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The purpose of this paper is two fold. First, we investigate estimation for varying coefficient partially linear models in which covariates in the nonparametric part are measured with errors. As there would be some spurious covariates in the linear part, a penalized profile least squares estimation is suggested with the assistance from smoothly clipped absolute deviation penalty. However, the estimator is often biased due to the existence of measurement errors, a bias correction is proposed such that the estimation consistency with the oracle property is proved. Second, based on the estimator, a test statistic is constructed to check a linear hypothesis of the parameters and its asymptotic properties are studied. We prove that the existence of measurement errors causes intractability of the limiting null distribution that requires a Monte Carlo approximation and the absence of the errors can lead to a chi-square limit. Furthermore, confidence regions of the parameter of interest can also be constructed. Simulation studies and a real data example are conducted to examine the performance of our estimators and test statistic.

In this paper, we focus on the partially linear varying-coefficient quantile regression model when the data are right censored and the censoring indicator is missing at random. Based on the calibration and imputation methods, a three-stage approach is proposed to construct the estimators of the linear part and the nonparametric varying-coefficient function for this model . At the same time, we discuss the variable selection of the covariates in the linear part by adopting adaptive LASSO penalty. Under appropriate assumptions, the asymptotic normality of the proposed estimators is established, and the penalized estimators are proven to have the oracle property. Simulation study and a real data analysis are conducted to evaluate the performance of the proposed estimators. [ABSTRACT FROM AUTHOR]

This paper presents a general result that allows for establishing a link between the Kolmogorov–Marcinkiewicz– Zygmund strong law of large numbers and Feller’s strong law of large numbers in a Banach space setting. Let { X , X n ; n ≥ 1 } be a sequence of independent and identically distributed Banach space valued random variables and set S n = ∑ i = 1 n X i , n ≥ 1 . Let { a n ; n ≥ 1 } and { b n ; n ≥ 1 } be increasing sequences of positive real numbers such that lim n → ∞ a n = ∞ and b n ∕ a n ; n ≥ 1 is a nondecreasing sequence. We show that S n − n E X I { ‖ X ‖ ≤ b n } b n → 0 almost surely for every Banach space valued random variable X with ∑ n = 1 ∞ P ( ‖ X ‖ > b n ) < ∞ if S n ∕ a n → 0 almost surely for every symmetric Banach space valued random variable X with ∑ n = 1 ∞ P ( ‖ X ‖ > a n ) < ∞ . To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables. [ABSTRACT FROM AUTHOR]

In this note, general symmetrization procedures for both the weak and strong laws of large numbers concerning the row sums from arrays of rowwise independent Banach space valued random variables are established. Necessary and sufficient conditions are provided for the weak and strong laws of large numbers to hold in terms of corresponding laws of large numbers for a symmetrized version of the array. Several corollaries of the main result are provided. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper by Chen (in press) the limit law of the iterated logarithm for the partial sums of i.i.d. real-valued random variables has been established. In this note we look at the corresponding problem in Banach space setting. Let be a real separable Banach space with topological dual . Let be a sequence of i.i.d. -valued random variables and set and . We show that provided that where and is the unit ball of . [Copyright &y& Elsevier]

Abstract: Let be a field of negatively associated random variables. Set , . Under some suitable conditions, we show that is a necessary and sufficient criteria for the almost sure central limit theorem, i.e.  where is the standard normal distribution function,  and  , , . In particular, we obtain here the almost sure central limit theorem for negatively associated fields that assures the usual central limit theorem. [Copyright &y& Elsevier]

Let { X n, n≥1} be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f( x), the recursive kernel estimate of f( x) is defined by where h n is a sequence of positive bandwidths tending to 0, as n→∞, K(·) is a univariate kernel function. In this note, we discuss the point asymptotic normality for f n( x).

Suppose the observations ( X i, Y i) taking values in R d× R, are f-mixing. Compared with the i.i.d. case, some known strong uniform convergence results for the estimators of the regression function r( x)= E( Y i| X i= x) need strong moment conditions under f-mixing setting. We consider the following improved kernel estimators of r( x) suggested by Cheng (1983): Qian and Mammitzsch (2000) investigated the strong uniform convergence and convergence rate for to r( x) under weaker moment conditions than those of the others in the literature, and the optimal convergence rate can be attained under almost the same conditions as stated in Theorem 3.3.2 of Györfi et al. (1989). In this paper, under the similar conditions of Qian and Mammitzsch (2000), we study the strong uniform convergence and convergence rates for ( j=2,3) to r( x), which have not been discussed by Qian and Mammitzsch (2000). In contrast to , our estimators and are recursive, which is highly desirable for practical computation.

Abstract: Under very general weight function, we discuss the convergence of Jamison-type weighted sums of pairwise negatively quadrant dependent (NQD) r.v.'s. The results on i.i.d. setting of [3] and [1] are extended and generalized. As corollaries, we obtain some results of [11].

In this paper we are concerned with the heteroscedastic regression model yi=xiβ+g(ti)+σiei, 1≤i≤n under correlated errors ei , where it is assumed that σi2=f(ui) , the design points (xi,ti,ui) are known and nonrandom, and g and f are unknown functions. The interest lies in the slope parameter β . Assuming the unobserved disturbance ei are negatively associated, we study the issue of strong consistency for two different slope estimators: the least squares estimator and the weighted least squares estimator.