hierarchical bayesian inference for ill posed problems via variational method this paper investigates a novel approximate bayesian inference procedure for numerically solving inverse problems a hierarchical formulation which determines automatically the regularization parameter and the noise level together with the inverse solution is adopted the framework is of variational type and it can deliver the inverse solution and regularization parameter together with their uncertainties calibrated it approximates the posteriori probability distribution by separable distributions based on kullback leibler divergence two approximations are derived within the framework and some theoretical properties e g variance estimate and consistency are also provided algorithms for their efficient numerical realization are described and their convergence properties are also discussed extensions to nonquadratic regularization nonlinear forward models are also briefly studied numerical results for linear and nonlinear cauchy type problems arising in heat conduction with both smooth and nonsmooth solutions are presented for the proposed method and compared with that by markov chain monte carlo the results illustrate that the variational method can faithfully capture the posteriori distribution in a computationally efficient way

on the removal of skewness by transformation summary we suggest empirical transformations for removing most of the skewness of an asymmetric statistic these transformations may be used as the basis for accurate confidence procedures or hypothesis tests and can be employed in conjunction with either a normal approximation or the bootstrap our approach differs from those of other researchers in that the transformation is monotone and invertible in this paper we suggest elementary but general transformations for eliminating skewness from the distribution of a studentized statistic the transformations are monotone and have simple explicit inversion formulae and so are readily applied to confidence interval problems when the transformations are used in conjunction with the bootstrap they can produce confidence intervals with particularly low levels of coverage error that confidence intervals for symmetric statistics have lower coverage error than their counterparts in the case of asymmetry may be deduced from edgeworth expansions the first term in an expansion of size n 1 2 where n is the sample size describes the error in the usual normal approximation and is due entirely to skewness if it can be eliminated for example by transformation then the normal approxima tion will be in error by only o n 1 thus our method involves transforming one statistic into another where the distribution is virtually symmetric applying the normal approximation or the bootstrap to the new statistic and then regaining the asymmetry of the original problem by inverse transformation at the same time retain ing the high coverage accuracy conferred by applying confidence interval procedures to a symmetric statistic

bootstrapping in multiplicative models abstract this paper develops the bootstrap theory to obtain a confidence interval for the constant term of the cobb douglas multiplicative model this also proposes a simple and easily computable estimator of the constant term along with its mean square error

differentiability of statistical functionals and consistency of the jackknife in statistical applications the unknown parameter of interest can frequently be defined as a functional 0 t f where f is an unknown population statistical inferences about 0 are usually made based on the statistic t fn where fn is the empirical distribution assessing t fn as an estimator of 0 or making large sample inferences usually requires a consistent estimator of the asymptotic variance of t fn asymptotic behaviour of the jackknife variance estimator is closely related to the smoothness of the functional t this paper studies the smoothness of t through the differentiability of t and establishes some general results for the consistency of the jackknife variance estimators the results are applied to some examples in which the statistics t fn are l m estimators and some test statistics

interpolation rules suggested by asymptotic expansions summary we suggest a simple interpolation rule based on edgeworth and cornish fisher expansions which permits highly accurate numerical approximation in problems involving calculation of distributions or quantiles without requiring explicit calculation of terms in the expansions edgeworth expansions are often used to provide explicit corrections for asymptotic approxima tions as in recent work in testing and confidence procedures involving nonparametric methods pfanzagl 1979 hall 1983 abramovitch singh 1985 and in time series econometrics and related fields sargan 1976 phillips 1978 durbin 1980 taniguchi 1985 taniguchi maekawa 1990 in the present note we point out a useful application of edgeworth expansion methods in producing simple but accurate interpolation rules for deriving information from or for constructing statistical tables suppose the function f is tabulated for discrete values ni of its argument and that we seek the value off for an untabulated point say n we suggest that shouldf admit an edgeworth expansion in increasing powers of n 2 or n1 then interpolation should be among values of nwf nj or nf ni respectively rather than values of f ni this recommendation applies to approximations of both distribution functions and percentage points in the latter case our reference to edgeworth expansions should really be to cornish fisher expansions we should stress that our method does not require calculation of any terms of an edgeworth or cornish fisher expansion it uses only existing information available in statistical tables and knowledge that the edgeworth expansion exists this existence has been established for large classes of statistical problems see for example sargan 1975 phillips 1977 bhattacharya ghosh 1978 durbin 1980 and sargan satchell 1986 aside from advantages of greater accuracy compared with traditional interpolation rules our method allows linear and quadratic

improvements of jackknife confidence limit methods summary edgeworth expansion is applied to studentized parameter estimates when the standard error has been computed by a jackknife method adjustments to the usual jackknife confidence limit formulae are obtained this approach is contrasted with a bootstrap approach in numerical illustrations for estimation of a ratio jackknife methods are nonparametric methods for estimating the bias and standard error of an estimate t approximate confidence limits for the estimand can be found by using a large sample normal approximation for t somewhat curiously little is known about possible improvements to the normal approximation in this context although improvements based on edgeworth expansions are familiar in other problems in this paper we show that edgeworth expansion methods can be applied in the jackknife context numerical results for a particular application raise the possibility that the resulting improvements of jackknife methods can be matched by a suitable use of bootstrap methods efron 1982

nodal domain distributions for quantum maps the statistics of the nodal lines and nodal domains of the eigenfunctions of quantum billiards have recently been observed to be fingerprints of the chaoticity of the underlying classical motion by blum et al 2002 phys rev lett 88 114101 and by bogomolny and schmit 2002 phys rev lett 88 114102 these statistics were shown to be computable from the random wave model of the eigenfunctions we here study the analogous problem for chaotic maps whose phase space is the two torus we show that the distributions of the numbers of nodal points and nodal domains of the eigenvectors of the corresponding quantum maps can be computed straightforwardly and exactly using random matrix theory we compare the predictions with the results of numerical computations involving quantum perturbed cat maps

normalizing transformatins and bootstrap confidence intervals this paper considers the problem of constructing approximate confidence intervals for functional parameters in the nonparametric case the approach based on transformation theory is applied to improve standard confidence intervals the accelerated bias corrected percentile interval introduced by efron relies on the existence of a normalizing transformation with bias and skewness corrections although calculation does not require explicit knowledge of its functional form we formally construct such a transformation and estimate bias and skewness correction factors for nonparametric situations the resulting interval is shown to be second order accurate to this end edgeworth expansions for the distributions of transformed statistics are derived using the von mises expansion

hierarchical mixtures of experts and the em algorithm we present a tree structured architecture for supervised learning the statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models glim s learning is treated as a maximum likelihood problem in particular we present an expectation maximization em algorithm for adjusting the parameters of the architecture we also develop an on line learning algorithm in which the parameters are updated incrementally comparative simulation results are presented in the robot dynamics domain

nodal densities of gaussian random waves satisfying mixed boundary conditions nodal statistics are derived for boundary adapted gaussian random waves u i v in th ep lane satisfying the helmholtz equation and mixed boundary conditions in which a linear combination of and its normal derivative characterized by a parameter a v anishes on aline for both the density of nodal lines for re u and the density of nodal points for complex e ffects of the boundary persist surprisingly infinitely far from the boundary and also surprisingly are independent of a a sa increases from the dirichlet value a 0 nodal line and point structures migrate from the boundary and can be described analytically

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

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

bootstrapping the distance between smooth bootstrap and sample quantile distribution the normalized kolmogorov smirnov and variational distance between the distribution of the sampleq quantile and the pertaining smooth bootstrap distribution are asymptotically distributed like the absolute value of a normal random variable the distribution functions of these random distances may serve as measures of the accuracy of the bootstrap procedure

asymptotic iterated bootstrap confidence intervals an iterated bootstrap confidence interval requires an additive correction to be made to the nominal coverage level of an uncorrected interval such correction is usually performed using a computationally intensive monte carlo simulation involving two nested levels of bootstrap sampling asymptotic expansions of the required correction and the iterated interval endpoints are used to provide two new computationally efficient methods for constructing an approximation to the iterated bootstrap confidence interval the first asymptotic interval replaces the need for a second level of bootstrap sampling with a series of preliminary analytic calculations which are readily automated and from which an approximation to the coverage correction is easily obtained the second interval directly approximates the endpoints of the iterated interval and yields for the first time the possibility of constructing an approximation to an iterated bootstrap confidence interval which does not require any resampling the theoretical properties of the two intervals are considered the computation required for their construction is detailed and has been coded in a fully automatic user friendly fortran program which may be obtained by anonymous ftp a simulation study which illustrates their effectiveness on three examples is presented

calculation of power and sample size with bounded outcome scores the two sample wilcoxon rank sum test is the most popular non parametric test for the comparison of two samples when the underlying distributions are not normal although the underlying distributions need not be known in detail to calculate the null distribution of the test statistic parametric assumptions are often made to determine the power of the test or the sample size we encountered difficulties with this approach in the planning of a recent clinical trial in stroke patients it is shown that for power and sample size estimation it can be dangerous to apply the classical formulae routinely especially with outcome scores having a u shaped or a j shaped distribution as an example we have taken the barthel index a quality of life outcome measure in stroke patients further we have investigated alternative methods by means of monte carlo simulation the distributional characteristics of the estimated powers were compared our findings suggest more appropriate computer software is necessary for the calculation of power and sample size when efficacy is measured by a non parametric method

calibrating confidence coefficients abstract two approaches for dealing with the problem of poor coverage probabilities of certain standard confidence intervals are proposed the first is a recommendation that the actual coverage be estimated directly from the data and its value reported in addition to the nominal level this is achieved through a combination of computer simulation and density estimation the asymptotic validity of the procedure is proved for a number of common situations a classical example is the nonparametric estimation of the variance of a population using the normal theory interval here it is shown that the estimated coverage probability consistently estimates the true coverage probability if the population distribution possesses a finite sixth moment the second approach is more traditional it is a procedure for modifying an interval to yield improved coverage properties given a confidence interval its estimated coverage probability obtained in the first approach is used to alter the nominal level of the interval

solving inverse problems by decomposition classification and simple modeling inverse problems appear in many areas ranging from microwave circuits to environmental studies and to robotics just to mention a few in this paper we propose a new approach to solving inverse problems based on decomposition of output space into cells with the corresponding regions in the input space solutions are identified using a clustering method and the relationship between data in an output cell and the corresponding input region is modeled by a simple polynomial it is shown that the proposed method achieves very high accuracy even with relatively high number of inputs and outputs it is also extremely fast and is suitable for real time control where needed the method is applied to a highly complex inverse problem in robot kinematics and its performance is demonstrated

hecke theory and equidistribution for the quantization of linear maps of the torus 1 1 background one of the key issues of quantum chaos is the nature of the semiclassical limit of eigenstates of classically chaotic systems when the classical system is given by the geodesic flow on a compact riemannian manifoldm or rather on its cotangent bundle one can formulate the problem as follows the quantum hamiltonian is in suitable units represented by the positive laplacian on m to measure the distribution of its eigenstates we start with a smooth classical observable that is a smooth function on the unit cotangent bundle s m via some choice of quantization from symbols to pseudodifferential operators we form its quantization op f this is a zero order pseudodifferential operator with principal symbol f the expectation value of op f in the eigenstate is op f let j be a sequence of normalized eigenfunctions j j j 0 m j 2 1 the problem then is to understand the possible limits as j of the distributions f c s m op f j j 1 1

histospline smoothing the bayesian bootstrap summary this paper describes a version of the bayesian bootstrap that assigns random dirichlet mass uniformly across statistically equivalent blocks the method requires prior knowledge that the underlying distribution is continuous with known compact support under these conditions the resulting analysis has three advantages over traditional bootstrap competitors first it takes account of the probability integral transformation secondly it often enables exact expressions for the bootstrap distributions of common statistical functionals thus small sample properties are known precisely in contrast to the usual bootstrap situation thirdly it shows consistent superiority in large scale simulation experiments that compare the global accuracy of confidence intervals for the mean median variance and distribution function the simulation experiments use a goodnessof fit test as the basis for comparing the performance of competing bootstrap techniques

chi squared approximations to the distribution of a sum of independent random variables we suggest several chi squared approximations to the distribution of a sum of independent random variables and derive asymptotic expansions which show that the error of approximation is of order n 1 as n rightarrow infty the error may be reduced to n 3 2 by making a simple secondary approximation

a review of bootstrap confidence intervals on passe en revue plusieurs methodes distinctes basees sur le bootstrap pour construire des intervalles de confiance

non parametric analysis of treatment covariate interaction in the presence of censoring the demonstration of varying treatment efficacy among different subsets of patients is an important part of the analysis of clinical trials the paper commences by clarifying the meaning of interaction and by reviewing valid procedures of analysis in the presence of interaction suitable descriptive measures of treatment covariate interactions are ratios of hazard ratios to which generalized patel hoel tests for qualitative or ordinal categorical covariates are related both the hazard ratios and the tests are given in a formulation for use with different scoring systems such as u or e furthermore the procedures are related to common k sample tests and thus are suitable for proportional and for converging hazards a worked example is included

calibrating prediction regions abstract suppose that the variable x to be predicted and the learning sample yn that was observed have a joint distribution which depends on an unknown parameter the parameter can be finite or infinite dimensional a prediction region dn for x is a random set depending on yn that contains x with prescribed probability this article studies methods for controlling simultaneously the conditional coverage proability of dn given yn and the overall unconditional coverage probability of dn the basic construction yields a prediction region dn which has the following properties in regular models both the conditional and overall coverage probabilities of dn converge to as the size n of the learning sample increases the convergence of the former is in probability moreover the asymptotic distribution of the conditional coverage probability about is typically normal and the overall coverage probability tends to at rate n 1 can one reduce the dispersion of the conditional coverage pr

statistics of nodal lines and points in chaotic quantum billiards perimeter corrections fluctuations curvature for real time reversal symmetric quantum billiards the mean length l of nodal line is calculated for the nth mode n 1 with wavenumber k using a gaussian random wave model adapted locally to satisfy dirichlet or neumann boundary conditions the leading term is of order k i e n and the first perimeter correction dominated by an unanticipated long range boundary effect is of order log k i e log n with the same sign negative for both boundary conditions the leading order state to state fluctuations l are of order log k for the curvature of nodal lines and 2 are of order k but 3 and higher moments diverge for complex e g aharonov bohm billiards the mean number n of nodal points phase singularities in the mode has a leading term of order k2 i e n the perimeter correction again a long range effect is of order klog k i e nlog n and positive notwithstanding nodal depletion near the boundary and the fluctuations n are of order k log k generalizations of the results for mixed robin boundary conditions are stated

uniform coverage bounds for confidence intervals and berry esseen theorems for edgeworth expansion we derive upper bounds for the coverage error of confidence intervals for a population mean uniformly over large classes of populations and different types of confidence intervals it is shown that the order of these bounds is achieved by the normal approximation method for constructing confidence intervals uniformly over distributions with finite third moment and by an empirical edgeworth correction of this approach uniformly over smooth distributions with finite fourth moments these results have straightforward extensions to higher orders of edgeworth correction and higher orders of moments our upper bounds to coverage accuracy are based on berry esseen theorems for edgeworth expansions of the distribution of the studentized mean

artificially augmented samples shrinkage and mean squared error reduction an inequality is provided that determines when shrinkage reduces the mean squared error mse of an unbiased estimate artificially augmented samples are then used to obtain among others shrinkage estimates of the population s variance and covariance which improve the unbiased estimates for all parameter values and for all probability models with marginals having finite second moments and alternative jackknife estimates that complement the usual jackknife estimates in reducing the mse

normal scores normal plots tests for normality abstract in this article we develop new plotting positions for normal plots the use of the plots usually centers on detecting irregular tail behavior or outliers along with the normal plot we develop tests for various departures from normality especially for skewness and heavy tails the tests can be considered as components of a shapiro wilk type test that has been decomposed into different sources of nonnormality convergence to the limiting distributions is slow so finite sample corrections are included to make the tests useful for small sample sizes

on the bootstrap and continuity correction on propose une correction de continuite utile lorsque le bootstrap est utilise pour approcher des distributions de moyennes d echantillons a valeurs sur un treillis

a comparative study of ordinary cross validation v fold cross validation and the repeated learning testing methods summary concepts of v fold cross validation and repeated learning testing methods have been introduced here in many problems these methods are computationally much less expensive than ordinary cross validation and can be used in its place a comparative study of these three methods has been carried out in detail

choice of test for comparing two groups with particular application to skewed outcomes we consider a clinical trial where a skewed outcome variable is to be compared between two groups while comparison of sample means may lack power we show that power also depends on the nature of the anticipated treatment effect for any given distribution in the control arm there is a family of true distributions in the intervention arm for which the most powerful test is a comparison of arithmetic means similar results hold for a comparison of geometric means and approximately for the wilcoxon rank sum test and a comparison of medians we discuss how these methods could be used in planning the analysis of a clinical trial in which the intervention effect alters the shape of the distribution these ideas are illustrated by a trial in community psychiatry where the primary outcome days in hospital was highly skewed but the intervention was mainly expected to reduce the frequency of values in the tail we show that a comparison of sample means is a reasonable choice in this case despite the skewness

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

on the edgeworth expansion and the bootstrap approximation for a studentized u statistic the asymptotic accuracy of the estimated one term edgeworth expansion and the bootstrap approximation for a studentized u statistic is investigated it is shown that both the edgeworth expansion estimate and the bootstrap approximation are asymptotically closer to the exact distribution of a studentized u statistic than the normal approximation the conditions needed to obtain these results are weak moment assumptions on the kernel h of the u statistic and a nonlattice condition for the distribution of g x1 e h xl x2 1x11 as an application improved edgeworth and bootstrap based confidence intervals for the mean of a u statistic are obtained

some power considerations when deciding to use transformations conventional wisdom suggests that for small data sets having substantial skew one should attempt to determine the correct distributional form if possible and apply statistical methods appropriate for that distribution transformations such as the log or square root are often used if an appropriate distributional form cannot be determined a distribution free procedure such as a rank transformation or a randomization test procedure can be used to better appreciate the effect of such alternatives on both the type i error and power of detecting differences between treatment groups simulation studies were conducted for responses having specific gamma g r theta and log normal ln m v distributions the gamma and log normal distributions were selected so that they had the same first two moments a simple two group design was assumed the reference group always had an average disease level mu 3 0 mu r theta for gamma mu m for log normal and the treatment group always had means whose reductions ranged from 0 per cent to 50 per cent the effect of distributional type and the degree of skewness was investigated by varying the population parameter values six statistical test procedures were compared for the gamma distributions all test procedures were robust relative to the type i error the ump test based on a ratio of sample means produced the greatest power for all combinations of n r and rt the power losses associated with the randomization test the t test on original scale and the t test on the square root scale were very small 3 per cent to 6 per cent in absolute value for n 10 and 15 and less than 2 per cent for group sizes of 25 or more the power loss associated with the t test on the log scale was much larger ranging from 5 per cent to 10 per cent smaller power than the t test on original scale the wilcoxon rank test produced similar results to that of the log t test for small samples the power for the shifted log x c test increased monotonically to the asymptotic value of the orig t test the same five test procedures based on differences in sample means were then compared for the corresponding log normal distributions the ump test that is log x produced the highest power there was very little power lost for the sqrt t test the loss in power varied between 2 per cent and 5 per cent for the rank test abstract truncated at 400 words

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

asymptotic expansions for confidence limits in the presence of nuisance parameters with applications summary series expansions for confidence limits based on consistent estimators are obtained in the presence of nuisance parameters the maximum likelihood estimator is treated as a special case and the necessary characteristics of its sampling distribution are obtained in terms of basic quantities derivable from the likelihood function the theory is applied to the problem of setting separate confidence limits for the scale and shape parameters of the weibull distribution comparisons are made with earlier simulation studies

on the number of bootstrap simulations required to construct a confidence interval abstract the purpose of this document is to make two points about the effect of the number of bootstrap simulations b on percentile t bootstrap confidence intervals the first point concerns coverage probability the second distance of the simulated critical point form the true critical point derived with b infinity in both cases the author has in mind applications to smooth statistics such as the studentized mean of a sample drawn from a continuous distribution he indicates the change that have to be made if the distribution of the statistic is not smooth additional keywords exponentive functions

testing the mean of skewed distributions abstract johnson proposed some modified t tests to assess the mean of asymmetrical distributions the tests were suggested for any parent distribution ranging from the normal to a distribution as asymmetric as an exponential distribution for sample sizes as small as 13 in many practical situations the skewness of the parent distribution can be greater than those studied by johnson and the sample sizes can be quite small possibly as small as 10 due to the cost of the sampling procedures in these cases johnson s test can be quite inaccurate sutton suggested a composite test to improve johnson s upper tailed t test in this article a new test procedure is proposed for the upper tailed test for the mean of positively skewed distributions which requires only a little more effort than johnson s test a monte carlo study investigates the new procedure s properties for a variety of positively skewed distributions with small sample sizes it is shown that the new test procedure is more accurate and more powe

comment on estimate of standard deviation for a log transformed variable using arithmetic means and standard deviations by quan hui and zhang ji statistics in medicine 2003 22 2723 2736

on the nodal count statistics for separable systems in any dimension we consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in an arbitrary dimension we give an explicit expression for the limiting distribution of normalized nodal counts and analyse some of its universal properties our results are illustrated by the detailed discussion of simple examples and numerical nodal count distributions

computer intensive methods for tests about the mean of an asymmetrical distribution abstract for one sided tests about the mean of a skewed distribution the t test is asymptotically robust for validity however it can be quite inaccurate and inefficient with small sample sizes results presented here confirm that a procedure due to johnson should be preferred to the t test when the parent distribution is asymmetrical because it reduces the probability of type i error in cases where the t test has an inflated type i error rate and it is more powerful in other situations but if the skewness is severe and the sample size is small then johnson s test can also be appreciably inaccurate for such situations computer intensive test procedures using bootstrap resampling are proposed and with an extensive monte carlo study it is shown that these procedures are remarkably robust and can result in reduced probabilities of type i and type ii errors compared to johnson s test

nodal domains of a non separable problem the right angled isosceles triangle we study the nodal set of eigenfunctions of the laplace operator on the right angled isosceles triangle a local analysis of the nodal pattern provides an algorithm for computing the number n of nodal domains for any eigenfunction in addition an exact recursive formula for the number of nodal domains is found to reproduce all existing data eventually we use the recursion formula to analyse a large sequence of nodal counts statistically our analysis shows that the distribution of nodal counts for this triangular shape has a much richer structure than the known cases of regular separable shapes or completely irregular shapes furthermore we demonstrate that the nodal count sequence contains information about the periodic orbits of the corresponding classical ray dynamics

coverage probabilities of bootstrap confidence intervals for quantiles an asymptotic expansion of length 2 is established for the coverage probabilities of confidence intervals for the underlying q quantile which are derived by bootstrapping the sample q quantile the corresponding level error turns out to be of order o n 1 2 which is unexpectedly low a confidence interval of even more practical use is derived by using backward critical points the resulting confidence interval is of the same length as the one derived by ordinary bootstrap but it is distribution free and has higher coverage probability

prepivoting test statistics a bootstrap view of asymptotic refinements abstract approximate tests for a composite null hypothesis about a parameter may be obtained by referring a test statistic to an estimated critical value either asymptotic theory or bootstrap methods can be used to estimate the desired quantile the simple asymptotic test a refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated the bootstrap approach used here is based on the concept of prepivoting prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution the simple bootstrap test b refers the prepivoted test statistic to a quantile of the uniform 0 1 distribution under regularity conditions the bootstrap test b has a smaller asymptotic order of error in level than does the asymptotic test a provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters in the contrary case both a and b have the same order of level error certain class

universality in the flooding of regular islands by chaotic states we investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori depending on dynamical tunneling rates and the heisenberg time regular states disappear and chaotic states flood the regular tori for a quantitative understanding we introduce a random matrix model the resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system we discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

the log transformation is special the logarithmic log transformation is a simple yet controversial step in the analysis of positive continuous data measured on an interval scale situations where a log transformation is indicated will be reviewed this paper contends that the log transformation should not be classed with other transformations as it has particular advantages problems with using the data themselves to decide whether or not to transform will be discussed it is recommended that log transformed analyses should frequently be preferred to untransformed analyses and that careful consideration should be given to use of a log transformation at the protocol design stage