the trace of spatial brownian motion is capacity equivalent to the unit square we show that with probability 1 the trace b 0 1 of brownian motion in space has positive capacity with respect to exactly the same kernels as the unit square more precisely the energy of occupation measure on b 0 1 in the kernel f x y is bounded above and below by constant multiples of the energy of lebesgue measure on the unit square the constants are random but do not depend on the kernel as an application we give almost sure asymptotics for the probability that an stable process approaches within of b 0 1 conditional on b 0 1

the multifractal spectrum of brownian intersection local times let l be the projected intersection local time of two independent brownian paths in d for d 2 3 we determine the lower tail of the random variable ell mathbb u where mathbb u is the unit ball the answer is given in terms of intersection exponents which are explicitly known in the case of planar brownian motion we use this result to obtain the multifractal spectrum or spectrum of thin points for the intersection local times

on latent trait estimation in multidimensional compensatory item response models making inferences from irt based test scores requires accurate and reliable methods of person parameter estimation given an already calibrated set of item parameters the latent trait could be estimated either via maximum likelihood estimation mle or using bayesian methods such as maximum a posteriori map estimation or expected a posteriori eap estimation in addition warm s psychometrika 54 427 450 1989 weighted likelihood estimation method was proposed to reduce the bias of the latent trait estimate in unidimensional models in this paper we extend the weighted mle method to multidimensional models this new method denoted as multivariate weighted mle mwle is proposed to reduce the bias of the mle even for short tests mwle is compared to alternative estimators i e mle map and eap and shown both analytically and through simulations studies to be more accurate in terms of bias than mle while maintaining a similar variance in contrast bayesian estimators i e map and eap result in biased estimates with smaller variability

multiple points in the sample paths of a levy process we obtain a sufficient condition for the sample paths of a levy process to contain multiple points our condition is close to one conjectured by hendricks and taylor

sets avoided by brownian motion a fixed two dimensional projection of a three dimensional brownian motion is almost surely neighborhood recurrent is this simultaneously true of all the two dimensional projections with probability 1 equivalently three dimensional brownian motion hits any infinite cylinder with probability 1 does it hit all cylinders this papers shows that the answer is no brownian motion in three dimensions avoids random cylinders and in fact avoids bodies of revolution that grow almost as fast as cones

the multivariate skew normal distribution summary the paper extends earlier work on the so called skew normal distribution a family of distributions including the normal but with an extra parameter to regulate skewness the present work introduces a multivariate parametric family such that the marginal densities are scalar skew normal and studies its properties with special emphasis on the bivariate case

intersections of random walks in four dimensions ii letf n be the probability that the paths of two simple random walks of lengthn starting at the origin in 4 have no intersection it has previously been shown thatf n c logn 1 2 here it is proved that for allr 1 2 mathop lim limits n to infty log n r f n infty

galton watson trees with the same mean have the same polar sets evans defined a notion of what it means for a set b to be polar for a process indexed by a tree the main result herein is that a tree picked from a galton watson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate this implies that deterministic and nondeterministic trees behave identically in a variety of probability models mapping subsets of euclidean space to trees and polar sets to capacity criteria it follows that certain random cantor sets are capacity equivalent to each other and to deterministic cantor sets an extension to branching processes in varying environment is also obtained

on the structure of mandelbrot s percolation process and other random cantor sets we consider generalizations of mandelbrot s percolation process for the process which we call the random sierpinski carpet we show that it passes through several different phases as its parameter increases from zero to one the final section treats the percolation phase

item selection in multidimensional computerized adaptive testing gaining information from different angles over the past thirty years obtaining diagnostic information from examinees item responses has become an increasingly important feature of educational and psychological testing the objective can be achieved by sequentially selecting multidimensional items to fit the class of latent traits being assessed and therefore multidimensional computerized adaptive testing mcat is one reasonable approach to such task this study conducts a rigorous investigation on the relationships among four promising item selection methods d optimality kl information index continuous entropy and mutual information some theoretical connections among the methods are demonstrated to show how information about the unknown vector can be gained from different perspectives two simulation studies were carried out to compare the performance of the four methods the simulation results showed that mutual information not only improved the overall estimation accuracy but also yielded the smallest conditional mean squared error in most region of in the end the overlap rates were calculated to empirically show the similarity and difference among the four methods

random walks capacity and percolation on trees a collection of several different probabilistic processes involving trees is shown to have an unexpected unity this makes possible a fruitful interplay of these probabilistic processes the processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well our work has five specific aims first an exact correspondence between random walks and percolation on trees is proved extending and sharpening previous work of the author this is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes second we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation this capacitary formulation extends and sharpens work of fan on random interval coverings third we show how this formulation also applies to generalize work of evans on random labelling of trees fourth the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a s in particular we resolve a conjecture of griffeath on the necessity of the nash williams criterion fifth for this last purpose we establish several new basic results on branching processes in varying environments

investigating the impact of uncertainty about item parameters on ability estimation asymptotic expansions of the maximum likelihood estimator mle and weighted likelihood estimator wle of an examinee s ability are derived while item parameter estimators are treated as covariates measured with error the asymptotic formulae present the amount of bias of the ability estimators due to the uncertainty of item parameter estimators a numerical example is presented to illustrate how to apply the formulae to evaluate the impact of uncertainty about item parameters on ability estimation and the appropriateness of estimating ability using the regular mle or wle method

capacity and principal eigenvalues the method of enlargement of obstacles revisited we describe a coarse graining method which provides lower bounds on the principal dirichlet eigenvalue of the laplacian in regions receiving small obstacles and sharpens the previous method of enlargement of obstacles based on a quantitative wiener criterion one replaces the actual obstacles by obstacles of a much larger size controls on the shift of principal eigenvalues and capacity estimates on the locus where the wiener criterion breaks down are derived the results are written in a self contained fashion

a new fractal dimension the topological hausdorff dimension we introduce a new concept of dimension for metric spaces the so called topological hausdorff dimension it is defined by a very natural combination of the definitions of the topological dimension and the hausdorff dimension the value of the topological hausdorff dimension is always between the topological dimension and the hausdorff dimension in particular this new dimension is a non trivial lower estimate for the hausdorff dimension we examine the basic properties of this new notion of dimension compare it to other well known notions determine its value for some classical fractals such as the sierpinski carpet the von koch snowflake curve kakeya sets the trail of the brownian motion etc as our first application we generalize the celebrated result of chayes chayes and durrett about the phase transition of the connectedness of the limit set of mandelbrot s fractal percolation process they proved that certain curves show up in the limit set when passing a critical probability and we prove that actually thick families of curves show up where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large hausdorff dimension the proof of this is basically a lower estimate of the topological hausdorff dimension of the limit set for the sake of completeness we also give an upper estimate and conclude that in the non trivial cases the topological hausdorff dimension is almost surely strictly below the hausdorff dimension finally as our second application we show that the topological hausdorff dimension is precisely the right notion to describe the hausdorff dimension of the level sets of the generic continuous function in the sense of baire category defined on a compact metric space

ability estimation for conventional tests five different ability estimators maximum likelihood mle weighted likelihood wle bayesian modal bme expected a posteriori eap and the standardized number right score z were used as scores for conventional multiple choice tests the bias standard error and reliability of the five ability estimators were evaluated using monte carlo estimates of the unknown conditional means and variances of the estimators the results indicated that ability estimates based on bme eap or wle were reasonably unbiased for the range of abilities corresponding to the difficulty of a test and that their standard errors were relatively small also they were as reliable as the old standby the number right score

bias reduction of maximum likelihood estimates summary it is shown how in regular parametric problems the first order term is removed from the asymptotic bias of maximum likelihood estimates by a suitable modification of the score function in exponential families with canonical parameterization the effect is to penalize the likelihood by the jeffreys invariant prior in binomial logistic models poisson log linear models and certain other generalized linear models the jeffreys prior penalty function can be imposed in standard regression software using a scheme of iterative adjustments to the data

intersection equivalence of brownian paths and certain branching processes we show that sample paths of brownian motion and other stable processes intersect the same sets as certain random cantor sets constructed by a branching process with this approach the classical result that two independent brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process and estimates due to lawler 1982 for the long range intersection probability of several random walk paths reduce to kolmogrov s 1938 law for the lifetime of a critical branching process extensions to random walks with long jumps and applications to hausdorff dimension are also derived

statistical applications of the multivariate skew normal distribution azzalini and dalla valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter the first part of the present paper examines further probabilistic properties of the distribution with special emphasis on aspects of statistical relevance inferential and other statistical issues are discussed in the following part with applications to some multivariate statistics problems illustrated by numerical examples finally a further extension is described which introduces a skewing factor of an elliptical density

connectivity properties of mandelbrot s percolation process in 1974 mandelbrot introduced a process in 0 1 2 which he called canonical curdling and later used in this book s on fractals to generate self similar random sets with hausdorff dimension d 0 2 in this paper we will study the connectivity or percolation properties of these sets proving all of the claims he made in sect 23 of the fractal geometry of nature and a new one that he did not anticipate there is a probability pc 0 1 so that if p pc then the set is duslike i e the largest connected component is a point whereas if p pc notice the opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles more succinctly put the system has a first order phase transition

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

harmonic analysis of additive levy processes let x1 xn denote n independent d dimensional levy processes and consider the n parameter random field mathfrak x t x 1 t 1 cdots x n t n first we demonstrate that for all nonrandom borel sets f subseteq bf r d the minkowski sum mathfrak x bf r n oplus f of the range mathfrak x bf r n of mathfrak x with f can have positive d dimensional lebesgue measure if and only if a certain capacity of f is positive this improves our earlier joint effort with yuquan zhong by removing a certain condition of symmetry in khoshnevisan et al ann probab 31 2 1097 1141 2003 moreover we show that under mild regularity conditions our necessary and sufficient condition can be recast in terms of one potential densities this rests on developing results in classical non probabilistic harmonic analysis that might be of independent interest as was shown in khoshnevisan et al ann probab 31 2 1097 1141 2003 the potential theory of the type studied here has a large number of consequences in the theory of levy processes presently we highlight a few new consequences

strong law of large numbers for the capacity of the wiener sausage in dimension four we prove a strong law of large numbers for the newtonian capacity of a wiener sausage in the critical dimension four where a logarithmic correction appears in the scaling the main step of the proof is to obtain precise asymptotics for the expected value of the capacity this requires a delicate analysis of intersection probabilities between two independent wiener sausages

the probability of intersection of independent random walks in four dimensions lets1 ands2 be independent simple random walks of lengthn inz4 starting at 0 andx0 respectively if x0 2 n it is shown that the probability that the paths intersect is of order logn 1 ifx0 0 it is shown that the probability of no intersection of the paths decays no faster than logn 1 and no slower than logn 1 2 it is conjectured that logn 1 2 is the actual decay rate

intersection properties of simple random walks a renormalization group approach we study estimates for the intersection probability g m of two simple random walks on lattices of dimensiond 4 4 e as a problem in euclidean field theory we rigorously establish a renormalization group flow equation forg m and bounds on the function which show that ind 4 g m tends to zero logarithmically as the killing rate mass m tends to zero and that the fixed point g ind 4 e is bounded by const e g conste our methods also yield estimates on the intersection probability of three random walks ind 3 3 e for e 0 these results were first obtained by lawler 1

stable processes on the boundary of a regular tree we define a class of processes on the boundary of a regular tree that can be viewed as stable levy processes on z n 0 z n we show that the range of these processes can be compared with a bernoulli percolation as in peres which easily leads to various results on the intersection properties we develop an alternative approach based on the comparison with a branching random walk by this method we establish the existence of points of infinite multiplicity when the index of the process equals the dimension of the state space as for planar brownian motion

what is the probability of intersecting the set of brownian double points we give potential theoretic estimates for the probability that a set a contains a double point of planar brownian motion run for unit time unlike the probability for a to intersect the range of a markov process this cannot be estimated by a capacity of the set a instead we introduce the notion of a capacity with respect to two gauge functions simultaneously we also give a polar decomposition of a into a set that never intersects the set of brownian double points and a set for which intersection with the set of brownian double points is the same as intersection with the brownian path

direct estimates on intersection probabilities of random walks a study is made of intersection properties of independent random walks in d dimensional lattice space z sup d a simple method is developed which makes it possible to estimate intersection probabilities of two random walks with killing rate m directly it is expected that the method can be generalized and extended to other issues

martin capacity for markov chains the probability that a transient markov chain or a brownian path will ever visit a given set is classically estimated using the capacity of with respect to the green kernel g x y we show that replacing the green kernel by the martin kernel g x y g 0 y yields improved estimates which are exact up to a factor of 2 these estimates are applied to random walks on lattices and also to explain a connection found by lyons between capacity and percolation on trees

level sets of additive levy processes we provide a probabilistic interpretation of a class of natural capacities on euclidean space in terms of the level sets of a suitably chosen multiparameter additive levy process x we also present several probabilistic applications of the aforementioned potential theoretic connections they include areas such as intersections of levy processes and level sets as well as hausdorff dimension computations

kullback leibler information and its applications in multi dimensional adaptive testing this paper first discusses the relationship between kullback leibler information kl and fisher information in the context of multi dimensional item response theory and is further interpreted for the two dimensional case from a geometric perspective this explication should allow for a better understanding of the various item selection methods in multi dimensional adaptive tests mat which are based on these two information measures the kl information index ki method is then discussed and two theorems are derived to quantify the relationship between ki and item parameters due to the fact that most of the existing item selection algorithms for mat bear severe computational complexity which substantially lowers the applicability of mat two versions of simplified kl index ski built from the analytical results are proposed to mimic the behavior of ki while reducing the overall computational intensity

exceptional times and invariance for dynamical random walks consider a sequence open image in new window of i i d random variables associate to each xi 0 an independent mean one poisson clock every time a clock rings replace that x variable by an independent copy and restart the clock in this way we obtain i i d stationary processes xi t t 0 i 1 2 whose invariant distribution is the law of x1 0

fractal percolation and branching cellular automata branching cellular automata bca are introduced as generalisations of fractal percolation by admitting neighbour dependence we associate sequences of random sets with bca s and study their convergence in case of convergence we derive the hausdorff dimension of the limit set and of its boundary to accomplish the latter we proof that the boundary of a set generated by a bca is again a set generated by a bca

weighted likelihood estimation of ability in item response theory applications of item response theory which depend upon its parameter invariance property require that parameter estimates be unbiased a new method weighted likelihood estimation wle is derived and proved to be less biased than maximum likelihood estimation mle with the same asymptotic variance and normal distribution wle removes the first order bias term from mle two monte carlo studies compare wle with mle and bayesian modal estimation bme of ability in conventional tests and tailored tests assuming the item parameters are known constants the monte carlo studies favor wle over mle and bme on several criteria over a wide range of the ability scale

hausdorff dimension of the contours of symmetric additive levy processes let x 1 x n denote n independent symmetric levy processes on r d the corresponding additive levy process is defined as the following n parameter random field on r d mathfrak x t x 1 t 1 cdots x n t n quad t in bf r n khoshnevisan and xiao ann probab 30 1 62 100 2002 have found a necessary and sufficient condition for the zero set mathfrak x 1 0 of mathfrak x to be non trivial with positive probability they also provide bounds for the hausdorff dimension of mathfrak x 1 0 which hold with positive probability in the case that mathfrak x 1 0 can be non void here we prove that the hausdorff dimension of mathfrak x 1 0 is a constant almost surely on the event mathfrak x 1 0 neq varnothing moreover we derive a formula for the said constant this portion of our work extends the well known formulas of horowitz israel j math 6 176 182 1968 and hawkes j lond math soc 8 517 525 1974 both of which hold for one parameter levy processes more generally we prove that for every nonrandom borel set f in 0 n the hausdorff dimension of mathfrak x 1 0 cap f is a constant almost surely on the event mathfrak x 1 0 cap f neq varnothing this constant is computed explicitly in many cases

on the finiteness of the weighted likelihood estimator of ability the purpose of this note is to focus on the finiteness of the weighted likelihood estimator wle of ability in the context of dichotomous and polytomous item response theory irt models it is established that the wle always returns finite ability estimates this general result is valid for dichotomous one two three and four parameter logistic irt models the class of polytomous difference models and divide by total models independently of the number of items the item parameters and the response patterns further implications of this result are outlined

brownian sheet and capacity the main goal of this paper is to present an explicit capacity estimate for hitting probabilities of the brownian sheet as applications we determine the escape rates of the brownian sheet and also obtain a local intersection equivalence between the brownian sheet and the additive brownian motion other applications concern quasisure properties in wiener space

a new online calibration method for multidimensional computerized adaptive testing multidimensional method a m method a has been proposed as an efficient and effective online calibration method for multidimensional computerized adaptive testing mcat chen xin paper presented at the 78th meeting of the psychometric society arnhem the netherlands 2013 however a key assumption of m method a is that it treats person parameter estimates as their true values thus this method might yield erroneous item calibration when person parameter estimates contain non ignorable measurement errors to improve the performance of m method a this paper proposes a new mcat online calibration method namely the full functional mle m method a ffmle m method a this new method combines the full functional mle jones jin in psychometrika 59 59 75 1994 stefanski carroll in annals of statistics 13 1335 1351 1985 with the original m method a in an effort to correct for the estimation error of ability vector that might otherwise adversely affect the precision of item calibration two correction schemes are also proposed when implementing the new method a simulation study was conducted to show that the new method generated more accurate item parameter estimation than the original m method a in almost all conditions

superbranching processes and projections of random cantor sets we study sequences x 0 x 1 of random variables taking values in the positive integers which grow faster than branching processes in the sense that x m n geqq sum limits i 1 x m x n m i for m n 0 where the x n m i are distributed as x n and have certain properties of independence we prove that under appropriate conditions x 1 almost surely and in l 1 where sup e x n 1 n our principal application of this result is to study the lebesgue measure and hausdorff dimension of certain projections of sets in a class of random cantor sets being those obtained by repeated random subdivisions of the m adic subcubes of 0 1 d we establish a necessary and sufficient condition for the lebesgue measure of a projection of such a random set to be non zero and determine the box dimension of this projection

an approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three parameter logistic model using taylor s expansion of the likelihood equation he obtained an equation that includes the conditional expectation given true ability of the discrepancy between the maximum likelihood estimate and true ability all terms of orders higher thann 1 are ignored wheren indicates the number of items lord assumed that all item and individual parameters are bounded all item parameters are known or well estimated and the number of items is reasonably large in the present paper an approximation for the bias function of the maximum likelihood estimate of the latent trait or ability will be developed using the same assumptions for the more general case where item responses are discrete this will include the dichotomous response level for which the three parameter logistic model has been discussed the graded response level and the nominal response level some observations will be made for both dichotomous and graded response levels

wiener s test for super brownian motion and the brownian snake we prove a wiener type criterion for super brownian motion and the brownian snake if f is a borel subset of d and x d we provide a necessary and sufficientcondition for super brownian motion started at x to immediately hit the set f equivalently this condition is necessary and sufficient for the hitting time of f by thebrownian snake with initial point x to be 0 a key ingredient of the proof isan estimate showing that the hitting probability of f is comparable up to multiplicative constants to the relevant capacity of f this estimate which is of independent interest refines previous results due to perkins and dynkin an important role is played by additivefunctionals of the brownian snake which are investigated here via the potentialtheory of symmetric markov processes as a direct application of our probabilisticresults we obtain a necessary and sufficient condition for the existence in a domain d of a positivesolution of the equation u u 2 which explodes at a given point of d

a hierarchical framework for modeling speed and accuracy on test items current modeling of response times on test items has been strongly influenced by the paradigm of experimental reaction time research in psychology for instance some of the models have a parameter structure that was chosen to represent a speed accuracy tradeoff while others equate speed directly with response time also several response time models seem to be unclear as to the level of parametrization they represent a hierarchical framework for modeling speed and accuracy on test items is presented as an alternative to these models the framework allows a plug and play approach with alternative choices of models for the response and response time distributions as well as the distributions of their parameters bayesian treatment of the framework with markov chain monte carlo mcmc computation facilitates the approach use of the framework is illustrated for the choice of a normal ogive response model a lognormal model for the response times and multivariate normal models for their parameters with gibbs sampling from the joint posterior distribution

the intersection of brownian paths as a case study of a renormalization group method for quantum field theory a new approach is presented for the study of the probability that the random paths generated by two independent brownian motions in d intersect or more generally are within a short distance a of each other the well known behavior of that function of a above below and at the critical dimension d 4 as well as further corrections are derived here by means of a single renormalization group equation the equation s derivation is expected to shed some light on the function of the o d 4 quantum field theory

quadratic forms in skew normal variates in this paper first a characterization of the multivariate skew normal distribution is given then the joint moment generating functions of two quadratic forms and a linear compound and a quadratic form in skew normal variates have been derived and conditions for their independence are given distribution of the ratios of quadratic forms in skew normal variates has also been studied

boundary trace of reflecting brownian motions we establish a uniform dimensional result for normally reflected brownian motion rbm in a large class of non smooth domains hausdorff dimensions for the boundary occupation time and the boundary trace of rbm are determined extensions to stable like jump processes and to symmetric reflecting diffusions are also given

unbiased estimators of ability parameters of their variance and of their parallel forms reliability given known item parameters unbiased estimators are derived i for an examinee s ability parameter and for his proportion correct true score ii for the variances of and across examinees in the group tested and iii for the parallel forms reliability of the maximum likelihood estimator hat theta

bias correction in maximum likelihood logistic regression this paper provides an expression for bias of the maximum likelihood logistic regression estimates for use with small sample sizes this bias correction is based on an expansion of the maximum likelihood equation simulation results show these corrections to be highly effective in small samples

no directed fractal percolation in zero area we consider the fractal percolation process on the unit square with fixed decimation parametern and level dependent retention parameters pk that is for allk 1 at thek th stage every retained square of side lengthn1 k is partitioned inton2 congruent subsquares and each of these is retained with probabilitypk independent of all others we show that if kpk 0 i e if the area of the limiting set vanishes a s then a s the limiting set contains no directed crossings of the unit square a directed crossing is a path that crosses the unit square from left to right and moves only up down and to the right

thick points for spatial brownian motion multifractal analysis of occupation measure let t x r denote the total occupation measure of the ball of radius r centered at x for brownian motion in ir we prove that sup x 1 t x r r log r 16 a s as r 0 thus solving a problem posed by taylor in 1974 furthermore for any a 0 16 the hausdorff dimension of the set of thick points x for which lim sup r 0 t x r r log r a is almost surely 2 a 8 this is the correct scaling to obtain a nondegenerate multifractal spectrum for brownian occupation measure analogous results hold for brownian motion in any dimension d 3 these results are related to the lil of ciesielski and taylor 1962 for the brownian occupation measure of small balls in the same way that levy s uniform modulus of continuity and the formula of orey and taylor 1974 for the dimension of fast points are related to the usual lil we also show that the lim inf scaling of t x r is quite different we exhibit non random c1 c2 0 such that c1 supx lim infr 0 t x r r 2 c2 a s in the course of our work we provide a general framework for obtaining lower bounds on the hausdorff dimension of random fractals of limsup type

critical brownian sheet does not have double points we derive a decoupling formula for the brownian sheet which has the following ready consequence an n parameter brownian sheet in r d has double points if and only if d 4n in particular in the critical case where d 4n the brownian sheet does not have double points this answers an old problem in the folklore of the subject we also discuss some of the geometric consequences of the mentioned decoupling and establish a partial result concerning k multiple points in the critical case k d 2n d

high dimensional exploratory item factor analysis by a metropolis hastings robbins monro algorithm a metropolis hastings robbins monro mh rm algorithm for high dimensional maximum marginal likelihood exploratory item factor analysis is proposed the sequence of estimates from the mh rm algorithm converges with probability one to the maximum likelihood solution details on the computer implementation of this algorithm are provided the accuracy of the proposed algorithm is demonstrated with simulations as an illustration the proposed algorithm is applied to explore the factor structure underlying a new quality of life scale for children it is shown that when the dimensionality is high mh rm has advantages over existing methods such as numerical quadrature based em algorithm extensions of the algorithm to other modeling frameworks are discussed

multidimensional adaptive testing with optimal design criteria for item selection several criteria from the optimal design literature are examined for use with item selection in multidimensional adaptive testing in particular it is examined what criteria are appropriate for adaptive testing in which all abilities are intentional some should be considered as a nuisance or the interest is in the testing of a composite of the abilities both the theoretical analyses and the studies of simulated data in this paper suggest that the criteria of a optimality and d optimality lead to the most accurate estimates when all abilities are intentional with the former slightly outperforming the latter the criterion of e optimality showed occasional erratic behavior for this case of adaptive testing and its use is not recommended if some of the abilities are nuisances application of the criterion of as optimality or ds optimality which focuses on the subset of intentional abilities is recommended for the measurement of a linear combination of abilities the criterion of c optimality yielded the best results the preferences of each of these criteria for items with specific patterns of parameter values was also assessed it was found that the criteria differed mainly in their preferences of items with different patterns of values for their discrimination parameters