linear or generalised linear models.     

Least squares estimation of the parameters of a stochastic difference equation with polynomial regression component

In this paper, strong consistency of least squares estimates of the coefficients of stochastic difference equations with polynomial regression components, has been established, using martingale arguments, under autoregressive, partially explosive and purely explosive situations. The asymptotic normality of these estimates have also been discussed in this paper.     

Weak and strong uniform consistency rates of kernel density estimates for randomly censored data

Let X₁,., X_n, be i.i.d. random variables with distribution function F, and let Y₁,.,.,Y_n be i.i.d. with distribution function G. For i = 1, 2,.,., n set δ_i, = 1 if X_i ≤ Y_i, and 0 otherwise, and X_i, = min{X_i, K_i}. A kernel-type density estimate of f, the density function of F w.r.t. Lebesgue measure on the Borel o-field, based on the censored data (δ_i, X_i), i = 1,.,.,n, is considered. Weak and strong uniform consistency properties over the whole real line are studied. Rates of convergence results are established under higher-order differentiability assumption on f. A procedure for relaxing such assumptions is also proposed.     

Kernel type smoothed quantile estimation under long memory

This paper studies nonparametric kernel type (smoothed) estimation of quantiles for long memory stationary sequences. The uniform strong consistency and asymptotic normality of the estimates with rates are established. Finite sample behaviors are investigated in a small Monte Carlo simulation study.     

On drift parameter estimation in models with fractional Brownian motion

We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein–Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behaviour of the fractional derivative of the fractional Brownian motion is established.     

Estimation in multiple linear regression Berkson model for processes with uncorrelated increments

This paper presents a method of estimating the regression and variance parameters in the multiple linear regression Berkson model for a continuous-time stochastic process with uncorrelated increments. Under minimal conditions, we establish (i) the Gauss–Markov theorem and the quadratic mean—as well as the strong consistency of the proposed estimate of the regression parameter and (ii) the weak consistency of the proposed estimate of the variance parameter.     

Complete and complete moment convergence with applications to the EV regression models

In this paper, the complete convergence and complete moment convergence for arrays of rowwise m-extended negatively dependent (m-END, for short) random variables are established. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for m-END random variables is also achieved. By using the results that we established, we further investigate the strong consistency of the least square estimator in the simple linear errors-in-variables models, and provide some simulations to verify the validity of our theoretical results.     

Estimation and Asymptotic Properties in Periodic GARCH(1, 1) Models

This article studies the probabilistic structure and asymptotic inference of the first-order periodic generalized autoregressive conditional heteroscedasticity (PGARCH(1, 1)) models in which the parameters in volatility process are allowed to switch between different regimes. First, we establish necessary and sufficient conditions for a PGARCH(1, 1) process to have a unique stationary solution (in periodic sense) and for the existence of moments of any order. Second, using the representation of squared PGARCH(1, 1) model as a PARMA(1, 1) model, we then consider Yule-Walker type estimators for the parameters in PGARCH(1, 1) model and derives their consistency and asymptotic normality. The estimator can be surprisingly efficient for quite small numbers of autocorrelations and, in some cases can be more efficient than the least squares estimate (LSE). We use a residual bootstrap to define bootstrap estimators for the Yule-Walker estimates and prove the consistency of this bootstrap method. A set of numerical experiments illustrates the practical relevance of our theoretical results.     

On the Strong Consistency of M‐Estimates in Linear Models for Negatively Superadditive Dependent Errors

In this paper we discuss the strong consistency of M‐estimates of the regression parameters in a linear model with negatively superadditive dependent (NSD) random errors. The result improves the moment condition and generalises the case of independent random errors to that of NSD random errors.     

Uniform strong consistency of histogram density estimation for dependent process

Histogram density estimator is very intuitive and easy to compute and has been widely adopted. Especially in today's big data environment, people pay more attention to the computational cost and are more willing to choose estimators with less to compute. And so, many scholars have been interested in the various estimates based on the histogram technique. Under strong mixing process, this article studies the uniform strong consistency of histogram density estimator and the convergence rate. Our conditions on the mixing coefficient and the bin width are very mild.