Berry–Esseen type bounds in heteroscedastic errors-in-variables model

ABSTRACT

Consider the heteroscedastic partially linear errors-in-variables (EV) model y_i = x_iβ + g(t_i) + ε_i, ξ_i = x_i + μ_i (1 ? i ? n), where ε_i = σ_ie_i are random errors with mean zero, σ²_i = f(u_i), (x_i, t_i, u_i) are non random design points, x_i are observed with measurement errors μ_i. When f( · ) is known, we derive the Berry–Esseen type bounds for estimators of β and g( · ) under {e_i,?1 ? i ? n} is a sequence of stationary α-mixing random variables, when f( · ) is unknown, the Berry–Esseen type bounds for estimators of β, g( · ), and f( · ) are discussed under independent errors.     

Change-Point Detection With Non-Parametric Regression

We consider the regression model y_i = ?(x_i ) + ε in which the function ? or its pth derivative ?^(p) may have a discontinuity at some unknown point τ. By fitting local polynomials from the left and right, we test the null that ?^(p) is continuous against the alternative that ?^(p)(τ?) ≠ ?^(p)(τ+). We obtain Darling-Erdös type limit theorems for the test statistics under the null hypothesis of no change, as well as their limits in probability under the alternative. Consistency of the related change-point estimators is also established.     

D-Optimal Axial Designs for Quadratic and Cubic Additive Mixture Models

The paper discusses D-optimal axial designs for the additive quadratic and cubic mixture models σ_1≤i≤q(β_ix_i + β_iix²_i) and σ_1≤i≤q(β_ix_i + β_iix²_i + β_iiix³_i), where x_i≥ 0, x₁ + . . . + x_q = 1. For the quadratic model, a saturated symmetric axial design is used, in which support points are of the form (x₁, . . . , x_q) = [1 ? (q?1)δ_i, δ_i, . . . , δ_i], where i = 1, 2 and 0 ≤δ₂ <_{δ1 ≤} 1/(q ?1). It is proved that when 3 ≤q≤ 6, the above design is D-optimal if δ₂ = 0 and δ₁ = 1/(q?1), and when q≥ 7 it is D-optimal if δ₂ = 0 and δ₁ = [5q?1 ? (9q²?10q + 1)^1/2]/(4q²). Similar results exist for the cubic model, with support points of the form (x₁, . . . , x_q) = [1 ? (q?1)δ_i, δ_i, . . . , δ_i], where i = 1, 2, 3 and 0 = δ₃ <_δ2 < δ_{1 ≤}1/(q?1). The saturated D-optimal axial design and D-optimal design for the quadratic model are compared in terms of their efficiency and uniformity.     

Inequalities for the partial sums of elliptical order statistics related to genetic selection

Let X₁,…,X_n be exchangeable normal variables with a common correlation p, and let X₍₁₎ > … > X_(n) denote their order statistics. The random variable σⁿ_i=_n—_k+1x_i, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on the distribution of this random variable as a function of ρ. The same technique used can be applied to yield more general results for linear combinations of order statistics from elliptical distributions.     

Confidence Intervals for Nonparametric Regression Functions with Missing Data

Suppose that we have a nonparametric regression model Y = m(X) + ε with X ∈ R^p, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x₀) for fixed x₀ ∈ R^p are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x₀). We also construct an empirical likelihood (EL) statistic for m(x₀) with limiting distribution of χ²₁, which is used to construct an EL confidence interval for m(x₀).     

ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS

Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (q_ij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., q_ij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability p_ij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p_0N(t) (p^(m)_(N0)(t)), 0 ≤ m ≤N, where f^(m)(t) denotes the mth derivative of a function f(t) with f⁽⁰⁾(t) =f(t). If X(t) is a birth-death process, then p_ij(t) is represented as a linear combination of p_0N^(m)(t), 0 ≤m≤N - |i-j|.     

Nonparametric Estimation of a Regression Function: Limiting Distribution2

Consider the regression model Y_i= g(x_i) + e_i, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x₀ < x₁ < … < x_n≤ 1 are chosen so that max_1≤i≤n(x_i-x_{i- 1}) = 0(n^-1), and where {e_i} are i.i.d. with Ee₁= 0 and Var e₁ - s?². In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g_1n of Cheng & Lin (1979), g_2n of Clark (1977), and g_3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &_in, i = 1, 2, 3, and that of the isotonic estimator g**_n are considered.     

GEOMETRIC DECAY OF THE STEADY-STATE PROBABILITIES IN A QUASI-BIRTH-AND-DEATH PROCESS WITH A COUNTABLE NUMBER OF PHASES

A sufficient condition is proved for geometric decay of the steady-state probabilities in a quasi-birth-and-death process having a countable number of phases in each level. If there is a positive number η and positive vectors x = (x _i) and y = (y _j ) satisfying some equations and inequalities, the steady-state probability π _mi decays geometrically with rate η in the sense π _mi ～ cη ^m x _i as m → ∞. As an example, the result is applied to a two-queue system with shorter queue discipline.     

ON TESTING AGAINST RESTRICTED ALTERNATIVES FOR THE VARIANCES OF GAUSSIAN MODELS

Let π₁,…,π_p be p independent normal populations with means μ₁…, μ_p and variances σ²₁,…, σ²_p respectively. Let X(n_i) be a simple random sample of size n_i from π_i, i = 1,…,p. Given the simple random samples X(n₁),…, X(n_p) from π₁,…,π_p respectively, a test has been proposed for testing the homogeneity of variances H₀: σ²₁=…σ²_p, against the restricted alternative, H₁: σ²₁≥…≥σ²_p, with at least one strict inequality. Some properties of the test are discussed and critical values are tabulated.     

Distributions minimizing fisher information for scale in kolmogorov neighbourhoods

We construct those distributions minimizing Fisher information for scale in Kolmogorov neighbourhoods K_?(G) = {F|sup_x|F(x) - G(x| ? ?} of d.f.'s G satisfying certain mild conditions. The theory is sufficiently general to include those cases in which G is normal, Laplace, logistic, Student's t, etc. As well, we consider G(x) = 1 - e^-x, ? 0, and correct some errors in the literature concerning this case.     

Pseudo New Modified Latin Square (NML m (m)) Type PBIB Designs

A New Modified Latin square [NML _i (m)] association scheme with i constraints for v = m ² treatments was introduced by Garg (2008 Garg , D. K. ( 2008 ). New modified Latin square (NMLi) type PBIB designs . J. Math. Syst. Sci. 1 ( 4 ): 83 – 89 . [Google Scholar]). In this article, a new association scheme known as Pseudo New Modified Latin square [Pseudo NML _m (m)] type association scheme is defined. The parameters of Pseudo NML _m (m) association scheme turned out to be parameters of NML _i (m) association scheme by taking i = m in NML _i (m) association scheme. The Pseudo NML _m (m) association scheme will be the usual NML _m (m) association scheme when m is a prime or a prime power. The PBIB designs following Pseudo NML _m (m) association scheme will be called the Pseudo NML _m (m) type PBIB designs. Analysis of Pseudo NML _m (m) designs along with a construction method of these designs is also given in this article.     

Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X₁, …, X_N a random sample from a distribution F, let the process S^δ_N(t) be defined as where K²_N = σ^N_i=1(c_i ? c?)² and R x_i, + Δd, is the rank of X_i + Δd_i, among X₁ + Δd₁, …, X_N + Δd_N. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c_i and d_i, S^Δ_N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.     

Combining unbiased estimates —a further examination of some old estimators

This paper considers the problem of combining k unbiased estimates, x _i of a parameter,μ, where each estimate, x _i is the average of n _i + l independent normal observations with unknown mean, μ, and unknown variance, σ _i ². The behavior of several commonly used estimators of μ is studied by means of an empirical sampling study, and the empirical results of this experiment are interpreted in terms of previous theoretical results. Finally, some extrapolations are made as to how these estimators might behave under varying conditions, and some new estimators are proposed which might have higher efficiencies under certain conditions than those which are generally used.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

The Characterization of the Symmetric Lorenz Curves by Doubly Truncated Mean Function

We characterize symmetric Lorenz curves by the relation m(x, μ²/x) = μ (where μ =E(X) and m(x, y) = E(X | x ≤ X ≤ y) is the doubly truncated mean function). We establish that the points of the r.v. which generate the symmetric points on the Lorenz curve are x and μ²/x, and that all the distribution functions defined on the same support which are generators of the symmetric Lorenz curves have the same mean. We obtain the conditions under which doubly truncated distributions generate symmetrical Lorenz curves.     

Adaptive parametric test in a semiparametric regression model

Consider the semiparametric regression model Y_i = x′_iβ +g(t_i)+e_i for i=1,2, …,n. Here the design points (x_i,t_i) are known and nonrandom and the e_i are iid random errors with Ee₁ = 0 and Ee² ₁ = α²<∞. Based on g(.) approximated by a B-spline function, we consider using atest statistic for testing H₀ : β = 0. Meanwhile, an adaptive parametric test statistic is constructed and a large sample study for this adaptive parametric test statistic is presented.     

Geometric ergodicity of nonlinear autoregressive models with changing conditional variances

The authors give easy‐to‐check sufficient conditions for the geometric ergodicity and the finiteness of the moments of a random process x_t = ?(x_t‐1,…, x_t‐p) + ?_tσ(x_t‐1,…, x_t‐q) in which ?: R^p → R, σ R^q → R and (?_t) is a sequence of independent and identically distributed random variables. They deduce strong mixing properties for this class of nonlinear autoregressive models with changing conditional variances which includes, among others, the ARCH(p), the AR(p)‐ARCH(p), and the double‐threshold autoregressive models.     

Precise large deviations for the difference of two sums of WUOD and non identically distributed random variables with dominatedly varying tails

In this article, we study large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j ? ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j ? ∑^N₂(t)_{j = 1}X_2j, where {X_1j, j ? 1} is a sequence of widely upper orthant dependent (WUOD) random variables with non identical distributions {F_1j(x), j ? 1}, {X_2j, j ? 1} is a sequence of independent identically distributed random variables, n₁(t) and n₂(t) are two positive integer-valued functions, and {N_i(t), t ? 0}²_{i = 1} with EN_i(t) = λ_i(t) are two counting processes independent of {X_ij, j ? 1}²_{i = 1}. Under several assumptions, some results of precise large deviations for non random difference and random difference are derived, and some corresponding results are extended.     

Bootstrap test of goodness of fit to a linear model when errors are correlated

Given the regression model Y_i = m(x_i) +ε_i (x_i ε C, i = l,…,n, C a compact set in R) where m is unknown and the random errors {ε_i} present an ARMA structure, we design a bootstrap method for testing the hypothesis that the regression function follows a general linear model: H_o : m ε {mθ(.) = A^t(.)θ : θ ε ? ? R^q} with A a functional from R to R^q. The criterion of the test derives from a Cramer-von-Mises type functional distance D = d²([mcirc]_n, A^t(.)θ_n), between [mcirc]_n, a Gasser-Miiller non-parametric estimator of m, and the member of the class defined in H_o that is closest to m_n in terms of this distance. The consistency of the bootstrap distribution of D and θ_n is obtained under general conditions. Finally, simulations show the good behavior of the bootstrap approximation with respect to the asymptotic distribution of D = d².     

Selecting the normal population with the smallest variance: A restricted subset selection rule

Consider k( ? 2) normal populations whose means are all known or unknown and whose variances are unknown. Let σ²_[1] ? ??? ? σ_[k]² denote the ordered variances. Our goal is to select a non empty subset of the k populations whose size is at most m(1 ? m ? k ? 1) so that the population associated with the smallest variance (called the best population) is included in the selected subset with a guaranteed minimum probability P* whenever σ²_[2]/σ_[1]² ? δ* > 1, where P* and δ* are specified in advance of the experiment. Based on samples of size n from each of the populations, we propose and investigate a procedure called R_BCP. We also derive some asymptotic results for our procedure. Some comparisons with an earlier available procedure are presented in terms of the average subset sizes for selected slippage configurations based on simulations. The results are illustrated by an example.