The Recursive Kernel Distribution Function Estimator Based on Negatively and Positively Associated Sequences

Let {X _j , j ≥ 1} be a strictly stationary negatively or positively associated sequence of real valued random variables with unknown distribution function F(x). On the basis of the random variables {X _j , j ≥ 1}, we propose a smooth recursive kernel-type estimate of F(x), and study asymptotic bias, quadratic-mean consistency and asymptotic normality of the recursive kernel-type estimator under suitable conditions.     

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.     

Conditional estimation of average on the basis of weighting data

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples: % MathType!End!2!1! and % MathType!End!2!1!.Let % MathType!End!2!1! and % MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination % MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived. The variance of the statistic % MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.     

THE CONVERGENCE RATE OF SEQUENTIAL FIXED-WIDTH CONFIDENCE INTERVALS FOR REGULAR FUNCTIONALS

Let X₁X₂,.be i.i.d. random variables and let U_n= (n r)^-1S?^(n,r) h (X_i1,., X_ir,) be a U-statistic with EU_n= v, v unknown. Assume that g(X₁) =E[h(X₁,.,X_r) - v |X₁]has a strictly positive variance s?². Further, let a be such that φ(a) - φ(-a) =α for fixed α, 0 < α < 1, where φ is the standard normal d.f., and let S²_n be the Jackknife estimator of n Var U_n. Consider the stopping times N(d)= min {n: S²_n: + n^-1≤²a^-2},d > 0, and a confidence interval for v of length 2d,of the form I_n,d= [U_n,-d, Un + d]. We assume that Var U_n is unknown, and hence, no fixed sample size method is available for finding a confidence interval for v of prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, let I_N(d),d be a sequence of sequential confidence intervals for v. The asymptotic consistency of this procedure, i.e. lim_{d → 0}P(v ∈ I_N(d),d)=α follows from Sproule (1969). In this paper, the rate at which |P(v ∈ I_N(d),d) converges to α is investigated. We obtain that |P(v ∈ I_N(d),d) - α| = 0 (d^{1/2-(1+k)/2(1+m)}), d → 0, where K = max {0,4 - m}, under the condition that E|h(X₁, X_r)|^m < ∞m > 2. This improves and extends recent results of Ghosh & DasGupta (1980) and Mukhopadhyay (1981).     

CHARACTERIZATIONS OF SHIFT-INVARIANT DISTRIBUTIONS BASED ON SUMMATION MODULO ONE

Let X₁Y_1,…, Y_n be independent random variables. We characterize the distributions of X and Y_j satisfying the equation {X+Y₁++Y_n}=_dX, where {Z} denotes the fractional part of a random variable Z. In the case of full generality, either X is uniformly distributed on [0,1), or Y_j has.a shifted lattice distribution and X is shift-invariant. We also give a characterization of shift-invariant distributions. Finally, we consider some special cases of this equation.     

Confidence intervals for the largest mean from k correlated normal populations

Let X= (X₁,…, X_k)’ be a k-variate (k ≥ 2) normal random vector with unknown population mean vector μ = (μ₁ ,…, μ_k)’ and covariance matrix Σ of order k and let μ_[1] ≤ … ≤ μ_[k] be the ordered values of the μ ’ s. No prior knowledge of the pairing of the μ_[i] with the X_j. (or μ_[i] with the σ_j ²) is assumed for any i and j (1 ≤ i, j ≤ k). Based on a random sample of N independent vector observations on X, this paper considers both upper and lower (one-sided) and two-sided 100γ% (0 < γ < 1) confidence intervals for μ_[k] and μ_[1], the largest and the smallest mean, respectively, when Σ is known and when Σ is equal to σ²R with common unknown variance σ² > 0 and correlation matrix R known, respectively. An optimum two-sided confidence interval via finding the shortest length from this class is also considered. Necessary tables and computer program to actually apply these procedures are provided.     

A remark on estimating the mean of a normal distribution with known coefficient of variation

Let X₁, X₂, …, X_n be iid N(μ, aμ²) (a>0) random variables with an unknown mean μ>0 and known coefficient of variation (CV) √a. The estimation of μ is revisited and it is shown that a modified version of an unbiased estimator of μ [cf. Khan RA. A note on estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1968;63:1039–1041] is more efficient. A certain linear minimum mean square estimator of Gleser and Healy [Estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1976;71:977–981] is also modified and improved. These improved estimators are being compared with the maximum likelihood estimator under squared-error loss function. Based on asymptotic consideration, a large sample confidence interval is also mentioned.     

Estimated Generalized Least Squares for a Heteroscedastic Regression Model

This paper considers the general linear regression model yc = X₁β+u_t under the heteroscedastic structure E(u_t) = 0, E(u²) =σ²- (X_tβ)², E(u_t u_s) = 0, tæs, t, s= 1, T. It is shown that any estimated GLS estimator for β is asymptotically equivalent to the GLS estimator under some regularity conditions. A three-step GLS estimator, which calls upon the assumption E(u_t²) =s?²(X,β)² for the estimation of the disturbance covariance matrix, is considered.     

Nonparametric estimation of a multivariate multiple regression function

Let (X₁, X₂, Y₁, Y₂) be a four dimensional random variable having the joint probability density function f(x₁, x₂, y₁, y₂). In this paper we consider the problem of estimating the regression function \({{E[(_{Y_2 }^{Y_1 } )} \mathord{\left/ {\vphantom {{E[(_{Y_2 }^{Y_1 } )} {_{X_2 = X_2 }^{X_1 = X_1 } }}} \right. \kern-0em} {_{X_2 = X_2 }^{X_1 = X_1 } }}]\) on the basis of a random sample of size n. We have proved that under certain regularity conditions the kernel estimate of this regression function is uniformly strongly consistent. We have also shown that under certain conditions the estimate is asymptotically normally distributed.     

Variants of the graph dependent model in extreme value theory

With a set X₁, X₂, .... X_n n random variables, a graph is associated whose vertices are the integers 1,2,..., n and whose edges represent those pairs i and j for which the events {X_i>X} and {X_j>X} do not become “almost independent” for “large X”. With a variety of assumption on the edge set of the graph, the asymptotic distribution of the extremes of the X_j, when properly normalized, is determined. This refines the earlier result of the present author on this kind of dependence, and extends and unifies several known dependent extreme value models.     

Penalized regression with model‐based penalties

Nonparametric regression techniques such as spline smoothing and local fitting depend implicitly on a parametric model. For instance, the cubic smoothing spline estimate of a regression function ∫ μ based on observations ti, Yi is the minimizer of Σ{Yi ‐ μ(ti)}² + λ∫(μ′′)². Since ∫(μ″)² is zero when μ is a line, the cubic smoothing spline estimate favors the parametric model μ(t) = α_o + α₁t. Here the authors consider replacing ∫(μ″)² with the more general expression ∫(Lμ)² where L is a linear differential operator with possibly nonconstant coefficients. The resulting estimate of μ performs well, particularly if Lμ is small. They present an O(n) algorithm for the computation of μ. This algorithm is applicable to a wide class of L's. They also suggest a method for the estimation of L. They study their estimates via simulation and apply them to several data sets.     

Inverse Regression in Binary Response LDV Model

Sliced inverse regression, a link-free and distribution-free method, is applied to binary response limited dependent variable models. An inverse regression property of binary response LDV model is found. Based on this property, if the distributions of X _j (j = 1, 2,…, p) satisfy the linearity condition, then β can be estimated up to a positive multiplicative scalar without any assumptions on the distribution of error ε. Moreover, the estimator can be proved to be asymptotically normal based on which testing hypotheses are considered. Simulations results are reported.     

A distribution function estimator for the difference of order statistics from two independent samples

For two independent populations X and Y we develop the empirical distribution function estimator for the difference of order statistics of the form X _(i)−Y _(j). The key practical application for this estimator pertains to inference between quantiles from two independent populations.     

Exact Moment Convergence Rates of U-Statistics

Let U _n be a U-statistic based on a symmetric kernel h(x, y) and i.i.d. samples {X, X _n ; n ≥ 1}. In this article, the exact moment convergence rates in the first moment of U _n are obtained, which extend previous results concerning partial sums.     

Changepoint Problems Under Random Censorship

Let X ₁, …, X _[nθ], X _{[nθ] + 1}, …, X _n be a sequence of independent random variables (the “lifetimes”) such that X _j ? F ₁ for 1 ≤ j ≤ [nθ] and X _j ? F ₂ for [nθ] + 1 ≤ j ≤ n, with F ₁ ≠ F ₂ unknown. In this paper we investigate an estimator θ _n for the changepoint θ if the X's are subject to censoring. The rate of almost sure convergence of θ _n to θ is established and a test for the hypothesis θ = 0, i.e. “no change”, is proposed.     

Best quadratic unbiased estimation of the variance matrix in normal regression

In 1973 Balestra examined the linear model y=XB+u, where u is a normally distributed disturbance vector, with variance matrix Ω. Ω has spectral decomposition \(\sum\limits_{i = 1}^r {\lambda _i M_i } \) , and the matrices M_i are known. Estimation of ω is thus equivalent with estimation of the λ_i. Balestra presented the best quadratic unbiased estimator of λ_i. In the present paper a derivation will be given which is based on a procedure developed by this writer (1980).     

Identification of power distribution mixtures through regression of exponentials

Given two independent non-degenerate positive random variables X and Y, Lukacs (Ann Math Stat 26:319–324, 1955) proved that X/(X + Y) and X + Y are independent if and only if X and Y are gamma distributed with the same scale parameter. In this work, under the assumption X/U and U are independent, and X/U has a ${{\mathcal Be}(p,\,q)}$ distribution, we characterize the distribution of (U, X) by the condition E(h(U ? X)|X) = b, where h is allowed to be a linear combination of exponential functions. Since the assumption for X and U above is equivalent to X|U being ${\mathcal{B}e(p,\,1)}$ scaled by U, hence our results can be viewed as identification of a power distribution mixture.     

Characterizations of the exponential distribution by some properties of the record values

A sequence {X_n, n≥1} of independent and identically distributed random variables with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max {X₁, X₂, ..., X_j?1}. We define L(n)=min {j|j>L(n?1), X_j>X_L(n?1)}, with L(0)=1. Let Z_n,m=X_L(n)?X_L(m), n>m≥0. Some characterizations of the exponential distribution are considered in terms of Z_n,m and X_L(m).