continuous phase modulation part i full response signaling the continuous phase modulation cpm signaling scheme has gained interest in recent years because of its attractive spectral properties data symbol pulse shaping has previously been studied with regard to spectra for binary data and modulation index 0 5 in this paper these results have been extended to the m ary case where the pulse shaping is over a one symbol interval the so called full response systems results are given for modulation indexes of practical interest concerning both performance and spectrum comparisons are made with minimum shift keying msk and systems have been found which are significantly better in e b n 0 for a large signal to noise ratio snr without expanded bandwidth schemes with the same bit error probability as msk but with considerably smaller bandwidth have also been found significant improvement in both power and bandwidth are obtained by increasing the number of levels m from 2 to 4

a decomposition theorem for polymeasures we prove that every countably additive polymeasure can be decomposed in a unique way as the sum of a discrete polymeasure plus a continuous polymeasure this generalizes a previous result of saeki

lattice preserving maps on lattices of continuous functions abstract lattice preserving maps between sublattices of continuous functions on locally compact spaces e g the functions with compact support are studied these maps though not necessarily linear are related to continuous functions between the underlying locally compact spaces consequences for first countable spaces including an automatic continuity for these lattice maps are provided

characterization of dual mixed volumes via polymeasures we prove a characterization of the dual mixed volume in terms of functional properties of the polynomial associated with it to do this we use tools from the theory of multilinear operators on spaces of continuous functions along the way we reprove with these same techniques a recently found characterization of the dual mixed volume

a general framework for updating belief distributions we propose a framework for general bayesian inference we argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function which is recovered as a special case modern application areas make it increasingly challenging for bayesians to attempt to model the true data generating mechanism for instance when the object of interest is low dimensional such as a mean or median it is cumbersome to have to achieve this via a complete model for the whole data distribution more importantly there are settings where the parameter of interest does not directly index a family of density functions and thus the bayesian approach to learning about such parameters is currently regarded as problematic our framework uses loss functions to connect information in the data to functionals of interest the updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions importantly the procedure coincides with bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings connections to other inference frameworks are highlighted

statistical methods for eliciting probability distributions elicitation is a key task for subjectivist bayesians although skeptics hold that elicitation cannot or perhaps should not be done in practice it brings statisticians closer to their clients and subject matter expert colleagues this article reviews the state of the art reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information in a discussion of the elicitation process the first issue to address is what it means for an elicitation to be successful that is what criteria should be used our answer is that a successful elicitation faithfully represents the opinion of the person being elicited it is not necessarily true in some objectivistic sense and cannot be judged in that way we see that elicitation as simply part of the process of statistical modeling indeed in a hierarchical model at which point the likelihood ends and the pr

goodness of fit and related inference processes for quantile regression abstract we introduce a goodness of fit process for quantile regression analogous to the conventional r2 statistic of least squares regression several related inference processes designed to test composite hypotheses about the combined effect of several covariates over an entire range of conditional quantile functions are also formulated the asymptotic behavior of the inference processes is shown to be closely related to earlier p sample goodness of fit theory involving bessel processes the approach is illustrated with some hypothetical examples an application to recent empirical models of international economic growth and some monte carlo evidence

a factorization for orthogonally additive operators on banach lattices abstract factorizations for certain orthogonally additive monotone operators on a banach lattice e are provided given such an operator t it shown that t can be expressed as l composed with s where l is linear and s though not necessarily linear is related to an orthomorphism in various settings this factorization is valid on the cone of a e or of a sublattice of e the nature of the generalized orthomorphisms are related to operators that are pointwise or locally determined when e is viewed as a space of extended real valued functions

orthogonally additive polynomials on banach function algebras abstract for a banach function algebra a we consider the problem of representing a continuous d homogeneous polynomial p a x where x is an arbitrary banach space that satisfies the property p f g p f p g whenever f g a are such that supp f supp g we show that such a polynomial can be represented as p f t f d f a for some continuous linear map t a x for a variety of banach function algebras such as the algebra of continuous functions c 0 for any locally compact hausdorff space the algebra of lipschitz functions lip k for any compact metric space k and 0 1 the figa talamanca herz algebra a p g for some locally compact groups g and p 1 the algebras a c a b and b v c a b of absolutely continuous functions and of continuous functions of bounded variation on the interval a b in the case where a c n a b p can be represented as p f t n 1 n d f n 1 f n d where the sum is taken over n 1 n d z d with 0 n 1 n d n for appropriate continuous linear maps t n 1 n d c n n d a b x

on extensions of some nonlinear maps in vector lattices abstract we show that an urysohn lattice pre homomorphism defined on a normal sublattice d of a vector lattice e can be extended to the whole space e and the extended operator is an urysohn lattice homomorphism we introduce a new class of nonlinear operators which called operators and describe some of their properties finally we investigate a structure of positive orthogonally additive operators dominated by an urysohn lattice homomorphism defined on a vector lattice e and taking value in dedekind complete vector lattice f

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

polynomials on banach lattices and positive tensor products abstract this paper is the first systematic study of homogeneous polynomials on banach lattices a variety of new banach spaces and banach lattices of multilinear maps homogeneous polynomials and operators are introduced the main technique is to employ positive tensor products and quotients of positive tensor products our theorems generalize the results on orthogonally additive polynomials by benyamini lassalle and llavona 2006 in 4 the results by grecu and ryan 2005 in 14 and the results by sundaresan 1991 in 23

a general method for comparing probability assessors a probability assessor or forecaster is a person who assigns subjective probabilities to events which will eventually occur or not occur in this paper a general method for comparing forecasters after a finite number of trials is introduced the general method is proven to include calculating all proper scoring rules as special cases it also includes comparison of forecasters in all simple two decision problems as special cases the relationship between the general method and calibration is also explored the general method is also translated into a method for deciding who will give better forecasts in the future

nonlinear continuous phase frequency shift keying a novel signal space coded cpm continuous phase modulated signaling technique called nonlinear cpfsk continuous phase frequency shift keying signaling which guarantees the maximum constraint length allowed by the number of states is introduced nonlinear cpfsk signals use a suitably selected state transition matrix during all signaling intervals the selected state transition matrix is realized by changing the modulation index of the signals hence the modulation index used during any interval depends on the transmitted symbol as well as on the state of the system binary nonlinear cpfsk signaling schemes are constructed and analyzed numerical results indicate that nonlinear cpfsk signals can achieve attractive minimum distances compared with other existing cpm signaling formats

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

asymptotic equivalence of bayes cross validation and widely applicable information criterion in singular learning theory in regular statistical models the leave one out cross validation is asymptotically equivalent to the akaike information criterion however since many learning machines are singular statistical models the asymptotic behavior of the cross validation remains unknown in previous studies we established the singular learning theory and proposed a widely applicable information criterion the expectation value of which is asymptotically equal to the average bayes generalization loss in the present paper we theoretically compare the bayes cross validation loss and the widely applicable information criterion and prove two theorems first the bayes cross validation loss is asymptotically equivalent to the widely applicable information criterion as a random variable therefore model selection and hyperparameter optimization using these two values are asymptotically equivalent second the sum of the bayes generalization error and the bayes cross validation error is asymptotically equal to 2 n where is the real log canonical threshold and n is the number of training samples therefore the relation between the cross validation error and the generalization error is determined by the algebraic geometrical structure of a learning machine we also clarify that the deviance information criteria are different from the bayes cross validation and the widely applicable information criterion

weather forecasting for weather derivatives we take a simple time series approach to modeling and forecasting daily average temperature in u s cities and we inquire systematically as to whether it may prove useful from the vantage point of participants in the weather derivatives market the answer is perhaps surprisingly yes time series modeling reveals conditional mean dynamics and crucially strong conditional variance dynamics in daily average temperature and it reveals sharp differences between the distribution of temperature and the distribution of temperature surprises as we argue it also holds promise for producing the long horizon predictive densities crucial for pricing weather derivatives so that additional inquiry into time series weather forecasting methods will likely prove useful in weather derivatives contexts

bayesian model comparison with the hyvarinen score computation and consistency the bayes factor is a widely used criterion in model comparison and its logarithm is a difference of out of sample predictive scores under the logarithmic scoring rule however when some of the ca

powers of quasi banach lattices and orthogonally additive polynomials abstract the paper aims to examine convexity and concavity conditions for homogeneous orthogonally additive polynomials in quasi banach lattices we characterize p q convex quasi banach lattice by the property that every positive orthogonally additive polynomial defined on such lattice is p q convex as well as obtain some versions of krivine factorization result and grothendieck s inequality for orthogonally additive polynomials in quasi banach lattices

assigning a value to a power likelihood in a general bayesian model summarybayesian robustness under model misspecification is a current area of active research among recent ideas is that of raising the likelihood function to a power in this paper we discuss the choice of appropriate power and provide examples

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

polynomials with an integral representation abstract we introduce a large class of polynomials between banach spaces which admit an integral representation this class coincides with the class of integral polynomials in the scalar valued case but it is larger in the vector valued case by comparing this class with other well known classes we obtain characterizations of banach spaces containing no copy of c 0 of grothendieck spaces and of l spaces among some other results we show that a polynomial has an integral representation if and only if it admits an orthogonally additive extension to c b e moreover if the range space is complemented in its bidual every polynomial with an integral representation is extendible to every superspace

on a notion of data depth based on random simplices for a distribution f on mathbb r p and a point x in mathbb r p the simplical depth d x is introduced which is the probability that the point x is contained inside a random simplex whose vertices are p 1 independent observations from f mathematically and heuristically it is argued that d x indeed can be viewed as a measure of depth of the point x with respect to f an empirical version of d cdot gives rise to a natural ordering of the data points from the center outward the ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and l statistics this generalized sample median and l statistics are affine equivariant

strict definiteness of integrals via complete monotonicity of derivatives let k be a nonnegative integer and let 0 r be a c function with k k completely monotone and not constant if 6 0 is a signed measure on any euclidean space rd with vanishing moments up to order k 1 then the integral rd rd x y 2 d x d y is strictly positive whenever it exists for general d no larger class of continuous functions seems to admit the same conclusion examples and applications are indicated a section on bilinear integrability might be of independent interest

a predictive approach to the bayesian design problem with application to normal regression models a predictive decision theoretic approach is developed for the bayesian design problem the loss functions used are fair bayes or proper scoring rules and are quadratic measures of distance between probability measures optimal bayesian designs are those which minimise the preposterior risk for the decision problem such designs typically depend on both the prior distribution and the loss function the results are applied to certain normal regression models where explicit optimal designs are constructed

game theory maximum entropy minimum discrepancy and robust bayesian decision theory we describe and develop a close relationship between two problems that have customarily been regarded as distinct that of maximizing entropy and that of minimizing worst case expected loss using a formulation grounded in the equilibrium theory of zero sum games between decision maker and nature these two problems are shown to be dual to each other the solution to each providing that to the other although topsoe described this connection for the shannon entropy over 20 years ago it does not appear to be widely known even in that important special case we here generalize this theory to apply to arbitrary decision problems and loss functions we indicate how an appropriate generalized definition of entropy can be associated with such a problem and we show that subject to certain regularity conditions the above mentioned duality continues to apply in this extended context this simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust bayes acts we also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions this leads to an extension to arbitrary discrepancies of a well known minimax theorem for the case of kullback leibler divergence the redundancy capacity theorem of information theory for the important case of families of distributions having certain mean values specified we develop simple sufficient conditions and methods for identifying the desired solutions we use this theory to introduce a new concept of generalized exponential family linked to the specific decision problem under consideration and we demonstrate that this shares many of the properties of standard exponential families finally we show that the existence of an equilibrium in our game can be rephrased in terms of a pythagorean property of the related divergence thus generalizing previously announced results for kullback leibler and bregman divergences

evaluating probabilities asymmetric scoring rules proper scoring rules are over evaluation measures that reward accurate probabilities specific rules encountered in the literature and used in practice are invariably symmetric in the sense that the expected score for a perfectly calibrated probability assessor or model generating probabilities is minimized at a probability of one half a family of asymmetric scoring rules that provide better measures of the degree of skill inherent in the probabilities and render scores that are more comparable in different situations is developed here one member of this family a quadratic asymmetric rule is applied to evaluate an extensive set of precipitation probability forecasts from the u s national weather service connections to previous characterizations of proper scoring rules are investigated and some relevant issues pertaining to the design of specific asymmetric rules for particular inferential and decision making problems are discussed briefly

effective scoring rules for probabilistic forecasts this paper studies the use of a scoring rule for the elicitation of forecasts in the form of probability distributions and for the subsequent evaluation of such forecasts given a metric distance function on a space of probability distributions a scoring rule is said to be effective if the forecaster s expected score is a strictly decreasing function of the distance between the elicited and true distributions two simple well known rules the spherical and the quadratic are shown to be effective with respect to suitable metrics examples and a practical application in foreign exchange rate forecasting are also provided

model choice a minimum posterior predictive loss approach summary model choice is a fundamental and much discussed activity in the analysis of datasets nonnested hierarchical models introducing random effects may not be handled by classical methods bayesian approaches using predictive distributions can be used though the formal solution which includes bayes factors as a special case can be criticised we propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values we obtain this criterion by minimising posterior loss for a given model and then for models under consideration selecting the one which minimises this criterion for a broad range of losses the criterion emerges as a form partitioned into a goodness of fit term and a penalty term we illustrate its performance with an application to a large dataset involving residential property transactions

should scoring rules be effective a scoring rule is a reward function for eliciting or evaluating forecasts expressed as discrete or continuous probability distributions a rule is strictly proper if it encourages the forecaster to state his true subjective probabilities and effective if it is associated with a metric on the set of probability distributions recently the property of effectiveness which is stronger than strict properness has been proposed as a desideratum for scoring rules for continuous forecasts for reasons of monotonicity in keeping the forecaster close to his true probabilities since in practice the forecast must be chosen from a low dimensional set of admissible distributions it is shown in this paper that what effectiveness implies beyond strict properness is not a monotonicity property but a transitivity property which is difficult to justify behaviorally the logarithmic scoring rule is shown to violate the transitivity property and hence is not effective the l1 and l metrics are shown to allow no effective scoring rules some potential difficulties in interpreting admissible forecasts are also discussed

discrimination slope and integrated discrimination improvement properties relationships and impact of calibration discrimination slope defined as the slope of a linear regression of predicted probabilities of event derived from a prognostic model on the binary event status has recently gained popularity as a measure of model performance it is as a building block for the integrated discrimination improvement that equals the difference in discrimination slopes between the two models being compared several authors have pointed out that it does not make sense to apply the integrated discrimination improvement and discrimination slope when working with mis calibrated models whereas others have raised concerns about the ability of improving discrimination slope without adding new information in this paper we show that under certain assumptions the discrimination slope is asymptotically related to two other r squared measures one of which is a rescaled version of the brier score known to be proper furthermore we illustrate how a simple recalibration makes the slope equal to the rescaled brier r squared metric we also show that the discrimination slope can be interpreted as a measure of reduction in expected regret for the gini brier regret function using theoretical and practical examples we illustrate how all of these metrics are affected by different levels of model mis calibration in particular we demonstrate that simple recalibration ascertaining calibration in the large and calibration slope equal to 1 are not sufficient to correct for some forms of mis calibration we conclude that r squared metrics including the discrimination slope offer an attractive choice for quantifying model performance as long as one accounts for their sensitivity to model calibration copyright 2016 john wiley sons ltd

orthogonally additive polynomials on convolution algebras associated with a compact group abstract let g be a compact group let x be a banach space and let p l 1 g x be an orthogonally additive continuous n homogeneous polynomial then we show that there exists a unique continuous linear map l 1 g x such that p f f n f for each f l 1 g we also seek analogues of this result about l 1 g for various other convolution algebras including l p g for 1 p and c g

generalized nonlinear continuous phase modulation a class of generalized nonlinear full response cpm signals is constructed by extending the known ordinary nonlinear cpm signals numerical results presented at different numbers of states indicate that generalized nonlinear cpm signals achieve higher minimum euclidean distances than ordinary nonlinear cpm signals

orthogonally additive polynomials on spaces of continuous functions we show that for every orthogonally additive homogeneous polynomial p on a space of continuous functions c k with values in a banach space y there exists a linear operator s c k y such that p f s f n this is the c k version of a related result of sundaresam for polynomials on l p spaces

integral representation of product factorable bilinear operators and summability of bilinear maps on c k spaces abstract we present a constructive technique to represent classes of bilinear operators that allow a factorization through a bilinear product providing a general version of the well known characterization of integral bilinear forms as elements of the dual of an injective tensor product we show that this general method fits with several known situations coming from different contexts harmonic analysis c algebras c k spaces operator theory polynomials providing a unified approach to the integral representation of a broad class of bilinear operators some examples and applications are also shown regarding for example operator spaces and summability properties of bilinear maps

strictly proper scoring rules prediction and estimation scoring rules assess the quality of probabilistic forecasts by assigning a numerical score based on the predictive distribution and on the event or value that materializes a scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionf if he or she issues the probabilistic forecast f rather than g f it is strictly proper if the maximum is unique in prediction problems proper scoring rules encourage the forecaster to make careful assessments and to be honest in estimation problems strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand this article reviews and develops the theory of proper scoring rules on general probability spaces and proposes and discusses examples thereof proper scoring rules derive from convex functions and relate to information measures entropy functions and bregman divergences in the case of categorical variables we prove a rigorous version of the

coherent dispersion criteria for optimal experimental design we characterize those coherent design criteria which depend only on the dispersion matrix assumed proper and nonsingular of the state of nature which may be a parameter vector or a set of future observables and describe the associated decision problems connections are established with the classical approach to optimal design theory for the normal linear model based on concave functions of the information matrix implications of the theory for more general models are also considered

nonlinear continuous phase modulation a class of nonlinear cpm signals is introduced by extending nonlinear cpfsk signals to other forms of frequency pulses the distance results and spectral properties of nonlinear cpm signals are reported bit error probabilities of nonlinear cpm signals are simulated in presence of adjacent channel interference

orthogonally additive polynomials on fourier algebras abstract we show that an n homogeneous polynomial p on the fourier algebra a g of a locally compact group g can be represented in the form p f t f n f a g for some t in the group von neumann algebra vn g of g if and only if it is orthogonally additive and completely bounded

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

linear model selection by cross validation abstract we consider the problem of selecting a model having the best predictive ability among a class of linear models the popular leave one out cross validation method which is asymptotically equivalent to many other model selection methods such as the akaike information criterion aic the c p and the bootstrap is asymptotically inconsistent in the sense that the probability of selecting the model with the best predictive ability does not converge to 1 as the total number of observations n we show that the inconsistency of the leave one out cross validation can be rectified by using a leave n v out cross validation with n v the number of observations reserved for validation satisfying n v n 1 as n this is a somewhat shocking discovery because nv n 1 is totally opposite to the popular leave one out recipe in cross validation motivations justifications and discussions of some practical aspects of the use of the leave n v out cross validation method are provided and results