Showing papers in "Statistics in 1995"
TL;DR: This is an attempt to discuss various approaches developed in experimental design when constraints are imposed, and the basic idea of the paper is that all corresponding optimization problems can be imbedded in the convex theory of experimental design.
Abstract: This is an attempt to discuss various approaches developed in experimental design when constraints are imposed. These constraints may be on the total cost of the experiment, the location of the supporting point, the value of auxiliary objective functions, and so on. The basic idea of the paper is that all corresponding optimization problems can be imbedded in the convex theory of experimental design. Part 1 is concerned with the properties of optimal designs, while Part 2 is devoted mainly to numerical methods. We have tried to avoid details, emphasizing ideas rather than technicalities. This is not intended as a literature review. The authors subjectively surely left many excellent papers behind.
71 citations
TL;DR: A penalized likelihood estimator is proposed based on the m-dimensional distribution of HMM, and it is shown that in the limit it does not underestimate the order.
Abstract: Hidden Markov models (HMMs) have during the last decade become a widely spread tool for modelling sequences of dependent random variables. Inference for HMMs has been considered by several authors, but so far no work has been done on estimating their order. In this paper we propose a penalized likelihood estimator for this purpose. This estimator is based on the m-dimensional distribution of HMM, and it is shown that in the limit it does not underestimate the order.
67 citations
TL;DR: The aim of this paper is to present a nonparametric approach that allows to estimate the autoregression order without limiting oneself to any restrictive parametric class of processes.
Abstract: An extensive literature has been devoted to the problem of order choice in autoregressive models. Most of alternative methods to hypothesis tests are based on the minimization of the Akaike Information Criterion (AIC) or on some of its variants. These methods have the main drawback to have to assume a parametric form for the autoregression function. The aim of this paper is to present a nonparametric approach that allows to estimate the autoregression order without limiting ourself to any restrictive parametric class of processes. Our technique is in the same spirit as AIC criterion, in the sense that it is based on the minimization of some prediction error. Both theoretical and computational aspects of this method are discussed in this paper.
51 citations
TL;DR: In this article, the authors obtained the asymptotic distribution of the maximum likelihood ratio test when they test for possible changes in the regression coefficients, and they also proved that the MRL test is almost surely consistent.
Abstract: We obtain the asymptotic distribution of the maximum likelihood ratio test when we test for possible changes in the regression coefficients. We also prove that the maximum likelihood ratio tests is asymptotically consistent. We propose an estimator for the time of change and show that it is almost surely consistent. We consider the cases when the variance remains constant and when it may change at an unknown time.
51 citations
TL;DR: In this paper, the bias corrected largest order statistic (LOS) is used to compare the BLUE estimate of α, b and θ in an ideal model with the UMVU estimate of θ.
Abstract: Consider n observations on, ,generated independently by a distribution function F which is only partially specified: We require that F has in its upper tail a density f such that as x tends to the right endpoint θ of F. The parameters of interest are a > 0 and b, . Based on the largest order statistics in the sample, we can define in case α > 1 /2 estimates of α, b and θ that behave asymptotically like the BLUE of θ as if α was known and like the UMVU estimates of and θin some ideal model as if θ was known. We show in addition that the BLUE estimate of θ can be outperformed by a bias corrected largest order statistic.
35 citations
TL;DR: In this paper, the central limit theorem for the L 1 error was proved for the histogram estimate constructed from a sample of i.i.d. real-valued random variables with common continuously differentiable density f.
Abstract: Let f n be a histogram estimate constructed from a sample of i.i.d. real-valued random variables with common continuously differentiable density f. In this paper we prove a central limit theorem for the L 1 error . We determine a positive constant 0 < σ 2 ≤ 1 − 2/π in order that, under the usual conditions of consistency, the law of be asymptotically Gaussian , N(0, 1).
30 citations
TL;DR: In this paper, algebraic properties of the Hadamard product are used to establish some statistical properties, which are then used for statistical analysis of the hadamard products and their properties.
Abstract: Algebraic properties of the Hadamard product are used to establish some statistical properties.
26 citations
TL;DR: In this article, the authors considered the M-estimators for the Partially Linear Model (PLM) and obtained the convergence rate n -1/2 under some regular conditions, which is Stone's optimal global rate of convergence of least square estimators.
Abstract: This paper concerns with M-estimators for the partly linear model , where are i.i.d. random (d + 2)-vectors such that Y i , is real-valued, X i ∊ R d , and T i ranges over a nondegenerate compact interval; u i , is a random error; β o is a d-vector of parameters; and g o (·) is an unknown function. A piecewise polynomial is used to approximate g o (·). The estimators of β o and g o (t) considered are and respectively, where and are the solutions of the minimization problem and ϕ(·) is a vector of the basis functions of a piecewise polynomial space and ρ(·) is a function chosen suitably. Under some regular conditions, it is shown that achieves the convergence rate which is Stone’s optimal global rate of convergence of least square estimators for nonparametric regression and achieves the convergence rate n -1/2.
26 citations
TL;DR: In this paper, the authors give asymptotic expansions for the expected number of local extremes and inflection points of kernel density estimates, and argue that these numbers can be used as an indicator for the qualitative smoothness of the density estimate.
Abstract: In this paper we give asymptotic expansions for the expected number of local extremes and inflection points of kernel density estimates. We argue that these numbers can be used as an indicator for the “qualitative” smoothness of the density estimate.
22 citations
TL;DR: In this paper, the necessary and sufficient conditions for any real, continuous and strictly monotonic function ξ(x) to be equal to or equal to any real function x is given.
Abstract: Let be the order statistics of a sample of size n≥ 2 from a population with continuous distribution function F. In this paper, we obtain the distribution function F from conditional expectation or . where h is a real, continuous and strictly monotonic function. We give the necessary and sufficient conditions so that any real function ξ(x) is equal to or is equal to . Different continuous distributions are also characterized using our results.
21 citations
TL;DR: In this article, a test for whether an immune proportion is indeed present in the population, i.e., for H 0:p = 1, is presented, which involves testing at the boundary of the parameter space for p.
Abstract: Models for populations with immune or cured individuals but with others subject to failure are important in many areas, such as medical statistics and criminology. One method of analysis of data from such populations involves estimating an immune proportion 1 − p and the parameter(s) of a failure distribution for those individuals subject to failure. We use the exponential distribution with parameter λ for the latter and a mixture of this distribution with a mass 1 − p at infinity to model the complete data. This paper develops the asymptotic theory of a test for whether an immune proportion is indeed present in the population, i.e., for H 0:p = 1. This involves testing at the boundary of the parameter space for p. We use a likelihood ratio test for H 0. and prove that minus twice the logarithm of the likelihood ratio has as an asymptotic distribution, not the chi-square distribution, but a 50–50 mixture of a chi-square distribution with 1 degree of freedom, and a point mass at 0. The result is proved und...
TL;DR: It is shown that explicit bias estimation even with heteroscedastic variance structure leads to an improvement of coverage accuracy, when compared to undersmoothing, in curve estimation.
Abstract: Bootstrap methods in curve estimation have been introduced for smoothing parameter selection and for construction of confidence intervals. Most of the papers on confidence intervals use explicit bias estimation or the technique of "undersmoothing" to deal with bias. Coverage accuracy has only been considered for curve estimates with constant variance function. In this paper we show that explicit bias estimation even with heteroscedastic variance structure leads to an improvement of coverage accuracy, when compared to undersmoothing. Bootstrapping with this bias correction using the so-called wild bootstrap leads to an improved coverage accuracy.
TL;DR: In this paper, the convergence conditions for the convergence of the finite-dimensional distributions of the associated empirical process were studied and the convergence in probability and the almost complete convergence of this general estimator were obtained.
Abstract: We note that some classical functional estimation problems may be reduced to a general unique framework and study an estimator within this general framework that reduces to the classical histogram type estimators in various examples presented The convergence in probability and the almost complete convergence of this general estimator are studied obtaining convergence conditions which reduce to the classical conditions in each case Finally, this general framework provides conditions for the convergence of the finite dimensional distributions of the associated empirical process
TL;DR: In this article, recursive estimates of the regression function f(x) = E(Y/X = x) and of the variance function Var (Y, X) were used to prove pointwise mean square convergence under appropriate conditions.
Abstract: Consider an i.i.d. sample for a random vector (X, Y)is an element of R(d) x R. We are interested in recursive estimates of the regression function f(x) = E(Y/X = x) and of the variance function Var (Y/X = x). We use recursive kernel estimates for both problems and prove pointwise mean square convergence with rates under appropriate conditions. The results are compared with nonrecursive estimators, which have recently been suggested by various authors.
TL;DR: In this paper, the authors derived the best lower triangular equivariant estimators of the normal covariance matrix and the precision matrix Σ-1 simultaneously, where the estimators are of the form, where Tis a lower triangular matrix and Dis a diagonal matrix of constants.
Abstract: In this paper we derive the best lower triangular equivariant estimators of the normal covariance matrix Σ and the precision matrix Σ-1 simultaneously. Our estimators are of the form , where Tis a lower triangular matrix and Dis a diagonal matrix of constants. We derive improved estimators which dominate . The risk of the derived estimators for p= 2 and 3 is also given.
TL;DR: In this paper, a multivariate normal random matrix X, where X (p × r) is distributed with independent columns X _{ i }\sim N _{ p }\left ( \zeta i },\sum \right ), i = 1, 2, …, r, and S has Wishart distribution W p(n, Σ).
Abstract: Consider a multivariate normal random matrix X, where X (p × r) is distributed with independent columns X _{ i }\sim N _{ p }\left ( \zeta _{ i },\sum \right ) , i = 1, 2, …, r, and S has Wishart distribution W p(n, Σ). Estimators of |Σ| and |Σ−1| based on X and S are derived under the loss function L \left (\left |\sum \right |,\left |\circ {>\sum }\right. \right )=\left |\circ {>\sum }\sum ^{-1}\right |+\left |\sum \circ {>\sum }^{-1}\right |-2 . The risks of the improved estimators are evaluated in terms of an incomplete beta function of matrix argument. Numerical results for the risk of the improved estimator for selected values of r, p and n indicate a reduction in risk over the best affine equivariant estimator c 0|S|, where for n > p +1, c _{ o }=\left {\left (\left ( n - p -1\right )!\left ( n - p \right )!\right )/\left ( n -2\right )! n !\right }^{1/2} .
TL;DR: In this paper, the authors discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples, and derive approximate maximum likelihood estimators for the two parameters.
Abstract: In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.
TL;DR: In this article, the authors developed estimators of θ combining both X and U, which are similar to the James-Stein estimator but can give substantial risk improvement over it when θ is assumed to be in a neighborhood of a known value.
Abstract: Let X be a (p × 1) random vector following a multivariate normal distribution with mean vector 0 and a known dispersion matrix. Also suppose we have an extra observation U independent of X, whose distribution is completely known. In this article we first develop estimators of θ combining both X and U, which are similar to the James-Stein estimator but can give substantial risk improvement over it when θ is assumed to be in a neighborhood of a known value. Estimators based only on Xare then developed which can give further improvements.
TL;DR: In this paper, the authors consider the simple linear calibration problem where only the response y of the regression line y = β0 + β1 t is observed with errors. But they assume that there may be some gross errors providing outlying observations.
Abstract: We regard the simple linear calibration problem where only the response y of the regression line y = β0 + β1 t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.
TL;DR: In this article, maximum likelihood estimators for a broad class of matrix variate elliptically contoured distributions have been derived and the optimality properties of estimators in the multivariate regression models are studied when the error term has a matrix-varying elliptical contoured distribution.
Abstract: In this paper, maximum likelihood estimators for a broad class of matrix variate elliptically contoured distributions have been derived. Optimality properties of estimators in the multivariate regression models are studied when the error term has a matrix variate elliptically contoured distribution. Least-square estimators are also obtained for the multivariate random effect regression model when the underlying distribution is elliptically contoured.
TL;DR: In this paper, the authors consider the experimental design problem for the Gaus-Markov estimator or least squares estimator when the observations are correlated and derive lower bounds for the efficiency.
Abstract: In the general linear model consider the experimental design problem for the Gaus-Markov estimator or least squares estimator when the observations are correlated. We prove new formulas for the efficiency of an exact design with respect to the D-criterion. For models with intercept term, for example, these formulas are useful to derive better lower bounds for the efficiency than the bounds recently given for an arbitrary linear model. These bounds are applied in examples to symmetrical regular circulants as covariance matrices. A byproduct of the investigations is some insight as to what kinds of designs might retain their optimality or high efficiency (for the uncorrelated homoscedastic case) under correlated observations.
TL;DR: In this article, consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available, and convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principle.
Abstract: Consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available. Convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principles. As an application, a change point test in a simple AMOC renewal model is briefly discussed, where variance estimators possessing good enough convergence rates are required.
TL;DR: In this article, the authors apply the information inequality for the Bayes risk (global Cramer-Rao inequality) to nonexponential estimation problems, and propose a new methodology of proving minimaxity by considering the example of estimating the scale or location parameter under one-sided truncation of the parameter space.
Abstract: The paper is concerned with an application of the information inequality for the Bayes risk (global Cramer-Rao inequality) to nonexponential estimation problems. A new methodology of proving minimaxity is presented by considering the example of estimating the scale or location parameter under one-sided truncation of the parameter space.
TL;DR: In this article, the authors present some considerations on inferences based on the use of asymptotical normality of the maximum likelihood estimators of the BVED of Block and Basu (1974) considering accelerated life tests with a power rule model.
Abstract: In this paper, we present some considerations on inferences based on the use of asymptotical normality of the maximum likelihood estimators of the bivariate exponential distribution (BVED) of Block and Basu (1974) considering accelerated life tests with a power rule model. We show in one numerical example, the effect of a different parametrization to improve the accuracy of the obtained inferences on the parameters of the model. We also consider Bayesian methods to get inferences on the parameters of the BVED applied to accelerated life tests.
TL;DR: In this article, a pseudo-likelihood estimation method is applied to forest data consisting of so-called INKA sample plots, where the pair potential function of Gibbs point process through field observations is derived in unmarked and multitype cases.
Abstract: Pseudo-likelihood equations to estimate the pair potential function of Gibbs point process through field observations are derived in unmarked and multitype cases. The requirements the pair potential function sets our for observations are also considered. The pseudo-likelihood estimation method is applied to forest data consisting of so-called INKA sample plots.
TL;DR: In this article, a repairable system modelled by a semi-Markov process whose finite state space is partitioned into the set of up states U and down states D is considered.
Abstract: We consider a repairable system modelled by a semi-Markov process whose finite state space is partitioned into the set of up states U, and the set of down states D. The mission availability is defined as the probability that the system will not experience a down period exceeding, say, t f units of time before, say, T 0 units of time in U have accrued. A system of integral equations is established for the mission availability under the semi-Markov assumption. A closed form solution is also derived, which, however, is not suitable for computational work. The 9-state model of a two-unit maintained system is analysed by numerically solving the system of integral equations using a two-point trapezoidal rule. The system of implementation is the matrix computation package MATLAB on the Apple Macintosh Quadra 610. The method is shown to compare favourably with simulation.
TL;DR: In this paper, the problem of estimating a finite population total of a variable of sensitive in nature is studied under randomized response (RR) surveys and some optimal sampling strategies are presented under different superpopulation models.
Abstract: Problems of estimation of a finite population total of a variable of sensitive in nature are studied under randomized response (RR) surveys. Some optimal sampling strategies are presented under different superpopulation models.
TL;DR: In this article, the authors considered a general growth curve model with known n × k, p × l, n × s matrices and showed that the MINQE(U, I) of an estimable parameter function tr (CΣ) is a UMVIQUE.
Abstract: Consider a general Growth Curve Model as follows where are known n × k, p × l, n × s matrices respectively, B is an unknown k × l matrix of regression coefficients, and n × p matrix of observations and s × p matrix of random errors respectively, such that are independent random vectors with , (it's existent and finite) i = 1, … ,s, where Σ and Ψ and are matrices of unknown parameters. In the section 2 of this paper, for any given p × p matrix C = C′ ≠ 0, we give the MINQE(U, I) of an estimable parameter function tr (CΣ). Section 3 gives the n.s. conditions for tr (CΣ)'s MINQE((U, I) to be a UMVIQUE. Section 4 gives the n.s. conditions for tr (CΣ)'s UMVIQUE to exist, and shows the MINQE(U, I) of tr (CΣ) is just the UMVIQUE of tr(CΣ) provided a UMVIQUE exists.
TL;DR: In this article, a method of estimation of variance components in a random effect linear model is presented. But this method is mainly a resampling method and relies on the Jackknife principle, and the derived estimators are presented as least squares estimators in an appropriate linear model, and one of them appears as a MINQUE estimator.
Abstract: This paper concerns a method of estimation of variance components in a random effect linear model. It is mainly a resampling method and relies on the Jackknife principle. The derived estimators are presented as least squares estimators in an appropriate linear model, and one of them appears as a MINQUE (Minimum Norm Quadratic Unbiased Estimation) estimator. Our resampling method is illustrated by an example given by C. R. Rao [7] and some optimal properties of our estimator are derived for this example. In the last part, this method is used to derive an estimation of variance components in a random effect linear model when one of the components is assumed to be known.
TL;DR: In this article, the expectation of various covariogram estimators is derived assuming a polynomial trend with second-order stationary errors, and the goal of this paper is to compare the estimator based on ordinary-least-squares residuals, with the estimation based on recursive residuals.
Abstract: Assuming a polynomial trend with second-order stationary errors, expressions for the expectation of various covariogram estimators are derived. The goal of this article is to compare the estimator based on ordinary-least-squares residuals, with the estimator based on recursive residuals.