adaptive estimation of regression models via moment restrictions abstract this paper considers adaptive estimation of regression models by means of generalized method of moments estimators two models are considered one with an i i d disturbance that is independent of the regressors and one with a conditionally symmetric but possibly heteroskedastic disturbance for both cases the paper develops linearized estimators that are asymptotically efficient if the number and variety of moment conditions are allowed to grow at an appropriate rate with the sample size in the general symmetric case no other adaptive estimator has yet been proposed also results of a small monte carlo study indicate that in the independence case the small sample performance of the generalized method of moments estimator can be quite good vis a vis other estimators previously proposed

quantiles expectiles and splines abstract a time varying quantile can be fitted by formulating a time series model for the corresponding population quantile and iteratively applying a suitably modified state space signal extraction algorithm it is shown that such quantiles satisfy the defining property of fixed quantiles in having the appropriate number of observations above and below like quantiles time varying expectiles can be estimated by a state space signal extraction algorithm and they satisfy properties that generalize the moment conditions associated with fixed expectiles because the state space form can handle irregularly spaced observations the proposed algorithms can be adapted to provide a viable means of computing spline based non parametric quantile and expectile regressions

m quantile models for small area estimation small area estimation techniques are employed when sample data are insufficient for acceptably precise direct estimation in domains of interest these techniques typically rely on regression models that use both covariates and random effects to explain variation between domains however such models also depend on strong distributional assumptions require a formal specification of the random part of the model and do not easily allow for outlier robust inference we describe a new approach to small area estimation that is based on modelling quantile like parameters of the conditional distribution of the target variable given the covariates this avoids the problems associated with specification of random effects allowing inter domain differences to be characterized by the variation of area specific m quantile coefficients the proposed approach is easily made robust against outlying data values and can be adapted for estimation of a wide range of area specific parameters including that of the quantiles of the distribution of the target variable in the different small areas results from two simulation studies comparing the performance of the m quantile modelling approach with more traditional mixed model approaches are also provided

classroom note finding the center of a circular starting line in an ancient greek stadium two methods for finding the center and radius of a circular starting line of a racetrack in an ancient greek stadium are presented and compared the first is a method employed by the archaeologists who surveyed the starting line and the second is a least squares method leading to a maximum likelihood circle we show that the first method yields a circle whose radius is somewhat longer than the radius determined by the least squares method and propose reasons for this difference a knowledge of the center and radius of the starting line is useful for determining units of length and angle used by the ancient greeks in addition to providing information on how ancient racetracks were laid out

a tutorial history of least squares with applications to astronomy and geodesy this article surveys the history development and applications of least squares including ordinary constrained weighted and total least squares the presentation includes proofs of the basic theory in particular unitary factorizations and singular value decompositions of matrices numerical examples with real data demonstrate how to set up and solve several types of problems of least squares the bibliography lists comprehensive sources for more specialized aspects of least squares

consistent least squares fitting of ellipsoids a parameter estimation problem for ellipsoid fitting in the presence of measurement errors is considered the ordinary least squares estimator is inconsistent and due to the nonlinearity of the model the orthogonal regression estimator is inconsistent as well i e these estimators do not converge to the true value of the parameters as the sample size tends to infinity a consistent estimator is proposed based on a proper correction of the ordinary least squares estimator the correction is explicitly given in terms of the true value of the noise variance

estimation of tail risk based on extreme expectiles we use tail expectiles to estimate value at risk var expected shortfall es and marginal expected shortfall mes three instruments of risk protection of utmost importance in actuarial science and statistical finance the concept of expectiles is a least squares analogue of quantiles both expectiles and quantiles were embedded in the more general class of m quantiles as the minimizers of an asymmetric convex loss function it has been proved very recently that the only m quantiles that are coherent risk measures are the expectiles moreover expectiles define the only coherent risk measure that is also elicitable the elicitability corresponds to the existence of a natural backtesting methodology the estimation of expectiles did not however receive yet any attention from the perspective of extreme values the first estimation method that we propose enables the usage of advanced high quantile and tail index estimators the second method joins together the least asymmetrically weighted squares estimation with the tail restrictions of extreme value theory we establish the limit distributions of the proposed estimators when they are located in the range of the data or near and even beyond the maximum observed loss a main tool is to first estimate the intermediate large expectile based var es and mes and then extrapolate these estimates to the very far tails we show through a detailed simulation study the good performance of the procedures and also present concrete applications to medical insurance data and three large us investment banks

optimal forecast combinations under general loss functions and forecast error distributions existing results on the properties and performance of forecast combinations have been derived in the context of mean squared error loss under this loss function empirical studies have generally found that equally weighted combined forecasts lead to better performance than estimates of optimal forecast combination weights which in turn outperform the best individual predictions we show that this and other results can be overturned when asymmetries are introduced in the loss function and the forecast error distribution is skewed we characterize the optimal combination weights for the most commonly used alternatives to mean squared error loss and demonstrate how the degree of asymmetry in the loss function and skews in the underlying forecast error distribution can significantly change the optimal combination weights we also propose estimation methods and investigate their small sample properties in simulations and in an inflation forecasting exercise

on the early history of the singular value decomposition this paper surveys the contributions of five mathematicians eugenio beltrami 1835 1899 camille jordan 1838 1921 james joseph sylvester 1814 1897 erhard schmidt 1876 1959 and hermann weyl 1885 1955 who were responsible for establishing the existence of the singular value decomposition and developing its theory

assessing value at risk with care the conditional autoregressive expectile models in this paper we propose a downside risk measure the expectile based value at risk evar which is more sensitive to the magnitude of extreme losses than the conventional quantile based var qvar the index of an evar is the relative cost of the expected margin shortfall and hence reflects the level of prudentiality it is also shown that a given expectile corresponds to the quantiles with distinct tail probabilities under different distributions thus an evar may be interpreted as a flexible qvar in the sense that its tail probability is determined by the underlying distribution we further consider conditional evar and propose various conditional autoregressive expectile models that can accommodate some stylized facts in financial time series for model estimation we employ the method of asymmetric least squares proposed by newey and powell newey w k powell j l 1987 asymmetric least squares estimation and testing econometrica 55 819 847 and extend their asymptotic results to allow for stationary and weakly dependent data we also derive an encompassing test for non nested expectile models as an illustration we apply the proposed modeling approach to evaluate the evar of stock market indices 2008 elsevier b v all rights reserved

saddlepoint approximations for regression models summary this paper uses the techniques of saddlepoint or tilted exponential approximation to develop an approximation to the small sample distribution of estimators defined by a system of estimating equations when observations are independently but not identically distributed this allows for the explicit treatment of models with explanatory variables the approximation is applied to l1 regression and the accuracy of the approximation is demonstrated in a case in which the usual normal approximation is very poor an analysis of the example indicates the usefulness of a new variant of exponential tilting in calculating the transformation measure a quantity which plays an important role in assuring the accuracy of the approximation

stretched exponential fixation in stochastic ising models at zero temperature we study a class of continuous time markov processes which describes 1 spin flip dynamics on the hypercubic lattice d d 2 with initial spin configurations chosen according to the bernoulli product measure with density p of spins 1 during the evolution the spin at each site flips at rate c 0 or 0 0 q t exp t 1 d for large t in d 2 we obtain the complementary bound for arbitrary 0 q t exp t 1 2 for large t

pairwise difference estimators of censored and truncated regression models abstract this paper proposes a class of estimators of the semiparametric censored regression model under the assumption that the error terms are i i d and independent of the covariates the estimators exploit the fact that for a pair of observations a particular transformation of the censored variables depending upon the parameter vector and both covariate vectors will be identically distributed therefore the difference in the transformed dependent variables will be symmetrically distributed around zero when the transformation is evaluated at the true parameter vector an analogous class of estimators for the truncated regression model includes the bhattacharya chernoff and yang 1983 estimator as a special case the estimators are defined as minimizers of u processes so the large sample theory for classical m estimators is extended to this case conditions are given to ensure root n consistency and asymptotic normality of the estimators and a small scale simulation study for the censored regression estimator suggest that it will be well behaved in finite samples

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

quantiles for counts this article studies the estimation of conditional quantiles of counts given the discreteness of the data some smoothness must be artificially imposed on the problem we show that it is possible to smooth the data in a way that allows inference to be performed using standard quantile regression techniques the performance and implementation of the estimators are illustrated by simulations and an application

estimation of type 3 tobit models using symmetric trimming and pairwise comparisons abstract this paper considers estimation of the semiparametric type 3 tobit model we construct two step estimators that use the ideas of symmetric trimming and pairwise comparisons which have been proposed for the censored regression model by powell 1986 and honore and powell 1994 the estimators are consistent and asymptotically normal under two identifying assumptions conditional symmetry and independence between the errors and regressors the ideas and techniques are also implementable in other selection models that are variations of the basic type 3 tobit model we demonstrate this by applying the proposed method in estimating an empirical model considered by udry 1994

making and evaluating point forecasts typically point forecasting methods are compared and assessed by means of an error measure or scoring function with the absolute error and the squared error being key examples the individual scores are averaged over forecast cases to result in a summary measure of the predictive performance such as the mean absolute error or the mean squared error i demonstrate that this common practice can lead to grossly misguided inferences unless the scoring function and the forecasting task are carefully matched effective point forecasting requires that the scoring function be specified ex ante or that the forecaster receives a directive in the form of a statistical functional such as the mean or a quantile of the predictive distribution if the scoring function is specified ex ante the forecaster can issue the optimal point forecast namely the bayes rule if the forecaster receives a directive in the form of a functional it is critical that the scoring function be consistent for it in the sense that t

flexible expectile regression in reproducing kernel hilbert spaces abstractexpectile first introduced by newey and powell in 1987 in the econometrics literature has recently become increasingly popular in risk management and capital allocation for financial institutions due to its desirable properties such as coherence and elicitability the current standard tool for expectile regression analysis is the multiple linear expectile regression proposed by newey and powell in 1987 the growing applications of expectile regression motivate us to develop a much more flexible nonparametric multiple expectile regression in a reproducing kernel hilbert space the resulting estimator is called kere which has multiple advantages over the classical multiple linear expectile regression by incorporating nonlinearity nonadditivity and complex interactions in the final estimator the kernel learning theory of kere is established we develop an efficient algorithm inspired by majorization minimization principle for solving the entire solution path of kere it is shown that the algori

phase transitions in the quadratic contact process on complex networks the quadratic contact process qcp is a natural extension of the well studied linear contact process where infected 1 individuals infect susceptible 0 neighbors at rate ensuremath lambda and infected individuals recover 1 0 at rate 1 in the qcp a combination of two 1 s is required to effect a 0 1 change we extend the study of the qcp which so far has been limited to lattices to complex networks we define two versions of the qcp vertex centered vqcp and edge centered eqcp with birth events 1 ensuremath 0 ensuremath 1 1 ensuremath 1 ensuremath 1 and 1 ensuremath 1 ensuremath 0 1 ensuremath 1 ensuremath 1 respectively where ensuremath represents an edge we investigate the effects of network topology by considering the qcp on random regular erd ifmmode mbox h o else h o fi s r enyi and power law random graphs we perform mean field calculations as well as simulations to find the steady state fraction of occupied vertices as a function of the birth rate we find that on the random regular and erd ifmmode mbox h o else h o fi s r enyi graphs there is a discontinuous phase transition with a region of bistability whereas on the heavy tailed power law graph the transition is continuous the critical birth rate is found to be positive in the former but zero in the latter

panel data discrete choice models with lagged dependent variables in this paper we consider identification and estimation in panel data discrete choice models when the explanatory variable set includes strictly exogenous variables lags of the endogenous dependent variable as well as unobservable individual specific effects for the binary logit model with the dependent variable lagged only once chamberlain 1993 gave conditions under which the model is not identified we present a stronger set of conditions under which the parameters of the model are identified the identification result suggests estimators of the model and we show that these are consistent and asymptotically normal although their rate of convergence is slower than the inverse of the square root of the sample size we also consider identification in the semiparametric case where the logit assumption is relaxed we propose an estimator in the spirit of the conditional maximum score estimator manski 1987 and we show that it is consistent in addition we discuss an extension of the identification result to multinomial discrete choice models and to the case where the dependent variable is lagged twice finally we present some monte carlo evidence on the small sample performance of the proposed estimators for the binary response model

total least squares state of the art regression in numerical analysis total least squares regression tls fits a line to data where errors may occur in both the dependent and independent variables in higher dimensions tls fits a hyperplane to such data the elementary algorithm presented here fits readily in a first course in numerical linear algebra

a test for asymmetry associated with the hodges lehmann estimator abstract one version of the hodges lehmann estimator median xi xj 2 1 i j n is also the minimum distance estimator derived from a particular cramer von mises distance this distance evaluated at that is the minimized distance provides a natural statistic for testing asymmetry of the underlying distribution

threshold θ 2 contact processes on homogeneous trees we study the threshold 2 contact process on a homogeneous tree mathbb t b of degree b 1 with infection parameter 0 and started from a product measure with density p the corresponding mean field model displays a discontinuous transition at a critical point lambda rm c rm mf kappa theta and for lambda geq lambda rm c rm mf kappa theta it survives iff p geq p rm c rm mf kappa theta lambda where this critical density satisfies 0 0 we also show that lim b to infty b lambda rm c mathbb t b theta phi theta where 1 2 3 lim theta to infty phi theta infty and 0 lim inf b to infty b theta theta 1 p rm c mathbb t b theta infty leq lim sup b to infty b theta theta 1 p rm c mathbb t b theta infty infty

the survival of large dimensional threshold contact processes we study the threshold theta contact process on z d with infection parameter lambda we show that the critical point lambda c defined as the threshold for survival starting from every site occupied vanishes as d infinity this implies that the threshold theta voter model on z d has a nondegenerate extremal invariant measure when d is large

elicitation complexity of statistical properties a property or statistical functional is said to be elicitable if it minimizes expected loss for some loss function the study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization while recent work asks which properties are elicitable we instead advocate for a more nuanced question how many dimensions are required to indirectly elicit a given property this number is called the elicitation complexity of the property we lay the foundation for a general theory of elicitation complexity including several basic results about how elicitation complexity behaves and the complexity of standard properties of interest building on this foundation our main result gives tight complexity bounds for the broad class of bayes risks we apply these results to several properties of interest including variance entropy norms and several classes of financial risk measures we conclude with discussion and open directions

poisson overdispersion estimates based on the method of asymmetric maximum likelihood abstract a common difficulty in regression problems with poisson or binomial or other exponential family response variables is over dispersion the scatter around the fitted regression is too large by the standards of poisson variability this article concerns the description estimation and testing of various patterns of overdispersion with particular emphasis on the poisson case asymmetric maximum likelihood aml is a method of fitting regressions for the conditional percentiles of the response variable as a function of the predictors e g the conditional 90th percentile of y given x distances between the various regression percentiles give a direct assessment of overdispersion the discussion is carried through in terms of an archaeological data set where we see that the counts are overdispersed by a factor of 1 35 in one part of the covariate space but not at all in another moreover the overdispersion is about 40 larger in the positive response direction than in the negative the aml es

a consistent test for conditional symmetry in time series models the assumption of conditional symmetry is often invoked to validate adaptive estimation and consistent estimation of arch garch models by quasi maximum likelihood imposing conditional symmetry can increase the efficiency of bootstraps if the symmetry assumption is valid this paper proposes a procedure for testing conditional symmetry the proposed test does not require the data to be stationary or i i d and the dimension of the conditional variables can be infinite the proposed test is consistent and is asymptotically distribution free in addition the test is shown to have nontrivial power against root t local alternatives the size and power of the test are satisfactory even for small samples applying the test to various time series we reject conditional symmetry in inflation exchange rate and stock returns among the nonfinancial time series considered we find that investment the consumption of durables and manufacturing employment also reject conditional symmetry

optimal stopping under probability distortions in this paper we study optimal stopping problems with respect to distorted expectations with concave distortion functions our starting point is a seminal work of xu and zhou in 2013 who gave an explicit solution of such a stopping problem under a rather large class of distortion functionals in this paper we continue this line of research and prove a novel representation which relates the solution of an optimal stopping problem under distorted expectation to the sequence of standard optimal stopping problems and hence makes the application of the standard dynamic programming based approaches possible furthermore by means of the well known kusuoka representation we extend our results to optimal stopping under general law invariant coherent risk measures finally based on our representations we develop several monte carlo approximation algorithms and illustrate their power for optimal stopping under absolute semideviation risk measures

of quantiles and expectiles consistent scoring functions choquet representations and forecast rankings summary in the practice of point prediction it is desirable that forecasters receive a directive in the form of a statistical functional for example forecasters might be asked to report the mean or a quantile of their predictive distributions when evaluating and comparing competing forecasts it is then critical that the scoring function used for these purposes be consistent for the functional at hand in the sense that the expected score is minimized when following the directive we show that any scoring function that is consistent for a quantile or an expectile functional can be represented as a mixture of elementary or extremal scoring functions that form a linearly parameterized family scoring functions for the mean value and probability forecasts of binary events constitute important examples the extremal scoring functions admit appealing economic interpretations of quantiles and expectiles in the context of betting and investment problems the choquet type mixture representations give rise to simple checks of whether a forecast dominates another in the sense that it is preferable under any consistent scoring function in empirical settings it suffices to compare the average scores for only a finite number of extremal elements plots of the average scores with respect to the extremal scoring functions which we call murphy diagrams permit detailed comparisons of the relative merits of competing forecasts

a finite algorithm to fit geometrically all midrange lines circles planes spheres hyperplanes and hyperspheres the algorithm proved here solves the problem of orthogonal distance regression for the maximum norm with hyperplanes and hyperspheres for each finite set of points in a euclidean space of any dimension the algorithm determines through finitely many arithmetic operations all the hyperplanes and hyperspheres that minimize the maximum euclidean distance measured perpendicularly from the data the algorithm finds all the slabs bounded by parallel hyperplanes and all the spherical shells bounded by concentric hyperspheres that contain all the data and are rigidly supported by the data for which there does not exist any other pair of parallel hypersurfaces of the same type that intersect the data at the same points the computational complexity of the algorithm increases as the number of data points raised to the dimension of the ambient space the solutions are then the midrange hyperplanes in the thinnest slabs and the midrange hyperspheres in the thinnest shells their sensitivity to perturbations of the data is of the order of a power of the reciprocal of the smallest angle between two median hyperplanes separating two pairs of data points the methods of proof consist in showing that if a pair of parallel hyperplanes or hyperspheres is not rigidly supported but encompasses all the data then there exists a projective shift of their common projective center producing a thinner slab or shell that still contains all the data

higher order elicitability and osband s principle a statistical functional such as the mean or the median is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score such scoring functions are called strictly consistent for the functional the elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores in this paper we explore the notion of elicitability for multi dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions we cover the case of functionals with elicitable components but we also show that one dimensional functionals that are not elicitable can be a component of a higher order elicitable functional in the case of the variance this is a known result however an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the correct quantiles a direct consequence of applied interest is that the pair value at risk expected shortfall is jointly elicitable under mild conditions that are usually fulfilled in risk management applications

on the stability of a population growth model with sexual reproduction on z d d geq 2 we continue our study on the stability properties of a population growth model with sexual reproduction on z d d 2 in the author s previous work it was proved that in the type iv process the two dimensional symmetric model on z 2 the vacant state o is stable under perturbation of the initial state the first kind of perturbation and it is unstable under perturbation of the birth rate the second kind of perturbation in this paper we prove that in the type iii process on z 2 the vacant state o is stable under the second kind of perturbation and in three or higher dimensional symmetric models the vacant state o is unstable under the first kind of perturbation these results combined with the results obtained earlier provide a fairly complete picture concerning the stability properties of these models

estimation of best predictors of binary response abstract this paper investigates analog estimation of predictors of binary response given semiparametric restrictions on the response process results are presented for the isotonic response model and for the weaker single crossing model the isotonic model assumes that the response probability is monotone in a linear function of the conditioning variables in the single crossing model the response probability is above a given value when a linear function of the conditioning variables is positive and below that value otherwise it is found that analog estimators computed under the isotonicity restriction are generally consistent estimation under the single crossing restriction is consistent only for special loss functions

simulation based finite sample tests for heteroskedasticity and arch effects tests for heteroskedasticity in linear regressions are typically based on asymptotic approximations we show that the size of such tests can be perfectly controlledin nite samples through monte carlo test techniques with both gaussian and non gaussian heavy tailed disturbance distributions the procedures studied include standard heteroskedasticity tests e g glejser bartlett cochran hartley breusch pagan godfrey white szroeter as well as tests for arch type heterosked asticity sup type andcombinedtests are also proposedto allow for unknown breakpoints in the variance the fact that the proposedproced ures achieve size control andhave goodpower is demonstrated in a monte carlo simulation c 2003 elsevier b v all rights reserved

on m estimators and normal quantiles this paper explores a class of robust estimators of normal quantiles filling the gap between maximum likelihood estimators and empirical quantiles our estimators are linear combinations of m estimators their asymptotic variances can be arbitrarily close to variances of the maximum likelihood estimators compared with empirical quantiles the new estimators offer considerable reduction of variance at near normal probability distributions

locating objects in the plane using global optimization techniques we address the problem of locating objects in the plane such as segments arcs of circumferences arbitrary convex sets their complements or their boundaries given a set of points we seek the rotation and translation for such an object optimizing a very general performance measure which includes as a particular case the classical objectives in semi obnoxious facility location in general the above mentioned model yields a global optimization problem whose resolution is dealt with using difference of convex dc techniques such as outer approximation or branch and bound

glejser s test revisited godfrey 1996 journal of econometrics 72 275 299 has shown that the glejser test for heteroskedasticity is valid only under conditional symmetry here modifications of the glejser test are suggested the proposed test statistics are asymptotically valid even when the disturbances are not symmetrically distributed and can be used to test for heteroskedasticity when conditional location functions other than the conditional mean are estimated

asymmetric least squares estimation and testing this paper considers estimation and hypothesis tests for coefficients of linear regression models where the coefficient estimates are based on location measures defined by an asymmetric least squares criterion function these asymmetric least squares estimators have properties which are analogous to regression quantile estimators but are much simpler to calculate as are the corresponding test statistics the coefficient estimators can be used to construct test statistics for homoskedasticity and conditional symmetry of the error distribution and we find these tests compare quite favorably with other commonly used tests of these null hypotheses in terms of local relative efficiency consequently asymmetric least squares estimation provides a convenient and relatively efficient method of summarizing the conditional distribution of a dependent variable given the regressors and a means of testing whether a linear model is an adequate characterization of the typical value for this conditional distribution

fitting cylinders to data the problem of fitting cylinders to data arises in science and industry this article proves the existence of generalized cylinders cartesian products of generalized spheres and affine manifolds fitted to data by using many criteria including generalized least squares weighted median and midrange regressions

a simple consistent test of conditional symmetry in symmetrically trimmed tobit models we propose a weighted and sample size adjusted kolmogorov smirnov type statistic to test the assumption of conditional symmetry maintained in the symmetrically trimmed least squares stls approach of powell 1986b which is widely used to estimate censored or truncated regression models without making distributional assumptions our statistic is consistent and does not require any nonparametric smoothing although we test the validity of a conditional feature we also propose a bootstrap procedure to obtain the p values and critical values that are required to carry out the test in practical applications results from a simulation study suggest that our test can work very well even in small to moderately sized samples as an empirical illustration we apply our test to two datasets that have been used in the literature to estimate censored regression models using powell s stls approach to check whether the assumption of conditional symmetry is supported by these datasets

median spheres theory algorithms applications a generalized hypersphere is either a hyperplane or a hypersphere which consists of all points equidistant from a center geometrically a weighted median hypersphere minimizes a weighted average of the distances from it to finitely many data points as proved here for each finite data set there exists at least one weighted median generalized hypersphere moreover denote the sums of the weights of the data points inside by w outside by w and on the hypersphere by w 0 the present results show that each weighted median hypersphere is a weighted pseudo halving hypersphere in the sense that w w w 0 and passes through at least two distinct data points combinatorically a hypersphere is blocked if and only if it passes through data points in general position in the sense that no other hypersphere passes through the same data points a hypersphere is a halving hypersphere if and only if it is blocked contains exactly k data points inside confines exactly l data points outside and k l 1 in the plane the present results also show that if a median circle is not a halving circle then moving its center along a median between two data points on it until it passes through the next data point yields a halving circle relative to the center if the direction cosines of the external and internal data points have the same mean and variance then the median circle must be blocked and stays so under sufficiently small perturbations of the data moreover for every set of four points at least one unweighted median circle is blocked these results lend credence to a variant of a method used by archaeologists and explain some findings from operations research

contact processes in several dimensions this paper deals primarily with the basic contact processes introduced and studied by harris 12t4 these are random evolutions on the state space s a l l subsets of z a zd the d dimensional integer lattice with extremely simple local dynamics namely if we think of the contact process as representing the spread of an infection t es denoting the set of infected sites at time t then x e it becomes healthy at exponential rate 1 while x z a t is infected at a rate proport ional to the number of sites neighboring x where infection is present the proportionali ty constant 2 is called the infection parameter contact processes are perhaps the simplest s valued markov processes which exhibit a critical phenomenon infection emanating from a single site dies out with probabili ty one for small positive 2 but has positive probability of surviving for all time when is large there is a critical value 2 d where the transition occurs principal objectives of analysis are the precise formulation of the critical phenomenon and detailed description of the ergodic behavior both below and above a c our recent survey articles 10 and i 6 provide introductions to contact processes and to the general field of interacting s valued systems respectively we will make constant use of notation and techniques from the surveys assuming that the reader is familiar with thern the article 10 describes in some detail the current state of knowledge concerning one dimensional contact processes the theory for d 1 is now fairly complete with two major exceptions 1 is not even determined to one decimal place 1 18 2 e there is an

trimmed lad and least squares estimation of truncated and censored regression models with fixed effects this paper considers estimation of truncated and censored regression models with fixed effects up until now no estimator has been shown to be consistent as the cross section dimension increases with the time dimension fixed trimmed least absolute deviations and trimmed least squares estimators are proposed for the case where the panel is of length two and it is proven that they are consistent and asymptotically normal it is not necessary to maintain parametric assumptions on the error terms to obtain this result a small scale monte carlo study demonstrates that these estimators can perform well in small samples copyright 1992 by the econometric society