finite size scaling in the driven lattice gas we present a monte carlo study of the high temperature phase of the two dimensional driven lattice gas at infinite driving field we define a finite volume correlation length verify that this definition has a good infinite volume limit independent of the lattice geometry and study its finite size scaling behavior the results for the correlation length are in good agreement with the predictions based on the field theory proposed by janssen schmittmann leung and cardy the theoretical predictions for the susceptibility and the magnetization are also well verified we show that the transverse binder parameter vanishes at the critical point in all dimensions d geqslant 2 and discuss how such result should be expected in the theory of janssen et al in spite of the existence of a dangerously irrelevant operator our results confirm the gaussian nature of the transverse excitations

comparing variances of correlated variables a test is proposed for the equality of the variances ofk 2 correlated variables pitman s test fork 2 reduces the null hypothesis to zero correlation between their sum and their difference its extension eliminating nuisance parameters by a bootstrap procedure is valid for any correlation structure between thek normally distributed variables a monte carlo study for several combinations of sample sizes and number of variables is presented comparing the level and power of the new method with previously published tests some nonnormal data are included for which the empirical level tends to be slightly higher than the nominal one the results show that our method is close in power to the asymptotic tests which are extremely sensitive to nonnormality yet it is robust and much more powerful than other robust tests

the canonical bicoherence part i definition multitaper estimation and statistics the central theme of this pair of papers parts i and ii ieee transactions on signal processing vol 57 no 4 april 2009 is a new definition the canonical bicoherence a combination of the canonical coherence and the bicoherence the canonical bicoherence is an effective tool for analyzing quadratic nonlinearity in multivariate signals in this first part the definition and properties of the canonical bicoherence are presented the feasibility of the canonical bicoherence in detecting quadratic phase coupling qpc of multivariate signals is explained theoretically illustrated by an example and verified by numerical simulations multitaper methods and a sequence of three singular value decompositions svd s are used to estimate canonical bicoherences to achieve reliable estimates with a reasonable amount of memory and computation time finally we show that the canonical bicoherence estimate has an approximate asymptotic kappachinu 2 distribution and the weighted jackknife method used over tapers and segments is applied to estimate variances of multitaper canonical bicoherence estimates

analytic smoothing for equipercentile equating under the common item nonequivalent populations design a cubic spline method for smoothing equipercentile equating relationships under the common item nonequivalent populations design is described statistical techniques based on bootstrap estimation are presented that are designed to aid in choosing an equating method degree of smoothing these include a asymptotic significance tests that compare no equating and linear equating to equipercentile equating b a scheme for estimating total equating error and for dividing total estimated error into systematic and random components the smoothing technique and statistical procedures are explored and illustrated using data from forms of a professional certification test

reliability inference for a multicomponent stress strength model based on kumaraswamy distribution abstract in this paper inference for a multicomponent stress strength mss model is studied under censored data when both latent strength and stress random variables follow kumaraswamy distributions with common shape parameters the maximum likelihood estimate of mss reliability is established and associated approximate confidence interval is constructed using the asymptotic distribution theory and delta method moreover pivotal quantities based generalized point and confidence interval estimates are presented for the mss reliability furthermore likelihood and generalized pivotal based estimates are also presented when the strength and stress variables have unequal shape parameters for complementary and comparison bootstrap confidence intervals are provided as well under common and unequal parameter cases in addition to compare the equivalence between strength and stress shape parameters the likelihood ratio test for hypothesis of interest is also discussed finally simulation study and a real data example are provided to investigate the performance of proposed procedures

temporal moments revisited why there is no better way for physically based model reduction in time 1 many hydro geo logical problems are highly complex in space and time coupled with scale issues variability and uncertainty especially time dependent models often consume enormous computational resources but model reduction techniques can alleviate this problem temporal moments tm offer an approach to reduce the time demands of transient hydro geo logical simulations tm reduce transient governing equations to steady state and directly simulate the temporal characteristics of the system if the equations are linear and coefficients are time independent this is achieved by an integral transform projecting the dynamic system response onto monomials in time in comparison to classical approaches of model reduction that involve orthogonal base functions however the monomials for tm are nonorthogonal which might impair the quality and efficiency of model reduction thus we raise the question of whether there are more suitable temporal base functions than the monomials that lead to tm in this work we will derive theoretically that there is only a limited class of temporal base functions that can reduce hydro geo logical models by comparing those to tm we conclude that in terms of gained efficiency versus maintained accuracy tm are the best possible choice while our theoretical results hold for all systems of linear partial or ordinary differential equations pdes odes with any order of space and time derivatives we illustrate our study with an example of pumping tests in a confined aquifer for that case we demonstrate that two four tm are sufficient to represent more than 80 90 of the dynamic behavior and that the information content strictly increases with increasing tm order

assessing the performance of an allocation rule the problem of estimating the error rates of a sample based rule on the basis of the same sample used in its construction is considered the apparent error rate is an obvious nonparametric estimate of the conditional error rate of a sample rule but unfortunately it provides too optimistic an assessment attention is focussed on the formation of improved estimates mainly through appropriate bias correction of the apparent error rate in this respect the role of the bootstrap a computer based methodology is highlighted

inferential methods for elasticity estimates abstract elasticities are often estimated from the results of demand analysis however drawing inferences from them may involve assumptions that could influence the outcome in this paper we investigate one of the most common forms of elasticity which is defined as a ratio of estimated relationships and demonstrate how the fieller method for the construction of confidence intervals can be used to draw inferences we estimate the elasticities of expenditure from engel curves using a variety of estimation models parametric engel curves are modelled using ols mm robust regression and tobit semiparametric engel curves are estimated using a penalized spline regression we demonstrate the construction of confidence intervals of the expenditure elasticities for a series of expenditure levels as well as the estimated cumulative density function for the elasticity evaluated for a particular household

confidence intervals for densities in line transect sampling we consider the problem of estimating the density d of a population of objects or animals with ungrouped data from line transect sampling we present a method of deriving an approximate confidence interval for d this confidence interval requires the estimation of a certain standard deviation by the bootstrap method it also requires the estimation with negligible bias of the value f o of a density functions we present a method of bias reduction in the case wheref o is estimated by the fourier series method monte carlo simulations indicate that the coverage rates are smaller than their nominal values but the bias reduction offers an improvement especially when f is peaked at 0 or when it is long tailed

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

occupational segregation in the multidimensional case decomposition and tests of significance a new multidimensional version of the g segregation index is developed and applied to the study of occupational segregation u s current population survey data are used to measure the difference in occupational segregation between races as well as the change between time periods decomposition of the difference change into occupation mix and gender composition components indicates the contribution of each factor because these inequality measures are computed from sample data distributional information required to test hypotheses is lacking two computer intensive methods for estimating the distributional properties are demonstrated the approximate randomization and bootstrap methodologies are used to test for statistically significant differences in segregation between races and for changes over time in addition the components of the decomposition are examined for statistical significance

large sample theory of intrinsic and extrinsic sample means on manifolds ii this article develops nonparametric inference procedures for estimation and testing problems for means on manifolds a central limit theorem for frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on riemannian manifolds central limit theorems are also obtained for extrinsic sample means w r t an arbitrary embedding of a differentiable manifold in a euclidean space bootstrap methods particularly suitable for these problems are presented applications are given to distributions on the sphere s d directional spaces real projective space rp n 1 axial spaces complex projective space cp k 2 planar shape spaces w r t veronese whitney embeddings and a three dimensional shape space 4 3

the analysis of repeated categorical measurements subject to nonignorable nonresponse abstract conditional likelihood methods have been proposed for analyzing the repeated measurement of categorical responses this article extends the methods to include partially classified subjects the extension allows for an ignorable or nonignorable nonresponse mechanism and uses standard statistical software for the computations two examples of incomplete longitudinal categorical data illustrate the method

structural analysis of subjective categorical data a general approach to the analysis of subjective categorical data is considered in which agreement matrices of two or more raters are directly expressed in terms of error and agreement parameters the method provides focused analyses of ratings from several raters for whom ratings have measurement error distributions that may induce bias in the evaluation of substantive questions of interest each rater s judgment process is modeled as a mixture of two components an error variable that is unique for the rater in question as well as an agreement variable that operationalizes the true values of the units of observation the statistical problems of identification estimation and testing of such measurement models are discussed

assessment of individual and population bioequivalence using the probability that bioavailabilities are similar in this paper a new method for the assessment of both individual and population bioequivalence is proposed that is derived from the approach of schall and luus 1993 statistics in medicine 12 1109 1124 the method considers the probability that the bioavailability of the test product is close to the bioavailability of the reference product against the probability that the bioavailability of the reference product is close to itself when administered on different occasions the method can accommodate unequal variances of the bioavailabilities of the test and reference formulations as well as formulation by subject interaction it is thus as general and comprehensive as the methods proposed by sheiner 1992 statistics in medicine 11 1777 1778 and schall and luus 1993 but the resulting bioequivalence criteria are simple and easy to interpret

double bootstrap for shrinkage estimators computer intensive bootstrap methods are potentially problematic when applied to biased estimators similar to ridge regression because biased estimators may lack a pivot we propose a modification for efron s bias corrected percentiles and suggest an approximate pivoting transformation for ridge regression we indicate practical aspects of beran s double bootstrap designed specifically for situations where there is nonnormality and a lack of a pivot econometric applications to estimating consumption functions a simulation with multicollinear data and new graphical tools are included

conditional bootstrap methods in the mean shift model summary bootstrap methods are not inherently conditional but they can be made so by appropri ate stratification of the simulated samples which bootstrap produces we show how stratification can work in a bootstrap analysis of mean shift in nile river flow data the results are compared with both parametric and semiparametric likelihood analyses the paper ends with some general remarks on conditional bootstraps bootstrap methods efron 1982 are simulation methods for assessing frequency properties of statistical analyses the simulation model being that which is fitted to the actual data often the simulation involves monte carlo rather than theoretical evaluation unfortunately the bootstrap as usually described is inherently unconditional rubin s 1981 discussion bears on this point however if the bootstrap simulation results can be properly stratified then it is possible to estimate relevant conditional probabilities stratification can sometimes be effected in the model itself as when fixing the values of sample sizes or explanatory variables otherwise one must run the unconditional bootstrap and stratify its output in the present paper we illustrate the latter situation in the analysis of a mean shift model section 2 describes the application and its analysis by parametric methods sections 3 and 4 produce alternative bootstrap analyses the first of which uses the stratification of an unconditional monte carlo simulation the second bootstrap analysis involves estimation of a likelihood function the final section contains some brief general remarks about conditional bootstraps

regression analysis for categorical variables with outcome subject to nonignorable nonresponse abstract we develop a log linear model for categorical response subject to nonignorable nonresponse the paper differs from fay 1986 in its focus on estimation and hypothesis testing in a regression setting as opposed to imputation in a multivariate setting we present several new results concerning the existence of solutions on the boundary of the parameter space and the construction of confidence intervals for estimates we illustrate the method by estimating the proportion of voters preferring truman in a 1948 preelection poll mosteller hyman mccarthy marks and truman 1949 results may depend strongly on the model assumed for nonresponse goodness of fit tests are available for comparing alternative models

strong approximation of density estimators from weakly dependent observations by density estimators from independent observations we derive an approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors we construct on a sufficiently rich probability space such a pairing of the random variables of both experiments that the set of observations x 1 x n from the time series model is nearly the same as the set of observations y 1 y n from the i i d model with a high probability all sets of the form x 1 x n delta y 1 y n boolean and a 1 b 1 x x a d b d contain no more than o n 1 2 pi b i a i 1 log n elements respectively although this does not imply very much for parametric problems it has important implications in nonparametric statistics it yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i i d model moreover it is shown that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap using these results we derive simultaneous confidence bands as well as supremum type nonparametric tests based on reasoning for the i i d model

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

estimating the power of the two sample wilcoxon test for location shift summary traditional methods for calculating the power of a statistical test for location shift require knowledge of the shape of the underlying probability distribution the distribution shape however may be unknown this paper describes a bootstrap method for using observed data or pilot data to approximate the power no assumptions need be made about the shape of the underlying continuous probability distribution simulation evidence shows that when applied to the wilcoxon two sample test for location shift the suggested method is reliable the evidence also shows that it is more accurate than a benchmark traditional approach the bootstrap method is applied to a real data example the analysis demonstrates how the method can be used to determine sample sizes and how to choose the more powerful of two alternative tests for location shift

causal models for patterns of nonresponse abstract the problem of missing data for categorical variables is examined from the perspective of modeling the mechanisms of nonresponse log linear causal models as formulated by goodman are studied for the relationship of the survey variables to response under some conditions several such models are estimable from the observed data for nested patterns of nonresponse a specific causal model represents exactly the assumption of ignorable response most causal models however imply nonignorable response mechanisms and yield alternative estimates for the distribution of the survey variables

bootstrap method for measuring accuracy of probabilistic choice models abstract probabilistic choice models that include choice attributes are increasingly being utilized by decision makers particularly in the business area the output of these models are point estimates of the coefficients of the choice attributes and rate of correct prediction accuracy measures of these point estimates are either based upon the asymptotic properties of the estimation procedures or are not provided this paper indicates that bootstrap is a method by which the assumption of asymptotic properties can be empirically validated for a given model and can provide measures of accuracy on point estimates where currently no such measure exists the empirical application of bootstrap to a real world data set indicates that decision makers will have considerably more information about their problem than is currently provided in the point estimates of the choice model managerial strategy and actions taken based upon the additional information may be quite different from actions suggested by the point estimates alone

evaluating screening for the early detection and treatment of cancer without using a randomized control group abstract sometimes the only data available for evaluating screening for the early detection and treatment of cancer come from studies lacking a randomized control group to evaluate these data various approaches have been proposed they include constructing a model to estimate the duration of preclinical cancer and the sensitivity of the screen formulating a model to estimate rates of cancer mortality and using a nonrandomized control group to estimate age specific incidence and mortality rates these methods have various limitations either the estimated quantities are insufficient for a complete evaluation strong parametric assumptions are required or an assumption must be made that the nonrandomized control group yields unbiased estimates of cancer incidence to bypass some of these limitations a new method has been developed for analyzing screening data without a randomized control group the method is called periodic screening evaluation pse because it requires data from periodic screens rath

distributed estimation and detection for sensor networks using hidden markov random field models we develop a hidden markov random field hmrf framework for distributed signal processing in sensor network environments under this framework spatially distributed observations collected at the sensors form a noisy realization of an underlying random field that has a simple structure with markovian dependence we derive iterated conditional modes icm algorithms for distributed estimation of the hidden random field from the noisy measurements we consider both parametric and nonparametric measurement error models the proposed distributed estimators are computationally simple applicable to a wide range of sensing environments and localized implying that the nodes communicate only with their neighbors to obtain the desired results we also develop a calibration method for estimating markov random field model parameters from training data and discuss initialization of the icm algorithms the hmrf framework and icm algorithms are applied to event region detection numerical simulations demonstrate the performance of the proposed approach

quadrature receiver mismatch calibration this article introduces nonlinear regression techniques to estimate gain and phase mismatches between the in phase i and quadrature q branches of a quadrature receiver under modest assumptions the system intrinsically follows a nonlinear regression model the algorithm is effective easily implemented customizable and requires few assumptions large sample jackknife and bootstrap techniques provide on line error assessment and parameter inference

robust predictive distributions for exponential families summary the bootstrap predictive distribution considered by harris 1989 is modified by replacing the maximum likelihood estimator of the unknown parameter by its minimum hellinger distance estimator the predictive distribution thus obtained is asymptotically superior to the corresponding estimative distribution in terms of the average kullback leibler divergence when the true distribution is in the natural exponential family simulation results are provided for the binomial and normal distributions which suggest that the proposed predictive distributions are robust and can perform better than the likelihood based methods under data contamination

asymptotics for high dimensional regression m estimates fixed design results we investigate the asymptotic distributions of coordinates of regression m estimates in the moderate p n regime where the number of covariates p grows proportionally with the sample size n under appropriate regularity conditions we establish the coordinate wise asymptotic normality of regression m estimates assuming a fixed design matrix our proof is based on the second order poincare inequality and leave one out analysis some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices we also show a counterexample namely an anova type design to emphasize that the technical assumptions are not just artifacts of the proof finally numerical experiments confirm and complement our theoretical results

cross validation the jackknife and the bootstrap excess error estimation in forward logistic regression abstract given a prediction rule based on a set of patients what is the probability of incorrectly predicting the outcome of a new patient call this probability the true error an optimistic estimate is the apparent error or the proportion of incorrect predictions on the original set of patients and it is the goal of this article to study estimates of the excess error or the difference between the true and apparent errors i consider three estimates of the excess error cross validation the jackknife and the bootstrap using simulations and real data the three estimates for a specific prediction rule are compared when the prediction rule is allowed to be complicated overfitting becomes a real danger and excess error estimation becomes important the prediction rule chosen here is moderately complicated involving a variable selection procedure based on forward logistic regression

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

evaluating prediction rules for t year survivors with censored regression models suppose that we are interested in establishing simple but reliable rules for predicting future t year survivors through censored regression models in this article we present inference procedures for evaluating such binary classification rules based on various prediction precision measures quantified by the overall misclassification rate sensitivity and specificity and positive and negative predictive values specifically under various working models we derive consistent estimators for the above measures through substitution and cross validation estimation procedures furthermore we provide large sample approximations to the distributions of these nonsmooth estimators without assuming that the working model is correctly specified confidence intervals for example for the difference of the precision measures between two competing rules can then be constructed all of the proposals are illustrated with real examples and their finite sample properties are evaluated through a simulation study

nonparametric estimation of standard errors in covariance analysis using the infinitesimal jackknife the infinitesimal jackknife provides a simple general method for estimating standard errors in covariance structure analysis beyond its simplicity and generality what makes the infinitesimal jackknife method attractive is that essentially no assumptions are required to produce consistent standard error estimates not even the requirement that the population sampled has the covariance structure assumed commonly used covariance structure analysis software uses parametric methods for estimating parameters and standard errors when the population sampled has the covariance structure assumed but fails to have the distributional form assumed the parameter estimates usually remain consistent but the standard error estimates do not this has motivated the introduction of a variety of nonparametric standard error estimates that are consistent when the population sampled fails to have the distributional form assumed the only distributional assumption these require is that the covariance structure be correctly specified as noted even this assumption is not required for the infinitesimal jackknife the relation between the infinitesimal jackknife and other nonparametric standard error estimators is discussed an advantage of the infinitesimal jackknife over the jackknife and the bootstrap is that it requires only one analysis to produce standard error estimates rather than one for every jackknife or bootstrap sample

inference from stratified samples second order analysis of three methods for nonlinear statistics abstract for stratified samples and nonlinear statistics that can be expressed as functions of estimated totals second order asymptotic expansions of the linearization jackknife and balanced repeated replication variance estimators are obtained based on these comparisons are made in terms of their biases some higher order asymptotic equivalence results are also established the special case of a combined ratio estimator is investigated in detail some results on bias reduction achieved by the jack knife and balanced repeated replication estimators of a nonlinear function of totals are also given

a simulation study of cross validation for selecting an optimal cutpoint in univariate survival analysis continuous measurements are often dichotomized for classification of subjects this paper evaluates two procedures for determining a best cutpoint for a continuous prognostic factor with right censored outcome data one procedure selects the cutpoint that minimizes the significance level of a logrank test with comparison of the two groups defined by the cutpoint this procedure adjusts the significance level for maximal selection the other procedure uses a cross validation approach the latter easily extends to accommodate multiple other prognostic factors we compare the methods in terms of statistical power and bias in estimation of the true relative risk associated with the prognostic factor both procedures produce approximately the correct type i error rate use of a maximally selected cutpoint without adjustment of the significance level however results in a substantially elevated type i error rate the cross validation procedure unbiasedly estimated the relative risk under the null hypothesis while the procedure based on the maximally selected test resulted in an upward bias when the relative risk for the two groups defined by the covariate and true changepoint was small the cross validation procedure provided greater power than the maximally selected test the cross validation based estimate of relative risk was unbiased while the procedure based on the maximally selected test produced a biased estimate as the true relative risk increased the power of the maximally selected test was about 10 per cent greater than the power obtained using cross validation the maximally selected test overestimated the relative risk by about 10 per cent the cross validation procedure produced at most 5 per cent underestimation of the true relative risk finally we report the effect of dichotomizing a continuous non linear relationship between covariate and risk we compare using a linear proportional hazard model to using models based on optimally selected cutpoints our simulation study indicates that we can have a substantial loss of statistical power when we use cutpoint models in cases where there is a continuous relationship between covariate and risk

interim analyses based on median survival times summary for a typical two arm survival study analyses are usually performed periodically to evaluate for example the treatment difference in this paper we propose to use differences or ratios of estimated median survival times repeatedly during a study to evaluate the relative merit of two treatment groups this new procedure provides quantitative assessments of the treatment difference and is completely model free our method is illustrated with a recent aids clinical study which evaluates the efficacy of a reduced dose of azt

parameter estimation by ensemble kalman filters with transformed data approach and application to hydraulic tomography 1 ensemble kalman filters enkfs are a successful tool for estimating state variables in atmospheric and oceanic sciences recent research has prepared the enkf for parameter estimation in groundwater applications enkfs are optimal in the sense of bayesian updating only if all involved variables are multivariate gaussian subsurface flow and transport state variables however generally do not show gaussian dependence on hydraulic log conductivity and among each other even if log conductivity is multi gaussian to improve enkfs in this context we apply nonlinear monotonic transformations to the observed states rendering them gaussian gaussian anamorphosis ga similar ideas have recently been presented by beal et al 2010 in the context of state estimation our work transfers and adapts this methodology to parameter estimation additionally we address the treatment of measurement errors in the transformation and provide several multivariate analysis tools to evaluate the expected usefulness of ga beforehand for illustration we present a first time application of an enkf to parameter estimation from 3 d hydraulic tomography in multi gaussian log conductivity fields results show that 1 ga achieves an implicit pseudolinearization of drawdown data as a function of log conductivity and 2 this makes both parameter identification and prediction of flow and transport more accurate combining enkfs with ga yields a computationally efficient tool for nonlinear inversion of data with improved accuracy this is an attractive benefit given that linearization free methods such as particle filters are computationally extremely demanding

on the bootstrap and confidence intervals we derive an explicit formula for the first term in an unconditional edgeworth type expansion of coverage probability for the nonparametric bootstrap technique applied to a very broad class of studentized statistics the class includes sample mean k sample mean sample correlation coefficient maximum likelihood estimators expressible as functions of vector means etc we suggest that the bootstrap is really an empiric one term edgeworth inversion with the bootstrap simulations implicitly estimating the first term in an edgeworth expansion this view of the bootstrap is reinforced by our discussion of the iterated bootstrap which inverts an edgeworth expansion to arbitrary order by simulating simulations

on bootstrap resampling and iteration summary we propose a single unifying approach to bootstrap resampling applicable to a very wide range of statistical problems it enables attention to be focused sharply on one or more characteristics which are of major importance in any particular problem such as coverage error or length for confidence intervals or bias for point estimation our approach leads easily and directly to a very general form of bootstrap iteration unifying and generalizing present disparate accounts of this subject it also provides simple solutions to relatively complex problems such as a suggestion by lehmann 1986 for conditionally short confidence intervals we set out a single unifying principle guiding the operation of bootstrap resampling applicable to a very wide range of statistical problems including bias reduction shrinkage hypothesis testing and confidence interval construction our principle differs from other approaches in that it focuses attention directly on a measure of quality or accuracy expressed in the form of an equation whose solution is sought a very general form of bootstrap iteration is an immediate consequence of iterating the empirical solution to this equation so as to improve accuracy when employed for bias reduction iteration of the resampling principle yields a competitor to the generalized jackknife enabling bias to be reduced to arbitrarily low levels when applied to confidence intervals it produces the techniques of hall 1986 and beran 1987 the resampling principle leads easily to solutions of new complex problems such as empirical versions of confidence intervals proposed by lehmann 1986 lehmann argued that an ideal confidence interval is one which is short when it covers the true parameter value but not necessarily otherwise the resampling principle suggests a simple empirical means of constructing such intervals section 2 describes the general principle and 3 shows how it leads naturally to bootstrap iteration there we show that in many problems of practical interest such as bias reduction and coverage error reduction in two sided confidence intervals each iteration reduces error by the factor n 1 where n is sample size in the case of confidence intervals our result sharpens one of beran 1987 who showed that coverage error is reduced by the factor n 2 in two sided intervals the main exception to our n 1 rule is coverage error of one sided intervals where error is reduced by the factor n a at each iteration our approach to bootstrap iteration serves to unify not just the philosophy of iteration for different statistical problems but also different techniques of iteration for the same

estimation and accuracy after model selection classical statistical theory ignores model selection in assessing estimation accuracy here we consider bootstrap methods for computing standard errors and confidence intervals that take model selection into account the methodology involves bagging also known as bootstrap smoothing to tame the erratic discontinuities of selection based estimators a useful new formula for the accuracy of bagging then provides standard errors for the smoothed estimators two examples nonparametric and parametric are carried through in detail a regression model where the choice of degree linear quadratic cubic is determined by the cp criterion and a lasso based estimation problem

change point in stochastic design regression and the bootstrap in this paper we study the consistency of dierent bootstrap procedures for constructing condence intervals cis for the unique jump discontinuity change point in an otherwise smooth regression function in a stochastic design setting this problem exhibits nonstandard asymptotics and we argue that the standard bootstrap procedures in regression fail to provide valid condence intervals for the change point we propose a version of smoothed bootstrap illustrate its remarkable nite sample performance in our simulation study and prove the consistency of the procedure the m out of n bootstrap procedure is also considered and shown to be consistent we also provide sucient conditions for any bootstrap procedure to be consistent in this scenario

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

a note on the first integrals of the abc system without loss of generality the abc systems reduce to two cases either a 0 and b c 0 or a 1 and 0 0 sufficiently small has no real meromorphic first integrals we improve ziglin s result showing that there are no c1 first integrals under convenient assumptions

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

bootstrap model selection abstract in a regression problem typically there are p explanatory variables possibly related to a response variable and we wish to select a subset of the p explanatory variables to fit a model between these variables and the response a bootstrap variable model selection procedure is to select the subset of variables by minimizing bootstrap estimates of the prediction error where the bootstrap estimates are constructed based on a data set of size n although the bootstrap estimates have good properties this bootstrap selection procedure is inconsistent in the sense that the probability of selecting the optimal subset of variables does not converge to 1 as n this inconsistency can be rectified by modifying the sampling method used in drawing bootstrap observations for bootstrapping pairs response explanatory variable it is found that instead of drawing n bootstrap observations a customary bootstrap sampling plan much less bootstrap observations should be sampled the bootstrap selection p

copula modeling of receiver operating characteristic and predictiveness curves receiver operating characteristic roc and predictiveness curves are graphical tools to study the discriminative and predictive power of a continuous valued marker in a binary outcome in this paper a copula based construction of the joint density of the marker and the outcome is developed for plotting and analyzing both curves the methodology only requires a copula function the marginal distribution of the marker and the prevalence rate for the model to be characterized the adoption of the gaussian copula and the customization of the margin for the marker are proposed for such characterization the computation of both curves is numerically more feasible than methods that attempt to obtain one curve in terms of the other estimation is carried out using maximum likelihood and resampling based methods randomized quantile residuals from each conditional distribution are employed for both assessing the adequacy of the model and identifying outliers the performance of the estimators of both curves and their underlying quantities is evaluated in simulation studies that assume different dependence structures and sample sizes the methods are illustrated with an analysis of the level of progesterone receptor gene expression for the diagnosis and prediction of estrogen receptor positive breast cancer

classification of gene expression data the manifold based metric learning way classification of microarray gene expression data can potentially help medical diagnosis and becomes an important topic in bioinformatics however microarray data sets are usually of small sample size relative to an overwhelming number of genes this makes the classification problem fairly challenging instance based learning ibl algorithms such as nearest neighbor k nn are usually the baseline algorithm due to their simplicity however practices show that k nn performs not very well in this field this paper introduces manifold based metric learning to improve the performance of ibl methods a novel metric learning algorithm is proposed by utilizing both local manifold structural information and local discriminant information in addition a random subspace extension is also presented we apply the proposed algorithm to the gene classification problem in three ways one in the original feature space another in the reduced feature space and the third via the random subspace extension statistical evaluation shows that the proposed algorithm can achieve promising results and gain significant performance improvement over traditional ibl algorithms

bootstrap and wild bootstrap for high dimensional linear models in this paper two bootstrap procedures are considered for the estimation of the distribution of linear contrasts and of f test statistics in high dimensional linear models an asymptotic approach will be chosen where the dimension p of the model may increase for sample size n rightarrow infty the range of validity will be compared for the normal approximation and for the bootstrap procedures furthermore it will be argued that the rates of convergence are different for the bootstrap procedures in this asymptotic framework this is in contrast to the usual asymptotic approach where p is fixed

some contributions to efficient statistics in structural models specification and estimation of moment structures current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments e g means and covariances and to distribution theory based on multivariate normality in psychometrics the multinormality assumption is often incorrect so that statistical tests on parameters or model goodness of fit will frequently be incorrect as well it is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary structural representations are developed for generalizations of the bentler weeks joreskog keesling wiley and factor analytic models some asymptotically distribution free efficient estimators for such arbitrary structural models are developed limited information estimators are obtained as well the special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail the argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds many open research areas are described

numerical extension of cft amplitude universality to three dimensional systems conformal eld theory cft predicts universal relations between scaling amplitudes and scaling dimensions for two dimensional systems on innite length cylinders which hold true even independent of the model under consideration we discuss dierent possible generalizations of such laws to three dimensional geometries using a cluster update monte carlo algorithm we investigate the nite size scaling fss of the correlation lengths of several representatives of the class of three dimensional classical o n spin models we nd that choosing appropriate boundary conditions the two dimensional situation can be restored c 2000 elsevier science b v all rights reserved

jackknife bootstrap and other resampling methods in regression analysis motivated by a representation for the least squares estimator we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time they are unbiased for homoscedastic errors and a special case the delete one jackknife is almost unbiased for heteroscedastic errors the method is extended to cover nonlinear parameters regression m estimators nonlinear regression and generalized linear models interval estimators can be constructed from the jackknife histogram three bootstrap methods are considered two are shown to give biased variance estimators and one does not have the bias robustness property enjoyed by the weighted delete one jackknife a general method for resampling residuals is proposed it gives variance estimators that are bias robust several bias reducing estimators are proposed some simulation results are reported

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

estimating the asymptotic covariance matrix for quantile regression models a monte carlo study abstract this monte carlo study examines several estimation procedures of the asymptotic covariance matrix in the quantile and censored quantile regression models design matrix bootstrap error bootstrapping order statistic sigma bootstrap homoskedastic kernel and heteroskedastic kernel the monte carlo samples are drawn from two alternative data sets 1 a the unaltered current population survey cps for 1987 and 2 b this cps data with independence between error term and regressors imposed this special setup allows one to evaluate the estimators under various realistic scenarios the results favor the design bootstrap for the general case but also support the order statistic when the error term is independent of the regressors

spin models on random lattices under certain conditions phase transitions in systems with quenched disorder are expected to exhibit a dierent behaviour than in the corresponding pure system here we discuss a series of monte carlo studies of a special type of such disordered systems namely spin models dened on quenched random lattices exhibiting geometrical disorder in the connectivity of the lattice sites in two dimensions we present results for the q state potts model on random tri valent 3 planar graphs which appear quite naturally in the dynamically triangulated random surface dtrs approach to quantum gravity as well as on poissonian random lattices of voronoi delaunay type both cases q64 and 4 are discussed which in the pure model without disorder give rise to second and rst order phase transitions respectively in three dimensions results for the ising model on poissonian random lattices are briey described we conclude with a comparison of the two types of connectivity disorder with the more standard case of bond disorder and a discussion of the distinguishing dierences c 2000 elsevier science b v all rights reserved

a note on bootstrapping the correlation coefficient summary smoothed bootstrap estimation of the sampling standard deviation of the variance stabilized correlation coefficient is reconsidered an approximation to the mean squared error of the bootstrap estimator is obtained and an empirical procedure for choosing the degree of smoothing in the bootstrap estimation is presented performance of the procedure is examined in a simulation study silverman young 1987 discuss use of the smoothed bootstrap for estimation of the sampling standard deviation of the variance stabilized correlation coefficient in bivariate samples and examine from a theoretical viewpoint circumstances when some smoothing is advantageous in the bootstrap estimation in the present paper the analytical tools used in that investigation are applied to estimate the mean squared error of the bootstrap estimator and to define a procedure for empirical choice of the degree of smoothing the procedure is illustrated in a small simulation study

estimation in choice based sampling with measurement error and bootstrap analysis in this paper we discuss the estimation of a logit binary response model the sampling is choice based and is done in two stages we investigate a likelihood based estimator which reduces to the usual logistic estimator when there is no measurement error and which takes into account the constraints imposed by the structure of the problem estimated standard errors obtained by formulae for prospective analysis are asymptotically correct a robust estimation procedures is proposed and an asymptotic covariance matrix obtained several bootstrap methods are applied to this retrospective problem numerical results are presented to illustrate useful properties of the methods

a note on the jackknife the bootstrap and the delta method estimators of bias and variance three methods for the nonparametric reduction of bias and estimation of variance are in common use the jackknife the delta method i e the method of statistical differentials and the bootstrap in recent work efron 1981 a b 1982 has compared these three methods numerically for the case of the sample correlation coefficient for present purposes we define the procedures as follows consider a statistic tn thought of as an estimator of an unknown parameter 0 we wish to study the possibility of bias reduction by estimating the bias and subtracting off the estimate and variance estimation for g tj considered as an estimator of g 0 throughout we assume that we observe x1 xn independent and identically distributed with common cumulative distribution function f let tn denote the abovementioned statistic computed based on x1 xn we assume tn to be symmetric in its arguments furthermore let t 1 denote the above mentioned statistic computed based upon the sample of size n 1 consisting of x1 xi1 xi 1 xn the jackknife estimator of the bias of g tn is then given as b n n 1 1 g t 1 g tn n and the jackknife variance estimator is just

estimation of multiresponse simulation metamodels using control variates this paper provides a unified development of the method of control variates for simulation experiments in which the objective is estimation of a multiresponse metamodel that is a linear model for an output vector of simulation performance measures expressed in terms of an input vector of decision variables for the target system in contrast to previous treatments of this topic we allow both the input and output of the metamodel to be multidimensional so that control variates can be applied to multipopulation multiresponse simulation experiments assuming that the responses and controls are jointly normal with a homogeneous covariance structure across the points of the experimental design we develop control variates procedures for point and confidence region estimation and for hypothesis testing on the coefficients of a postulated metamodel we derive a generalized minimum variance ratio to quantify the maximum efficiency that is achievable with a given set of controls and we formulate a generalized l

analysis of cusum and other markov type control schemes by using empirical distributions the run length distribution of a cumulative sum control scheme when the underlying distribution of the incoming observations is unknown is discussed given a sample of size n from this distribution the estimators related to various characteristics of the run length can be obtained by using the empirical cdf instead of the true cdf in the standard analysis procedure the article discusses the properties of the resulting point estimators as well as interval estimators obtained by using resampling techniques applications of the technique for the purpose of design and analysis of control schemes are also discussed the proposed methodology can be easily adapted for other markov type control schemes

linear models random censoring and synthetic data summary estimators for the linear model in the presence of censoring are available a new extension of the least squares estimator to censored data is equivalent to applying the ordinary least squares estimator to synthetic times time constructed by magnifying the gaps between successive order statistics under suitable regularity conditions the synthetic data estimator is fisher consistent and asymptotically normal examples facilitate com parison of the synthetic data estimator with estimators proposed by buckley james 1979 and by koul susarla van ryzin 1981 the entire lifetime of a person a machine a plant or an animal is not always observable some lifetimes may be censored in that only a lower bound for the lifetime is recorded statisticians are often interested in modelling the distribution of the true lifetimes as a function of covariates when censoring is present a proportional hazards model is often used when there is no censoring linear models are often used for ad hoc modelling of dependent variables miller halpern 1982 survey the methods that have been proposed for the linear model in the presence of censoring this paper describes a closed form method which is consistent and asymptotically normal this method incorporates censor ing in a natural way section 2 formulates classical least squares estimation in a manner that generalizes to accommodate random right censoring section 3 gives some theoretical results for the simplest nontrivial linear model the two sample case section 4 contains examples and the final section discusses the role of this estimator the covariate vector of the ith person will be denoted by xi and usually includes one as the first component the true lifetime yi follows a linear model if its conditional distribution given xi is

three state potts antiferromagnet on the triangular lattice the three state potts antiferromagnetic model on the triangular lattice 3paft has a weak first order transition in the pure case the first order nature follows from analytic arguments and while it has been observed in some simulations the precision of and internal disagreement between these observations has left much to be desired the 3paft is the only two dimensional nearest neighbour potts model with an ordered phase that does not have an infinite degeneracy that remains without an exact value for its critical point this model is expected to exhibit interesting glassy behaviour and a possible crossover to a second order transition upon dilution a careful characterization of the nature of the pure transition is needed in order to explore the physics of the dilute model we find clear indication of a first order transition at tt 0 62731 or 0 00006 using a cluster method that can be extended to the dilute case and histogram reweighting analysis our estimate is two orders of magnitude more precise than previous simulations and falls within the error bounds of a reanalysis of existing series expansion tt 0 628 or 0 004 reported on herein

an approximate analytical approach to resampling averages using a novel reformulation we develop a framework to compute approximate resampling data averages analytically the method avoids multiple retraining of statistical models on the samples our approach uses a combination of the replica trick of statistical physics and the tap approach for approximate bayesian inference we demonstrate our approach on regression with gaussian processes a comparison with averages obtained by monte carlo sampling shows that our method achieves good accuracy

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

jackknife estimation of stationary autoregressive models abstract this paper explores the properties of jackknife methods of estimation in stationary autoregressive models some general results concerning the correct weights for bias reduction under various sampling schemes are provided and the asymptotic properties of a jackknife estimator based on non overlapping sub samples are derived for the case of a stationary autoregression of order p when the number of sub samples is either fixed or increases with the sample size at an appropriate rate the results of a detailed investigation into the finite sample properties of various jackknife and alternative estimators are reported and it is found that the jackknife can deliver substantial reductions in bias in autoregressive models this finding is robust to departures from normality arch effects and misspecification the median unbiasedness and mean squared error properties are also investigated and compared with alternative methods as are the coverage rates of jackknife based confidence intervals

norm approximation for many body quantum dynamics focusing case in low dimensions abstract we study the norm approximation to the schrodinger dynamics of n bosons in r d d 1 2 with an interaction potential of the form n d 1 w n x y here we are interested in the focusing case w 0 assuming that there is complete bose einstein condensation in the initial state we show that in the large n limit the evolution of the condensate is effectively described by a nonlinear schrodinger equation and the evolution of the fluctuations around the condensate is governed by a quadratic hamiltonian resulting from bogoliubov approximation our result holds true for all 0 when d 1 and for all 0 1 when d 2

empirical bayes methods for combining likelihoods abstract suppose that several independent experiments are observed each one yielding a likelihood l k k for a real valued parameter of interest k for example k might be the log odds ratio for a 2 2 table relating to the kth population in a series of medical experiments this article concerns the following empirical bayes question how can we combine all of the likelihoods l k to get an interval estimate for any one of the k s say 1 the results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis no special mathematical forms are required for the priors or the likelihoods this scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods l k k even in very complicated situations with all nuisance parameters eliminated the empirical bayes likelihood theory is extended to situations where the k s have a regression structure as well as an empiri

application of the bootstrap methods in factor analysis a monte carlo experiment is conducted to investigate the performance of the bootstrap methods in normal theory maximum likelihood factor analysis both when the distributional assumption is satisfied and unsatisfied the parameters and their functions of interest include unrotated loadings analytically rotated loadings and unique variances the results reveal that a bootstrap bias estimation performs sometimes poorly for factor loadings and nonstandardized unique variances b bootstrap variance estimation performs well even when the distributional assumption is violated c bootstrap confidence intervals based on the studentized statistics are recommended d if structural hypothesis about the population covariance matrix is taken into account then the bootstrap distribution of the normal theory likelihood ratio test statistic is close to the corresponding sampling distribution with slightly heavier right tail

exponential empirical likelihood is not bartlett correctable in a recent paper diciccio hall and romano established that owen s empirical likelihood is bartlett correctable this is an intriguing and perhaps surprising result and is the only nonparametric context in which bartlett correctability is known to hold an alternative closely related nonparametric likelihood referred to here as exponential empirical likelihood may be constructed using efron s method of nonparametric tilting the purpose of this note is to show that exponential empirical likelihood is not bartlett correctable

latent change classes in dichotomous data changes in dichotomous data caused by treatments can be analyzed by means of the so called linear logistic model with relaxed assumptions llra the llra does not require observable criteria representing a single underlying latent trait but it postulates the generalizability of the treatment effects over criteria and subjects to test this latter crucial assumption the mixture llra was proposed that allows directly unobservable types of subjects to have different treatment effects as the earlier methods for estimating the parameters of the mixture llra have specific drawbacks a further method based on the conditional maximum likelihood principle will be presented here in contrast to the earlier conditional methods it uses all of the dichotomous change data while having fewer parameters further its goodness of fit tests become more sensitive to a falsely specified number of change types even though the treatment effects are biased for typically occurring small to moderate sample sizes however parametric bootstrapping of the distributions of the fit statistics is recommended for performing hypotheses tests finally three applications of the new method to empirical data are described first about the effect of the so called trager psychophysical integration second about the effect of autogenic therapy on patients with psychosomatic symptoms and third about the effect of religious education on the attitude towards sects

superconsistent estimation and inference in structural econometric models using extreme order statistics abstract data generating processes whose distributions supports depend on unknown parameters arise naturally in empirical applications in such situations the maximum likelihood estimator is often difficult to calculate and usually has a nonstandard limiting distribution that depends on nuisance parameters we propose an alternative estimation strategy that is typically simpler to implement than the likelihood approach and allows one to conduct inference using simulation methods our proposed estimators are based on the analog estimation principle and bear a striking resemblance to generalized method of moments estimators although here the estimators are generally parameter consistent at rate t rather than the usual rate t

the moving blocks bootstrap and robust inference for linear least squares and quantile regressions abstract this paper presents the moving blocks bootstrap mbb as a new inference procedure for linear regression estimation which is robust to heteroskedasticity and autocorrelation of unknown forms the mbb covariance estimator is shown to provide heteroskedasticity and autocorrelation consistent hac standard errors for least squares ls and quantile regression qr coefficient estimators the mbb covariance estimator is shown to be asymptotically equivalent to the bartlett kernel estimator suggested by newey and west 1987 and the asymptotically optimal choice of the blocksize is discussed a monte carlo study is included showing that the mbb fares well in comparison to standard hac inference procedures considering strong mixing data generating processes the paper extends existing asymptotic results for the qr estimator the analogy to the ls case is stressed the paper analyzes both the cases of stochastic and of nonstochastic regressors and suggests two new grenander like conditions for the latter case the use of the mbb approach is illustrated for a practical example using a standard econometric package

the bootstrap to smooth or not to smooth summary the bootstrap and smoothed bootstrap are considered as alternative methods of estimating properties of unknown distributions such as the sampling error of parameter estimates criteria are developed for determining whether it is advantageous to use the smoothed bootstrap rather than the standard bootstrap key steps in the argument leading to these criteria include the study of the estimation of linear functionals of distributions and the approximation of general functionals by linear functionals consideration of an example the estimation of the standard error in the variance stabilized sample correlation coefficient elucidates previously published simulation results and also illustrates the use of computer algebraic manipulation as a useful technique in asymptotic statistics finally the various approximations used are vindicated by a simulation study

alternative estimation procedures for pr x y in categorized data summary consider two independent random variables x and y the functional r pr x y or x pr x y pr y x is of practical importance in many situations including clinical trials genetics and reliability in this paper several approaches to estimation of x when x and y are presented in discretized categorical form are analyzed and compared asymptotic formulas for the variances of the estimators are derived use of the bootstrap to estimate variances is also discussed computer simulations indicate that the choice of the best estimator depends on the value of x the underlying distribution and the sparseness of the data it is shown that the bootstrap provides a robust estimate of variance several examples are treated

a new approach to critical exponents in phase transitions of spin systems abstract we investigate a method to obtain critical exponents using standard numerical simulation techniques for spin systems we show that by analyzing the inverse of the logarithmic derivative of the susceptibility with respect to temperature the accuracy in determining can be increased significantly the method is tested on the two and three dimensional ising models but this alternative approach is thought to be most useful for spin systems on non periodic structures like quasi crystals and fractal lattices where the finite size scaling method is difficult to apply