designs for generalized linear models with several variables and model uncertainty standard factorial designs sometimes may be inadequate for experiments that aim to estimate a generalized linear model for example for describing a binary response in terms of several variables a method is proposed for finding exact designs for such experiments that uses a criterion allowing for uncertainty in the link function the linear predictor or the model parameters together with a design search designs are assessed and compared by simulation of the distribution of efficiencies relative to locally optimal designs over a space of possible models exact designs are investigated for two applications and their advantages over factorial and central composite designs are demonstrated

a note on the conditional approach to interval estimation in the calibration problem summary in the calibration problem the need to construct a confidence interval to estimate the unknown x0 arises when the null hypothesis of zero slope is rejected otherwise the resulting confidence interval will be infinite to reflect the fact that the slope of the regression line may be zero under the condition of rejecting the hypothesis of zero slope we study the properties of the conditional coverage rate of the calibration confidence interval the conditional coverage rate pf is a function of the slope distance between x0 and the mean of the trailing sample x the sum of squares of x and n when the true slope is close to 0 and x0 is away from x pi can go down to 0 on the other hand as the power of testing zero slope reaches 1 with or without x0 close to x pi will tend to the desired nominal coverage rate in summary one should choose a reasonably small a in testing zero slope to avoid constructing a confidence interval for x0 when the true slope is 0 in addition it is desirable to have high power in testing zero slope so that the resulting confidence interval will maintain the desired coverage rate when using the conditional approach in the calibration problem

reduced system algorithms for markov chains a reduced system is a smaller system derived in the process of analyzing a larger system in solving for steady state probabilities of a markov chain generally the solution can be found by first solving a reduced system of equations which is obtained by appropriately partitioning the transition probability or rate matrix following lal lal r 1981 a unified study of algorithms for steady state probabilities in makov chains ph d dissertation department of operations research and engineering management school of engineering and applied sciences southern methodist university dallas tx 75275 a markov chain can be categorized as standard or nonstandard depending on the location of an invertible submatrix necessary for an efficient solution in a transition probability or rate matrix in this paper algorithms for the determination of steady state probabilities are developed by using i a backward recursion which is efficient for standard systems and ii a forward recursion which is efficient for nonstandard systems it is also shown that the backward recursion can be used for finding the first passage time distribution and its mean and variance

estimating fetal age by ultrasound measurements an example of multivariate calibration a problem in the estimation of gestational age by ultrasound measurement of fetal bone lengths the femur length f and the biparietal diameter bpd is described by means of multivariate calibration techniques f and bpd are combined to give an index of gestational age which is significantly more accurate than existing indices based on f or bpd alone a detailed description of the statistical analysis is given ranging from the initial steps of data analysis and model choice to the final step of determining the index and its accuracy a number of interesting aspects of multivariate calibration theory are illustrated

optimal designs for binary data abstract this article is concerned with the problem of selecting the values of the explanatory variable in logistic regression to obtain likelihood based confidence regions of minimum area one way of finding analytic solutions is by replacing the log likelihood by a quadratic surface around the maximum approximating in this way the likelihood regions by ellipses the optimal allocation for this local approximation is derived without unnecessary restrictions since the implementation of the optimal allocation requires accurate initial estimates a two stage procedure that uses the second stage to complement the first is recommended an alternative approach is to deal directly with the likelihood regions in this case the identification of the globally optimal allocation calls for numerical integration and optimization the results presented here attempt to strengthen those introduced by abdelbasit and plackett 1983 who derived optimal allocation for the local criterion under the restriction of symmet

on distributions of ratios a large number of exact inferential procedures in statistics and econometrics involve the sampling distribution of ratios of random variables if the denominator variable is positive then tail probabilities of the ratio can be expressed as those of a suitably defined difference of random variables if in addition the joint characteristic function of numerator and denominator is known then standard fourier inversion techniques can be used to reconstruct the distribution function from it most research in this field has been based on this correspondence but which breaks down when both numerator and denominator are supported on the entire real line the present manuscript derives inversion formulae and saddlepoint approximations that remain valid in this case and reduce to known results when the denominator is almost surely positive applications include the iv est imator of a structural parameter in a just identified equation

kazhdan lusztig immanants and products of matrix minors abstract using the kazhdan lusztig basis c w 1 w s n for the symmetric group algebra we obtain nonnegativity properties of certain polynomials in matrix minors in particular we show that the application of these polynomials to jacobi trudi matrices yields symmetric functions which are equal to nonnegative linear combinations of schur functions

quantization of branching coefficients for classical lie groups abstract we study natural quantizations k of branching coefficients corresponding to the restrictions of the classical lie groups to the levi subgroups of their standard parabolic subgroups the polynomials obtained can be regarded as generalizations of lusztig q analogues of weight multiplicities for gl n they coincide with poincare polynomials previously studied by shimozono and weyman they also appear in the hilbert series of the euler characteristic of certain graded virtual g modules and by a result of broer admit nonnegative coefficients providing that restrictive conditions are verified when the levi subgroup considered is isomorphic to a direct product of linear groups we prove that these polynomials admit a stable limit k which decomposes as nonnegative integer combination of poincare polynomials for a general levi subgroup it is conjectured that the polynomials k have nonnegative coefficients when they are parametrized by two partitions when g gl n the polynomials k can be interpreted as quantizations of the littlewood richardson coefficients we show that there also exists a duality between tensor product coefficients for types b c d defined as the analogues of the littlewood richardson coefficients and branching coefficients corresponding to the restriction of so 2 n to subgroups defined from orthogonal decompositions of the root system d n which are not levi subgroups these coefficients can also be quantified but the q analogues obtained may have negative coefficients given a tensor product of irreducible gl n modules we then introduce for each classical group g so n or sp n some q analogues d of the multiplicities obtained by decomposing into its g irreducible components we establish a duality between the polynomials d and k according to a conjecture by shimozono the stable one dimensional sums for nonexceptional affine crystals are expected to coincide with the polynomials d associated to a sequence of rectangular partitions of decreasing heights

optimal design for elisa and other forms of immunoassay in enzyme linked immunosorbent assay elisa as well as in many other kinds of immunoassay a log logistic or similar shaped calibration curve is fit using standards at a series of known levels and then used to transform the measured values for the unknowns into estimated concentrations the choice of the number of standards the concentration of the standards and the number of replicates of the standards and of the unknowns all affect the precision of the measurement this article develops an optima1 design paradigm for this type of problem and shows how optimal choices can be calculated so that the system achieves the maximum precision of which it is capable although exact calculation of optimal designs requires use of a computer program close approximations to the optimum can be derived from simple rules for hand calculation

rules of three for commutation relations abstract we study the phenomenon in which commutation relations for sequences of elements in a ring are implied by similar relations for subsequences involving at most three indices at a time

variance functions and the minimum detectable concentration in assays abstract assay data are often fit by a nonlinear regression model incorporating heterogeneity of variance as ion radioimmunoassay for example typically the standard deviation of the response is taken to be proportional to a power theta of the mean there is considerable empirical evidence suggesting that for assays of a reseasonable size how one estimates the parameter theta does not greatly affect how well one estimates the the mean regression function an additional component of assay analysis is the estimation of auxillary constructs such as the minimum detectable concentration for which many definitions exist we focus on one such definition the minimum detectable concentration depends both on theta and the mean regression function we compare three standard method of estimating the parameter theta due to rodbard 1978 raab 1981a and carroll and ruppert 1982b when duplicate counts are taken at each concentration the first method is only 20 efficient asymptotically in comparison to the third and the resulting estimate of the minimum detectable concentration is asymptotically 3 3 times more variable for first than the third less dramatic results obtain for the second estimator compared to the third this estimator is still not efficient however simulation results and an example are supportive of the asymptotic theory keywords least squares method

vertex operators solvable lattice models and metaplectic whittaker functions we show that spherical whittaker functions on an n fold cover of the general linear group arise naturally from the quantum fock space representation of u q widehat mathfrak sl n introduced by kashiwara miwa and stern kms we arrive at this connection by reconsidering the solvable lattice models known as metaplectic ice whose partition functions are metaplectic whittaker functions first we show that a certain hecke action on metaplectic whittaker coinvariants agrees up to twisting with a hecke action of ginzburg reshetikhin and vasserot arising in quantum affine schur weyl duality this allows us to expand the framework of kms by drinfeld twisting to introduce gauss sums into the quantum wedge which are necessary for connections to metaplectic forms our main theorem interprets the row transfer matrices of this ice model as half vertex operators on quantum fock space that intertwine with the action of u q widehat mathfrak sl n in the process we introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to whittaker functions on an n fold metaplectic cover of text gl r these resemble llt polynomials or ribbon symmetric functions introduced by lascoux leclerc and thibon and in fact the metaplectic symmetric functions are up to twisting specializations of supersymmetric llt polynomials defined by lam indeed lam constructed families of symmetric functions from heisenberg algebra actions on the fock space commuting with the u q widehat mathfrak sl n action the heisenberg algebra is independent of drinfeld twisting of the quantum group we explain that half vertex operators agree with lam s construction and this interpretation allows for many new identities for metaplectic symmetric and whittaker functions including cauchy identities while both metaplectic symmetric functions and llt polynomials can be related to vertex operators on the quantum fock space only metaplectic symmetric functions are connected to solvable lattice models

a method for designing nonlinear univariate calibration a method is proposed for designing nonlinear univariate calibration of measuring instruments the problem addressed is how to select a set of design points standards or calibrators to minimize the errors in the inverse predictions the curve parameters are assumed to vary randomly between calibrations with known expected value and known covariance matrix a design criterion is suggested for analytical procedures according to which the coefficient of variation and the area under the precision profile are minimized

optimal designs for michaelis menten kinetic studies many reactions in enzymology are governed by the michaelis menten equation characterising these reactions requires the estimation of the parameters k m and v m a x which determine the michaelis menten equation and this is done by observing rates of reactions at a set of substrate concentrations the choice of substrate concentrations is investigated by determining bayesian d optimal designs for a model in which residuals have a normal distribution with constant variance designs which focus on alternative quantities such as k m or the ratio v m a x k m are also considered the effect on the optimal designs of alternative error distributions is also considered

confidence regions in multivariate calibration on propose une methode alternative qui donne une region a bon comportement correspondant a la region invariante par translation uniformement la plus precise

optimal design for goodness of fit of the michaelis menten enzyme kinetic function we construct efficient designs for the michaelis menten enzyme kinetic model capable of checking model assumptions an extended model called emax is also considered for this purpose this model is widely used in pharmacokinetics and reduces to the michaelis menten model for a specific choice of parameter settings our strategy is to find efficient designs for estimating the parameters in the emax model and at the same time test the validity of the michaelis menten model against the emax model by maximizing a minimum of the d or d1 efficiencies taken over a range of values for the nonlinear parameters in particular we show that such designs are a efficient for estimating parameters in the emax model b about 70 efficient for estimating parameters in the michaelis menten model c efficient for testing the michaelis menten model against the emax model and d robust with respect to misspecification of the unknown parameters in the nonlinear model

bootstrapping in controlled calibration experiments we consider the determination of an unknown quantity for example the concentration of a particular chemical in a given sample or samples using controlled calibration here several samples are prepared with concentrations chosen to cover a required range and these are used to establish the relationship between concentration and the measured response to an assay method this relationship is then used to estimate the concentration in the unknown samples from their measured responses confidence intervals for the estimated concentrations can usually be calculated by inverting a prediction interval but in some situations this method becomes intractable we explore the use of the bootstrap as an alternative in linear nonlinear and multivariate controlled calibration using both simulation and real datasets from the field of immunoassay we also discuss the alternatives afforded by replication of the design points the bootstrap is found to be comparable to the standard method in simple situations and is ea

a bayesian approach to nonlinear calibration problems abstract in bioassays enzyme assays or radioimmunoassays the concentration response relationships are mostly nonlinear very often because of physical restrictions or for biological reasons the prediction samples cannot be diluted and the determination of very low concentrations is of interest thus one cannot be restricted to the almost linear part of the response curve and standard linear calibration methods are not applicable the nonlinear calibration problem is of great importance in these types of applications to implement the bayesian paradigm in nonlinear calibration problems a substantial amount of numerical integration is necessary in practice when concentrations of hundreds of samples have to be determined routinely even with a highly efficient integration method naylor and smith 1982 the numerical effort becomes prohibitive an approximation method is proposed therefore to reduce the calculation the precision of the approximation method is examined by validation subsamples and

asymptotic behavior of the stationary distributions in the gi ph c queue with heterogeneous servers this paper deals with the stablec server queue with renewal input the service time distributions may be different for the various servers they are however all probability distributions of phase type it is shown that the stationary distribution of the queue length at arrivals has an exact geometric tail of rate 0 0 the quantities and may be evaluated together by an elementary algorithm for both distributions the multiplicative constants which arise in the asymptotic forms may be fully characterized these constants are however difficult to compute in general

haglund s conjecture on 3 column macdonald polynomials we prove a positive combinatorial formula for the schur expansion of llt polynomials indexed by a 3 tuple of skew shapes this verifies a conjecture of haglund proc natl acad sci usa 101 46 16127 16131 2004 the proof requires expressing a noncommutative schur function as a positive sum of monomials in lam s eur j combin 29 1 343 359 2008 algebra of ribbon schur operators combining this result with the expression of haglund et al j am math soc 18 3 735 761 2005 for transformed macdonald polynomials in terms of llt polynomials then yields a positive combinatorial rule for transformed macdonald polynomials indexed by a shape with 3 columns

on the calibration problem extended to nonlinear models the traditional calibration problem is concerned with estimating the unknown value of an explanatory variable x corresponding to an observed value of a response variable y that is assumed to be functionally related to x the problem for which y is related to x by a simple linear model has been discussed in the literature this paper is concerned with obtaining a confidence interval for an unknown value of x corresponding to a specified or fixed value of y where y and x are related through a known nonlinear function given a specified value y of a response variable y that is related to x through a known function linear or nonlinear a point estimate of xc can be found through the inverse function if the inverse exists or through an iterative search procedure two approaches are presented for determining a confidence interval for xc given a value y when the inverse function exists a confidence interval about xc can be obtained from the approximate distribution of kc which provides a symmetric confidence interval about xc confidence intervals based on the asymptotic distribution of f3 are not symmetric about kc which is particularly appropriate when considering nonlinear functions that have asymptotes the technique based on the distribution of f3 is more general because it does not depend on having a closed form of the inverse function a simulation study for sample sizes of 20 involving an exponential decay function indicates that both approaches attained the desired confidence coefficient a test procedure comparing two calibration points from different populations or treatment groups is presented a simulation study of sample sizes of 20 involving an exponential decay function indicates that the test procedure attains the desired level of significance with good power additional simulation studies may be necessary to determine the appropriateness of these techniques for different model types and or sample sizes an example from meat sciences research is presented demonstrating the techniques discussed

on bayesian methods for bioequivalence bayesian methods are presented for assessing bioequivalence for studies in which a new formulation and a standard are administered simultaneously and for latin square designs which compare two or more new formulations to a standard two examples illustrate the application of the methods

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

convergence properties of the nelder mead simplex method in low dimensions the nelder mead simplex algorithm first published in 1965 is an enormously popular direct search method for multidimensional unconstrained minimization despite its widespread use essentially no theoretical results have been proved explicitly for the nelder mead algorithm this paper presents convergence properties of the nelder mead algorithm applied to strictly convex functions in dimensions 1 and 2 we prove convergence to a minimizer for dimension 1 and various limited convergence results for dimension 2 a counterexample of mckinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the nelder mead algorithm converges to a nonminimizer it is not yet known whether the nelder mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions

design considerations for calibration this article examines the effects of the choice of design points on calibration in the presence of a simple linear regression model the distribution of the classical estimator is examined via modes median and moments and the asymptotic variance of this estimator is minimized the influence of the designs on confidence regions constructed via fieller s theorem is investigated through accuracy curves a numerical example is presented and some additional problems are suggested

a two component model for measurement error in analytical chemistry in this article we propose and test a new model for measurement error in analytical chemistry often the standard deviation of analytical errors is assumed to increase proportionally to the concentration of the analyte a model that cannot be used for very low concentrations for near zero amounts the standard deviation is often assumed constant which does not apply to larger quantities neither model applies across the full range of concentrations of an analyte by positing two error components one additive and one multiplicative we obtain a model that exhibits sensible behavior at both low and high concentration levels we use maximum likelihood estimation and apply the technique to toluene by gas chromatography mass spectrometry and cadmium by atomic absorption spectroscopy

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

errors in variables estimation in multivariate calibration a set of q responses y y 1 y 2 y q t is related to a set of p explanatory variables x x 1 x 2 x p t through the classical linear regression model y t a t x t b e t first the unknown parameters a and b are estimated using a calibrafion set the statistical problem that is considered here is that of estimating the vector x o that underlies a new observed vector of responses y o using the parameter estimates obtained from the first procedure these two procedures are commonly referred to as calibration and prediction or inverse prediction and sometimes jointly referred to as calibration the prediction procedure can be viewed as parameter estimation in errors in variables regression the maximum likelihood estimator assuming normally distributed measurement errors is proposed for the prediction procedure unlike the classical estimator used in the prediction procedure the proposed estimator is consistent with respect to the number of response variables the performan

estimation of relative potency with sequential dilution errors in radioimmunoassay summary sequential dilution is a very common procedure in radioimmunoassay in which the dilution error will be accumulated from the highest to the lowest concentration a simulated example in relative potency determination is used to demonstrate the potentially wrong conclusion that can be drawn when the dilution error is not properly included in the model a bayesian method is used and an alternative approximation via maximum likelihood is proposed an alternative experimental design is recommended to increase the precision of the inference

ribbon tableaux hall littlewood functions quantum affine algebras and unipotent varieties we introduce a new family of symmetric functions which are q analogs of products of schur functions defined in terms of ribbon tableaux these functions can be interpreted in terms of the fock space representation fq of uq sln and are related to hall littlewood functions via the geometry of flag varieties we present a series of conjectures and prove them in special cases the essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of fq using ribbon tableaux

calibration with randomly changing standard curves changes in calibration curves from one time to the next caused by drift often require measuring devices to be recalibrated at frequent intervals in such situations the usual practice is to estimate the unknown values of test samples using only data from the corresponding calibration period under a random coefficient regression model for the different calibration curves however it can be shown that it is more efficient to combine the data from all calibration periods to estimate the unknowns we consider a particular class of point estimators obtained by inverting suitable prediction functions and show that the estimator obtained from a best prediction function is optimal in a sense defined by godambe 1960 and durbin 1960 in the context of unbiased estimating equations we also compare the smallsample performance of this estimator with the usual estimator using the pitman closeness criterion

efficient design of experiments in the monod model summary estimation and experimental design in a non linear regression model that is used in microbiology are studied the monod model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics water research pharmacokinetics and plant physiology it is proved that least squares estimates are asymptotically unbiased and normally distributed the asymptotic covariance matrix of the estimator is the basis for the construction of efficient designs of experiments in particular locally d e and c optimal designs are determined and their properties are studied theoretically and by simulation if certain intervals for the non linear parameters can be specified locally optimal designs can be constructed which are robust with respect to a misspecification of the initial parameters and which allow efficient parameter estimation parameter variances can be decreased by a factor of 2 by simply sampling at optimal times during the experiment

bayesian inference for ratios of coefficients in a linear model a variety of statistical problems e g slope ratio and parallel line bioassay calibration bioequivalence can be viewed as questions of inference on the ratio of two coefflcients in a suitably constructed linear model this paper develops a bayesian approach for the general formulation of the problem using a class of priors which involve the unknown ratio explicitly the posterior distribution of the ratio is obtained analytically and its properties are investigated especially the sensitivity to the choice of the prior examples are given of applications to slope ratio bioassay comparison of the mean effects of two drugs and a bioequivalence problem

a bayesian analysis of the linear calibration problem as part of a bayesian analysis of the linear calibration problem a posterior distribution is derived that is mathematically identical to the structural distribution obtained by kalotay 1971 but different from the bayesian distribution obtained by hoadley 1970 useful approximations are given although this posterior distribution has infinite variance no practical problems arise in using it the controversy concerning infinite variances and the use of inverse regression is discussed

affine springer fibers of type a and combinatorics of diagonal coinvariants abstract we calculate the borel moore homology of affine springer fibers of type a associated with some regular semisimple nil elliptic elements as a result we obtain bigraded s n modules whose bigraded frobenius series are a generalization of the symmetric functions introduced by haglund haiman loehr remmel and ulyanov

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

the hopf algebra of odd symmetric functions abstract we consider a q analogue of the standard bilinear form on the commutative ring of symmetric functions the q 1 case leads to a z graded hopf superalgebra which we call the algebra of odd symmetric functions in the odd setting we describe counterparts of the elementary and complete symmetric functions power sums schur functions and combinatorial interpretations of associated change of basis relations

the state reduction and related algorithms and their applications to the study of markov chains graph theory and the optimal stopping problem abstract we discuss the state reduction gth grassmann taksar heyman algorithm for recursively finding invariant measure we demonstrate the relationship between this algorithm and the freidlin wentzell tree decomposition approach to study the characteristics of markov chains the structure of the state reduction gth algorithm suggests the natural idea for finding the distribution of a markov chain at the moment of first visit to a given set and some similar characteristics we study the possible range of such algorithms we also present a new algorithm for solving the classical problem of optimal stopping of a markov chain based on a similar idea of sequential elimination of some states we give shorter and more transparent proofs of some previously known results and improve the bounds of freidlin wentzell in the perturbation theory of markov chains some applications to graph theory are also discussed

robust and efficient designs for the michaelis menten model for the michaelis menten model we determine designs that maximize the minimum of the d efficiencies over a certain interval for the nonlinear parameter the best two point designs can be found explicitly and a characterization is given when these designs are optimal within the class of all designs in most cases of practical interest the determined designs are highly efficient and robust with respect to misspecification of the nonlinear parameter the results are illustrated and applied in an example of a hormone receptor assay

inverse estimates from a multiresponse model summary in a multiresponse model a multivariate y is modelled as a nonlinear function of the independent variable x the inverse estimate of x for a given observation on y is the object of study in this paper a straightforward method is proposed for estimating x and setting confidence limits to the estimate an example of determining age from the lengths of horns of a rhinoceros is given animal based on horn lengths only we specifically deal with the situation in which the response functions are nonlinear in the unknown parameters the method proposed is to augment the original data set with the additional measure ments lag and vpg of the extra animal whose age is needed and a guessed age xi varying values of xg will generate varying values of the maximized likelihood conditional on xg and these define the profile likelihood kalbfleisch and sprott 1970 for xg this profile likelihood can be used and interpreted in a number of ways all basically intended to illustrate the reliability of the age estimate a cautionary word is required the step of associating the profile likelihood with classical confidence limits for the true value of xg is difficult and only approximate solutions are available to that extent this paper should not be regarded as proposing a definitive answer there are open questions raised in particular it would be valuable to quantify more precisely the sampling distribution of the profile likelihood function the problem posed here is a version of the multivariate calibration problem the literature although sometimes couched in more general terms invariably deals in detail only with response models that are linear in the unknown parameters 0 brown 1982 gives a broad review of the linear multivariate calibration problem and outlines both bayesian and non bayesian solutions hunter and lamboy 1981 review the linear problem from a bayesian viewpoint brown and sundberg 1987 1989 propose using the profile likelihood of x minder and whitney 1975 derived a marginal likelihood approach the classic paper on univariate linear calibration is that of fieller 1940 and many modern ideas can be related back to the fieller fiducial limits oman and wax 1984 give an interesting application of linear multivariate calibration in estimating fetal age given two

robust estimation in heteroscedastic linear models we consider a heteroscedastic linear model in which the variances are given by a parametric function of the mean responses and a parameter theta we propose robust estimates for the regression parameter beta and show that as long as a reasonable starting estimate of theta is available our estimates of beta are asymptotically equivalent to the natural estimate obtained with known variances a particular method for estimating theta is proposed and shown by monte carlo to work quite well especially in power and exponential models for the variances we also briefly discuss a feedback estimate of beta

the delta conjecture we conjecture two combinatorial interpretations for the symmetric function eken where f is an eigenoperator for the modified macdonald polynomials defined by bergeron garsia haiman and tesler both interpretations can be seen as generalizations of the shuffle conjecture a statement originally conjectured by haglund haiman remmel loehr and ulyanov and recently proved by carlsson and mellit we show how previous work of the second and third authors on tesler matrices and ordered set partitions can be used to verify several cases of our conjectures furthermore we use a reciprocity identity and llt polynomials to prove another case finally we show how our conjectures inspire 4 variable generalizations of the catalan numbers extending work of garsia haiman and the first author

analyzing residuals in calibration problems the analysis of residuals in the calibration or inverse regression problem is considered a statistic is proposed that is appropriate to the specific use of the regression equation in this context namely estimating x from y the statistic similar in spirit to cook s distance may be used to measure the influence a particular observation may have on future estimates of x from the calibration curve it may also be used to measure the differences between estimates of x obtained using calibration curves based on different models an example is given

robust experimental design for multivariate generalized linear models a simple heuristic is proposed for constructing robust experimental designs for multivariate generalized linear models the method is based on clustering a set of local optimal designs a method for finding local d optimal designs using available resources is also introduced clustering with its simplicity and minimal computation needs is demonstrated to outperform more complex and sophisticated methods