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In the present paper, the sufficient and necessary conditions of the complete convergence and complete moment convergence for -mixing random variables are established, which extend some well-known results.

In this article, the complete convergence and the Kolmogorov strong law of large numbers for weighted sums of -mixing random variables are presented. An application to simple linear errors-in-variables model is provided. Simulation studies are also carried out to support the theoretical results.

In this paper, we estimate the difference | E h ( Z n ) − E h ( Y ) | between the expectations of real finite Lipschitz function h of the sum Z n = ( X 1 + ⋯ + X n ) /B n , where B n 2 = E ( X 1 + ⋯ + X n ) 2 > 0, and a standard normal random variable Y , where real centered random variables X 1 ,X 2 , … satisfy the φ -mixing condition, defined between the “past” and “ future”, or are m -dependent. In particular cases, under the condition ∑ r = 1 ∞ r φ ( r ) < ∞ or ∑ r = 1 ∞ r φ 1 / 2 ( r ) < ∞ , the obtained upper bounds for φ -mixing random variables are of order O ( n − 1 / 2 ). In addition, we refine the previously known upper bounds of order O (( m + 1) 1+ δ L 2+ δ,n ), where L 2+ δ,n is the Lyapunov fraction of order 2 + δ , for m -dependent random variables, supplementing them with explicit constants. We also separately present the case of independent r.v.s.

In this paper, we consider the partial linear regression model y i = x i β * + g ( t i ) + ε i , i = 1, 2, …, n , where ( x i , t i ) are known fixed design points, g (·) is an unknown function, and β* is an unknown parameter to be estimated, random errors ε i are ( α , β )-mixing random variables. The p -th ( p > 1) mean consistency, strong consistency and complete consistency for least squares estimators of β* and g (·) are investigated under some mild conditions. In addition, a numerical simulation is carried out to study the finite sample performance of the theoretical results. Finally, a real data analysis is provided to further verify the effect of the model.

The paper contains the comparative analysis of the efficiency of different qunatile estimators for various distributions. Additionally, we show strong consistency of different quantile estimators and we study the Bahadur representation for each of the quantile estimators, when the sample is taken from NA, φ, ρ∗, ρ-mixing population.

Let r≥1, 1≤p<2, and α,β>0 with 1/α+1/β=1/p. Let ank,1≤k≤n,n≥1 be an array of constants satisfying supn≥1n−1∑k=1n|ank|α<∞, and let Xn,n≥1 be a sequence of identically distributed ρ∗-mixing random variables. For each of the three cases αrp, we provide moment conditions under which ∑n=1∞nr−2Pmax1≤m≤n|∑k=1mankXk|>εn1/p<∞,∀ε>0. We also provide moment conditions under which ∑n=1∞nr−2−q/pE(max1≤m≤n|∑k=1mankXk|−εn1/p)+q<∞,∀ε>0, where q>0. Our results improve and generalize those of Sung (Discrete Dyn. Nat. Soc. 2010:630608, 2010) and Wu et al. (Stat. Probab. Lett. 127:55–66, 2017).

In this paper, we investigate the parametric component and nonparametric component estimators in a semiparametric regression model based on \\[ \\]-mixing random variables. The rth mean consistency, complete consistency, uniform rth mean consistency and uniform complete consistency are established under some suitable conditions. In addition, a simulation to study the numerical performance of the consistency of the nearest neighbor weight function estimators is provided. The results obtained in the paper improve the conditions in the literature and generalize the existing results of independent random errors to the case of \\[ \\]-mixing random errors.

In this paper, the Berry–Esseen type bounds of the weighted estimator in a nonparametric regression model are investigated under some mild conditions when random errors are from a linear process generated by φ-mixing random variables. In particular, the rate of uniform normal approximation is near to O(n-316) by the choice of some constants, which generalizes and improves the corresponding results of Li et al. (Stat Probab Lett 81:103–110, 2011) and Ding et al. (J Inequal Appl 2018:10, 2018). Finally, the simulation study is provided to verify the validity of the theoretical results.

This paper studies a heteroscedastic partially linear model based on ρ − -mixing random errors, stochastically dominated and with zero mean. Under some suitable conditions, the strong consistency and p -th ( p > 0 ) mean consistency of least squares (LS) estimators and weighted least squares (WLS) estimators for the unknown parameter are investigated, and the strong consistency and p -th ( p > 0 ) mean consistency of the estimators for the non-parametric component are also studied. These results include the corresponding ones of independent, negatively associated (NA), and ρ * -mixing random errors as special cases. At last, two simulations are presented to support the theoretical results.

For the semiparametric regression model: Y(j)(xin,tin)=tinβ+g(xin)+e(j)(xin),1≤j≤m,1≤i≤n, where tin∈R and xin∈Rp are known to be nonrandom, g is an unknown continuous function on a compact set A in Rp, ej(xin) are ρ∗-mixing random errors with mean zero, Y(j)(xin,tin) represent the j-th response variables which are observable at points xin,tin. In this paper, we study the strong consistency and r-th (1

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