Distribution-free Setting Research Articles

Orthogonal Inductive Matrix Completion.

We propose orthogonal inductive matrix completion (OMIC), an interpretable approach to matrix completion based on a sum of multiple orthonormal side information terms, together with nuclear-norm regularization. The approach allows us to inject prior knowledge about the singular vectors of the ground-truth matrix. We optimize the approach by a provably converging algorithm, which optimizes all components of the model simultaneously. We study the generalization capabilities of our method in both the distribution-free setting and in the case where the sampling distribution admits uniform marginals, yielding learning guarantees that improve with the quality of the injected knowledge in both cases. As particular cases of our framework, we present models that can incorporate user and item biases or community information in a joint and additive fashion. We analyze the performance of OMIC on several synthetic and real datasets. On synthetic datasets with a sliding scale of user bias relevance, we show that OMIC better adapts to different regimes than other methods. On real-life datasets containing user/items recommendations and relevant side information, we find that OMIC surpasses the state of the art, with the added benefit of greater interpretability.

Distribution-free robust linear regression

We study random design linear regression with no assumptions on the distribution of the covariates and with a heavy-tailed response variable. In this distribution-free regression setting, we show that boundedness of the conditional second moment of the response given the covariates is a necessary and sufficient condition for achieving non-trivial guarantees. As a starting point, we prove an optimal version of the classical in-expectation bound for the truncated least squares estimator due to Györfi, Kohler, Krzyżak, and Walk. However, we show that this procedure fails with constant probability for some distributions despite its optimal in-expectation performance. Then, combining the ideas of truncated least squares, median-ofmeans procedures, and aggregation theory, we construct a non-linear estimator achieving excess risk of order d/n with the optimal sub-exponential tail. While existing approaches to linear regression for heavy-tailed distributions focus on proper estimators that return linear functions, we highlight that the improperness of our procedure is necessary for attaining non-trivial guarantees in the distribution-free setting.

The limits of distribution-free conditional predictive inference

Abstract We consider the problem of distribution-free predictive inference, with the goal of producing predictive coverage guarantees that hold conditionally rather than marginally. Existing methods such as conformal prediction offer marginal coverage guarantees, where predictive coverage holds on average over all possible test points, but this is not sufficient for many practical applications where we would like to know that our predictions are valid for a given individual, not merely on average over a population. On the other hand, exact conditional inference guarantees are known to be impossible without imposing assumptions on the underlying distribution. In this work, we aim to explore the space in between these two and examine what types of relaxations of the conditional coverage property would alleviate some of the practical concerns with marginal coverage guarantees while still being possible to achieve in a distribution-free setting.

Chain Ladder Method: Bayesian Bootstrap Versus Classical Bootstrap

The intention of this paper is to analyse the mean square error of prediction (MSEP) under the distribution-free chain ladder (DFCL) claims reserving method. We compare the estimation obtained from the classical bootstrap method with the one obtained from a Bayesian bootstrap. To achieve this in the DFCL model we develop a novel approximate Bayesian computation (ABC) sampling algorithm to obtain the empirical posterior distribution. We need an ABC sampling algorithm because we work in a distribution-free setting. The use of this ABC methodology combined with bootstrap allows us to obtain samples from the intractable posterior distribution without the requirement of any distributional assumptions. This then enables us to calculate the MSEP and other risk measures like Value-at-Risk.

Risk management strategies for finding universal portfolios

We consider Cover’s universal portfolio and the problem of risk management in a distribution-free setting when learning from experts. We aim to find optimal portfolios without modelling the financial market at the outset. Although it exists, the price distribution of the constituent assets is neither known nor given as part of the input. We consider the portfolio selection problem from the perspective of online algorithms that process input piece-by-piece in a serial fashion. Under the minimax regret criterion, we propose two risk-adjusted algorithms that track the expert with the lowest maximum drawdown. We obtain upper bounds on the worst-case performance of our algorithms that equal the bounds obtained by Cover (Math Finance 1(1):1–29, 1991). We also present computational evidence using NYSE data over a 22-year period, which shows superior performance of investment strategies that take risk management into account.

Optimal unbiased estimation of some population central moments

In this paper, we have treated the problem of estimating some population central moments under distribution-free setting. Uniformly minimum variance unbiased estimators for some population central moments have been derived. Some examples of unbiased estimators of central moments have been given under various sampling designs such as simple random sampling with replacement (srsr) or without replacement (srs), probability proportional to size with replacement (ppsr) and probability graduated variable proportional to size without replacement (pgvps). An optimal unbiased estimator of the third population central moment is proposed and extended to some real situations. Some optimal unbiased estimators of the fourth population central moment are given. Several optimal unbiased estimators of the variance of the “sample quasivariance estimator” are identified. Finally, computer programs in R implementing all of the estimators are given.

Adaptive Generalized Likelihood Ratio Control Charts for Detecting Unknown Patterned Mean Shifts

Traditional statistical control schemes have been mainly focused on detecting constant mean shifts. In many practical applications, however, the mean of the observed sequence can exhibit a time-varying behavior after the fault occurrence. Shewhart control charts supplemented with sensitizing run rules have been suggested for detecting nonrandom dynamic mean patterns. However, the choice of a particular set of run rules requires some prior knowledge of the possible expected patterns. In addition, the resulting schemes can have poor performance against small shifts. When the mean shift pattern is known in advance, generalized likelihood ratio (GLR), cumulative score (CUSCORE), and optimal general linear filter (OGLF) control charts have also been proposed for change-point detection. Further, some adaptive CUSCORE schemes have been recently developed for detecting an unknown patterned mean shift. However, these adaptive CUSCOREs assume the occurrence of one-sided mean shifts. Hence, their performance may be poor in the case of an oscillatory behavior of the process mean. To overcome these limitations, we propose estimating a possible pattern in the mean using an exponentially weighted moving average (EWMA), which is especially effective with one-sided shifts in the mean, combined with a wavelet smoother, which is effective in estimating oscillatory mean patterns. The two estimates then drive two separate conventional GLR tests for significance. A control chart is proposed for observations from normal distributions and then extended to a completely distribution-free setting. Extensive simulation results demonstrate the efficiency of the proposed charts with a wide variety of patterned mean shifts in both the independent and autocorrelated scenarios and with a wide variety of distributions. In addition, the proposed scheme provides post-signal diagnostic information that estimates the change-point location and the shift pattern of the mean. An R package is available online as supplementary material.

Chain ladder method: Bayesian bootstrap versus classical bootstrap

The intention of this paper is to estimate a Bayesian distribution-free chain ladder (DFCL) model using approximate Bayesian computation (ABC) methodology. We demonstrate how to estimate quantities of interest in claims reserving and compare the estimates to those obtained from classical and credibility approaches. In this context, a novel numerical procedure utilizing a Markov chain Monte Carlo (MCMC) technique, ABC and a Bayesian bootstrap procedure was developed in a truly distribution-free setting. The ABC methodology arises because we work in a distribution-free setting in which we make no parametric assumptions, meaning we cannot evaluate the likelihood point-wise or in this case simulate directly from the likelihood model. The use of a bootstrap procedure allows us to generate samples from the intractable likelihood without the requirement of distributional assumptions; this is crucial to the ABC framework. The developed methodology is used to obtain the empirical distribution of the DFCL model parameters and the predictive distribution of the outstanding loss liabilities conditional on the observed claims. We then estimate predictive Bayesian capital estimates, the value at risk (VaR) and the mean square error of prediction (MSEP). The latter is compared with the classical bootstrap and credibility methods.

Abstract: P345 FEELING DISCONTENT PREDICTS CARDIOVASCULAR EVENTS IN 58-YEAR OLD HEALTHY MEN

The intention of this paper is to estimate a Bayesian distribution-free chain ladder (DFCL) model using approximate Bayesian computation (ABC) methodology. We demonstrate how to estimate quantities of interest in claims reserving and compare the estimates to those obtained from classical and credibility approaches. In this context, a novel numerical procedure utilizing a Markov chain Monte Carlo (MCMC) technique, ABC and a Bayesian bootstrap procedure was developed in a truly distribution-free setting. The ABC methodology arises because we work in a distribution-free setting in which we make no parametric assumptions, meaning we cannot evaluate the likelihood point-wise or in this case simulate directly from the likelihood model. The use of a bootstrap procedure allows us to generate samples from the intractable likelihood without the requirement of distributional assumptions; this is crucial to the ABC framework. The developed methodology is used to obtain the empirical distribution of the DFCL model parameters and the predictive distribution of the outstanding loss liabilities conditional on the observed claims. We then estimate predictive Bayesian capital estimates, the value at risk (VaR) and the mean square error of prediction (MSEP). The latter is compared with the classical bootstrap and credibility methods.

Optimal estimation of population variance using equilibrated stratified sampling from infinite populations

A class of unbiased estimators of the population variance is proposed when the coefficient of variation is known for each stratum. The technique of equilibrated stratification is used. It is shown that, with optimum allocation, the variance of the suggested estimator for sufficiently large sample size is less than that of the classical minimum variance unbiased estimator under distribution-free setting. Other classes of estimators are also given with similar or better practical properties.

Proper Learning Algorithm for Functions of k Terms under Smooth Distributions

In this paper, we introduce a probabilistic distribution, called a smooth distribution, which is a generalization of variants of the uniform distribution such as q -bounded distribution and product distribution. Then, we give an algorithm that, under the smooth distribution, properly learns the class of functions of k terms given as F k ∘ T k n ={ g ( f 1 ( v ), …, f k ( v )) | g ∈ F k , f 1 , …, f k ∈ T n } in polynomial time for constant k , where F k is the class of all Boolean functions of k variables and T n is the class of terms over n variables. Although class F k ∘ T k n was shown by Blum and Singh to be learned using DNF as the hypothesis class, it has remained open whether it is properly learnable under a distribution-free setting.

Characterizing Rational versus Exponential Learning Curves

We consider the standard problem of learning a concept from random examples. Here alearning curveis defined to be the expected error of a learner's hypotheses as a function of training sample size. Haussler, Littlestone, and Warmuth have shown that, in the distribution-free setting, the smallest expected error a learner can achieve in the worst case over a class of conceptsCconverges rationally to zero error; i.e.,Θ(t−1) in the training sample sizet. However, Cohn and Tesauro have recently demonstrated thatexponentialconvergence can often be observed in experimental settings (i.e., average error decreasing aseΘ−t)). By addressing a simple non-uniformity in the original analysis this paper shows how the dichotomy between rational and exponential worst case learning curves can be recovered in the distribution-free theory. In particular, our results support the experimental findings of Cohn and Tesauro: for finite concept classes any consistent learner achieves exponential convergence, even in the worst case, whereas for continuous concept classes no learner can exhibit sub-rational convergence for every target concept and domain distribution. We also draw a precise boundary between rational and exponential convergence for simple concept chains—showing that somewhere-dense chains always force rational convergence in the worst case, while exponential convergence can always be achieved for nowhere-dense chains.

A symptotic robustness of sequential confidence intervals and the connection with influence functions

The robustness of sequential confidence intervals is studied by considering contamination with probability ε of the basic underlying distribution in a so-called gross errors model. Asymptotic theory is considered when d → 0, where the prescribed length of the interval is 2d, and simultaneously ε ≏ ε(d) → 0. A general theorem, in a distribution free setting, is given which provides expressions for the asymptotic coverage probability and the asymptotic distribution of the stopping variable. The results depend on the rate of ε(d)/d as d → 0 and on the contaminating distribution. If the latter distribution is degenerate, it turns out that the influence functions of the above mentioned two estimators used in the construction of the procedure, appear in the expressions for the asymptotic coverage probability and the asymptotic distribution of the stopping variable respectively. This shows how the sequential procedure inherits the robustness properties of the estimators concerned and how this is quantified. The general theorem is specialized to two procedures for the estimation of the mean of a symmetric distribution. Results of Monte Carlo studies indicate agreement between the asymptotic theory and the actual behavior of the procedures.

Equilibrated strategy for population variance estimation

Under the hypothesis that a statistical population can be represented by a density function in a similar way to that of the optimum stratification problem dealt with in Dalenius (1957) and Dalenius and Hodges (1959) the estimation of the population variance is analysed by means of the technique of equilibrated stratification. An unbiased estimator is suggested and it is shown that, with optimum allocation, the precision of the estimator for sufficiently large sample size is always at least that of the classical minimum variance unbiased estimator for a distribution-free setting. A numerical study complete the article.