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In this work we develop a new algorithm for rating of teams (or players) in one-on-one games by exploiting the observed difference of the game-points (such as goals), also known as a margin of victory (MOV). Our objective is to obtain the Elo-style algorithm whose operation is simple to implement and to understand intuitively. This is done in three steps: first, we define the probabilistic model between the teams' skills and the discretized MOV variable: this generalizes the model underpinning the Elo algorithm, where the MOV variable is discretized into three categories (win/loss/draw). Second, with the formal probabilistic model at hand, the optimization required by the maximum likelihood rule is implemented via stochastic gradient; this yields simple on-line equations for the rating updates which are identical in their general form to those characteristic of the Elo algorithm: the main difference lies in the way the scores and the expected scores are defined. Third, we propose a simple method to estimate the coefficients of the model, and thus define the operation of the algorithm; it is done in a closed form using the historical data so the algorithm is tailored to the sport of interest and the coefficients defining its operation are determined in entirely transparent manner. The alternative, optimization-based strategy to find the coefficients is also presented. We show numerical examples based on the results of the association football of the English Premier League and the American football of the National Football League.

In this paper, we introduce reduced-bias estimators for the estimation of the tail index of a Pareto-type distribution. This is achieved through the use of a regularised weighted least squares with an exponential regression model for log-spacings of top order statistics. The asymptotic properties of the proposed estimators are investigated analytically and found to be asymptotically unbiased, consistent and normally distributed. Also, the finite sample behaviour of the estimators are studied through a simulations theory. The proposed estimators were found to yield low bias and MSE. In addition, the proposed estimators are illustrated through the estimation of the tail index of the underlying distribution of claims from the insurance industry.

The present paper continues our investigation of an implementation of a least-squares collocation method for higher-index differential-algebraic equations. In earlier papers, we were able to substantiate the choice of basis functions and collocation points for a robust implementation as well as algorithms for the solution of the discrete system. The present paper is devoted to an analytic estimation of condition numbers for different components of an implementation. We present error estimations, which show the sources for the different errors.

Missing data is frequently encountered in practice. Propensity score estimation is a popular tool for handling such missingness. The propensity score is often developed using a model for the response probability, which can be subject to model misspecification. In this paper, we consider an alternative approach of estimating the inverse of the propensity scores using the density ratio function. The smoothed density ratio function is obtained by the solution to the information projection onto the space satisfying the moment conditions on the balancing scores. By including the covariates for the outcome regression models only into the density ratio model, we can achieve efficient propensity score estimation. Penalized regression is used to identify important covariates. We further extend the proposed approach to the multivariate missing case. Some limited simulation studies are presented to compare with the existing methods.

We say that a continuous real-valued function $x$ admits the Hurst roughness exponent $H$ if the $p^{\text{th}}$ variation of $x$ converges to zero if $p>1/H$ and to infinity if $p<1/H$. For the sample paths of many stochastic processes, such as fractional Brownian motion, the Hurst roughness exponent exists and equals the standard Hurst parameter. In our main result, we provide a mild condition on the Faber--Schauder coefficients of $x$ under which the Hurst roughness exponent exists and is given as the limit of the classical Gladyshev estimates $\widehat H_n(x)$. This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because no assumption whatsoever is made on the possible dynamics of the function $x$. Nonetheless, our proof is probabilistic and relies on a martingale that is hidden in the Faber--Schauder expansion of $x$. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators that are derived from the sequence $(\widehat H_n)_{n\in\mathbb N}$. We also discuss how a dynamic change in the Hurst roughness parameter of a time series can be detected. Finally, we extend our results to the case in which the $p^{\text{th}}$ variation of $x$ is defined over a sequence of unequally spaced partitions. Our results are illustrated by means of high-frequency financial time series.

We develop a new Bayesian modelling framework for the class of higher-order, variable-memory Markov chains, and introduce an associated collection of methodological tools for exact inference with discrete time series. We show that a version of the context tree weighting algorithm can compute the prior predictive likelihood exactly (averaged over both models and parameters), and two related algorithms are introduced, which identify the a posteriori most likely models and compute their exact posterior probabilities. All three algorithms are deterministic and have linear-time complexity. A family of variable-dimension Markov chain Monte Carlo samplers is also provided, facilitating further exploration of the posterior. The performance of the proposed methods in model selection, Markov order estimation and prediction is illustrated through simulation experiments and real-world applications with data from finance, genetics, neuroscience, and animal communication. The associated algorithms are implemented in the R package BCT.

Models defined by moment conditions are at the center of structural econometric estimation, but economic theory is mostly agnostic about moment selection. While a large pool of valid moments can potentially improve estimation efficiency, in the meantime a few invalid ones may undermine consistency. This paper investigates the empirical likelihood estimation of these moment-defined models in high-dimensional settings. We propose a penalized empirical likelihood (PEL) estimation and establish its oracle property with consistent detection of invalid moments. The PEL estimator is asymptotically normally distributed, and a projected PEL procedure further eliminates its asymptotic bias and provides more accurate normal approximation to the finite sample behavior. Simulation exercises demonstrate excellent numerical performance of these methods in estimation and inference.

The optimal receiver operating characteristic (ROC) curve, giving the maximum probability of detection as a function of the probability of false alarm, is a key information-theoretic indicator of the difficulty of a binary hypothesis testing problem (BHT). It is well known that the optimal ROC curve for a given BHT, corresponding to the likelihood ratio test, is theoretically determined by the probability distribution of the observed data under each of the two hypotheses. In some cases, these two distributions may be unknown or computationally intractable, but independent samples of the likelihood ratio can be observed. This raises the problem of estimating the optimal ROC for a BHT from such samples. The maximum likelihood estimator of the optimal ROC curve is derived, and it is shown to converge to the true optimal ROC curve in the \levy\ metric, as the number of observations tends to infinity. A classical empirical estimator, based on estimating the two types of error probabilities from two separate sets of samples, is also considered. The maximum likelihood estimator is observed in simulation experiments to be considerably more accurate than the empirical estimator, especially when the number of samples obtained under one of the two hypotheses is small. The area under the maximum likelihood estimator is derived; it is a consistent estimator of the true area under the optimal ROC curve.

Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.

We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.