discrete differential geometry the nonplanar quadrilateral mesh we consider the problem of constructing a discrete differential geometry defined on nonplanar quadrilateral meshes physical models on discrete nonflat spaces are of inherent interest as well as being used in applications such as computation for electromagnetism fluid mechanics and image analysis however the majority of analysis has focused on triangulated meshes we consider two approaches discretizing the tensor calculus and a discrete mesh version of differential forms while these two approaches are equivalent in the continuum we show that this is not true in the discrete case nevertheless we show that it is possible to construct mesh versions of the levi civita connection and hence the tensorial covariant derivative and the associated covariant exterior derivative the torsion and the curvature we show how discrete analogs of the usual vector integral theorems are constructed in such a way that the appropriate conservation laws hold exactly on the mesh rather than only as approximations to the continuum limit we demonstrate the success of our method by constructing a mesh version of classical electromagnetism and discuss how our formalism could be used to deal with other physical models such as fluids

the effect of random measurement error on receiver operating characteristic roc curves in this paper confidence intervals for the area under the roc curve are adjusted for the presence of measurement error a parametric normal model is assumed the ratio of intra individual to inter individual variance provides a relative measure of the amount of measurement error an exact adjusted confidence interval is developed for the equal variance case and an approximation is derived for the unequal variance case we illustrate using an example the effect of ignoring the measurement error on the confidence interval for the area and show that the effect can be substantial on the decision as to the diagnostic effectiveness of the marker we also examine by means of a simulation study the coverage probability of the unadjusted confidence interval and show that it can be far from its nominal value when measurement error is present

confidence intervals for the generalized roc criterion receiver operating characteristic roc curves are frequently used to assess the usefulness of diagnostic markers when several diagnostic markers are available they can be combined by a best linear combination that is when the area under the roc curve of this combination is maximized among all possible linear combinations this maximal area is the generalized roc criterion which provides a measure of how effective the combination of the markers is this criterion needs to be estimated from the data and is usually evaluated against single markers in the present paper we provide confidence intervals for the generalized roc criterion under the assumption of homogeneous covariance matrices derive an approximation for the heterogeneous covariance matrices case and evaluate the approximation via a simulation study finally we present an illustrative example

interval criteria for forced oscillation with nonlinearities given by riemann stieltjes integrals abstract we study the oscillation of second order forced differential equations with nonlinearity given by a riemann stieltjes integral of the form p t x q t x 0 b r t s x t s sgn x t d s e t where b 0 c 0 b is strictly increasing such that 0 0 1 b p q e c 0 with p t 0 r c 0 0 b and 0 b r is nondecreasing interval oscillation criteria of the el sayed type and the kong type are obtained as a special case the work in this paper unifies and improves the existing results in the literature for equations with a finite number of nonlinear terms we also extend our results to equations with delays

on geometric discretization of elasticity this paper presents a geometric discretization of elasticity when the ambient space is euclidean this theory is built on ideas from algebraic topology exterior calculus and the recent developments of discrete exterior calculus we first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity similar to those of electromagnetism can be written in terms of a material hodge star operator in the discrete theory presented in this paper instead of referring to continuum quantities we postulate the existence of some discrete scalar valued and vector valued primal and dual differential forms on a discretized solid which is assumed to be a triangulated domain we find the discrete governing equations by requiring energy balance invariance under time dependent rigid translations and rotations of the ambient space there are several subtle differences between the discrete and continuous theories for example power of tractions in the discrete theory is written on a layer of cells with a nonzero volume we obtain the compatibility equations of this discrete theory using tools from algebraic topology we study a discrete cosserat medium and obtain its governing equations finally we study the geometric structure of linearized elasticity and write its governing equations in a matrix form we show that in addition to constitutive equations balance of angular momentum is also metric dependent all the other governing equations are topological

oscillation of second order delay differential equations with mixed nonlinearities abstract in this paper we study the following second order delay differential equation with mixed nonlinearities r t u t 1 u t q 0 t u 0 t 1 u 0 t q 1 t u 1 t 1 u 1 t q 2 t u 2 t 1 u 2 t 0 where 0 oscillation criteria for the equation are established which generalize the results by d urina and stavroulakis j d d urina i p stavroulakis oscillation criteria for second order delay differential equations appl math comput 140 2003 44 453 and by sun and meng y g sun f w meng note on the paper of d urina and stavroulakis appl math comput 174 2006 1634 1641

oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities abstract we present new oscillation criteria for the second order forced ordinary differential equation with mixed nonlinearities p t x q t x i 1 n q i t x i sgn x e t where p t q t q i t e t c 0 p t is positive and differentiable 1 m 1 m 1 n no restriction is imposed on the forcing term e t to be the second derivative of an oscillatory function when n 1 our results reduce to those of el sayed m a el sayed an oscillation criterion for a forced second order linear differential equation proc amer math soc 118 1993 813 817 wong j s w wong oscillation criteria for a forced second linear differential equations j math anal appl 231 1999 235 240 sun ou and wong y g sun c h ou j s w wong interval oscillation theorems for a linear second order differential equation comput math appl 48 2004 1693 1699 for the linear equation nazr a h nazr sufficient conditions for the oscillation of forced super linear second order differential equations with oscillatory potential proc amer math soc 126 1998 123 125 for the superlinear equation and sun and wong y g sun j s w wong note on forced oscillation of nth order sublinear differential equations j math anal appl 298 2004 114 119 for the sublinear equation

generalized treatment effects for clinical trials practice in the analysis of clinical trials with continuously measured endpoints is to focus on the difference or percentage change in mean or median response however treatments may have effects on the distribution of responses other than on the average response we sought an approach to such generalized treatment effects that i targets a parameter that is easily understood by our clinical colleagues and ii employs confidence intervals as the basis for inference we consider one such approach based on work in reliability theory namely setting pr y x as the target parameter and compare this approach to an earlier one due to o brien the two approaches have similar properties when they both seek to reject the null hypothesis of no effect due to different variances but differ when the larger variance corresponds to the larger mean in that case our approach views that the larger variance attenuates the effect of the larger mean out suggested approach applies easily to positive control clinical equivalence trials copyright 2000 john wiley sons ltd

statistical inference for p x y let x and y be two independent continuous random variables we make statistical inference about p x y the so called stress strength model through the edgeworth expansions and the bootstrap approximations of the studentized wilcoxon mann whitney statistics finite sample accuracy of the confidence intervals is assessed through a simulation study two real data sets are analysed to illustrate our methods copyright 2007 john wiley sons ltd

testing reliability in a stress strength model when x and y are normally distributed we consider the stress strength problem in which a unit of strength x is subjected to environmental stress y an important problem in stress strength reliability concerns testing hypotheses about the reliability parameter r p x yl in this article we consider situations in which x and y are independent and have normal distributions or can be transformed to normality we do not require the two population variances to be equal our approach leads to test statistics which are exact p values that are represented as one dimensional integrals on the basis of the p value one can also construct approximate confidence intervals for the parameter of interest we also present an extension of the testing procedure to the case in which both strength and stress depend on covariates for comparative purposes the bayesian solution to the problem is also presented we use data from a rocket motor experiment to illustrate the procedure

stress strength reliability for designs based on large historic values of stress abstract some new ideas and thoughts concerning the definition and calculation of reliability for stress strength models for failure are presented in particular calculations are carried out and statistical inference is made for systems whose design is made on the basis of the past data with emphasis on extremes and excesses this is done based on the observation that for such designs reliability estimation can be viewed as the statistical problem of comparing future values with large values of the past for a single distribution it is discussed that this approach could particularly prove useful when no basis exists for assuming any specific distributions for either stress or strength or both but when design is made or experimentation has been performed yielding sufficient information to assume a certain functional relationship between distributions for stress and strength some ideas from information theory are also brought in to provide a guideline for defining reliability on the basis of an equivalent system finally a simple demonstrating example is also included using a set of published data

a note on nasr s and wong s papers abstract in the case of oscillatory potentials we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form x t q t x t sgn x t f t in the linear 1 and the superlinear 1 cases

note on forced oscillation of nth order sublinear differential equations in the case of oscillatory potentials we establish an oscillation theorem for the forced sublinear differential equation x n q t x sgnx e t t t0 no restriction is imposed on the forcing term e t to be the nth derivative of an oscillatory function in particular we show that all solutions of the equation x t sint x sgnx mt cost t 0 0 2 1 this provides an analogue of a result of nasr proc amer math soc 126 1998 123 for the forced superlinear equation and answers a question raised in an earlier paper j s w wong siam j math anal 19 1988 673

on the laplace beltrami operator and brain surface flattening in this paper using certain conformal mappings from uniformization theory the authors give an explicit method for flattening the brain surface in a way which preserves angles from a triangulated surface representation of the cortex the authors indicate how the procedure may be implemented using finite elements further they show how the geometry of the brain surface may be studied using this approach

modeling and inference for an ordinal effect size measure an ordinal measure of effect size is a simple and useful way to describe the difference between two ordered categorical distributions this measure summarizes the probability that an outcome from one distribution falls above an outcome from the other adjusted for ties we develop and compare confidence interval methods for the measure simulation studies show that with independent multinomial samples confidence intervals based on inverting the score test and a pseudo score type test perform well this score method also seems to work well with fully ranked data but for dependent samples a simple wald interval on the logit scale can be better with small samples we also explore how the ordinal effect size measure relates to an effect measure commonly used for normal distributions and we consider a logit model for describing how it depends on explanatory variables the methods are illustrated for a study comparing treatments for shoulder tip pain

estimation of the area under the roc curve the area under the receiver operating characteristic curve is frequently used as a measure for the effectiveness of diagnostic markers in this paper we discuss and compare estimation procedures for this area these are based on i the mann whitney statistic ii kernel smoothing iii normal assumptions iv empirical transformations to normality these are compared in terms of bias and root mean square error in a large variety of situations by means of an extensive simulation study overall we find that transforming to normality usually is to be preferred except for bimodal cases where kernel methods can be effective

interval criteria for oscillation of second order linear ordinary differential equations abstract new oscillation criteria are established for the equation py qy 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem and sharper than some previous results and can be applied to extreme cases such as t 0 q t dt

comparison criteria for third order functional dynamic equations with mixed nonlinearities in this paper we investigate comparison criteria for third order nonlinear dynamic equations with mixed nonlinearities on time scales our results are essentially new some applications illustrating the importance of our results are included and these applications solve a problem posed in 2 remark 3 3

oscillation of second order functional differential equations with mixed nonlinearities and oscillatory potentials in this paper we study oscillation of second order functional differential equations with mixed nonlinearities p t x t q t x t t i 1nq i t x t t a isgnx t t e t where t 0 p t c 1 0 q t q i t e t c 0 p t 0 a 1 a m 1 a m 1 a n 0 n m 1 without assuming that q t q i t and e t are nonnegative the results given in y g sun f w meng interval criteria for oscillation of second order differential equations with mixed nonlinearities appl math comput 198 2008 375 381 have been extended to the aforementioned functional differential equation

a family of nonparametric statistics for comparing diagnostic markers with paired or unpaired data summary in this paper we study a broad class of nonparametric statistics for comparing two diagnostic markers one can compare the sensitivities of these diagnostic markers over restricted ranges of specificity by selecting an appropriate statistic from this class as special cases one can compare the entire area under the receiver operator curve hanley mcneil 1982 or one can compare the sensitivities at a fixed common specificity usually we would recommend a comparison based on an average of sensitivities over a restricted high level of specificities test procedures and confidence intervals are based on asymptotic normality these procedures are applicable for paired data in which both diagnostic markers are performed on each subject and for unpaired data the procedures may be used to compare two real functions of multiple diagnostic markers as well as to compare individual markers

vector difference calculus for physical lattice models a vector difference calculus is developed for physical models defined on a general triangulating graph g which may be a regular or an extremely irregular lattice using discrete field quantities roughly analogous to differential forms the role of the space ensuremath lambda p of p forms at a point is taken on by the linear space generated at a graph vertex by the geometrical p simplices which contain it the vector operations divergence gradient and curl are developed using the boundary ensuremath partial and coboundary d dot cross and scalar products are defined in such a way that discrete analogs of the vector integral theorems including theorems of gauss ostrogradski stokes and green as well as most standard vector identities hold exactly not as approximations to a continuum limit physical conservation laws for the models become theorems satisfied by the discrete fields themselves three discrete lattice models are constructed as examples namely a discrete version of the maxwell equations the navier stokes equation for incompressible flow and the navier linearized model for a homogeneous isotropic elastic medium weight factors needed for obtaining quantitative agreement with continuum calculations are derived for the special case of a regular triangular lattice green functions are developed using a generalized helmholtz decomposition of the fields

measuring the effectiveness of diagnostic markers in the presence of measurement error through the use of roc curves the area under the receiver operating characteristic curve is frequently used to assess the effectiveness of diagnostic markers in distinguishing between diseased and healthy individuals these markers are generally subject to measurement error in this paper confidence intervals for the area under the curve are developed which take measurement error into account these intervals depend on the availability of replicated observations for the subject both equal and unequal numbers of replicates per subject are considered the practice of averaging over replicates and then ignoring measurement error is examined and found wanting

estimation of reliability for exponential stress strength models with explanatory variables abstract confidence bounds for p y x z w are obtained where x and y are independent exponential variables with explanatory variables z and w respectively the fisher scoring method as an iteration method is used to obtain estimates of regression parameters simulation results are presented to verify the distributional assumptions of estimators

a parametric solution for simple stress strength model of failure with an application abstract estimation of p r s is considered for the simple stress strength model of failure using the pareto and power distributions together with their combined form a useful parametric solution is obtained and is illustrated numerically it is shown that these models are also applicable when only the tails of distributions for r and s are considered an application to the failure study concerning the fractures is also included

deformable templates using large deformation kinematics a general automatic approach is presented for accommodating local shape variation when mapping a two dimensional 2 d or three dimensional 3 d template image into alignment with a topologically similar target image local shape variability is accommodated by applying a vector field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth globally positive definite jacobian smoothness is guaranteed without specifically penalizing large magnitude deformations of small subvolumes by constraining the transformation on the basis of a stokesian limit of the fluid dynamical navier stokes equations this differs fundamentally from quadratic penalty methods such as those based on linearized elasticity or thin plate splines in that stress restraining the motion relaxes over time allowing large magnitude deformations kinematic nonlinearities are inherently necessary to maintain continuity of structures during large magnitude deformations and are included in all results after initial global registration final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability

spherical demons fast diffeomorphic landmark free surface registration we present the spherical demons algorithm for registering two spherical images by exploiting spherical vector spline interpolation theory we show that a large class of regularizors for the modified demons objective function can be efficiently approximated on the sphere using iterative smoothing based on one parameter subgroups of diffeomorphisms the resulting registration is diffeomorphic and fast the spherical demons algorithm can also be modified to register a given spherical image to a probabilistic atlas we demonstrate two variants of the algorithm corresponding to warping the atlas or warping the subject registration of a cortical surface mesh to an atlas mesh both with more than 160 k nodes requires less than 5 min when warping the atlas and less than 3 min when warping the subject on a xeon 3 2 ghz single processor machine this is comparable to the fastest nondiffeomorphic landmark free surface registration algorithms furthermore the accuracy of our method compares favorably to the popular freesurfer registration algorithm we validate the technique in two different applications that use registration to transfer segmentation labels onto a new image 1 parcellation of in vivo cortical surfaces and 2 brodmann area localization in ex vivo cortical surfaces

on some simple estimates of an exponent of regular variation summary we present a class of simple estimates of an exponent of regular variation unlike those proposed recently by de haan and resnick 1980 ours converge at an algebraic rather than a logarithmic rate and unlike those considered by teugels 1980 their form does not depend on the unknown exponent

interval criteria for oscillation of second order differential equations with mixed nonlinearities abstract in this paper we establish interval criteria for oscillation of the second order forced ordinary differential equations with mixed nonlinearities p t x q t x i 1 n q i t x i sgn x e t where p t q t qi t e t are continuous functions defined on 0 p t is positive and differentiable 1 m 1 m 1 n 0 n m 1 no restriction is imposed on the potentials q t qi t and e t to be nonnegative

confidence intervals for p y1 y2 with normal outcomes in linear models recently there is an emerging interest in the inference of p y1 y2 where y1 and y2 stand for two independent continuous random variables so far most of the research in this field focuses on simply comparing two outcomes without adjusting for covariates this paper mainly presents a large sample approach based on a noncentral t distribution for the confidence interval estimation of p y1 y2 with normal outcomes in linear models furthermore the performance of the proposed large sample approach is compared with that of a generalized variable approach and a bootstrap approach simulation studies demonstrate that for small to medium sample sizes both the large sample approach and the generalized variable approach provide confidence intervals with satisfying coverage probabilities whereas the bootstrap approach can be slightly liberal for certain scenarios the proposed approaches are applied to three real life data sets copyright 2008 john wiley sons ltd

generalized p values in significance testing of hypotheses in the presence of nuisance parameters abstract this article examines some problems of significance testing for one sided hypotheses of the form h 0 0 versus h 1 0 where is the parameter of interest in the usual setting let x be the observed data and let t x be a test statistic such that the family of distributions of t x is stochastically increasing in define cx as x t x t x 0 the p value is p x sup 0 pr x cx in the presence of a nuisance parameter there may not exist a nontrivial cx with a p value independent of we consider tests based on generalized extreme regions of the form cx x t x x t x x and conditions on t x x are given such that the p value p x sup 0 pr x cx is free of the nuisance parameter where t is stochastically increasing in we provide a solution to the problem of testing hypotheses about the differences in means of two independent exponential distributions a problem for which the fixed level testing approach

interval criteria for oscillation of a forced impulsive differential equation with riemann stieltjes integral in this paper both el sayed type and kamenev type oscillation criteria for a forced impulsive differential equation with riemann stieltjes integral are established by using a generalized el sayed type function and kong s technique in terms of the number of impulse moments on a series of intervals we not only drop the restriction on the impulse constants c k and d k that d k c k in the literature but also extend some existing results to the case of riemann stieltjes integral and improve some results in the case of riemann integral two examples are also considered to illustrate the main results

interval estimation for the area under the receiver operating characteristic curve when data are subject to error the area a under the receiver operating characteristic curve is commonly used to quantify the ability of a biomarker to correctly classify individuals into two populations however many markers are subject to measurement error which must be accounted for to prevent understating their effectiveness in this paper we develop a new confidence interval procedure for a which is adjusted for measurement error using either external or internal replicated measurements based on the observation that a is a function of normal means and variances we develop the procedure by recovering variance estimates needed from confidence limits for normal means and variances simulation results show that the procedure performs better than the previous ones based on the delta method in terms of coverage percentage balance of tail errors and interval width two examples are presented

comparing two samples extensions of the t rank sum and log rank tests abstract i consider the problem of testing that two populations are identical with respect to the distribution of a continuous variable against the alternative that values tend to be larger in one population i observe that the t rank sum and log rank tests are insensitive for a large class of alternatives that may be expected to occur commonly in practice propose a criterion for identifying this problem in a particular data set and propose corresponding extensions of the conventional methods the proposed methods should be useful for both identifying and interpreting group differences

confidence limits for stress strength models with explanatory variables a lower confidence bound is obtained for pr y x z 1 z 2 where x and y are independent normal variables with explanatory variables z 1 and z 2 respectively for equal residual variances an exact solution is obtained but for the unequal variance case an approximate lower confidence bound is developed examples of the use of these procedures are given

tests of equivalence and non inferiority for diagnostic accuracy based on the paired areas under roc curves assessment of equivalence or non inferiority in accuracy between two diagnostic procedures often involves comparisons of paired areas under the receiver operating characteristic roc curves with some pre specified clinically meaningful limits the current approach to evaluating equivalence is to perform the two one sided tests tost based on the difference in paired areas under roc curves estimated by the non parametric method we propose to use the standardized difference for assessing equivalence or non inferiority in diagnostic accuracy based on paired areas under roc curves between two diagnostic procedures the bootstrap technique is also suggested for both non parametric method and the standardized difference approach a simulation study was conducted empirically to investigate the size and power of the four methods for various combinations of distributions data types sample sizes and different correlations simulation results demonstrate that the bootstrap procedure of the standardized difference approach not only can adequately control the type i error rate at the nominal level but also provides equivalent power under both symmetrical and skewed distributions a numerical example using published data illustrates the proposed methods

roc curve analysis for biomarkers based on pooled assessments interleukin 6 is a biomarker of inflammation which has been suggested as having potential discriminatory ability for myocardial infarction because of its high assaying cost it is very expensive to evaluate this marker in order to reduce this cost we propose pooling the specimens in this paper we examine the efficiency of roc curve analysis specifically the estimation of the area under the roc curve when dealing with pooled data we study the effect of pooling when there are only a fixed number of individuals available for testing and pooling is carried out to save on the number of assays alternatively we examine how many pooled assays of size g are necessary to provide essentially the same information as n individual assays we measure loss of information by means of the change in root mean square error of the estimate of the area under the roc curve and study the extent of this loss via a simulation study

estimation of p z y for correlated stochastic time series models let z and y represent two time series that are not necessarily independent and z n l y m k denote their values respectively at future times n l and m k where n l m k autoregressive ar moving average ma and autoregressive moving average arma models are employed both under stationary and non stationary conditions to estimate p z n l

triangle lattice green functions for vector fields all the identities and integral theorems of vector calculus are contained in the calculus of differential forms the analogy between the exterior calculus of forms and the homology theory of a cell complex yields discrete lattice models for an array of interesting physical phenomena these models based on arbitrary combinations of coupled scalar and polar or axial vector field quantities can be manipulated as conveniently as the standard scalar tight binding models we develop green functions gfs for an infinite hierarchy of such models expressed using the four fundamental operators of the triangle lattice the triangle lattice is distinguished among two dimensional grid types for having the highest possible isotropy closed formulae are derived for gfs of the scalar and vector models that belong to a hierarchy generated by the four fundamental operators of the triangle family of lattices the particular example of lattice electromagnetism coupled to an elastic distortion field is treated in detail topological properties not dependent upon symmetry split the response functions into plasmon and polariton like parts since the fundamental vector operators of the triangle lattice are related simply to adjacency matrices of the kagome lattice scalar gfs for this lattice are found also as a byproduct

interval oscillation criteria for second order forced delay dynamic equations with mixed nonlinearities interval oscillation criteria are established for second order forced delay dynamic equations on time scales containing mixed nonlinearities of the form r t f a x d t d p 0 t f a x t 0 t i 1np i t f b i x t i t e t t t 0 t where t is a time scale t 0 t a fixed number t 0 t is a time scale interval f u u 1u the functions r p i e t 0 t r are right dense continuous with r 0 nondecreasing t k t t are nondecreasing right dense continuous with t k t t lim t t k t and the exponents satisfy b 1 b m a b m 1 b n 0 all results are new even for t r and t z analogous results for related advance type equations are also given as well as extended delay and advance equations the theory can be applied to second order dynamic equations regardless of the choice of delta or nabla derivatives two examples are provided to illustrate one of the theorems

statistical inference for pr y x the normal case this article examines statistical inference for pr y x where x and y are independent normal variates with unknown means and variances the case of unequal variances is stressed x can be interpreted as the strength of a component subjected to a stress y and pr y x is the component s reliability two approximate methods for obtaining confidence intervals and an approximate bayesian probability interval are obtained the actual coverage probabilities of these intervals are examined by simulation

confidence intervals for the log normal mean in this paper we conduct a stimulation study to evaluate coverage error interval width and relative bias of four main methods for the construction of confidence intervals of log normal means the naive method cox s method a conservative method and a parametric bootstrap method the simulation study finds that the naive method is inappropriate that cox s method has the smallest coverage error for moderate and large sample sizes and that the bootstrap method has the smallest coverage error for small sample sizes in addition cox s method produces the smallest interval width among the three appropriate methods we also apply the four methods to a real data set to contrast the differences

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

estimating the stable index alpha in order to measure tail thickness a critique stable laws are often fit to outlier prone data and if the index alpha is estimated to be much less than two then the normal law is rejected in favor of an infinite variance stable law this paper derives the theoretical properties of such a procedure when the true distribution is stable the distribution of the m l e of alpha is non regular if alpha 2 when the true distribution is not stable the estimate of alpha is not a robust measure of the rate of decrease of the tail probabilities a more robust procedure is developed and a statistic for describing and comparing the tail shapes of arbitrary samples is proposed

normal based methods for a gamma distribution prediction and tolerance intervals and stress strength reliability in this article we propose inferential procedures for a gamma distribution using the wilson hilferty wh normal approximation specifically using the result that the cube root of a gamma random variable is approximately normally distributed we propose normal based approaches for a gamma distribution for a constructing prediction limits one sided tolerance limits and tolerance intervals b for obtaining upper prediction limits for at least l of m observations from a gamma distribution at each of r locations and c assessing the reliability of a stress strength model involving two independent gamma random variables for each problem a normal based approximate procedure is outlined and its applicability and validity for a gamma distribution are studied using monte carlo simulation our investigation shows that the approximate procedures are very satisfactory for all of these problems for each problem considered the results are illustrated using practical examples our overall conclusion is tha