multiple probability assessments by dependent experts abstract when two or more information sources experts provide a decision maker with information on two or more random variables the decision maker using bayes s rule has an opportunity to a update a prior about the random variables and b calibrate the experts calibration is the process of adjusting the decision maker s likelihood about the experts assessments this article presents a model for this two way process and specializes to the case in which the experts assessment errors have a multivariate normal density in general we find that variables which the decision maker and the experts regard as independent a priori will be dependent a posteriori because of dependence in the assessment errors formulas for posterior densities are given for the normal model in this model the posterior density of the random variables depends on only a weighted average of the expert s means with weights that depend on the experts assessments of previously known quantities i also present a special case o

strategic manipulation of empirical tests theories can be produced by experts seeking a reputation for having knowledge hence a tester could anticipate that theories may have been strategically produced by uninformed experts who want to pass an empirical test we show that with no restriction on the domain of permissible theories strategic experts cannot be discredited for an arbitrary but given number of periods no matter which test is used provided that the test does not reject the actual data generating process natural ways around this impossibility result include 1 assuming that unbounded data sets are available and 2 restricting the domain of permissible theories opening the possibility that the actual data generating process is rejected out of hand in both cases it is possible to dismiss strategic experts but only to a limited extent these results show significant limits on what data can accomplish when experts produce theories strategically

on testing the validity of sequential probability forecasts events are observed sequentially and at each stage a probability for the next event is assessed we consider the relationship between the sequence of probability forecasts and the sequence of outcomes we argue that the forecasts may be considered empirically valid when both these sequences are consistent with a common joint distribution for the events to aid in assessing validity we introduce various test statistics that measure in a natural way the empirical performance of the probability forecasts in the light of the outcomes obtained it is shown that under the null hypothesis of forecast validity such statistics will after suitable normalization have a standard normal or chi squared distribution virtually irrespective of the common joint distribution supposed to underlie both sequences

further thoughts on testing for causality with econometric models abstract the ontological basis for causality testing must be some empirically interpretable system leading one to specify a logically valid a priori law that can be shown to exist such paradigms as conventional linear and non linear models be they structural or time series tend to embody contradictory assumptions that unfortunately make causal interpretability problematic the paper demonstrates how these difficulties can be avoided using a stochastic coefficient approach the final part of this article is devoted to a discussion of probabilistic logic as a valid tool for scientific analysis and interpretation of causal relationships

bounded analytic functions in the dirichlet space in this paper we study the hilbert space of analytic functions with finite dirichlet integral in a connected open set c2 in the complex plane we show that every such function can be represented as a quotient of two bounded analytic functions each of which has a finite dirichlet integral this has several consequences for the structure of invariant subspaces of the algebra of multiplication operators on the dirichlet space in case f2 is simply connected namely we show that every nontrivial invariant subspace contains a nontrivial bounded function that each two nontrivial invariant subspaces have a nontrivial intersection that is the algebra is cellular indecomposable and that each nontrivial invariant subspace has the codimension one property with respect to certain multiplication operators

invariant subspaces of the dirichlet shift and pseudocontinuations in this paper we study extremal functions for invariant subspaces x of the dirichlet shift i e solutions v of the extremal problem sup if n 0 1 1f lid f e x f o here n is the smallest nonnegative integer such that the sup is positive it is known that such a function v generates x we show that the derivative z o has a pseudocontinuation to the exterior disc this pseudocontinuation is an analytic continuation exactly near those points of the unit circle where 0 is bounded away from zero we also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace

cyclic polynomials in two variables we give a complete characterization of polynomials in two com plex variables that are cyclic with respect to the coordinate shifts acting ondirichlet type spaces in the bidisk which include the ha

an analogue of a hardy littlewood fejer inequality for upper triangular trace class operators let h denote a separable infinite dimensional hilbert space we shall consider operators that is bounded linear transformations on an operator t is said to be in the schatten class cp if t is compact and if the eigenvalues of t t t 2 are in 1 p 0 p oo the norm in cp is the i p norm of these eigenvalues the class c 2 is called the hilbert schmidt class and c 1 is called the trace class by c o we denote the class of all compact operators for information on these classes see for example 23 4 chap xi 8 chap iii 19 13 although there is an obvious analogy with the sequence spaces i p there is a deeper analogy with the function spaces l p on the unit circle however the containment relations are reversed from the l p case namely cp c cr for p r this analogy serves as a source for new conjectures in operator theory motivated by this we establish an operator analgue of a coefficient inequality for the function space h 1 first we survey some known results in order to emphasize the nature and depth of the analogy fix an orthonormal basis en n l 2 in h and let en span e i en each operator t on has a matrix representation

a nonmanipulable test a test is said to control for type i error if it is unlikely to reject the data generating process however if it is possible to produce stochastic processes at random such that for all possible future realizations of the data the selected process is unlikely to be rejected then the test is said to be manipulable so a manipulable test has essentially no capacity to reject a strategic expert many tests proposed in the existing literature including calibration tests control for type i error but are manipulable we construct a test that controls for type i error and is nonmanipulable

a representation theorem for cyclic analytic two isometries a bounded linear operator t on a complex separable hilbert space z is called a 2 isometry if t 2t2 2t t i o we say that t is analytic if nn 0 tnt 0 in this paper we show that every cyclic analytic 2 isometry can be represented as multiplication by z on a dirichlet type space d 8 here ze denotes a finite positive borel measure on the unit circle for two measures u and v the 2 isometries obtained as multiplication by z on d 8 and d v are unitarily equivalent if and only if u v we also investigate similarity and quasisimilarity of these 2 isometries and we apply our results to the invariant subspaces of the dirichlet shift

weak products of dirichlet functions abstract for a hilbert space h of functions let h h be the space of weak products of functions in h i e all functions h that can be written as h i 1 f i g i for some f i g i h with i 1 f i g i let d denote the dirichlet space of the unit circle d i e the nontangential limits of functions f hol d with d f 2 d a and let d h be the harmonic dirichlet space which consists of functions of the form f g for f g d in this paper we show that every real valued function in d h d h is a single product of two functions in d h and that the cauchy projection is a bounded operator from d h d h onto d d it follows that d d consists exactly of the h 1 functions whose real and imaginary parts are single products of d h functions the dual space of d d was characterized by arcozzi rochberg sawyer and wick in 3 and the result implies the characterization of the dual of d h d h we will show that the characterization of the dual of d h d h also follows from results of maz ya and verbitsky 17 thus we will establish a precise connection between the results of 3 and 17

cyclicity and invariant subspaces in dirichlet spaces let be a positive finite measure on the unit circle and d the associated dirichlet space the generalized brown shields conjecture asserts that an outer function f d is cyclic if and only if c z f 0 where c is the capacity associated with d and z f is the zero set of f in this paper we prove that this conjecture is true for measures with countable support we also give in this case a complete and explicit characterization of invariant subspaces

cracking node method for dynamic fracture with finite elements a new method for modeling discrete cracks based on the extended finite element method is described in the method the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements the method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path a set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled the effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching copyright 2008 john wiley sons ltd

calibration sufficiency and domination considerations for bayesian probability assessors abstract degroot and fienberg 1982a recently considered various aspects of the problem of evaluating the performance of probability appraisers after briefly reviewing their notions of calibration and sufficiency we introduce related ideas of semicalibration and domination and consider their relationship to the earlier concepts we then discuss some simple bayesian mechanisms for making probability assessments and study their calibration semicalibration sufficiency and domination properties finally several results concerning the comparison of finite dichotomous experiments relevant to the present work are collected in an appendix

maximal minimal and primary invariant subspaces abstract let x be a complex infinite dimensional banach space an operator l on x is called of subcritical class if n 1 n 3 2 log l n t is an operator on x whose iterates have norms of polynomial growth we prove that if t has a range of finite codimension and a left inverse of subcritical class then every maximal invariant subspace of t has codimension one and if t has a finite dimensional kernel and a right inverse of subcritical class then every minimal invariant subspace of t is one dimensional using these results we obtain new information on the invariant subspace lattices of shifts and backward shifts on a wide class of banach spaces of analytic functions on the unit disc we also introduce the notion of primary invariant subspaces and determine their structure for a large class of shifts

kernel and capacity estimates in dirichlet spaces abstract we study the capacity in the sense of beurling deny associated with the dirichlet space d where is a finite positive borel measure on the unit circle first we obtain a sharp asymptotic estimate of the norm of the reproducing kernel of d it allows us to give an estimate of the capacity of points and arcs of the unit circle we also provide new conditions on closed sets to be polar our method is based on sharp estimates of norms of some outer functions which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted sobolev space

hankel matrices acting on dirichlet spaces abstract we give a connection between the hankel matrix acting on dirichlet spaces d 0 2 and the carleson measure supported on 1 1 as an application we prove that the generalized hilbert operators h are always bounded on dirichlet spaces d for 0 2 and that the range 0 2 of in our results is the best possible

normal and tangential compliances of interface of rough surfaces with contacts of elliptic shape abstract the incremental compliances normal and tangential of an interface between rough surfaces are considered contacts are assumed to be elliptic the shape of hertzian contacts between any two locally smooth asperities the ellipses may have diverse eccentricities and random or non random orientation distribution in the latter case the tangential compliance is anisotropic it is found that the hertzian contacts and welded zones of the same geometry produce the same incremental compliances microstructural characteristics of the interface that controls its incremental compliances is identified for the circular contacts is the sum a k where a k is k th contact area in the more general case of elliptic contacts each a k enters in product with its shape factor dependent on the ellipse aspect ratio the mentioned in plane tangential anisotropy is relatively mild even for parallel strongly elongated elliptic contacts this is due to weakness of the anisotropy for a single elliptic contact or external elliptic crack comparison of the latter to the internal crack of the same elliptic geometry shows that whereas the anisotropy is mild for both crack types it is weaker for the external crack in which case it is also less sensitive to poisson s ratio

orbits of non elliptic disc automorphisms on h p motivated by the invariant subspace problem we describe explicitly the closed subspace h 2 generated by the limit points in the h 2 norm of the orbit of a thin blaschke product b under composition operators c phi induced by non elliptic automorphisms this description exhibits a surprising connection to model spaces finally we give a constructive characterization of the c phi eigenfunctions in h p for 1 p infinity c 2011 elsevier inc all rights reserved

a score regression approach to assess calibration of continuous probabilistic predictions calibration the statistical consistency of forecast distributions and the observations is a central requirement for probabilistic predictions calibration of continuous forecasts is typically assessed using the probability integral transform histogram in this article we propose significance tests based on scoring rules to assess calibration of continuous predictive distributions for an ideal normal forecast we derive the first two moments of two commonly used scoring rules the logarithmic and the continuous ranked probability score this naturally leads to the construction of two unconditional tests for normal predictions more generally we propose a novel score regression approach where the individual scores are regressed on suitable functions of the predictive variance this conditional approach is applicable even for certain nonnormal predictions based on the dawid sebastiani score two case studies illustrate that the score regression approach has typically more power in detecting miscalibrated forecasts than the other approaches considered including a recently proposed technique based on conditional exceedance probability curves

cantor sets and cyclicity in weighted dirichlet spaces we treat the problem of characterizing the cyclic vectors in the weighted dirichlet spaces extending some of our earlier results in the classical dirichlet space the absence of a carleson type formula for weighted dirichlet integrals necessitates the introduction of new techniques

on optimal sequential prediction for general processes this paper considers several aspects of the sequential prediction problem for unbounded nonstationary processes under pth power loss spl lscr sub p u v u v sup p 1 p spl infin in the first part it is shown that bayes prediction schemes are cesaro optimal under general conditions that cesaro optimal prediction schemes are unique in a natural sense and that cesaro optimality is equivalent to a form of weak calibration connections between calibration and stronger forms of optimality are considered extensions of the existence and uniqueness results to generalized prediction and prediction from observations with additive noise are established for binary processes it is shown that thresholding an optimal prediction scheme for the squared loss yields an optimal binary prediction scheme for the hamming loss in the second part of the paper it is shown how to construct from a countable family of prediction schemes a single composite scheme whose asymptotic performance on any suitable process dominates the performance of each member of the family the construction is based on aggregating methods for individual binary sequences using the construction some results of algoet 1994 1995 on the existence of cesaro optimal schemes for families of ergodic processes are rederived in a direct way and extended to unbounded processes

on the dirichlet space for finitely connected regions this paper is devoted to the study of the dirichlet space dir g for finitely connected regions g we are particularly interested in the algebra of bounded multiplication operators on this space results in different directions are obtained one direction deals with the structure of closed subspaces invariant under all bounded multiplication operators in particular we show that each such subspace contains a bounded function for regions with circular boundaries we prove that a finite codimensional closed subspace invariant under multiplication by z must be invariant under all bounded multiplication operators and furthermore it is of the form p dir g where p is a polynomial with all its roots lying in g another direction is to study cyclic and noncyclic vectors for the algebra of all bounded multiplication operators typical results are if f e dir g and f is bounded away from zero then f is cyclic on the other hand if the zero set of the radial limit function of f on the boundary has positive logarithmic capacity then f is not cyclic also some other sufficient conditions for a function to be cyclic are given lastly we study transitive operator algebras containing all bounded multiplication operators we prove that they are dense in the algebra of all bounded operators in the strong operator topology

the well calibrated bayesian abstract suppose that a forecaster sequentially assigns probabilities to events he is well calibrated if for example of those events to which he assigns a probability 30 percent the long run proportion that actually occurs turns out to be 30 percent we prove a theorem to the effect that a coherent bayesian expects to be well calibrated and consider its destructive implications for the theory of coherence

contact impact modeling in explicit transient dynamics abstract in this article we discuss the treatment of contact impact modeling in a large deformation explicit dynamic setting such problems are posed mathematically by demanding the usual satisfaction momentum balance equations and boundary conditions for each body separately while imposing an additional set of constraints that govern the interaction of these bodies with each other in applying such methods to contact problems we emphasize two requirements first that the treatment of contact should follow from the development of the local weak and global strong forms of the contact equations in the continuum setting algorithms developed from this framework are seen to readily handle the three dimensional multi body friction case second and equally important especially in large applications is the practical aspect of computing the closest point projection for contact nodes an essential component of defining the contact constraints this calculation is global in nature accounting for the fact that contact impact involves multiple bodies interacting in unforeseeable ways these requirements have guided the development of algorithms capable of treating contact impact in pronto3d m w heinstein s w attaway j w swegle f j mello a general purpose contact detection algorithm for nonlinear structural analysis sodes sand92 2141 sandia national laboratories albuquerque n 87185 1992 are discussed in this paper

the game of normal numbers we introduce a two player game where at each period one player say player 2 chooses a distribution and the other player player 1 chooses a realization player 1 wins the game if the sequence of realized outcomes is normal with respect to the sequence of distributions we present a pure winning strategy of player 1 and thereby provide a universal algorithm that generates a normal sequence for any discrete stochastic process it turns out that to select thenth digit the algorithm conductso n2 calculations the proof uses approachability in infinite dimensional spaces lehrer 2002

non stochastic infinite and finite sequences abstract combining outcomes of coin tossing and transducer algorithms it is possible to generate with probability close to 1 very pathological sequences for which computable probabilistic forecasting is impossible these sequences are not random with respect to any reasonable probability distribution a natural consequence from the definition of such sequences is that each simple measure of the set of all such sequences is equal to 0 it was kolmogorov s and levin s idea to estimate the probability of generating of such sequences in the combinations of probabilistic and algorithmic processes 8 14 21 we collect several results in this direction for infinite sequences and asymptotic results for finite sequences including estimation of space and time of losing randomness for time bounded forecasting systems a correction to 22

hypercyclic operators that commute with the bergman backward shift the backward shift b on the bergman space of the unit disc is known to be hypercyclic meaning it has a dense orbit here we ask which operators that commute with b inherit its hypercyclicity we show that the problem reduces to the study of operators of the form 9p b where so is a holomorphic self map of the unit disc that multiplies the dirichlet space into itself and that the question of hypercyclicity for such an operator depends on how freely 9p z is allowed to approach the unit circle as lz s 1

some stochastic properties of memoryless individual sequences an individual sequence of real numbers is memoryless if no continuous markov prediction scheme of finite order can outperform the best constant predictor under the squared loss it is established that memoryless sequences satisfy an elementary law of large numbers and sliding block versions of hoeffding s inequality and the central limit theorem it is further established that memoryless binary sequences have convergent sample averages of every order and that their limiting distributions are bernoulli several examples and sources of memoryless sequences are given and it is shown how memoryless binary sequences may be constructed from aggregating methods for sequential prediction

on the brown shields conjecture for cyclicity in the dirichlet space abstract let d be the dirichlet space namely the space of holomorphic functions on the unit disk whose derivative is square integrable we establish a new sufficient condition for a function f d to be cyclic i e for p f p a polynomial to be dense in d this allows us to prove a special case of the conjecture of brown and shields that a function is cyclic in d iff it is outer and its zero set defined appropriately is of capacity zero

fuzzy set theory observer bias and probability theory abstract it is known that given certain seemingly implausible assumption probability functions act like fuzzy set membership functions this paper shows that these seemingly implausible assumptions are satisfied in probabilistic situations where the sole uncertainty arises from unknown but systematic observer bias now such situations could be considered as on the borderline between the domain of applicability of probability functions and the domain of fuzzy set membership functions hence the convergence between the rules for combining probabilistic functions and those for combining fuzzy set membership functions in these borderline instances is intuitively quite satisfying this convergence also suggests a number of other ways in which possibility theory and probability theory are interrelated

simple common plane contact algorithm summary the common plane cp algorithm is widely used in the discrete element method to model contact forces between interacting particles or blocks of rock a new simple contact algorithm similar to the cp algorithm is proposed to model discontinuities such as joints faults and material interfaces in an explicit finite difference code the cp is defined as a plane separating interacting faces of grid cells instead of blocks or particles used in the original cp method the new method does not require iterations even for very stiff contacts it is very robust and easy to implement both in 2d and 3d parallel codes copyright 2011 john wiley sons ltd

bayesian nonparametric calibration and combination of predictive distributions abstractwe introduce a bayesian approach to predictive density calibration and combination that accounts for parameter uncertainty and model set incompleteness through the use of random calibration functionals and random combination weights building on the work of ranjan and gneiting we use infinite beta mixtures for the calibration the proposed bayesian nonparametric approach takes advantage of the flexibility of dirichlet process mixtures to achieve any continuous deformation of linearly combined predictive distributions the inference procedure is based on combination gibbs and slice sampling we provide some conditions under which the proposed probabilistic calibration converges in terms of weak posterior consistency to the true underlying density for both cases of iid and markovian observations this calibration property improves upon the earlier calibration approaches we study the methodology in simulation examples with fat tails and multimodal densities and apply it to density forecasts of dail

a new goodness of fit test for event forecasting and its application to credit defaults we develop a new goodness of fit test for validating the performance of probability forecasts our test statistic is particularly powerful under sparseness and dependence in the observed data to build our test statistic we start from a formal definition of calibrated forecasts which we operationalize by introducing two components the first component tests the level of the estimated probabilities the second validates the shape measuring the differentiation between high and low probability events after constructing test statistics for both level and shape we provide a global goodness of fit statistic which is asymptotically 2 distributed in a simulation exercise we find that our approach is correctly sized and more powerful than alternative statistics in particular our shape statistic is significantly more powerful than the kolmogorov smirnov test under independence our global test has significantly greater power than the popular hosmer lemeshow s 2 test moreover even under dependence our global test remains correctly sized and consistent as a timely and important empirical application of our method we study the validation of a forecasting model for credit default events this paper was accepted by wei xiong finance

a zk invariant subspace without the wandering property abstract we study operators of multiplication by z k in dirichlet type spaces d we establish the existence of k and for which some z k invariant subspaces of d do not satisfy the wandering property as a consequence of the proof any dirichlet type space accepts an equivalent norm under which the wandering property fails for some space for the operator of multiplication by z k for any k 6

any inspection is manipulable consider a sequential forecast that may vary with time and history a calibration test see dawid 1982 foster and vohra 1997 and kalai lehrer and smorodinsky 1999 compares the observed empirical frequency over a set of times with the average prediction over the same set of times each test checks different events at different times a forecasting scheme passes a calibration test at a sequence of state realizations if the gap between these two figures diminishes to zero in other words each test defines what is consistent with the observed facts differently a forecast is consistent with the observed facts according to a given test if the empirical frequency of the events checked coincides asymptotically with the average prediction of these events when the forecast and the true distribution according to which the states are selected are the same the forecast passes any calibration test with probability one a calibration test is therefore type i error free the calibration tests treated in the literature are pure after any history it is known what specific event for instance rain foggy is being considered here we introduce mixed calibration tests where after each history a randomly chosen event is checked a mixed test checks whether the average prediction of these randomly chosen events is asymptotically equal to the frequency of their realizations like a deterministic test a mixed calibration test is also type i error free the main result of this paper shows that there is no mixed calibration test that rejects any inaccurate forecast that is for any test of this kind there exists a forecasting scheme that is always consistent with the realized events a constructive method to produce this scheme is provided as well another interpretation of the main result involves a forecaster and an inspector the inspector randomly chooses a calibration test according to a known distribution it is shown that a forecaster without knowing anything regarding the way states are selected can pass almost every test chosen by the inspector independently of the sequence of realizations in other words the forecaster has a forecasting method that is consistent with the observed facts in the meaning induced by almost all tests instead of randomly choosing one test as discussed previously the inspector can employ a set of tests simultaneously the main result implies that if the inspector employs countably many calibration tests for instance all those computable by a turing machine a forecaster can likewise pass them all2 simultaneously it means that a consultant or a scientist e g a weather forecaster who knows the countable set of

discrete and continuum methods for numerical simulations of non linear wave propagation in discontinuous media the main objective of this paper is to develop a methodology to model dynamic loading of various discontinuous media two methods for modeling non linear waves in a media with multiple discontinuities are considered the first one a discrete method is based on the simple common plane contact algorithm this method can be applied both to compliant contacts characterized by finite thickness and elastic moduli such as joints in geomechanics as well as to non compliant frictional contacts traditionally described by the slide lines in finite element finite difference codes the second one a continuum method assumes that the contacts are not compliant and can be modeled as one or several weakness planes cutting through the elements of the computational mesh both discrete and continuum methods described in the paper can be applied to derive equivalent continuum properties for media with multiple discontinuities an example of such application for a randomly jointed media is given in the paper copyright 2010 john wiley sons ltd

a parallel two sided contact algorithm in ale3d a scalable parallel algorithm for treating two sided contact in a finite element multi physics code ale3d is presented this method assumes that proximity between the two sides changes continuously and uses a local search to update proximity relations each cycle the evolving communication pattern is treated entirely by local nearest neighbor communication there is no global communication opening and closing voids overlapping and intersecting contact surfaces and a number of other special features are supported

combining probability forecasts summary linear pooling is by far the most popular method for combining probability forecasts however any non trivial weighted average of two or more distinct calibrated probability forecasts is necessarily uncalibrated and lacks sharpness in view of this linear pooling requires recalibration even in the ideal case in which the individual forecasts are calibrated towards this end we propose a beta transformed linear opinion pool for the aggregation of probability forecasts from distinct calibrated or uncalibrated sources the method fits an optimal non linearly recalibrated forecast combination by compositing a beta transform and the traditional linear opinion pool the technique is illustrated in a simulation example and in a case study on statistical and national weather service probability of precipitation forecasts

toward a reconciliation of the bayesian and frequentist approaches to point estimation abstract the bayesian and frequentist approaches to point estimation are reviewed the status of the debate regarding the use of one approach over the other is discussed and its inconclusive character is noted a criterion for comparing bayesian and frequentist estimators within a given experimental framework is proposed the competition between a bayesian and a frequentist is viewed as a contest with the following components a random observable a true prior distribution unknown to both statisticians an operational prior used by the bayesian a fixed frequentist rule used by the frequentist and a fixed loss criterion this competition is studied in the context of exponential families conjugate priors and squared error loss the class of operational priors that yield bayes estimators superior to the best frequentist estimator is characterized the implications of the existence of a threshold separating the space of operational priors into good and bad priors are explored and their relevance in ar

invariant subspaces in banach spaces of analytic functions we study the invariant subspace structure of the operator of multiplication by z mz on a class of banach spaces of analytic functions for operators on hilbert spaces our class coincides with the adjoints of the operators in the cowendouglas class 1 q we say that an invariant subspace x satisfies cod x 1 if zx has codimension one in x we give various conditions on invariant subspaces which imply that codx 1 in particular we give a necessary and sufficient condition on two invariant subspaces x x with cod x cod x 1 so that their span again satisfies cod x v x 1 this result will be used to show that any invariant subspace of the bergman space l p 1 which is generated by functions in l2p must satisfy cod x 1 for an invariant subspace x we then consider the operator s m 1xl under some extra assumption on the domain of holomorphy we show that the spectrum of s coincides with the approximate point spectrum iff cod x 1 finally in the last section we obtain a structure theorem for invariant subspaces with cod x 1 this theorem applies to dirichlet type spaces

a two step bayesian approach for propensity score analysis simulations and case study a two step bayesian propensity score approach is introduced that incorporates prior information in the propensity score equation and outcome equation without the problems associated with simultaneous bayesian propensity score approaches the corresponding variance estimators are also provided the two step bayesian propensity score is provided for three methods of implementation propensity score stratification weighting and optimal full matching three simulation studies and one case study are presented to elaborate the proposed two step bayesian propensity score approach results of the simulation studies reveal that greater precision in the propensity score equation yields better recovery of the frequentist based treatment effect a slight advantage is shown for the bayesian approach in small samples results also reveal that greater precision around the wrong treatment effect can lead to seriously distorted results however greater precision around the correct treatment effect parameter yields quite good results with slight improvement seen with greater precision in the propensity score equation a comparison of coverage rates for the conventional frequentist approach and proposed bayesian approach is also provided the case study reveals that credible intervals are wider than frequentist confidence intervals when priors are non informative

cyclic vectors in the dirichlet space we study the hilbert space of analytic functions with finite dinchlet integral in the open unit disc we try to identify the functions whose polynomial multiples are dense in this space theorems 1 and 2 confirm a special case of the following conjecture if ij z i ig z l at all points and if g is cyclic thenjis cyclic theorems 3 5 give a sufficient condition t is an outer function with some smoothness and the boundary zero set is at most countable and a necessary condition the radial limit can vanish only for a set of loganthmic capacity zero for a function j to be cyclic introduction in this paper we shall study the hilbert space of analytic functions in the open unit disc a in the complex plane that have a finite dirichlet integral jj if t12 dx dy g z l for some cyclic g and all z question 4 asks if f must be cyclic whenever f and l f are both in the space no examples are known where either of these questions has a negative answer in 2 we begin the study of cyclic vectors in the dirichlet space d theorems 1 and 2 give a partial answer to question 3 above for this space this section also contains 2 propositions 10 11 and 4 questions 7 10 proposition 11 says that if f and g are bounded functions in d whose product is cyclic then bothf and g must be cyclic theorem 2 gives a partial converse we require that gl be dini continuous on received by the editors july 21 1983 1980 mathematics subject classification primary 30h05 secondary 46e15 46e20 47b37

on calibration error of randomized forecasting algorithms it has been recently shown that calibration with an error less than d 0 is almost surely guaranteed with a randomized forecasting algorithm where forecasts are obtained by random rounding the deterministic forecasts up to d we show that this error cannot be improved for a vast majority of sequences we prove that using a probabilistic algorithm we can effectively generate with probability close to one a sequence resistant to any randomized rounding forecasting with an error much smaller than d we also reformulate this result by means of a probabilistic game

regularity for generators of invariant subspaces of the dirichlet shift abstract let d denote the classical dirichlet space of analytic functions on the open unit disc whose derivative is square area integrable for a set e d we write d e f d lim r 1 f r e i t 0 q e where q e stands for except possibly for e i t in a set of logarithmic capacity 0 we show that if e is a carleson set then there is a function f d e that is also in the disc algebra and that generates d e in the sense that d e clos p f p is a polynomial we also show that if d is an extremal function i e p p 0 for every polynomial p then the limits of z exist for every e i t d as z approaches e i t from within any polynomially tangential approach region

inner functions and cyclic vectors in the bloch space in this paper we construct a singular inner function whose polynomial multiples are dense in the little bloch space qo to do this we construct a singular measure on the unit circle with best possible control of both the first and second differences

internal regret with partial monitoring calibration based optimal algorithms we provide consistent random algorithms for sequential decision under partial monitoring when the decision maker does not observe the outcomes but receives instead random feedback signals those algorithms have no internal regret in the sense that on the set of stages where the decision maker chose his action according to a given law the average payoff could not have been improved in average by using any other fixed law they are based on a generalization of calibration no longer defined in terms of a voronoi diagram but instead of a laguerre diagram a more general concept this allows us to bound for the first time in this general framework the expected average internal as well as the usual external regret at stage n by o n 1 3 which is known to be optimal

probabilistic forecasts calibration and sharpness summary probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions we propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only a simple theoretical framework allows us to distinguish between probabilistic calibration exceedance calibration and marginal calibration we propose and study tools for checking calibration and sharpness among them the probability integral transform histogram marginal calibration plots the sharpness diagram and proper scoring rules the diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the stateline wind energy centre in the us pacific northwest in combination with cross validation or in the time series context our proposal provides very general nonparametric alternatives to the use of information criteria for model diagnostics and model selection

calibration with many checking rules each period an outcome out of finitely many possibilities is observed for simplicity assume two possible outcomes a and b each period a forecaster announces the probability of a occurring next period based on the past consider an arbitrary subsequence of periods e g odd periods even periods all periods in which b is observed etc given an integer n divide any such subsequence into associated sub subsequences in which the forecast for a is between i n i 1 n i 0 1 n we compare the forecasts and the outcomes realized next period separately in each of these subsubsequences given any countable partition of 0 1 and any countable collection of subsequences we construct a forecasting scheme such that for all infinite strings of data the long run average forecast for a matches the long run frequency of realized a s

on norms of composition operators acting on bergman spaces for arbitrary composition operators acting on a general bergman space we improve the known lower bound for the norm and also generalize a related recent theorem of d g pokorny and j e shapiro next we obtain a geometric formula for the norms of composition operators with linear fractional symbols thus extending a result of c cowen and p hurst and revealing the meaning of their computation finally we obtain a lower bound for essential norm of an arbitrary composition operator related to the well known criterion of b maccluer and j h shapiro as a corollary norms and essential norms are obtained for certain univalently induced noncompact composition operators in terms of the minimum of the angular derivative of the symbol

evaluating probability forecasts probability forecasts of events are routinely used in climate predictions in forecasting default probabilities on bank loans or in estimating the probability of a patient s positive response to treatment scoring rules have long been used to assess the efficacy of the forecast probabilities after observing the occurrence or nonoccurrence of the predicted events we develop herein a statistical theory for scoring rules and propose an alternative approach to the evaluation of probability forecasts this approach uses loss functions relating the predicted to the actual probabilities of the events and applies martingale theory to exploit the temporal structure between the forecast and the subsequent occurrence or nonoccurrence of the event