logarithmic transformations in anova a method is presented for choosing an additive constant c when transforming data x to y log x c the method preserves type i error probability and power in anova under the assumption that the x c for some c are log normally distributed the method has advantages similar to those of rank transformations namely it is easy to use and is resistant to extreme observations since the special case c infinity corresponds in anova to y x the method is a useful generalization of least squares

arctic boundaries of the ice model on three bundle domains in this paper we consider the six vertex model at ice point on an arbitrary three bundle domain which is a generalization of the domain wall ice model on the square or equivalently of a uniformly random alternating sign matrix we show that this model exhibits the arctic boundary phenomenon whose boundary is given by a union of explicit algebraic curves this was originally predicted by colomo and sportiello j stat phys 164 1488 1523 2016 as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries called the geometric tangent method our proof uses a probabilistic analysis of non crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case which might be of independent interest

finite traces and representations of the group of infinite matrices over a finite field abstract the article is devoted to the representation theory of locally compact infinite dimensional group glb of almost upper triangular infinite matrices over the finite field with q elements this group was defined by s k a v and andrei zelevinsky in 1982 as an adequate n analogue of general linear groups gl n q it serves as an alternative to gl q whose representation theory is poor our most important results are the description of semifinite unipotent traces characters of the group glb via certain probability measures on the borel subgroup b and the construction of the corresponding von neumann factor representations of type ii as a main tool we use the subalgebra a glb of smooth functions in the group algebra l 1 glb this subalgebra is an inductive limit of the finite dimensional group algebras c gl n q under parabolic embeddings as in other examples of the asymptotic representation theory we discover remarkable properties of the infinite case which does not take place for finite groups like multiplicativity of indecomposable characters or connections to probabilistic concepts the infinite dimensional iwahori hecke algebra h q plays a special role in our considerations and allows to understand the deep analogy of the developed theory with the representation theory of infinite symmetric group s which had been intensively studied in numerous previous papers

quantized vershik kerov theory and quantized central measures on branching graphs abstract we propose a natural quantized character theory for inductive systems of compact quantum groups based on kms states on af algebras following stratila voiculescu s work 24 or 8 and give its serious investigation when the system consists of quantum unitary groups u q n with q 0 1 the key features of this work are the quantized trace of a unitary representation of a compact quantum group can be understood as a quantized character associated with the unitary representation and its normalized one is captured as a kms state with respect to a certain one parameter automorphism group related to the so called scaling group in this paper we provide a vershik kerov type approximation theorem for extremal quantized characters called the ergodic method and also compare our quantized character theory for the inductive system of u q n with gorin s theory on q gelfand tsetlin graphs 11

a quantization of the harmonic analysis on the infinite dimensional unitary group abstract the present work stemmed from the study of the problem of harmonic analysis on the infinite dimensional unitary group u that problem consisted in the decomposition of a certain 4 parameter family of unitary representations which replace the nonexisting two sided regular representation olshanski 31 the required decomposition is governed by certain probability measures on an infinite dimensional space which is a dual object to u a way to describe those measures is to convert them into determinantal point processes on the real line it turned out that their correlation kernels are computable in explicit form they admit a closed expression in terms of the gauss hypergeometric function f 1 2 borodin and olshanski 8 in the present work we describe a nonevident q discretization of the whole construction this leads us to a new family of determinantal point processes we reveal its connection with an exotic finite system of q discrete orthogonal polynomials the so called pseudo big q jacobi polynomials the new point processes live on a double q lattice and we show that their correlation kernels are expressed through the basic hypergeometric function 1 2 a crucial novel ingredient of our approach is an extended version g of the gelfand tsetlin graph the conventional graph describes the gelfand tsetlin branching rule for irreducible representations of unitary groups we find the q boundary of g thus extending previously known results gorin 17

the moyal bracket and the dispersionless limit of the kp hierarchy a new lax equation is introduced for the kp hierarchy which avoids the use of pseudo differential operators as used in the sate approach this lax equation is closer to that used in the study of the dispersionless kp hierarchy and is obtained by replacing the poisson bracket with the moyal bracket the dispersionless limit under which the moyal bracket collapses to the poisson bracket is particularly simple

infinite dimensional algebras and a trigonometric basis for the classical lie algebras this paper explores features of the infinite dimensional algebras that have been previously introduced in particular it is shown that the classical simple lie algebras an bn cn dn may be expressed in an egalitarian basis with trigonometric structure constants the transformation to the standard cartan weyl basis and the particularly transparent n limit that this formulation allows is provided

gues and queues consider the process d k k 1 2 given by b i being independent standard brownian motions this process describes the limiting behavior near the edge in queues in series totally asymmetric exclusion processes or oriented percolation the problem of finding the distribution of d was posed in gw the main result of this paper is that the process d has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from gaussian unitary ensemble

lozenge tilings and hurwitz numbers we give a new proof of the fact that near a turning point of the frozen boundary the vertical tiles in a uniformly random lozenge tiling of a large sawtooth domain are distributed like the eigenvalues of a gue random matrix our argument uses none of the standard tools of integrable probability in their place it uses a combinatorial interpretation of the harish chandra itzykson zuber integral as a generating function for desymmetrized hurwitz numbers

transformations to symmetry and homoscedasticity abstract this article considers transformations in regression to eliminate skewness and heteroscedasticity of the response we work with the transform both sides model where the relationship between the median response and the independent variables has been identified at least tentatively to preserve this relationship the response and the regression model are transformed in the same way extending the work of others for the location parameter case we propose an estimator that eliminates skewness we also develop an estimator to eliminate heteroscedasticity and an estimator that attempts to induce both symmetry and homoscedasticity both and appear new by comparing and we develop a test of the null hypothesis that there exists a transformation to both symmetry and homoscedasticity we study the question when does the estimator of behave in terms of asymptotic variance as if the regression parameter were known and vice versa the results are of use for telling when the optimal estimator of do

the q gelfand tsetlin graph gibbs measures and q toeplitz matrices abstract the problem of the description of finite factor representations of the infinite dimensional unitary group investigated by voiculescu in 1976 is equivalent to the description of all totally positive toeplitz matrices vershik kerov showed that this problem is also equivalent to the description of the simplex of central i e possessing a certain gibbs property measures on paths in the gelfand tsetlin graph we study a quantum version of the latter problem we introduce a notion of a q centrality and describe the simplex of all q central measures on paths in the gelfand tsetlin graph conjecturally q central measures are related to representations of the quantized universal enveloping algebra u gl we also define a class of q toeplitz matrices and show that extreme q central measures correspond to q toeplitz matrices with non negative minors finally our results can be viewed as a classification theorem for certain gibbs measures on rhombus tilings of the halfplane we use a class of q interpolation polynomials related to schur functions one of the key ingredients of our proofs is the binomial formula for these polynomials proved by okounkov

power transformations to symmetry summary power transformations for achieving distributional symmetry are discussed estimates of the transformation power are based on general measures of symmetry they are shown to be consistent and asymptotically normal use of the skewness coefficient as a measure of symmetry is shown to be optimal in an important special case the methods are compared to the likelihood methods of box cox 1964 and alternative methods of hinkley 1975 1977 the box cox method or a robust adaptation of it carroll 1980 bickel doksum 1981 is found to be the generally most suitable method

the boundary of the gelfand tsetlin graph a new approach the gelfand tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups the boundary of the gelfand tsetlin graph has at least three incarnations as a discrete potential theory boundary as the set of finite indecomposable characters of the infinite dimensional unitary group and as the set of doubly infinite totally positive sequences an old deep result due to albert edrei and dan voiculescu provides an explicit description of the boundary it can be realized as a region in an infinite dimensional coordinate space the paper contains a novel approach to the edrei voiculescu theorem it is based on a new explicit formula for the number of semi standard young tableaux of a given skew shape or of gelfand tsetlin schemes of trapezoidal shape the formula is obtained via the theory of symmetric functions and new schur like symmetric functions play a key role in the derivation

bulk universality for random lozenge tilings near straight boundaries and for tensor products we prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains the result applies to uniformly random lozenge tilings of large polygonal domains on triangular lattice and to the probability measures describing the decomposition in gelfand tsetlin bases of tensor products of representations of unitary groups in a weaker form our theorem also applies to random domino tilings

asymptotics of random lozenge tilings via gelfand tsetlin schemes a gelfand tsetlin scheme of depth n is a triangular array with m integers at level m m 1 ldots n subject to certain interlacing constraints we study the ensemble of uniformly random gelfand tsetlin schemes with arbitrary fixed n th row we obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble and also of its q deformation this provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice or equivalently of random stepped surfaces we work with a class of polygons which allows arbitrarily large number of sides we show that the local limit behavior of random tilings as all dimensions of the polygon grow is directed by ergodic translation invariant gibbs measures the slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by kenyon and okounkov acta math 199 2 263 302 2007 we also prove that at the edge of the limit shape the asymptotic behavior of random tilings is given by the airy process in particular our results cover the most investigated case of random boxed plane partitions when the polygon is a hexagon

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

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

a fourier view on the r transform and related asymptotics of spherical integrals abstract we estimate the asymptotics of spherical integrals of real symmetric or hermitian matrices when the rank of one matrix is much smaller than its dimension we show that it is given in terms of the r transform of the spectral measure of the full rank matrix and give a new proof of the fact that the r transform is additive under free convolution these asymptotics also extend to the case where one matrix has rank one but complex eigenvalue a result related with the analyticity of the corresponding spherical integrals

tests following transformations chen and loh showed that the box cox transformed two sample t test is more powerful than the ordinary t test under pitman alternatives where the location shifts appear in the untransformed scale in this article we prove that chen and loh s result also holds for a general family of transformations an upper bound on the asymptotic relative efficiency are is obtained in addition we investigate bounds on the are under pitman location shift alternatives in the transformed scale we find that when the estimate for a is consistent a lower bound on the are is the reciprocal of fisher information of the standard transformed distribution this lower bound is close to 1 for commonly used symmetric distributions

limit shapes and the complex burgers equation in this paper we study surfaces in r 3 that arise as limit shapes in random surface models related to planar dimers these limit shapes are surface tension minimizers that is they minimize a functional of the form h dx dy among all lipschitz functions h taking given values on the boundary of the domain the surface tension has singularities and is not strictly convex which leads to formation of facets and edges in the limit shapes we find a change of variables that reduces the euler lagrange equation for the variational problem to the complex inviscid burgers equation complex hopf equation the equation can thus be solved in terms of an arbitrary holomorphic function which is somewhat similar in spirit to weierstrass parametrization of minimal surfaces we further show that for a natural dense set of boundary conditions the holomorphic function in question is in fact algebraic the tools of algebraic geometry can thus be brought in to study the minimizers and especially the formation of their singularities this is illustrated by several explicitly computed examples

the retransformed mean after a fitted power transformation abstract an approximate method is given to estimate the mean of a dependent variable after a linear model is fitted to the box cox power transformation of this variable the estimate is accurate except when the transformation power parameter is near zero the properties of the estimate in the one sample and regression cases are considered by both asymptotic calculations and monte carlo simulations and comparisons are made with the smearing estimate duan 1983 it is shown that there can be some cost due to estimating the power transformation as opposed to assuming it is known however this cost is not severe

approximate median regression for complex survey data with skewed response the ready availability of public use data from various large national complex surveys has immense potential for the assessment of population characteristics using regression models complex surveys can be used to identify risk factors for important diseases such as cancer existing statistical methods based on estimating equations and or utilizing resampling methods are often not valid with survey data due to complex survey design features that is stratification multistage sampling and weighting in this article we accommodate these design features in the analysis of highly skewed response variables arising from large complex surveys specifically we propose a double transform both sides dtbs based estimating equations approach to estimate the median regression parameters of the highly skewed response the dtbs approach applies the same box cox type transformation twice to both the outcome and regression function the usual sandwich variance estimate can be used in our approach whereas a resampling approach would be needed for a pseudo likelihood based on minimizing absolute deviations mad furthermore the approach is relatively robust to the true underlying distribution and has much smaller mean square error than a mad approach the method is motivated by an analysis of laboratory data on urinary iodine ui concentration from the national health and nutrition examination survey

estimation for the box cox transformation model without assuming parametric error distribution box and cox proposed a power transformation for the response variable that yields a linear regression model with normal error and constant variance inference procedures for the regression coefficients and transformation parameter under this model setting have been studied extensively in this article we propose a simple semiparametric estimation method for the box cox transformation model with no specific parametric assumption on the distribution of the error term the resulting estimators are strongly consistent and asymptotically normal their covariance matrix can be estimated through a novel resampling method without involving nonparametric function estimates of the underlying unknown density function for the error term the new proposal is illustrated with a well known dataset in the literature of transformation and its efficiency and robustness are closely examined through numerical studies

asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory we develop a new method for studying the asymptotics of symmetric polynomials of representation theoretic origin as the number of variables tends to infinity several applications of our method are presented we prove a number of theorems concerning characters of infinite dimensional unitary group and their q deformations we study the behavior of uniformly random lozenge tilings of large polygonal domains and find the gue eigenvalues distribution in the limit we also investigate similar behavior for alternating sign matrices equivalently six vertex model with domain wall boundary conditions finally we compute the asymptotic expansion of certain observables in theo n 1 dense loop model

fluctuations of particle systems determined by schur generating functions abstract we develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems we introduce and study the notion of the schur generating function of a random discrete configuration our main result provides a central limit theorem clt for such a configuration given certain conditions on the schur generating function as applications of this approach we prove clt s for several probabilistic models coming from asymptotic representation theory and statistical physics including random lozenge and domino tilings non intersecting random walks decompositions of tensor products of representations of unitary groups

a one parameter family of hamiltonian structures for the kp hierarchy and a continuous deformation of the nonlinear wkp algebra the kp hierarchy is hamiltonian relative to a one parameter family of poisson structures obtained from a generalized adler map in the space of formal pseudodifferential symbols with noninteger powers the resulting w algebra is a one parameter deformation of wkp admitting a central extension for generic values of the parameter reducing naturally to w n for special values of the parameter and contracting to the centrally extended w1 w and further truncations in the classical limit all algebras in the one parameter family are equivalent and isomorphic towkp the reduction induced by setting the spin one field to zero yields a one parameter deformation of hat w infty which contracts to a new nonlinear algebra of the w type

measures of location of skew distributions obtained through box cox transformations abstract box cox power transformations are used to transform a set of observations to symmetry a measure of location of the original data is obtained by applying the inverse transformation to the center of the transformed data its properties are discussed and it is compared with other measures of location