On bootstrapping the mode in the nonparametric regression model with random design

In the nonparametric regression model with random design and based on i.i.d. pairs of observations (X _i, Y _i), where the regression function m is given by m(x)=?(Y _i|X _i=x), estimation of the location θ (mode) of a unique maximum of m by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve m is considered. In order to obtain asymptotic confidence intervals for θ, the suitably normalized distribution of is bootstrapped in two ways: we present a paired bootstrap (PB) where resampling is done from the empirical distribution of the pairs of observations and a smoothed paired bootstrap (SPB) where the bootstrap variables are generated from a smooth bivariate density based on the pairs of observations. While the PB requires only relatively small computational effort when carried out in practice, it is shown to work only in the case of vanishing asymptotic bias, i.e. of “undersmoothing” when compared to optimal smoothing for mode estimation. On the other hand, the SPB, although causing more intricate computations, is able to capture the correct amount of bias if the pilot estimator for m oversmoothes. Received: May 2000     

Verbesserung des nicht-randomisierten exakten Testes von R.A. Fisher

Summary The so-called Exact Test of R. A. Fisher for comparing two probabilitiesp ₁ andp ₂ in a Fourfold-Table with small cell frequencies is known as a UMPU-Test. But in practice the test is used in a nonrandomized, often tabulated version. Given a certain level of significanceα it is shown: the critical region of this nonrandomized test, referred to as “Fisher 1”, can be enlarged considerably. For instance for all sample-size-sums up to 20 andα=0.01 the total number of points in the critical regions of “Fisher 1” is 552 whereas the analogous number of the new version “Fisher 2” is 788. The size of tables for “Fisher 2” can be reduced considerably because the main parts of the critical regions can be described by the aid of some Chi-square-test versions. In particular Yates’ continuity-correction turns out to be always conservative in the above mentioned region relative to “Fisher 2” whereas this is not strictly true relative to “Fisher 1”.      

A general class of estimators for estimating population mean using auxiliary information

Summary A general class of estimators for estimating the population mean of the character under study which make use of auxiliary information is proposed. Under simple random sampling without replacement (SRSWOR), the expressions of Bias and Mean Square Error (MSE), up to the first and the second degrees of approximation are derived. General conditions, up to the first order approximation, are also obtained under which any member of this class performs more efficiently than the mean per unit estimator, the ratio estimator and the product estimator. The class of estimators in its optimum case, under the first degree approximation, is discussed. It is shown that it is not possible to obtain optimum values of parameters “a”, “b” and “p”, that are independent of each other. However, the optimum relation among them is given by (b−a)p=ρ C _y/C _x. Under this condition, the expression of MSE of the class is that of the linear regression estimator.     

Increasing failure rate and decreasing reversed hazard rate properties of the minimum and maximum of multivariate distributions with log-concave densities

For a multivariate random vector X = (X ₁,...,X _n ) with a log-concave density function, it is shown that the minimum min{X ₁,...,X _n } has an increasing failure rate, and the maximum max{X ₁,...,X _n } has a decreasing reversed hazard rate. As an immediate consequence, the result of Gupta and Gupta (in Metrika 53:39–49, 2001) on the multivariate normal distribution is obtained. One error in Gupta and Gupta method is also pointed out.      

Multivariate multistage methodologies for simultaneous all pairwise comparisons

We consider the problem of constructing simultaneous fixed-width confidence intervals for all pairwise treatment differences μ₁−μ _J , in the presence ofk(≥2) independent populationsN _p (μ₁,Σ), 1≤i≠j≤k. Appropriate purely sequential, accelerated sequential and three-stage sampling strategies have been developed and variousfirst-order asymptotic properties are then derived when Σ _pxp is completely unknown, but positive definite (p.d.). In the two special cases when the largest component variance in Σ is a known multiple of one of the variances or Σ=σ² H where σ(>0) is unknown, butH _pxp is known and p.d., the original multistage sampling strategies are specialized. Under such special circumstances, associatedsecond-order characteristics are then developed. It is to be noted that our present formulation and the methodologies fill important voids in the context of multivariate multiple comparisons which is a challenging area that has not yet been fully explored. Moderate sample performances of the proposed techniques were very encouraging and detailed remarks on these were included in Mukhopadhyay and Aoshima (1997).     

A new interpretation of optimality forE-optimal designs in linear regression models

The optimal design problem for the estimation of several linear combinationsc′ _l ϑ (l=1, …,m) is considered in the usual linear regression modely=f′(x)ϑ (f(x) ∈ ℝ ^k ,ϑ ∈ ℝ ^k ). An optimal design minimizes a (weighted)p-norm of the variances of the least squares estimates for the different linear combinationsc′ _l ϑ. A generalized Elfving theorem is used to derive the relation of the new optimality criterion to theE-optimal design problem. It is shown that theE-optimal design for the parameterϑ minimizes such a (weighted)p-norm whenever the vectorc=(c′ ₁, …, c′_k)′ is an inball vector of a symmetric convex and compact “Elfving set” in.     

Density estimation for randomly distributed circular objects

Summary Nearest neighbour methods traditionally used to estimate density of a sessile biological population treat individuals as points. The present paper suggests distance-based density estimators which treat individuals as circles with variable areas. Distribution of distance between a sample point and thek-th (k = 1, 2, 3, …) nearest circle is derived. Maximum likelihood estimator of density is obtained from a random sample of point tok-th order distances. Assuming a skewed distribution for the circle radius, moment estimators of density and mean circle area are derived.     

On kalman filtering,posterior mode estimation and fisher scoring in dynamic exponential family regression

Summary Dynamic exponential family regression provides a framework for nonlinear regression analysis with time dependent parametersβ ₀,β ₁, …,β _t, …, dimβ _t=p. In addition to the familiar conditionally Gaussian model, it covers e.g. models for categorical or counted responses. Parameters can be estimated by extended Kalman filtering and smoothing. In this paper, further algorithms are presented. They are derived from posterior mode estimation of the whole parameter vector (β′₀, …,β′_t) by Gauss-Newton resp. Fisher scoring iterations. Factorizing the information matrix into block-bidiagonal matrices, algorithms can be given in a forward-backward recursive form where only inverses of “small”p×p-matrices occur. Approximate error covariance matrices are obtained by an inversion formula for the information matrix, which is explicit up top×p-matrices. Heinz Leo Kaufmann, my friend and coauthor for many years, died in a tragical rock climbing accident in August 1989. This paper is dedicated to his memory.     

Multivariate gamma distributions with one-factorial accompanying correlation matrices and applications to the distribution of the multivariate range

Summary A new representation for the characteristic function of the joint distribution of the Mahalanobis distances betweenk independentN(μ, Σ)-distributed points is given. Especially fork=3 the corresponding distribution function is obtained as a special case of multivariate gamma distributions whose accompanying normal distribution has a positive semidefinite correlation matrix with correlationsϱ _ij=−a _i a _j. These gamma distribution functions are given here by one-dimensional parameter integrals. With some further trivariate gamma distributions third order Bonferroni inequalities are derived for the upper tails of the distribution function of the multivariate range ofk independentN(μ, I)-distributed points. From these inequalities very accurate (conservative) approximations to upperα-level bounds can also be computed for studentized multivariate ranges.     

On the product of gamma random variables

We give expressions for the distribution and density of a product of gamma or equivalently chi-square random variables. In particular, we give the distribution of the product of two independent gamma variables of mean k in terms of the Bessel functions K ₁, … , K _k .     

Determining of threat perceptions affecting direction of the US defense expenditures by using ordinal regression models

The article examines whether the US threat perceptions defined in terms of federal government national defense outlays in billions of constant (FY 2000) dollars change along with periodical changes in international politics between 1945 and 2007. Three different models affecting direction of the US defense expenditures are developed. The first model are estimated by using five link functions even though results of only two of them, complementary log–log and cauchit, are presented. As complementary log–log produced the best results, others models are predicted by using only this function. The parameter estimates of complementary log–log function for the first model indicate that four of these variables (Ford, Carter, Reagan and Bush Sr.) out of eleven are significant in the category of presidents. “Truman Docrtrine/Cominform”, “Korean War”, “Vietnam War”, and “Invasion of Iraq” also seem to be the important independent variables on empirical grounds for the first model. While “Party”, “Invasion of Iraq”, “Vietnam War”, “Korean War”, and “Cuban Missile Crisis” constitute the important independent variables on empirical grounds for the second model, “Korean War”, “Vietnam War”, “Invasion of Iraq”, “Truman Docrtrine/Cominform”, “The Cold War and New World Order”, and “Cuban Missile Crisis” are important independent variables on empirical grounds for the third model. Estimations based on these three models therefore suggest that aforementioned independent variables do indeed have effect on the US defense expenditures.     

A note on reliability properties of k-record statistics

LetY _k,n denote the nth (upper) k-record value of an infinite sequence of independent, identically distributed random variables with common continuous distribution function F. We show that if the nth k-record valueY _k,n has an increasing failure rate (IFR), thenY _l,n (l<k) andY _k+1,n+1(n≤k+1) also have IFR distributions. On the other hand, ifY _k,n has a decreasing failure rate (DFR), thenY _l,n (1>k) has also a DFR distribution. We also present some results concerning log convexity and log concavity ofY _k,n .     

Conditional nonparametric frontier models for convex and nonconvex technologies: a unifying approach

The explanation of productivity differentials is very important to identify the economic conditions that create inefficiency and to improve managerial performance. In the literature two main approaches have been developed: one-stage approaches and two-stage approaches. Daraio and Simar (2005, J Prod Anal 24(1):93–121) propose a fully nonparametric methodology based on conditional FDH and conditional order-m frontiers without any convexity assumption on the technology. However, convexity has always been assumed in mainstream production theory and general equilibrium. In addition, in many empirical applications, the convexity assumption can be reasonable and sometimes natural. Lead by these considerations, in this paper we propose a unifying approach to introduce external-environmental variables in nonparametric frontier models for convex and nonconvex technologies. Extending earlier contributions by Daraio and Simar (2005, J Prod Anal 24(1):93–121) as well as Cazals et al. (2002, J Econometrics 106:1–25), we introduce a conditional DEA estimator, i.e., an estimator of production frontier of DEA type conditioned to some external-environmental variables which are neither inputs nor outputs under the control of the producer. A robust version of this conditional estimator is proposed too. These various measures of efficiency provide also indicators of convexity which we illustrate using simulated and real data. Cinzia Daraio received Research support from the Italian Ministry of Education Research on Innovation Systems Project (iRis) “The reorganization of the public system of research for the technological transfer: governance, tools and interventions” and from the Italian Ministry of Educational Research Project (MIUR 40% 2004) “System spillovers on the competitiveness of Italian economy: quantitative analysis for sectoral policies” which are acknowledged. Léopold Simar received Research support from the “Interuniversity Attraction Pole”, Phase V (No. P5/24) from the Belgian Government (Belgian Science Policy) is acknowledged.     

Strong consistency of the MLE under random censoring

LetX ₁, ...,X _n be an i.i.d. sample from some parametric family {_θ :θ (Θ} of densities. In the random censorship model one observesZ _i =min (X _i ,Y _i ) andδ _i =1_{ x _i ≤Y _i}, whereY _i is a censoring variable being independent ofX _i . In this paper we investigate the strong consistency ofθ _n maximizing the modified likelihood function based on (Z _i ,δ _i , 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data. Work partially supported by the “Deutsche Forschungsgemeinschaft”.     

An empirical likelihood method for spatial regression

Properties of a “blockwise”empirical likelihood for spatial regression with non-stochastic regressors are investigated for spatial data on a lattice. The method enables nonparametric confidence regions for spatial trend parameters to be calibrated, even though non-random regressors introduce non-stationary forms of spatial dependence into the “blockwise” construction. Additionally, the regression results are valid in a general framework allowing for a variety of behavior in regressor variables as well as the underlying spatial error process. The same regression method also applies when the regressors are stochastic.     

Do components of smooth tests of fit have diagnostic properties?

Smooth goodness of fit tests were introduced by Neyman (1937). They can be regarded as a compromise between globally consistent (“omnibus”) tests of fit and procedures having high power in the direction of a specific alternative. It is commonly believed that components of smooth tests like, e.g., skewness and kurtosis measures in the context of testing for normality, have special diagnostic properties in case of rejection of a hypothesisH ₀ in the sense that they constitute direct measures of the kind of departure fromH ₀. Recent years, however, have witnessed a complete change of attitude towards the diagnostic capabilities of skewness and kurtosis measures in connection with normality testing. In this paper, we argue that any component of any smooth test of fit is strictly non-diagnostic when used conventionally. However, a proper rescaling of components does indeed achieve the desired “directed diagnosis”.     

Strong uniform convergence for the estimator of the regression function under ϕ-mixing conditions

Suppose the observations (X _i,Y _i), i=1,…, n, are ϕ-mixing. The strong uniform convergence and convergence rate for the estimator of the regression function was studied by serveral authors, e.g. G. Collomb (1984), L. Gy?rfi et al. (1989). But the optimal convergence rates are not reached unless the Y _i are bounded or the E exp (a|Y _i|) are bounded for some a>0. Compared with the i.i.d. case the convergence of the Nadaraya-Watson estimator under ϕ-mixing variables needs strong moment conditions. In this paper we study the strong uniform convergence and convergence rate for the improved kernel estimator of the regression function which has been suggested by Cheng P. (1983). Compared with Theorem A in Y. P. Mack and B. Silverman (1982) or Theorem 3.3.1 in L. Gy?rfi et al. (1989), we prove the convergence for this kind of estimators under weaker moment conditions. The optimal convergence rate for the improved kernel estimator is attained under almost the same conditions of Theorem 3.3.2 in L. Gy?rfi et al. (1989). Received: September 1999     

Simultaneous estimation after selection and ranking and other procedures: The negative exponential case

We considerk (≥2) independent negative exponential populations with unknown location parameters and unknown but equal scale parameter. We incorporate the existing purely sequential and three-stage sampling procedures for selecting the “best” population and study the asymptotic second-order characteristics of the proposed fixed-size simultaneous confidence regions for the location parameters constructed after selection and ranking. Some direct estimation procedures have also been discussed.     

On Minimally-supported <Emphasis Type="Italic">D</Emphasis>-optimal Designs for Polynomial Regression with Log-concave Weight Function

This paper studies minimally-supported D-optimal designs for polynomial regression model with logarithmically concave (log-concave) weight functions. Many commonly used weight functions in the design literature are log-concave. For example, and exp(−x ²) in Theorem 2.3.2 of Fedorov (Theory of optimal experiments, 1972) are all log-concave. We show that the determinant of information matrix of minimally-supported design is a log-concave function of ordered support points and the D-optimal design is unique. Therefore, the numerically D-optimal designs can be constructed efficiently by cyclic exchange algorithm.     

The queuing system GI/M/I with finite waiting space

Summary Time dependent solution of the queuing system characterized by a general independent input, exponential service time distribution and a finite waiting space, has been first investigated by using the “phase method”. On finding the waiting room full, the customers then arriving may be turned away or the first customer may wait outside and the input process may be stopped till the customer then being served, completes its service. Steady state solutions of both these problems have been obtained and the difference in the operational behaviour of the two systems has been pointed out. For a 2-Erlang arrival distribution, the queuing parameters have been evaluated for different values ofρ _r andN.