addendum to estimates of the determinant of a perturbed identity matrix linear algebra appl 558 2018 101 107 abstract many bounds for the determinant det i e of a perturbed identity matrix are known mostly the upper bound is inferior to the classical hadamard bound in this note we give simple and efficiently computable relative bounds differing by e f 3 where the upper bound is usually better than hadamard s bound

small domain estimation a conditional analysis abstract in estimating means totals and other parameters for small domains of a finite population the survey statistician is usually faced with a domain sample size that is random rather than controlled at the selection stage often a sensible approach is to make design based inference conditionally on the realized sample size in the domain nd in this article we suggest and analyze some small domain estimators and their design based conditional confidence intervals the conditional outlook leads us to some new small domain estimators that a are based on regression of pertinent auxiliary information and are therefore efficient to the extent that the auxiliary information is strong b are nearly design unbiased conditionally on nd as well as unconditionally and c give rise to design based confidence intervals that are valid conditionally as well as unconditionally the conditional properties of these new regression estimators are first derived theoretically then confirmed through a monte

verified bounds for the determinant of real or complex point or interval matrices abstract we discuss several methods to compute a verified inclusion of the determinant of a real or complex point or interval matrix for point matrices large condition number 1 0 15 and large dimension n 1000 still highly accurate inclusions are computed for real interval matrices we show that any vertex may be a unique extreme point for wide radii we show that preconditioning may widen an inclusion significantly and hadamard s bound may be much better

bias robust estimation in finite populations using nonparametric calibration a standard problem in sample survey inference is that of predicting the finite population total h of a function h y of a random variable y the model based approach to this problem first defines a working model for y and then predicts h by estimating its expectation under conditional on the sample values of y this approach leads to biased predictions if is incorrect we explore an automatic solution to this misspecification bias that uses nonparametric regression to define a robust but inefficient predictor of h and then calibrates this predictor for its bias under an application to prediction of the finite population distribution function of a population of australian beef farms is presented

higher order derivatives and perturbation bounds for determinants the first derivative of the determinant function is given by the well known jacobi s formula we obtain three different expressions for all higher order derivatives norms of these derivatives are then evaluated exactly

empirical likelihood estimation for finite populations and the effective usage of auxiliary information summary in finite population inference problems auxiliary population information is often available we show in this paper that the empirical likelihood method can be naturally applied to such problems to make effective use of the auxiliary information we prove that the resulting estimates have smaller asymptotic variances than the usual estimates which do not use auxiliary information a bahadur type representation for empirical likelihood sample quantiles is given simulation results show that the empirical likelihood estimates perform well among a number of competitors and are model robust

accurate floating point summation part i faithful rounding given a vector of floating point numbers with exact sum s we present an algorithm for calculating a faithful rounding of s i e the result is one of the immediate floating point neighbors of s if the sum s is a floating point number we prove that this is the result of our algorithm the algorithm adapts to the condition number of the sum i e it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number all statements are also true in the presence of underflow the algorithm does not depend on the exponent range our algorithm is fast in terms of measured computing time because it allows good instruction level parallelism it neither requires special operations such as access to mantissa or exponent it contains no branch in the inner loop nor does it require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithm are proved to be optimal

accurate sum and dot product algorithms for summation and dot product of floating point numbers are presented which are fast in terms of measured computing time we show that the computed results are as accurate as if computed in twice or k fold working precision k ge 3 for twice the working precision our algorithms for summation and dot product are some 40 faster than the corresponding xblas routines while sharing similar error estimates our algorithms are widely applicable because they require only addition subtraction and multiplication of floating point numbers in the same working precision as the given data higher precision is unnecessary algorithms are straight loops without branch and no access to mantissa or exponent is necessary

conditional coverage properties of finite population confidence intervals abstract standardized errors t v 1 2 were calculated for both ratio and regression estimators in each of 10 000 simple random samples of n 32 from each of six populations using four different variance estimators graphs show how the percentage of intervals t 1 96 v 1 2 that fail to contain t changes as a function of the average value of the auxiliary variable in the sample they reveal that a intervals using the variance estimators from standard linear regression theory were hopelessly unreliable b intervals using the conventional finite population variance estimators showed a striking excess of failures in badly balanced samples and c none of the four variance estimators produced satisfactory confidence intervals in populations arising from badly skewed prediction models

saddlepoint approximations in resampling methods summary saddlepoint approximations are shown to be easy to use and accurate in a variety of simple bootstrap and randomization applications examples include mean estimation ratio estimation two sample comparisons and autoregressive estimation

the weighted residual technique for estimating the variance of the general regression estimator of the finite population total summary the paper deals with design based estimation of the variance of the general regression estimator of the finite population total the usual taylor linearization variance estimator is an expression in the design weighted regression residuals in many applications the resulting expression is counterintuitive from a model based standpoint the improved variance estimator in this paper attaches another simple weight called g weight to each individual residual this new variance estimator i gives valid design based confidence intervals ii is nearly unbiased uiider a suitably chosen regression model and iii works well for conditional inference examples are given

conditioning ratio estimates under simple random sampling abstract the usual ratio estimates for a population mean use information on an auxiliary variable to improve the estimation the standard randomization analysis however does not take enough account of the observations on the auxiliary variable so estimates and estimated variates can be conditionally biased unless the sample is balanced with respect to the auxiliary variable i propose a method of adjustment for the bias of the estimate and its variance based only on the assumption of simple random sampling which uses an approximate conditional distribution of the estimate given the mean of the auxiliary variable roy all and cumberland 1981 illustrated empirically that the usual random sampling estimates and estimated variances could be badly conditionally biased when samples were obtained that were unbalanced in the auxiliary variate so if the sample mean and population mean both of which are known at the time of analysis are not close it is known that the results are biased but the usual rand

precision estimation in sample survey inference a criterion for choice between variance estimators summary we advocate the mean squared error of predicted squared error as a universal criterion for the choice between variance estimators in sample survey inference the predictive nature of the measure rewards variance estimators adapted to the amount of information in the actual sample this makes the new measure more satisfactory than the simpler mean squared error of the variance estimator the criterion turns out to be the same for design based as for model based inference and may also be used for comparison between the design and model based theoretical variances the theory is exemplified for the ratio estimator by a study of six variance estimators suggested in the literature these calculations are primarily made under a proportional regression model but consideration is also paid to model robustness three of the six estimators are shown to be inferior to the three others which are approximately equivalent

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

accurate floating point product and exponentiation several different techniques and softwares intend to improve the accuracy of results computed in a fixed finite precision here we focus on a method to improve the accuracy of the product of floating point numbers we show that the computed result is as accurate as if computed in twice the working precision the algorithm is simple since it only requires addition subtraction and multiplication of floating point numbers in the same working precision as the given data such an algorithm can be useful for example to compute the determinant of a triangular matrix and to evaluate a polynomial when represented by the root product form it can also be used to compute the integer power of a floating point number

estimates of the determinant of a perturbed identity matrix abstract recently brent et al presented new estimates for the determinant of a real perturbation i e of the identity matrix they give a lower and an upper bound depending on the maximum absolute value of the diagonal and the off diagonal elements of e and show that either bound is sharp their bounds will always include 1 and the difference of the bounds is at least tr e in this note we present a lower and an upper bound depending on the trace and frobenius norm e f of the real or complex perturbation e where the difference of the bounds is not larger than 2 o 3 provided that 1 moreover we prove a bound on the relative error between det i e and exp tr e of order 2

bounds for determinants of perturbed m matrices abstract a sharp upper bound for the relative error of the determinant of an m matrix in terms of the relative errors of the matrix entries is given it improves and completes a bound derived by jungong and jiang

estimating a finite population mean from unequal probability samples abstract two approaches to estimating a finite population mean from unequal probability samples are contrasted in the prediction approach a model is specified for the population values and is used to predict the nonsampled values in the generalized regression approach the horvitz thompson estimator of the population mean is modified to allow the introduction of covariate information sarndal 1980 generalized regression estimates have the desirable property of asymptotic design consistency which is not always enjoyed by estimates in the prediction class however it is suggested that the prediction approach is the more principled method of estimation if asymptotic design consistency is a desirable property in a given application then only models that yield asymptotically design consistent prediction estimates should be used two classes of models with this property are suggested namely fixed and random effects models that allow a separate intercept for subclasses of the population indexed by the pr

the prediction approach to robust variance estimation in two stage cluster sampling abstract robust statistics are derived for estimating the error variance when a finite population total or a ratio of totals is estimated from a two stage cluster sample in which the first stage sampling fraction is small the analysis is based on prediction superpopulation models that allow for very general correlation structure within clusters results are obtained for when all cluster sizes are known and for when sizes are known only for sample clusters ratio regression and mean of ratios statistics are in the class of population total estimators considered

the finite population linear regression estimator and estimators of its variance an empirical study abstract the usual variance estimator for the linear regression estimator of a finite population total is examined under some prediction superpopulation models its bias is compared with that of the least squares variance estimator also described are three bias robust alternatives one of which is the jackknife variance estimator the theoretical results are supported by an empirical study in which simple and restricted random samples as well as some purposive nonrandom samples are drawn from six real populations the results illustrate the need for prediction models and the inadequacy of randomization per se in providing a theoretical framework for robust inferences

poststratification and conditional variance estimation poststratification estimation is a technique used in sample surveys to improve efficiency of estimators survey weights are adjusted to force the estimated numbers of units in each of a set of estimation cells to be equal to known population totals the resulting weights are then used in forming estimates of means or totals of variables collected in the survey for example in a household survey the estimation cells may be based on age race sex categories of individuals and the known totals may come from the most recent population census although the variance of a poststratified estimator can be computed over all possible sample configurations inferences made conditionally on the achieved sample configuration are desirable theory and a simulation study using data from the u s current population survey are presented to study both the conditional bias and variance of the poststratified estimator of a total the linearization balanced repeated replication and jackknife variance estimators are also exa

recovery of minimal bases and minimal indices of rational matrices from fiedler like pencils abstract let g be an n n rational matrix by considering a minimal realization of g fiedler like pencils such as fiedler pencils generalized fiedler pencils and fiedler pencils with repetition of g have been proposed recently which are shown to be linearizations of g we show that the fiedler like pencils allow operation free recovery of eigenvectors and minimal bases of g that is eigenvectors and minimal bases of g can be recovered from those of the fiedler like pencils of g without performing any arithmetic operations further we show that the minimal indices of g can be easily recovered from those of the fiedler like pencils of g we also show that a fiedler pencil with repetition of g can be constructed directly from that of a matrix polynomial

does simple random sampling provide adequate balance une etude empirique montre que sous un echantillonnage aleatoire simple les intervalles de confiance du total peuvent avoir des probabilites de recouvrement conditionnelles qui sont imposantes de facon persistante lorsque la taille de l echantillon croit

outlier resistant alternatives to the ratio estimator abstract many techniques have been suggested to lower the impact of outliers on sample survey estimates outliers can be downweighted by winsorization that is by replacing extreme data points by a data dependent or a predetermined value before calculating estimates another approach is to reduce the weight of outliers from the inverse sampling fraction to 1 in the estimation of population characteristics in this article the problem of estimating the population mean using auxiliary information in the presence of outliers is considered a resistant version of the ratio estimator is introduced it is constructed with a m or a gm estimator of the slope of the regression model through the origin which is implicitly called on when considering the ratio estimator the asymptotic biases and asymptotic variances of the proposed alternatives to the ratio estimator are calculated with respect to the randomization of the sampling plan the selection of a resistant estimator is seen to involve a trade off betwee

recovery of eigenvectors of rational matrix functions from fiedler like linearizations abstract linearization is a standard method often used when dealing with matrix polynomials recently the concept of linearization has been extended to rational matrix functions and fiedler like matrix pencils for rational matrix functions have been constructed a linearization l of a rational matrix function g does not necessarily guarantee a simple way of recovering eigenvectors of g from those of l we show that fiedler like pencils of g allow an easy operation free recovery of eigenvectors of g that is eigenvectors of g are recovered from eigenvectors of fiedler like pencils of g without performing any arithmetic operations we also consider fiedler like pencils of the rosenbrock system polynomial s associated with an lti system in state space form ssf and show that the fiedler like pencils allow operation free recovery of eigenvectors of s the eigenvectors of s are the invariant zero directions of the lti system

local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems abstract this paper presents a definition for local linearizations of rational matrices and studies their properties this definition allows to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity this new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970 s until 2020 for computing zeros poles and eigenvalues of rational matrices particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation

simultaneously model unbiased design unbiased estimation this paper proposes a class of inferential procedures incorporating both design and estimation elements that yield estimates of means that are simultaneously model unbiased and design unbiased classical regression procedures yield conditionally unbiased estimators for the mean conditioning on the model and choice of observation points in contrast design based methods yield estimators that are unconditionally unbiased on matter what the form of the underlying model variance properties of the proposed class are examined and applications to bioavailability water quality from mine run off and finite population regression estimation are considered the proposed procedures perform well especially in the typical case where a model is only approximately correct

an empirical study of the ratio estimator and estimators of its variance abstract this paper reports results from an empirical study of the ratio estimator for a finite population total from each of six real populations 1 000 simple random samples 1 000 restricted random samples and three nonrandom samples of size 32 are drawn performance of the ratio estimator and of five estimators of its variance is compared with theoretical results generated using a prediction superpopulation models and b probability sampling distributions the results presented graphically show that theory based on prediction models can reveal relationships that are essential in making inferences but that are concealed in probability sampling analyses

median estimation in the presence of auxiliary information summary this paper deals with the estimation under simple random sampling of a finite population median in the presence of an auxiliary variable an estimator which is a simple modification of the ratio estimator for the mean is studied along with two other proposed estimators derived from a different approach these estimators are applicable in situations where only the population median or a grouped frequency distribution of the auxiliary variable is known on the basis of the asymptotic properties derived and some simulation results the efficiencies of these estimators are compared it is shown that the two estimators proposed unlike that obtained from modifying the ratio estimator always dominate the sample median in mean squared error and that their efficiencies depend directly on the probability of concordance rather than on the validity of a linearity assumption between the survey variable and the auxiliary variable when the linearity assumption is approximately satisfied the three estimators are seen to have comparable efficiencies while the modified ratio estimator is found to be quite sensitive to departures from the linearity assumnption