the berry esseen bound for student s statistic we prove the berry esseen bound for the student t statistic under the assumption of a third moment this bound coincides up to an absolute constant with the classical berry esseen bound for the mean in general the distribution of the student statistic converges to the standard normal distribution function at least as fast as the distribution of the mean and sometimes faster for example rates of convergence can be proved if the underlying distribution is in the domain of attraction of the normal law

efficient treatment allocation in 2 2 multicenter trials when costs and variances are heterogeneous at the design stage of a study it is crucial to compute the sample size needed for treatment effect estimation with maximum precision and power the optimal design depends on the costs which may be known at the design stage and on the outcome variances which are unknown a balanced design optimal for homogeneous costs and variances is typically used an alternative to the balanced design is a design optimal for the known and possibly heterogeneous costs and homogeneous variances called costs considering design both designs suffer from loss of efficiency compared with optimal designs for heterogeneous costs and variances for 2 2 multicenter trials we compute the relative efficiency of the balanced and the costs considering designs relative to the optimal designs we consider 2 heterogeneous costs and variance scenarios in 1 scenario 2 treatment conditions have small and 2 have large costs and variances in the other scenario 1 treatment condition has small 2 have intermediate and 1 has large costs and variances within these scenarios we examine the relative efficiency of the balanced design and of the costs considering design as a function of the extents of heterogeneity of the costs and of the variances and of their congruence congruent when the cheapest treatment has the smallest variance incongruent when the cheapest treatment has the largest variance we find that the costs considering design is generally more efficient than the balanced design and we illustrate this theory on a 2 2 multicenter trial on lifestyle improvement of patients in general practices

saddlepoint approximation for student s t statistic with no moment conditions a saddlepoint approximation of the student s t statistic was derived by daniels and young biometrika 78 1991 169 179 under the very stringent exponential moment condition that requires that the underlying density function go down at least as fast as a normal density in the tails this is a severe restriction on the approximation s applicability in this paper we show that this strong exponential moment restriction can be completely dispensed with that is saddlepoint approximation of the student s t statistic remains valid without any moment condition this confirms the folklore that the student s t statistic is robust against outliers the saddlepoint approximation neat only provides a very accurate approximation for the student s t statistic but it also can be applied much more widely in statistical inference as a result saddlepoint approximations should always be used whenever possible some numerical work will be given to illustrate these points

a simple procedure to compute the sample size needed to compare two independent groups when the population variances are unequal the usual methods of sample size determination which assume the homogeneity of within group variances are not sufficiently robust against the violation of this assumption a method using satterthwaite s correction for unequal variances is given it is shown that in the case of unequal variances the minimum total sample size is attained when the allocation ratio is equal to the proportion of the standard deviations

relative efficiency of unequal versus equal cluster sizes in cluster randomized and multicentre trials cluster randomized and multicentre trials evaluate the effect of a treatment on persons nested within clusters for instance patients within clinics or pupils within schools optimal sample sizes at the cluster centre and person level have been derived under the restrictive assumption of equal sample sizes per cluster this paper addresses the relative efficiency of unequal versus equal cluster sizes in case of cluster randomization and person randomization within clusters starting from maximum likelihood parameter estimation the relative efficiency is investigated numerically for a range of cluster size distributions an approximate formula is presented for computing the relative efficiency as a function of the mean and variance of cluster size and the intraclass correlation which can be used for adjusting the sample size the accuracy of this formula is checked against the numerical results and found to be quite good it is concluded that the loss of efficiency due to variation of cluster sizes rarely exceeds 10 per cent and can be compensated by sampling 11 per cent more clusters

precise rates in the law of the logarithm for negatively associated random variables let x n n 1 be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance set s email protected k 1 nx k m n max k n s k n 1 suppose s 2 ex 1 email protected k 2 ex 1x k we study the precise rates of a kind of weighted infinite series of p m n email protected snlogn and p s n email protected snlogn as email protected 0 and p m email protected email protected email protected 2n8logn as email protected the results are related to the convergence rates of the law of the logarithm and the chung type law of the logarithm

bootstrapping the student t statistic let x 1 x n n 1 be independent identically distributed random variables and consider the student t statistic t n based upon these random variables gine gotze and mason 1997 proved that t n converges in distribution to a standard normal random variable if and only if x is in the domain of attraction of a normal random variable and ex 0 we shall show that roughly the same holds true for the bootstrapped student t statistic t n in the process we shall disclose all the possible subsequential limiting laws of t n the proofs introduce a number of amusing tricks that may be of independent interest

precise asymptotics in the self normalized law of the iterated logarithm abstract let x x 1 x 2 be i i d nondegenerate random variables with zero means s n j 1 n x j and v n 2 j 1 n x j 2 we investigate the precise asymptotics in the law of the iterated logarithm for self normalized sums s n v n also for the maximum of self normalized sums max 1 k n s k v n when x belongs to the domain of attraction of the normal law

a noniterative sample size procedure for tests based on t distributions a noniterative sample size procedure is proposed for a general hypothesis test based on the t distribution by modifying and extending guenther s approach for the one sample and two sample t tests the generalized procedure is employed to determine the sample size for treatment comparisons using the analysis of covariance ancova and the mixed effects model for repeated measures in randomized clinical trials the sample size is calculated by adding a few simple correction terms to the sample size from the normal approximation to account for the nonnormality of the t statistic and lower order variance terms which are functions of the covariates in the model but it does not require specifying the covariate distribution the noniterative procedure is suitable for superiority tests noninferiority tests and a special case of the tests for equivalence or bioequivalence and generally yields the exact or nearly exact sample size estimate after rounding to an integer the method for calculating the exact power of the two sample t test with unequal variance in superiority trials is extended to equivalence trials we also derive accurate power formulae for ancova and mixed effects model for repeated measures and the formula for ancova is exact for normally distributed covariates numerical examples demonstrate the accuracy of the proposed methods particularly in small samples

optimum treatment allocation rules under a variance heterogeneity model we provide optimal treatment allocation schemes when the outcome variance varies across the treatment groups and our objectives are to estimate treatment effects with equal or unequal interest unlike other optimal designs such as a optimal designs the proposed designs can be found without an iterative scheme we evaluate robustness properties of the optimal designs to mis specification in the expected variance from each group and identify situations when popular allocation schemes have poor efficiencies an application to design a randomized rheumatoid arthritis trial is discussed along with a potential application to design a cancer screening trial when the main outcome is a continuous variable

self normalized large deviations let x x n n geq 1 be a sequence of independent and identically distributed random variables the classical cramer chernoff large deviation states that lim n to infty n 1 ln p sum i 1 n x i n geq x ln rho x if and only if the moment generating function of x is finite in a right neighborhood of zero this paper uses n p 1 p v n p n p 1 p sum i 1 n x i p 1 p p 1 as the normalizing constant to establish a self normalized large deviation without any moment conditions a self normalized moderate deviation that is the asymptotic probability of p s n v n p geq x n for x n o n p 1 p is also found for any x in the domain of attraction of a normal or stable law as a consequence a precise constant in the self normalized law of the iterated logarithm of griffin and kuelbs is obtained applications to the limit distribution of self normalized sums the asymptotic probability of the t statistic as well as to the erdos renyi shepp law of large numbers are also discussed

a general law of precise asymptotics for the counting process of record times abstract this paper achieves a general law of precise asymptotics for the counting process of record times of i i d absolutely continuous random variables it can describe the relations among the boundary function weighted function convergence rate and limit value in studies of complete convergence this extends and generalizes the corresponding results in stochastic process appl 101 2002 233 239

the precise asymptotic behavior of parameter estimators in ornstein uhlenbeck process abstract in the present paper we study the asymptotic behavior for estimator of the drift parameter in an ornstein uhlenbeck process the l r convergence rate and the precise asymptotics in the law of iterated logarithm and in the law of logarithm for the estimator are obtained moreover we also get the complete moment convergence of this estimator the main method of this paper is the deviation inequality for the quadratic functional

planning group sizes in clinical trials with a continuous outcome and repeated measures simple formulae have been derived for the accurate computation of sample sizes when the two sample t test is used required sample sizes may be reduced by taking repeated measures especially if correlations are about equal in a trial with two parallel groups repeated measures may be summarized by a pre treatment mean and a post treatment mean in a cross over trial measures may be repeated within each treatment period in case the difference between two independent groups increases in time individual slopes may be used if the correlation between two measures hardly depends on distance in time between these measures however it may be more efficient to use only the first and last measure in the statistical test if correlations strongly depend on time distance

nonlinear stability criteria for the hmf model we study the nonlinear stability of a large class of inhomogeneous steady state solutions to the hamiltonian mean field hmf model under a specific criterion we prove the nonlinear stability of steady states which are decreasing functions of the microscopic energy to achieve this task we extend to this context the strategy based on generalized rearrangement techniques which was developed recently for the gravitational vlasov poisson equation explicit stability inequalities are established and our analysis is able to treat non compactly supported steady states to hmf which are physically relevant in this context but induces additional difficulties compared to the vlasov poisson system

darling erdős theorem for self normalized sums let x x1 x2 be i i d nondegenerate random variables sn nj 1xj and v2n nj 1x2j we investigate the asymptotic vspace 1pt behavior in distribution of the maximum of self normalized sums max1 k nsk vk and the law of the iterated logarithm for self normalized sums sn vn when x belongs to the domain of attraction of the normal law in this context we establish a darling erd s type theorem as well as an erd s feller kolmogorov petrovski type test for self normalized sums

self normalized processes exponential inequalities moment bounds and iterated logarithm laws self normalized processes arise naturally in statistical applications being unit free they are not affected by scale changes moreover self normalization often eliminates or weakens moment assumptions in this paper we present several exponential and moment inequalities particularly those related to laws of the iterated logarithm for self normalized random variables including martingales tail probability bounds are also derived for random variables bt 0 and at let yt exp at 2bt2 2 we develop inequalities for the moments of at bt or supt 0at bt log logbt 1 2 and variants thereof when eyt 1 or when yt is a supermartingale for all belonging to some interval our results are valid for a wide class of random processes including continuous martingales with at mt and b t sqrt langle m rangle t and sums of conditionally symmetric variables di with at i 1tdi and b t sqrt sum i 1 t d i 2 a sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in rm m 1 is also established another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self normalized by the square root of the sum of squares the key ingredient in this development is a new exponential supermartingale involving i 1tdi and i 1tdi2 a compact law of the iterated logarithm for self normalized martingales is also derived in this connection

optimal design of multi subject blocked fmri experiments the design of a multi subject fmri experiment needs specification of the number of subjects and scanning time per subject for example for a blocked design with conditions a or b fixed block length and block order abn where n denotes a null block the optimal number of cycles of abn and the optimal number of subjects have to be determined this paper presents a method to determine the optimal number of subjects and optimal number of cycles for a blocked design based on the a optimality criterion and a linear cost function by which the number of cycles and the number of subjects are restricted estimation of individual stimulus effects and estimation of contrasts between stimulus effects are both considered the mixed effects model is applied and analytical results for the a optimal number of subjects and a optimal number of cycles are obtained under the assumption of uncorrelated errors for correlated errors with a first order autoregressive ar1 error structure numerical results are presented our results show how the optimal number of cycles and subjects depend on the within to between subject variance ratio our method is a new approach to determine the optimal scanning time and optimal number of subjects for a multi subject fmri experiment in contrast to previous results based on power analyses the optimal number of cycles and subjects can be described analytically and costs are considered

restricted unbiased iterative generalized least squares estimation summary it is shown that the iterative least squares procedure for estimating the parameters in a general multilevel random coefficients linear model can be modified to produce unbiased estimates of the random parameters in the multivariate normal case these are equivalent to restricted maximum likelihood estimates

precise rates in the law of logarithm for i i d random variables let x x n n 1 be a sequence of i i d random variables set s n x 1 x 2 x n and m n max k n s k n 1 by using the strong approximation method we obtain that for any 1 email protected email protected n 2e n 2 b 2 email protected k 0 1 k 2k 1 2 b 2 if and only if ex 0 and ex 2 which strengthen and extend the result of gut and email protected taru 1 where n is the standard normal random variable furthermore l 2 convergence and a s convergence are also discussed

efficient design of cluster randomized trials with treatment dependent costs and treatment dependent unknown variances cluster randomized trials evaluate the effect of a treatment on persons nested within clusters where treatment is randomly assigned to clusters current equations for the optimal sample size at the cluster and person level assume that the outcome variances and or the study costs are known and homogeneous between treatment arms this paper presents efficient yet robust designs for cluster randomized trials with treatment dependent costs and treatment dependent unknown variances and compares these with 2 practical designs first the maximin design mmd is derived which maximizes the minimum efficiency minimizes the maximum sampling variance of the treatment effect estimator over a range of treatment to control variance ratios the mmd is then compared with the optimal design for homogeneous variances and costs balanced design and with that for homogeneous variances and treatment dependent costs cost considered design the results show that the balanced design is the mmd if the treatment to control cost ratio is the same at both design levels cluster person and within the range for the treatment to control variance ratio it still is highly efficient and better than the cost considered design if the cost ratio is within the range for the squared variance ratio outside that range the cost considered design is better and highly efficient but it is not the mmd an example shows sample size calculation for the mmd and the computer code spss and r is provided as supplementary material the mmd is recommended for trial planning if the study costs are treatment dependent and homogeneity of variances cannot be assumed

an exponential nonuniform berry esseen bound for self normalized sums in this paper we shall derive exponential nonuniformberry esseen bounds in the central limit theorem for self normalized sums we show that the size of the error can be reduced considerably by replacing the usual standardization by self normalization in particular we establish the exponential bounds for the probability of the self normalized sums under the condition that the third moment is finite whereas an exponential moment assumption is required for the standardized sums applications to t statistics and the probabilities of moderate deviations of self normalized sums are also discussed

on landau damping going beyond the linearized study has been a longstanding problem in the theory of landau damping in this paper we establish exponential landau damping in analytic regularity the damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables rather than exchanges of energy phase mixing is the driving mechanism the analysis involves new families of analytic norms measuring regularity by comparison with solutions of the free transport equation new functional inequalities a control of non linear echoes sharp deflection estimates and a newton approximation scheme our results hold for any potential no more singular than coulomb or newton interaction the limit cases are included with specific technical effort as a side result the stability of homogeneous equilibria of the non linear vlasov equation is established under sharp assumptions we point out the strong analogy with the kam theory and discuss physical implications finally we extend these results to some gevrey non analytic distribution functions

limit distributions of studentized means let x xj j n be independent identically distributed random variables with probability distribution f it is shown that student s statistic of the sample xj nj 1 has a limit distribution g such that g 1 1 1 if and only if 1 x is in the domain of attraction of a stable law with some exponent 0 0 such that nx 2 1 x x y2f dy 1 if g 1 1 1 then student s statistic of the sample xj nj 1 has a limit distribution if and only if p x x x 0 is a slowly varying function at

when is the student t statistic asymptotically standard normal let x x i i in mathbb n be independent identically distributed random variables it is shown that the student t statistic based upon the sample x i i 1 n is asymptotically n 0 1 if and only if x is in the domain of attraction of the normal law it is also shown that for any x if the self normalized sums u n sum i 1 n x i sum i 1 n x i 2 1 2 n in mathbb n are stochastically bounded then they are uniformly subgaussian that is sup n mathbb e exp lambda u n 2 infty for some lambda 0

sample size formula with a continuous outcome for unequal group sizes and unequal variances guenther derived a simple formula for the accurate computation of equal sample sizes when the two sample t test with pooled variance estimate is used in the present paper the guenther formula is generalized to the case of unequal sample sizes in addition the case of unequal variances is considered

precise asymptotics in complete moment convergence for self normalized sums let x x 1 x 2 be a sequence of nondegenerate i i d random variables with zero means set s n x 1 x n and v n 2 x 1 2 x n 2 ex 2i x x is a slowly varying function at we prove that for any b 2 d 2 b 1 email protected email protected b d 1 email protected n 2 logn d 2 bne s n v n 2i s n email protected n logn 1 b email protected n b d 1 b d 1 2 where n is a standard normal random variable

precise asymptotics in the deviation probability series of self normalized sums abstract let x x 1 x 2 be a sequence of nondegenerate i i d random variables with zero means set s n x 1 x n and w n 2 x 1 2 x n 2 in the present paper we examine the precise asymptotic behavior for the general deviation probabilities of self normalized sums s n w n for positive functions g x x x and x we obtain the precise asymptotics for the following deviation probabilities of self normalized sums n 1 g n n e s n w n 2 i s n w n n n the results given can be considered as the generalization of that in the complete moment convergence law of iterated logarithm and large deviation for self normalized sums

precise asymptotics in the baum katz and davis laws of large numbers let x x1 x2 be a sequence of i i d random variables such that ex 0 let z be a random variable possessing a stable distribution g with exponent 1 2 assume the distribution of x is attracted to g and set sn x1 xn we prove that n 1nr p 2p sn en1 p e p p r p 1 pr pe z p p r p 1 as e 0 for 1 p r under the additional assumption that there is normal attraction to g and that n 11np sn en1 p p p logeas e 0 for 1 p we close with a related result for the limiting case r p 2

donsker s theorem for self normalized partial sums processes let x x1 x2 be a sequence of nondegenerate i i d random variables with zero means in this paper we show that a self normalized version of donsker s theorem holds only under the assumption that x belongs to the domain of attraction of the normal law a thus resulting extension of the arc sine law is also discussed we also establish that a weak invariance principle holds true for self normalized self randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero if and only if max1 j n xj vn p0 as n where v2n nj 1x2j

a modified approach to estimating sample size for simple logistic regression with one continuous covariate different methods for the calculation of sample size for simple logistic regression lr with one normally distributed continuous covariate give different results sometimes the difference can be large furthermore some methods require the user to specify the prevalence of cases when the covariate equals its population mean rather than the more natural population prevalence we focus on two commonly used methods and show through simulations that the power for a given sample size may differ substantially from the nominal value for one method especially when the covariate effect is large while the other method performs poorly if the user provides the population prevalence instead of the required parameter we propose a modification of the method of hsieh et al that requires specification of the population prevalence and that employs schouten s sample size formula for a t test with unequal variances and group sizes this approach appears to increase the accuracy of the sample size estimates for lr with one continuous covariate

exact convergence rate and leading term in central limit theorem for student s t statistic the leading term in the normal approximation to the distribution of student s t statistic is derived in a general setting with the sole assumption being that the sampled distribution is in the domain of attraction of a normal law the form of the leading term is shown to have its origin in the way in which extreme data influence properties of the studentized sum the leading term approximation is used to give the exact rate of convergence in the central limit theorem up to order n 1 2 where n denotes sample size it is proved that the exact rate uniformly on the whole real line is identical to the exact rate on sets of just three points moreover the exact rate is identical to that for the non studentized sum when the latter is normalized for scale using a truncated form of variance but when the corresponding truncated centering constant is omitted examples of characterizations of convergence rates are also given it is shown that in some instances their validity uniformly on the whole real line is equivalent to their validity on just two symmetric points

design and analysis of trials with a partially nested design and a binary outcome measure where treatments are administered to groups of patients or delivered by therapists outcomes for patients in the same group or treated by the same therapist may be more similar leading to clustering trials of such treatments should take account of this effect where such a treatment is compared with an un clustered treatment the trial has a partially nested design this paper compares statistical methods for this design where the outcome is binary investigation of consistency reveals that a random coefficient model with a random effect for group or therapist is not consistent with other methods for a null treatment effect and so this model is not recommended for this design small sample performance of a cluster adjusted test of proportions a summary measures test and logistic generalised estimating equations and random intercept models are investigated through simulation the expected treatment effect is biased for the logistic models empirical test size of two sided tests is raised only slightly but there are substantial biases for one sided tests three formulae are proposed for calculating sample size and power based on i the difference of proportions ii the log odds ratio or iii the arc sine transformation of proportions calculated power from these formulae is compared with empirical power from a simulations study logistic models appeared to perform better than those based on proportions with the likelihood ratio test performing best in the range of scenarios considered for these analyses the log odds ratio method of calculation of power gave an approximate lower limit for empirical power

sample size calculation in three level cluster randomized trials using generalized estimating equation models three level cluster randomized trials crts are increasingly used in implementation science where 2fold nested correlated data arise for example interventions are randomly assigned to practices and providers within the same practice who provide care to participants are trained with the assigned intervention teerenstra et al proposed a nested exchangeable correlation structure that accounts for two levels of clustering within the generalized estimating equations gee approach in this article we utilize gee models to test the treatment effect in a two group comparison for continuous binary or count data in three level crts given the nested exchangeable correlation structure we derive the asymptotic variances of the estimator of the treatment effect for different types of outcomes when the number of clusters is small researchers have proposed bias corrected sandwich estimators to improve performance in two level crts we extend the variances of two bias corrected sandwich estimators to three level crts the equal provider and practice sizes were assumed to calculate number of practices for simplicity however they are not guaranteed in practice relative efficiency re is defined as the ratio of variance of the estimator of the treatment effect for equal to unequal provider and practice sizes the expressions of res are obtained from both asymptotic variance estimation and bias corrected sandwich estimators their performances are evaluated for different scenarios of provider and practice size distributions through simulation studies finally a percentage increase in the number of practices is proposed due to efficiency loss from unequal provider and or practice sizes

asymptotic distribution of the largest off diagonal entry of correlation matrices suppose that we have observations from a dimensional population we are interested in testing that the variates of the population are independent under the situation where goes to infinity as a test statistic is chosen to be where is the sample correlation coefficient between the th coordinate and the th coordinate of the population under an independent hypothesis we prove that the asymptotic distribution of is an extreme distribution of type by using the chen stein poisson approximation method and the moderate deviations for sample correlation coefficients as a statistically more relevant result a limit distribution for where is spearman s rank correlation coefficient between the th coordinate and the th coordinate of the population is derived

self normalized cramer type large deviations for independent random variables let x 1 x 2 be independent random variables with zero means and finite variances it is well known that a finite exponential moment assumption is necessary for a cramer type large deviation result for the standardized partial sums in this paper we show that a cramer type large deviation theorem holds for self normalized sums only under a finite 2 th moment 0 1 in particular we show p s n v n x 1 x 1 o 1 1 x 2 d 2 n for 0 x d n where d n n i 1 ex 2 i 1 2 n i 1 e x i 2 1 2 and v n n i 1 x 2 i 1 2 applications to the studentized bootstrap and to the self normalized law of the iterated logarithm are discussed

self normalized laws of the iterated logarithm on demontre une loi du logarithme itere qui generalise la version classique pour toutes les distributions dans la classe de feller pour cette demonstration l auteur utilise une auto normalisation convenable pour les sommes partielles de variables aleatoires independantes et identiquement distribuees un cas particulier de ces resultats s applique a toute distribution dans le domaine de l attraction d une certaine loi stable

a kind of complete moment convergence for self normalized sums let x x n n 1 be a sequence of independent and identically distributed i i d random variables with x in the domain of attraction of the normal law and ex 0 in this note we obtain a new kind of complete moment convergence for self normalized sums

exact rates in log law for positively associated random variables abstract let x n n 1 be a strictly stationary sequence of positively associated random variables with mean zero and finite variance set s n k 1 n x k m n max k n s k n 1 suppose 2 e x 1 2 2 k 2 e x 1 x k in this paper we study the exact convergence rates of a kind of weighted infinite series of p m n e n log n p s n e n log n and i s n e n log n as e 0 respectively

spectral and formal stability criteria of spatially inhomogeneous stationary solutions to the vlasov equation for the hamiltonian mean field model stability of spatially inhomogeneous solutions to the vlasov equation is investigated for the hamiltonian mean field model to provide the spectral and formal stability criteria in the form of necessary and sufficient conditions these criteria determine stability of spatially inhomogeneous solutions whose stability has not been decided correctly by using a less refined formal stability criterion it is shown that some of such solutions can be found in a family of stationary solutions to the vlasov equation which is parametrized with macroscopic quantities and has a two phase coexistence region in the parameter space

orbital stability of spherical galactic models we consider the three dimensional gravitational vlasov poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters a well known conjecture is the stability of spherical models which are nonincreasing radially symmetric steady states solutions this conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by antonov in 1961 in a previous work we derived the stability of anisotropic models under it spherically symmetric perturbations using fundamental monotonicity properties of the hamiltonian under suitable generalized symmetric rearrangements first observed in the physics litterature in this work we show how this approach combined with a it new generalized antonov type coercivity property implies the orbital stability of spherical models under general perturbations

sample size adjustments for varying cluster sizes in cluster randomized trials with binary outcomes analyzed with second order pql mixed logistic regression adjustments of sample size formulas are given for varying cluster sizes in cluster randomized trials with a binary outcome when testing the treatment effect with mixed effects logistic regression using second order penalized quasi likelihood estimation pql starting from first order marginal quasi likelihood mql estimation of the treatment effect the asymptotic relative efficiency of unequal versus equal cluster sizes is derived a monte carlo simulation study shows this asymptotic relative efficiency to be rather accurate for realistic sample sizes when employing second order pql an approximate simpler formula is presented to estimate the efficiency loss due to varying cluster sizes when planning a trial in many cases sampling 14 per cent more clusters is sufficient to repair the efficiency loss due to varying cluster sizes since current closed form formulas for sample size calculation are based on first order mql planning a trial also requires a conversion factor to obtain the variance of the second order pql estimator in a second monte carlo study this conversion factor turned out to be 1 25 at most

on the neighborhood of an inhomogeneous stable stationary solution of the vlasov equation case of an attractive cosine potential we consider the one dimensional vlasov equation with an attractive cosine potential and its non homogeneous stable stationary states that are decreasing functions of the energy we show that in the sobolev space w1 p p 2 neighborhood of such a state all stationary states that are decreasing functions of the energy are stable this is in sharp contrast with the situation for homogeneous stationary states of a vlasov equation where a control over strictly more than one derivative is needed to ensure the absence of unstable stationary states in a neighborhood of a reference stationary state z lin and c zeng commun math phys 306 291 331 2011

large time behavior and asymptotic stability of the 2d euler and linearized euler equations abstract we study the asymptotic behavior and the asymptotic stability of the 2d euler equations and of the 2d linearized euler equations close to parallel flows we focus on flows with spectrally stable profiles u y and with stationary streamlines y y 0 such that u y 0 0 a case that has not been studied previously we describe a new dynamical phenomenon the depletion of the vorticity at the stationary streamlines an unexpected consequence is that the velocity decays for large times with power laws similarly to what happens in the case of the orr mechanism for base flows without stationary streamlines the asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations we argue on the asymptotic stability of this ensemble of flow profiles even in the absence of any dissipative mechanisms

on the effect of random norming on the rate of convergence in the central limit theorem it is shown that studentizing i e normalizing by the sample standard deviation rather than the population standard deviation can improve the rate of convergence in the central limit theorem this provides concise confirmation of one feature of the folklore that a studentized sum is in some sense more robust than a normed sum the case of infinite population standard deviation is also examined 1 introduction our purpose in this paper is to show that studentizing the operation of replacing a population standard deviation in a norming constant by its sample estimate can improve the rate of convergence in the central limit theorem thus we provide concise confirmation of one feature of the folklore that a studentized sum is in some sense more robust than a normed sum the reason is that large deviations of summands which are responsible for slow rates of convergence when ordinary norming constants are used cancel from the sample mean and sample standard deviation remark 3 7 in section 3 will give an intuitive explanation of the mechanism behind this cancellation note that the large deviations have a particularly subtle influence since under the condition that the distribution is attracted to the normal law the largest summand is negligible in comparison with the sum itself section 2 will briefly survey results on rates of convergence in the ordinary i e nonstudentized central limit theorem and will discuss their connection with our work section 3 will state our main results and summarize their consequences and section 4 will give proofs

a practical approach for comparing means of two groups without equal variance assumption h wang and s chow statistics in medicine 2002 21 3137 3151