Cramér‐von Mises regression

Consider a linear regression model with unknown regression parameters β₀ and independent errors of unknown distribution. Block the observations into q groups whose independent variables have a common value and measure the homogeneity of the blocks of residuals by a Cramér‐von Mises q‐sample statistic T_q(β). This statistic is designed so that its expected value as a function of the chosen regression parameter β has a minimum value of zero precisely at the true value β₀. The minimizer β of T_q(β) over all β is shown to be a consistent estimate of β₀. It is also shown that the bootstrap distribution of T_q(β₀) can be used to do a lack of fit test of the regression model and to construct a confidence region for β₀     

Failure–censored reliability sampling plans for the exponential distribution

In this paper we examine the failure-censored sampling plans for the two–parameter exponential distri- bution based on m random samples, each of size n. The suggested procedure is based on exact results and only the first failure time of each sample is needed. The values of the acceptability constant are also tabulated for selected values of p _α ¹ p _β ¹, α and β. Further, a comparison of the proposed sampling plans with ordinary sampling plans using a sample of size mn is made. When compared to ordinary sampling plans, the proposed plan has an advantage in terms of shorter test-time and a saving of resources.     

The Gauss—Markov Theorem and Random Regressors

In the standard linear regression model with independent, homoscedastic errors, the Gauss—Markov theorem asserts that = (X'X)^-1(X'y) is the best linear unbiased estimator of β and, furthermore, that is the best linear unbiased estimator of c'β for all p × 1 vectors c. In the corresponding random regressor model, X is a random sample of size n from a p-variate distribution. If attention is restricted to linear estimators of c'β that are conditionally unbiased, given X, the Gauss—Markov theorem applies. If, however, the estimator is required only to be unconditionally unbiased, the Gauss—Markov theorem may or may not hold, depending on what is known about the distribution of X. The results generalize to the case in which X is a random sample without replacement from a finite population.     

SOME ASYMPTOTIC PROPERTIES OF THE SIMPLE LOGLINEAR MODEL

Consistency and asymptotic normality of the maximum likelihood estimator of β in the loglinear model E(y_i) = e^α+βXi, where y_i are independent Poisson observations, 1 iaan, are proved under conditions which are near necessary and sufficient. The asymptotic distribution of the deviance test for β=β₀ is shown to be chi-squared with 1 degree of freedom under the same conditions, and a second order correction to the deviance is derived. The exponential model for censored survival data is also treated by the same methods.     

Box-Cox transformed linear models: A parameter-based asymptotic approach

A Box-Cox transformed linear model usually has the form y(λ) = μ + β₁x₁ +… + β_px_p + oe, where y(λ) is the power transform of y. Although widely used in practice, the Fisher information matrix for the unknown parameters and, in particular, its inverse have not been studied seriously in the literature. We obtain those two important matrices to put the Box-Cox transformed linear model on a firmer ground. The question of how to make inference on β = (β₁,…,β_p)^T when λ; is estimated from the data is then discussed for large but finite sample size by studying some parameter-based asymptotics. Both unconditional and conditional inference are studied from the frequentist point of view.     

ON THE USE OF X2 AS A TEST OF INDEPENDENCE IN CONTINGENCY TABLES WITH SMALL CELL EXPECTATIONS

When an r×c contingency table has many cells having very small expectations, the usual χ² approximation to the upper tail of the Pearson χ² goodness-of-fit statistic becomes very conservative. The alternatives considered in this paper are to use either a lognormal approximation, or to scale the usual χ² approximation. The study involves thousands of tables with various sample sizes, and with tables whose sizes range from 2×2 through 2×10×10. Subject to certain restrictions the new scaled χ² approximations are recommended for use with tables having an average cell expectation as small as 0·5.     

Efficient estimates in linear and nonlinear regression with heteroscedastic errors

In this paper we consider the heteroscedastic regression model defined by the structural relation Y = r(V, β) + s(W)ε, where V is a p-dimensional random vector, W is a q-dimensional random vector, β is an unknown vector in some open subset B of R^m, r is a known function from R^p × B into R, s is an unknown function on R^q, and ε is an unobservable random variable that is independent of the pair (V, W). We construct asymptotically efficient estimates of the regression parameter β under mild assumptions on the functions r and s and on the distributions of ε and (V, W).     

The Distribution of Sums of Certain I.I.D. Pareto Variates

ABSTRACT

Though the Pareto distribution is important to actuaries and economists, an exact expression for the distribution of the sum of n i.i.d. Pareto variates has been difficult to obtain in general. This article considers Pareto random variables with common probability density function (pdf) f(x) = (α/β) (1 + x/β)^α+1 for x > 0, where α = 1,2,… and β > 0 is a scale parameter. To date, explicit expressions are known only for a few special cases: (i) α = 1 and n = 1,2,3; (ii) 0 < α < 1 and n = 1,2,…; and (iii) 1 < α < 2 and n = 1,2,…. New expressions are provided for the more general case where β > 0, and α and n are positive integers. Laplace transforms and generalized exponential integrals are used to derive these expressions, which involve integrals of real valued functions on the positive real line. An important attribute of these expressions is that the integrands involved are non oscillating.     

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).     

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.     

EMPIRICAL LIKELIHOOD‐BASED INFERENCES FOR PARTIALLY LINEAR MODELS WITH MISSING COVARIATES

This paper considers statistical inference for partially linear models Y = X ^? β +ν(Z) +? when the linear covariate X is missing with missing probability π depending upon (Y, Z). We propose empirical likelihood‐based statistics to construct confidence regions for β and ν(z). The resulting empirical likelihood ratio statistics are shown to be asymptotically chi‐squared‐distributed. The finite‐sample performance of the proposed statistics is assessed by simulation experiments. The proposed methods are applied to a dataset from an AIDS clinical trial.     

Optimal quantization applied to sliced inverse regression

In this paper we consider a semiparametric regression model involving a d-dimensional quantitative explanatory variable X and including a dimension reduction of X via an index β′X. In this model, the main goal is to estimate the Euclidean parameter β and to predict the real response variable Y conditionally to X. Our approach is based on sliced inverse regression (SIR) method and optimal quantization in L^p-norm. We obtain the convergence of the proposed estimators of β and of the conditional distribution. Simulation studies show the good numerical behavior of the proposed estimators for finite sample size.     

On the asymptotic expectation and variance of the number of boundary crossings

Let X₁,X₂,… be independent and identically distributed nonnegative random variables with mean μ, and let S_n = X₁ + … + X_n. For each λ > 0 and each n ≥ 1, let A_n be the interval [λn^Y, ∞), where γ > 1 is a constant. The number of times that S_n is in A_n is denoted by N. As λ tends to zero, the asymtotic behavior of N is studied. Specifically under suitable conditions, the expectation of N is shown to be (μλ^?1)^β + o(λ^?β/2 where β = 1/(γ-1) and the variance of N is shown to be (μλ^?1)^β(βμ¹)²σ² + o(λ^?β) where σ² is the variance of X_n.     

Testing Serial Correlation in Partially Linear Single-Index Errors-in-Variables Models

In this article, we propose two test statistics for testing the underlying serial correlation in a partially linear single-index model Y = η(Z ^τα) + X ^τβ + ? when X is measured with additive error. The proposed test statistics are shown to have asymptotic normal or chi-squared distributions under the null hypothesis of no serial correlation. Monte Carlo experiments are also conducted to illustrate the finite sample performance of the proposed test statistics. The simulation results confirm that these statistics perform satisfactorily in both estimated sizes and powers.     

Locally minimax tests for multiple correlations

Let X be a normally distributed p-dimensional column vector with mean μ and positive definite covariance matrix σ. and let X α, α = 1,…, N, be a random sample of size N from this distribution. Partition X as ( X ₁, X ₍₂₎', X '₍₃₎)', where X₁ is one-dimension, X₍₂₎ is p₂- dimensional, and so 1 + p₁ + p₂ = p. Let ρ₁ and ρ be the multiple correlation coefficients of X₁ with X₍₂₎ and with ( X '₍₂₎, X '₍₃₎)', respectively. Write ρ2/2 = ρ² - ρ2/1. We shall cosider the following two problems     

CONDITIONING VS. STANDARDIZATION FOR CONTRASTS ON ERRORS ATTRACTED TO STABLE LAWS

In multiple regression and other settings one encounters the problem of estimating sampling distributions for contrast operations applied to i.i.d. errors. Permutation bootstrap applied to least squares residuals has been proven to consistently estimate conditionalsampling distributions of contrasts, conditional upon order statistics of errors, even for long-tailed error distributions. How does this compare with the unconditional sampling distribution of the contrast when standardizing by the sample s.d. of the errors (or the residuals)? For errors belonging to the domain of attraction of a normal we present a limit theorem proving that these distributions are far closer to one another than they are to the limiting standard normal distribution. For errors attracted to α-stable laws with α ≤ 2 we construct random variables possessing these conditional and unconditional sampling distributions and develop a Poisson representation for their a.s. limit correlation ρ_α. We prove that ρ₂= 1, ρ_α→ 1 for α → 0 + or 2 ?, and ρ_α< 1 a.s. for α < 2.     

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 ?}.     

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.