majorization as a tool for optimizing a class of matrix functions the problem of minimizing a general matrix trace function possibly subject to certain constraints is approached by means of majorizing this function by one having a simple quadratic shape and whose minimum is easily found it is shown that the parameter set that minimizes the majorizing function also decreases the matrix trace function which in turn provides a monotonically convergent algorithm for minimizing the matrix trace function iteratively three algorithms based on majorization for solving certain least squares problems are shown to be special cases in addition by means of several examples it is noted how algorithms may be provided for a wide class of statistical optimization tasks for which no satisfactory algorithms seem available

perturbation expansions of the axial next nearest neighbour ising and asymmetric four state clock models the axial next nearest neighbour ising annni and four state asymmetric clock 4 asc models are investigated by perturbing about points at which they reduce the simple nearest neighbour ising models these limits are a for the annni model i small nearest neighbour interaction j1 in the x direction and ii small next nearest neighbour interaction j2 in the x direction and b for the 4 asc model small asymmetry delta in the limit j1 to 0 the phase boundary is located to first order in j1 and found to be ising like in the limit j2 to 0 the behaviour to second order in j2 is more complicated but is identical to that found to second order in delta for the 4 asc model these results are discussed in the light of other recent work

simplimax oblique rotation to an optimal target with simple structure factor analysis and principal component analysis are usually followed by simple structure rotations of the loadings these rotations optimize a certain criterion e g varimax oblimin designed to measure the degree of simple structure of the pattern matrix simple structure can be considered optimal if a usually large number of pattern elements is exactly zero in the present paper a class of oblique rotation procedures is proposed to rotate a pattern matrix such that it optimally resembles a matrix which has an exact simple pattern it is demonstrated that this method can recover relatively complex simple structures where other well known simple structure rotation techniques fail

minimization of a class of matrix trace functions by means of refined majorization a procedure is described for minimizing a class of matrix trace functions the procedure is a refinement of an earlier procedure for minimizing the class of matrix trace functions using majorization it contains a recently proposed algorithm by koschat and swayne for weighted procrustes rotation as a special case a number of trial analyses demonstrate that the refined majorization procedure is more efficient than the earlier majorization based procedure

orthogonal rotations to maximal agreement for two or more matrices of different column orders methods for orthogonal procrustes rotation and orthogonal rotation to a maximal sum of inner products are examined for the case when the matrices involved have different numbers of columns an inner product solution offered by cliff is generalized to the case of more than two matrices a nonrandom start for a procrustes solution suggested by green and gower is shown to give better results than a random start the green gower procrustes solution with nonrandom start is generalized to the case of more than two matrices simulation studies indicate that both the generalized inner product solution and the generalized procrustes solution tend to attain their global optima within acceptable computation times a simple procedure is offered for approximating simple structure for the rotated matrices without affecting either the procrustes or the inner product criterion

acquiescent responding in balanced multidimensional scales and exploratory factor analysis personality tests often consist of a set of dichotomous or likert items these response formats are known to be susceptible to an agreeing response bias called acquiescence the common assumption in balanced scales is that the sum of appropriately reversed responses should be reasonably free of acquiescence however inter item correlation or covariance matrices can still be affected by the presence of variance due to acquiescence to analyse these correlation matrices we propose a method that is based on an unrestricted factor analysis and can be applied to multidimensional scales this method obtains a factor solution in which acquiescence response variance is isolated in an independent factor it is therefore possible without the potentially confounding effect of acquiescence to a examine the dominant factors related to content latent variables and b estimate participants factor scores on content latent variables this method which is illustrated by two empirical data examples has proved to be useful for improving the simplicity of the factor structure

component analysis in cross sectional and longitudinal data an extension of component analysis to longitudinal or cross sectional data is presented in this method components are derived under the restriction of invariant and or stationary compositing weights optimal compositing weights are found numerically the method can be generalized to allow differential weighting of the observed variables in deriving the component solution some choices of weightings are discussed an illustration of the method using real data is presented

the asymmetric clock model on a cayley tree the asymmetric clock model is studied on a cayley tree and commensurate and incommensurate phases are found the transition lines are obtained from stability conditions and characteristic points in the phase diagram are analysed by numerical iteration a critical endpoint is found for the case of three states per site and a lifshitz point for the case of four states per site

principal minor assignment isometries of hilbert spaces volumes of parallelepipeds and rescaling of sesqui holomorphic functions abstract in this article we consider the following equivalence relation on the class of all functions of two variables on a set x we will say that l m x x c are rescalings if there are non vanishing functions f g on x such that m x y f x g y l x y for any x y x we give criteria for being rescalings when x is a topological space and l and m are separately continuous or when x is a domain in c n and l and m are sesqui holomorphic a special case of interest is when l and m are symmetric and f g only has values 1 this relation between m and l in the case when x is finite and so l and m are square matrices is known to be characterized by the equality of the principal minors of these matrices we extend this result to the case when x is infinite as an application we characterize restrictions of isometries of hilbert spaces on weakly connected sets as the maps that preserve the volumes of parallelepipeds spanned by finite collections of vectors

on the matrix formulation of kaiser s varimax criterion the author provides a full fledged matrix derivation of sherin s matrix formulation of kaiser s varimax criterion he uses matrix differential calculus in conjunction with the hadamard or schur matrix product two results on hadamard products are presented

multidimensional rotation and scaling of configurations to optimal agreement an integrated method for rotating and rescaling a set of configurations to optimal agreement in subspaces of varying dimensionalities is developed the approach relates existing orthogonal rotation techniques as special cases within a general framework based on a partition of variation which provides convenient measures of agreement in addition to the well known procrustes and inner product optimality criteria a criterion which maximizes the consensus among subspaces of the configurations is suggested since agreement of subspaces of the configurations can be examined and compared rotation and rescaling is extended from a data transformation technique to an analytical method

an approach to n mode components analysis as an extension of lastovicka s four mode components analysis ann mode components analysis is developed using a convenient notation both a canonical and a least squares solution are derived the relation between both solutions and their computational aspects are discussed

uniqueness of three mode factor models with sparse cores the 3 x 3 x 3 case three mode factor analysis 3mfa and parafac are methods to describe three way data both methods employ models with components for the three modes of a three way array the 3mfa model also uses a three way core array for linking all components to each other the use of the core array makes the 3mfa model more general than the parafac model thus allowing a better fit but also more complicated moreover in the 3mfa model the components are not uniquely determined and it seems hard to choose among all possible solutions a particularly interesting feature of the parafac model is that it does give unique components the present paper introduces a class of 3mfa models in between 3mfa and parafac that share the good properties of the 3mfa model and the parafac model they fit almost as well as the 3mfa model they are relatively simple and they have the same uniqueness properties as the parafac model

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

the infinitesimal jackknife with exploratory factor analysis the infinitesimal jackknife a nonparametric method for estimating standard errors has been used to obtain standard error estimates in covariance structure analysis in this article we adapt it for obtaining standard errors for rotated factor loadings and factor correlations in exploratory factor analysis with sample correlation matrices both maximum likelihood estimation and ordinary least squares estimation are considered

quartic rotation criteria and algorithms most of the currently used analytic rotation criteria for simple structure in factor analysis are summarized and identified as members of a general symmetric family of quartic criteria a unified development of algorithms for orthogonal and direct oblique rotation using arbitrary criteria from this family is given these algorithms represent fairly straightforward extensions of present methodology and appear to be the best methods currently available

hierarchical relations among three way methods a number of methods for the analysis of three way data are described and shown to be variants of principal components analysis pca of the two way supermatrix in which each two way slice is strung out into a column vector the methods are shown to form a hierarchy such that each method is a constrained variant of its predecessor a strategy is suggested to determine which of the methods yields the most useful description of a given three way data set

orthogonal principal planes factor analysis and principal component analysis result in computing a new coordinate system which is usually rotated to obtain a better interpretation of the results in the present paper the idea of rotation to simple structure is extended to two dimensions while the classical definition of simple structure is aimed at rotating one dimensional factors the extension to a simple structure for two dimensions is based on the rotation of planes the resulting planes principal planes reveal a better view of the data than planes spanned by factors from classical rotation and hence allow a more reliable interpretation the usefulness of the method as well as the effectiveness of a proposed algorithm are demonstrated by simulation experiments and an example

oblique factors and components with independent clusters relationships between the results of factor analysis and component analysis are derived when oblique factors have independent clusters with equal variances of unique factors the factor loadings are analytically shown to be smaller than the corresponding component loadings while the factor correlations are shown to be greater than the corresponding component correlations the condition for the inequality of the factor component contributions is derived in the case with different variances for unique factors further the asymptotic standard errors of parameter estimates are obtained for a simplified model with the assumption of multivariate normality which shows that the component loading estimate is more stable than the corresponding factor loading estimate

simple structure in component analysis techniques for mixtures of qualitative and quantitative variables several methods have been developed for the analysis of a mixture of qualitative and quantitative variables and one called pcamix includes ordinary principal component analysis pca and multiple correspondence analysis mca as special cases the present paper proposes several techniques for simple structure rotation of a pcamix solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for pca in addition a recently developed technique for the analysis of mixtures of qualitative and quantitative variables called indomix is shown to construct component scores without rotational freedom maximizing the quartimax criterion over all possible sets of component scores a numerical example is used to illustrate the implication that when used for qualitative variables indomix provides axes that discriminate between the observation units better than do those generated from mca

newton algorithms for analytic rotation an implicit function approach in this paper implicit function based parameterizations for orthogonal and oblique rotation matrices are proposed the parameterizations are used to construct newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables the speed of the new algorithms is compared to that of existing algorithms and to that of newton algorithms based on alternative parameterizations several rotation criteria were examined and the algorithms were evaluated over a range of values for m initial guesses for newton algorithms were improved by subconvergence iterations of the gradient projection algorithm simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m among competing algorithms the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large in all other conditions however the implicit function algorithms were competitive with or faster than the fastest existing algorithms the new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion

star triangle equations and multicomponent chiral potts models abstract under certain conditions lipid bilayer membranes exhibit an incommensurate ripple phase pearce and scott have proposed lattice models with either next nearest neighbor interactions or with chiral nearest neighbor interactions to study this phenomenon after briefly discussing the equivalence of these two points of view we present the results of our first search for solutions of the star triangle equations for multicomponent chiral potts models generalizing some of the biomembrane models in the nonchiral limit the results apply also to the n n model of domany and riedel describing the adsorption of molecules on the surface of a crystal

a case of extreme simplicity of the core matrix in three mode principal components analysis in three mode principal components analysis thep q r core matrixg can be transformed to simple structure before it is interpreted it is well known that whenp qr g can be transformed to the identity matrix which implies that all elements become equal to values specified a priori in the present paper it is shown that whenp qr 1 g can be transformed to have nearly all elements equal to values spectified a priori a cllsed form solution for this transformation is offered theoretical and practical implications of this simple structure transformation ofg are discussed

numerical solution of the mean field theory of the annni model in an external field a comparison of two methods it is shown how the solution of the mean field equations for the annni model can be reduced to the iteration of an area preserving two dimensional mapping if an external magnetic field is applied this method is analysed in detail and compared with the exact numerical solution of the mean field equations for simple periodic magnetisations

suppressing permutations or rigid planar rotations a remedy against nonoptimal varimax rotations varimax rotation consists of iteratively rotating pairs of columns of a matrix to a maximal sum over columns of variances of squared elements of the matrix without loss of optimality the two rotated columns can be permuted and or reflected although permutations and reflections are harmless for each planar rotation per se they can be harmful in varimax rotation specifically they often give rise to the phenomenon that certain pairs of columns are consistently skipped in the iterative process whence varimax will be terminated at a nonstationary point the skipping phenomenon is demonstrated and it is shown how to prevent it

finite lattice methods in quantum hamiltonian field theory i o 2 and o 3 heisenberg models for pt i see ibid vol 14 p 241 57 1981 two efficient methods for finding the low lying states of hamiltonians on finite lattices are described the first involves constructing a finite representation of the hamiltonian using strong coupling eigenstates while the second is based on the lanczos recursion method the methods are used to determine the mass gap of the o 2 and o 3 heisenberg hamiltonians in 1 1 dimensions for a sequence of finite chains the critical behaviour of the infinite chain is then analysed by extrapolating the finite lattice estimates using finite size scaling a remarkably sensitive test is developed for the presence of a phase transition for the o 2 model data this test yields strong evidence for a phase transition with the weak coupling phase massless while in the o 3 case the test supports although more weakly the absence of any transition

a direct solution for pairwise rotations in kaiser s varimax method the present note contains a completing the squares type approach to the varimax rotation problem this approach yields a direct proof of global optimality of a solution for optimal rotation in a plane because varimax rotation can be interpreted as diagonalization of a set of symmetric matrices the present solution also applies to the diagonalization problem

errors in streamflow drought statistics reconstructed from tree ring data hydrologic variables such as streamflow may be reconstructed for years prior to the observational record through the use of historical tree ring data reconstructed records provide additional information for the estimation of statistics relating to rare events such as large and persistent water deficits however the reconstruction process can introduce bias and inflate the errors of estimated statistics the biases and standard errors of streamflow drought statistics derived from such reconstructed records are evaluated for idealized conditions by employing a monte carlo simulation method both the bias and the standard error depend to some degree on the goodness of fit of the reconstruction model the calibration sample length the reconstruction sample length and the autocorrelation of the reconstructed variable the bias is most dependent on the goodness of fit of the reconstruction model standard errors are often strongly dependent on several of these factors for a given drought statistic the sample length of an observed record which yields the same standard error as the longer reconstruction is called the effective sample length effective sample lengths are found to be less than 50 of the total reconstructed record length in many cases

a joint treatment of varimax rotation and the problem of diagonalizing symmetric matrices simultaneously in the least squares sense the present paper contains a lemma which implies that varimax rotation can be interpreted as a special case of diagonalizing symmetric matrices as discussed in multidimensional scaling it is shown that the solution by de leeuw and pruzansky is essentially equivalent to the solution by kaiser necessary and sufficient conditions for maxima and minima are derived from first and second order partial derivatives a counter example by gebhardt is reformulated and examined in terms of these conditions it is concluded that kaiser s method or equivalently the method by de leeuw and pruzansky is the most attractive method currently available for the problem at hand

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

the weighted oblimin rotation cureton mulaik 1975 proposed the weighted varimax rotation so that varimax kaiser 1958 could reach simple solutions when the complexities of the variables in the solution are larger than one in the present paper the weighting procedure proposed by cureton mulaik 1975 is applied to direct oblimin clarkson jennrich 1988 and the rotation method obtained is called weighted oblimin it has been tested on artificial complex data and real data and the results seem to indicate that even though direct oblimin rotation fails when applied to complex data weighted oblimin gives good results if a variable with complexity one can be found for each factor in the pattern although the weighting procedure proposed by cureton mulaik is based on landahl s 1938 expression for orthogonal factors weighted oblimin seems to be adequate even with highly oblique factors the new rotation method was compared to other rotation methods based on the same weighting procedure and whenever a variable with complexity one could be found for each factor in the pattern weighted oblimin gave the best results when rotating a simple empirical loading matrix weighted oblimin seemed to slightly increase the performance of direct oblimin

an efficient algorithm for the symmetric principal minor assignment problem abstract we consider the problem of reconstructing a symmetric matrix from its principal minors which has several applications in information theory and statistical modeling we develop a theory of symmetric matrices with equal corresponding principal minors based on a simple equivalent property due to oeding 2011 10 we then use this theory to provide a method for choosing a canonical representative from the class of all symmetric matrices with specified principal minors finally we provide an efficient algorithm for computing this canonical representative given its principal minors as input

skew symmetric matrices and their principal minors abstract in this article we study the following problem given a positive integer k what is the relationship between two matrices with entries in a field k and having equal corresponding principal minors of order at most k our main theorem improves a result of loewy 13 for skew symmetric matrices all of whose off diagonal entries are nonzero

hyperdeterminantal relations among symmetric principal minors abstract the principal minors of a symmetric n n matrix form a vector of length 2 n we characterize these vectors in terms of algebraic equations derived from the 2 2 2 hyperdeterminant

three mode orthomax rotation factor analysis and principal components analysis pca are often followed by an orthomax rotation to rotate a loading matrix to simple structure the simple structure is usually defined in terms of the simplicity of the columns of the loading matrix in three mode pca rotational freedom of the so called core a three way array relating components for the three different modes can be used similarly to find a simple structure of the core simple structure of the core can be defined with respect to all three modes simultaneously possibly with different emphases on the different modes the present paper provides a fully flexible approach for orthomax rotation of the core to simple structure with respect to three modes simultaneously computationally this approach relies on repeated two way orthomax applied to supermatrices containing the frontal lateral or horizontal slabs respectively the procedure is illustrated by means of a number of exemplary analyses as a by product application of the three mode orthomax procedures to two way arrays is shown to reveal interesting relations with and interpretations of existing two way simple structure rotation techniques

principal minors part i a method for computing all the principal minors of a matrix an order o 2 n algorithm for computing all the principal minors of an arbitrary n n complex matrix is motivated and presented offering an improvement by a factor of n 3 over direct computation the algorithm uses recursive schur complementation and submatrix extraction storing the answer in a binary order an implementation of the algorithm in matlab is also given and practical considerations are discussed and treated accordingly

the harris kaiser independent cluster rotation as a method for rotation to simple component weights procedures for oblique rotation of factors or principal components typically focus on rotating the pattern matrix such that it becomes optimally simple an important oblique rotation method that does so is harris and kaiser s 1964 independent cluster hkic rotation in principal components analysis a case can be made for interpreting the components on the basis of the component weights rather than on the basis of the pattern so it seems desirable to rotate the components such that the weights rather than the pattern become optimally simple in the present paper it is shown that hkic rotates the components such that both the pattern and the weights matrix become optimally simple in addition it is shown that the pattern resulting from hkic rotation is columnwise proportional to the associated weights matrix which implies that the interpretation of the components does not depend on whether it is based on the pattern or on the component weights matrix it is also shown that the latter result only holds for hkic rotation and slight modifications of it

techniques for rotating two or more loading matrices to optimal agreement and simple structure a comparison and some technical details matrices of factor loadings are often rotated to simple structure when more than one loading matrix is available for the same variables the loading matrices can be compared after rotating them all separately to simple structure an alternative procedure is to rotate them to optimal agreement and then compare them in the present paper techniques are described that combine these two procedures specifically five techniques that combine the ideals of rotation to optimal agreement and rotation to simple structure are compared on the basis of contrived and empirical data for the contrived data it is assessed to what extent the rotations recover the underlying common structure for both the contrived and the empirical data it is studied to what extent the techniques give well matching rotated matrices to what extent these have a simple structure and to what extent the most prominent parts of the different loading matrices agree it was found that the simple procedure of combining a generalized procrustes analysis gpa with varimax on the mean of the matched loading matrices performs very well on all criteria and for most purposes offers an attractive compromise of rotation to agreement and simple structure in addition to this comparison some technical improvements are proposed for bloxom s rotation to simple structure and maximum similarity

a simple general procedure for orthogonal rotation a very general algorithm for orthogonal rotation is identified it is shown that when an algorithm parameter is sufficiently large the algorithm converges monotonically to a stationary point of the rotation criterion from any starting value because a sufficiently large is in general hard to find a modification that does not require it is introduced without this requirement the modified algorithm is not only very general but also very simple its implementation involves little more than computing the gradient of the rotation criterion while the modified algorithm converges monotonically from any starting value it is not guaranteed to converge to a stationary point it however does so in all of our examples while motivated by the rotation problem in factor analysis the algorithms discussed may be used to optimize almost any function of a not necessarily square column wise orthonormal matrix a number of these more general applications are considered empirical examples show that the modified algorithm can be reasonably fast but its purpose is to save an investigator s effort rather than that of his or her computer this makes it more appropriate as a research tool than as an algorithm for established methods

a simple general method for oblique rotation a simple and very general algorithm for oblique rotation is identified while motivated by the rotation problem in factor analysis it may be used to minimize almost any function of a not necessarily square matrix whose columns are restricted to have unit length the algorithm has two steps the first is to compute the gradient of the rotation criterion and the second is to project this onto a manifold of matrices with unit length columns for this reason it is called a gradient projection algorithm because the projection step is very simple implementation of the algorithm involves little more than computing the gradient of the rotation criterion which for many applications is very simple it is proven that the algorithm is strictly monotone that is as long as it is not already at a stationary point each step will decrease the value of the criterion examples from a variety of areas are used to demonstrate the algorithm including oblimin rotation target rotation simplimax rotation and rotation to similarity and simplicity simultaneously while it may be the algorithm is not intended for use as a standard algorithm for well established problems but rather as a tool for investigating new methods where its generality and simplicity may save an investigator substantial effort

finite size scaling studies of the o 2 heisenberg spin model in d 1 dimensions the finite size scaling behaviour of the o 2 heisenberg spin model in d 1 dimensions d 1 and 2 is computed in the low temperature region using the stochastic truncation monte carlo method the numerical results so obtained agree very well with effective lagrangian theory and with spin wave perturbation theory

a factor simplicity index we propose an index for assessing the degree of factor simplicity in the context of principal components and exploratory factor analysis the new index which is called loading simplicity is based on the idea that the communality of each variable should be related to few components or factors so that the loadings in each variable are either zero or as far from zero as possible this index does not depend on the scale of the factors and its maximum and minimum are only related to the degree of simplicity in the loading matrix the aim of the index is to enable the degree of simplicity in loading matrices to be compared

an introduction to computational statistics regression analysis 1 a quick look at a typical regression program 2 simple linear regression 3 applying simple linear regression 4 multiple linear regression 5 computer assisted model building 6 the general linear model 7 analysis of variance and covariance 8 nonlinear regression 9 maximum likelihood analysis and robust estimation

principal minors part ii the principal minor assignment problem the inverse problem of finding a matrix with prescribed principal minors is considered a condition that implies a constructive algorithm for solving this problem will always succeed is presented the algorithm is based on reconstructing matrices from their principal submatrices and schur complements in a recursive manner consequences regarding the overdeterminancy of this inverse problem are examined leading to a faster polynomial time version of the algorithmic construction care is given in the matlab implementation of the algorithms regarding numerical stability and accuracy

alternative principal components regression procedures for dendrohydrologic reconstructions streamflow reconstruction using tree ring information dendrohydrology has traditionally used principal components analysis pca and stepwise regression to form a transfer function however pca has several procedural choices that may result in very different reconstructions this study assesses the different procedures in pca based regression and suggests alternative procedures for selection of variables and principal components cross validation statistics are presented as an alternative for independently testing and identifying the optimal model the objective is to use these statistics as a measure of the model s performance to find a conceptually acceptable model with a low prediction error and the fewest number of variables the results show that a parsimonious model with a low mean square error can be obtained by using strict rules for principal component selection and cross validation statistics additionally the procedure suggested in this study results in a model that is physically consistent with the relationship between the predictand and the predictor the alternative pca based regression models presented here are applied to the reconstruction of the upper colorado river basin streamflow and compared with results of a previous reconstruction using traditional procedures the streamflow reconstruction proposed in this study shows more intense drought periods which may influence the future allocation of water supply in the colorado river basin

some additional results on principal components analysis of three mode data by means of alternating least squares algorithms kroonenberg and de leeuw 1980 have developed an alternating least squares method tuckals 3 as a solution for tucker s three way principal components model the present paper offers some additional features of their method starting from a reanalysis of tucker s problem in terms of a rank constrained regression problem it is shown that the fitted sum of squares in tuckals 3 can be partitioned according to elements of each mode of the three way data matrix an upper bound to the total fitted sum of squares is derived finally a special case of tuckals 3 is related to the carroll harshman candecomp parafac model

more comments on c p we discuss the interpretation of c p plots and show how they can be calibrated in several ways we comment on the practice of using the display as a basis for formal selection of a subset regression model and extend the range of application of the device to encompass arbitrary linear estimates of the regression coefficients for example ridge estimates