Journal Description

Stats

Stats (ISSN 2571-905X) is an international peer-reviewed open access journal on statistical science. Stats is published quarterly online by MDPI. The journal focuses on methodological and theoretical papers in statistics, probability, stochastic processes and innovative applications of statistics in all scientific disciplines including biological and biomedical sciences, medicine, business, economics and social sciences, physics, data science and engineering.

Open Access—free for readers, with article processing charges (APC) paid by authors or their institutions.
Rapid Publication: manuscripts are peer-reviewed and a first decision provided to authors approximately 15.8 days after submission; acceptance to publication is undertaken in 3.6 days (median values for papers published in this journal in the first half of 2020).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.

Imprint Information Open Access

Latest Articles

Open AccessArticle

Local Processing of Massive Databases with R: A National Analysis of a Brazilian Social Programme

Marcos E. Barreto and

Anderson Ara

Stats 2020, 3(4), 444-464; https://doi.org/10.3390/stats3040028 - 19 Oct 2020

Abstract

The analysis of massive databases is a key issue for most applications today and the use of parallel computing techniques is one of the suitable approaches for that. Apache Spark is a widely employed tool within this context, aiming at processing large amounts of data in a distributed way. For the Statistics community, R is one of the preferred tools. Despite its growth in the last years, it still has limitations for processing large volumes of data in single local machines. In general, the data analysis community has difficulty to handle a massive amount of data on local machines, often requiring high-performance computing servers. One way to perform statistical analyzes over massive databases is combining both tools (Spark and R) via the sparklyr package, which allows for an R application to use Spark. This paper presents an analysis of Brazilian public data from the Bolsa Família Programme (BFP—conditional cash transfer), comprising a large data set with 1.26 billion observations. Our goal was to understand how this social program acts in different cities, as well as to identify potentially important variables reflecting its utilization rate. Statistical modeling was performed using random forest to predict the utilization rated of BFP. Variable selection was performed through a recent method based on the importance and interpretation of variables in the random forest model. Among the 89 variables initially considered, the final model presented a high predictive performance capacity with 17 selected variables, as well as indicated high importance of some variables for the observed utilization rate in income, education, job informality, and inactive youth, namely: family income, education, occupation and density of people in the homes. In this work, using a local machine, we highlighted the potential of aggregating Spark and R for analysis of a large database of 111.6 GB. This can serve as proof of concept or reference for other similar works within the Statistics community, as well as our case study can provide important evidence for further analysis of this important social support programme. Full article

(This article belongs to the Section Data Science)

► Show Figures

Figure 1

Open AccessArticle

Identification of Judicial Outcomes in Judgments: A Generalized Gini-PLS Approach

Gildas Tagny-Ngompé,

Stéphane Mussard,

Guillaume Zambrano,

Sébastien Harispe and

Jacky Montmain

Stats 2020, 3(4), 427-443; https://doi.org/10.3390/stats3040027 - 27 Sep 2020

Abstract

This paper presents and compares several text classification models that can be used to extract the outcome of a judgment from justice decisions, i.e., legal documents summarizing the different rulings made by a judge. Such models can be used to gather important statistics about cases, e.g., success rate based on specific characteristics of cases’ parties or jurisdiction, and are therefore important for the development of Judicial prediction not to mention the study of Law enforcement in general. We propose in particular the generalized Gini-PLS which better considers the information in the distribution tails while attenuating, as in the simple Gini-PLS, the influence exerted by outliers. Modeling the studied task as a supervised binary classification, we also introduce the LOGIT-Gini-PLS suited to the explanation of a binary target variable. In addition, various technical aspects regarding the evaluated text classification approaches which consists of combinations of representations of judgments and classification algorithms are studied using an annotated corpora of French justice decisions. Full article

(This article belongs to the Special Issue Interdisciplinary Research on Predictive Justice)

► Show Figures

Figure 1

Open AccessArticle

On the Use of the Cumulative Distribution Function for Large-Scale Tolerance Analyses Applied to Electric Machine Design

Edmund Marth and

Gerd Bramerdorfer

Stats 2020, 3(3), 412-426; https://doi.org/10.3390/stats3030026 - 22 Sep 2020

Abstract

In the field of electrical machine design, excellent performance for multiple objectives, like efficiency or torque density, can be reached by using contemporary optimization techniques. Unfortunately, highly optimized designs are prone to be rather sensitive regarding uncertainties in the design parameters. This paper introduces an approach to rate the sensitivity of designs with a large number of tolerance-affected parameters using cumulative distribution functions (CDFs) based on finite element analysis results. The accuracy of the CDFs is estimated using the Dvoretzky–Kiefer–Wolfowitz inequality, as well as the bootstrapping method. The advantage of the presented technique is that computational time can be kept low, even for complex problems. As a demanding test case, the effect of imperfect permanent magnets on the cogging torque of a Vernier machine with 192 tolerance-affected parameters is investigated. Results reveal that for this problem, a reliable statement about the robustness can already be made with 1000 finite element calculations. Full article

(This article belongs to the Special Issue Applied Statistics in Engineering)

► Show Figures

Figure 1

Open AccessArticle

Neural Legal Outcome Prediction with Partial Least Squares Compression

Charles Condevaux

Stats 2020, 3(3), 396-411; https://doi.org/10.3390/stats3030025 - 18 Sep 2020

Abstract

Predicting the outcome of a case from a set of factual data is a common goal in legal knowledge discovery. In practice, solving this task is most of the time difficult due to the scarcity of labeled datasets. Additionally, processing long documents often leads to sparse data, which adds another layer of complexity. This paper presents a study focused on the french decisions of the European Court of Human Rights (ECtHR) for which we build various classification tasks. These tasks consist first of all in the prediction of the potential violation of an article of the convention, using extracted facts. A multiclass problem is also created, with the objective of determining whether an article is relevant to plead given some circumstances. We solve these tasks by comparing simple linear models to an attention-based neural network. We also take advantage of a modified partial least squares algorithm that we integrate in the aforementioned models, capable of effectively dealing with classification problems and scale with sparse inputs coming from natural language tasks. Full article

(This article belongs to the Special Issue Interdisciplinary Research on Predictive Justice)

► Show Figures

Figure 1

Open AccessArticle

Bottleneck or Crossroad? Problems of Legal Sources Annotation and Some Theoretical Thoughts

Amedeo Santosuosso and

Giulia Pinotti

Stats 2020, 3(3), 376-395; https://doi.org/10.3390/stats3030024 - 09 Sep 2020

Abstract

So far, in the application of legal analytics to legal sources, the substantive legal knowledge employed by computational models has had to be extracted manually from legal sources. This is the bottleneck, described in the literature. The paper is an exploration of this obstacle, with a focus on quantitative legal prediction. The authors review the most important studies about quantitative legal prediction published in recent years and systematize the issue by dividing them in text-based approaches, metadata-based approaches, and mixed approaches to prediction. Then, they focus on the main theoretical issues, such as the relationship between legal prediction and certainty of law, isomorphism, the interaction between textual sources, information, representation, and models. The metaphor of a crossroad shows a descriptive utility both for the aspects inside the bottleneck and, surprisingly, for the wider scenario. In order to have an impact on the legal profession, the test bench for legal quantitative prediction is the analysis of case law from the lower courts. Finally, the authors outline a possible development in the Artificial Intelligence (henceforth AI) applied to ordinary judicial activity, in general and especially in Italy, stressing the opportunity the huge amount of data accumulated before lower courts in the online trials offers. Full article

(This article belongs to the Special Issue Interdisciplinary Research on Predictive Justice)

Open AccessArticle

Improving Access to Justice with Legal Chatbots

Marc Queudot,

Éric Charton and

Marie-Jean Meurs

Stats 2020, 3(3), 356-375; https://doi.org/10.3390/stats3030023 - 04 Sep 2020

Abstract

On average, one in three Canadians will be affected by a legal problem over a three-year period. Unfortunately, whether it is legal representation or legal advice, the very high cost of these services excludes disadvantaged and most vulnerable people, forcing them to represent themselves. For these people, accessing legal information is therefore critical. In this work, we attempt to tackle this problem by embedding legal data in a conversational interface. We introduce two dialog systems (chatbots) created to provide legal information. The first one, based on data from the Government of Canada, deals with immigration issues, while the second one informs bank employees about legal issues related to their job tasks. Both chatbots rely on various representations and classification algorithms, from mature techniques to novel advances in the field. The chatbot dedicated to immigration issues is shared with the research community as an open resource project. Full article

(This article belongs to the Special Issue Interdisciplinary Research on Predictive Justice)

► Show Figures

Figure 1

Open AccessArticle

Nonparametric Limits of Agreement for Small to Moderate Sample Sizes: A Simulation Study

Maria E. Frey,

Hans C. Petersen and

Oke Gerke

Stats 2020, 3(3), 343-355; https://doi.org/10.3390/stats3030022 - 28 Aug 2020

Abstract

The assessment of agreement in method comparison and observer variability analysis of quantitative measurements is usually done by the Bland–Altman Limits of Agreement, where the paired differences are implicitly assumed to follow a normal distribution. Whenever this assumption does not hold, the 2.5% and 97.5% percentiles are obtained by quantile estimation. In the literature, empirical quantiles have been used for this purpose. In this simulation study, we applied both sample, subsampling, and kernel quantile estimators, as well as other methods for quantile estimation to sample sizes between 30 and 150 and different distributions of the paired differences. The performance of 15 estimators in generating prediction intervals was measured by their respective coverage probability for one newly generated observation. Our results indicated that sample quantile estimators based on one or two order statistics outperformed all of the other estimators and they can be used for deriving nonparametric Limits of Agreement. For sample sizes exceeding 80 observations, more advanced quantile estimators, such as the Harrell–Davis and estimators of Sfakianakis–Verginis type, which use all of the observed differences, performed likewise well, but may be considered intuitively more appealing than simple sample quantile estimators that are based on only two observations per quantile. Full article

(This article belongs to the Special Issue Robust Statistics in Action)

► Show Figures

Figure 1

Open AccessBrief Report

Chi-Square and Student Bridge Distributions and the Behrens–Fisher Statistic

Wolf-Dieter Richter

Stats 2020, 3(3), 330-342; https://doi.org/10.3390/stats3030021 - 25 Aug 2020

Abstract

We prove that the Behrens–Fisher statistic follows a Student bridge distribution, the mixing coefficient of which depends on the two sample variances only through their ratio. To this end, it is first shown that a weighted sum of two independent normalized chi-square distributed random variables is chi-square bridge distributed, and secondly that the Behrens–Fisher statistic is based on such a variable and a standard normally distributed one that is independent of the former. In case of a known variance ratio, exact standard statistical testing and confidence estimation methods apply without the need for any additional approximations. In addition, a three pillar bridges explanation is given for the choice of degrees of freedom in Welch’s approximation to the exact distribution of the Behrens–Fisher statistic. Full article

(This article belongs to the Section Statistical Methods)

► Show Figures

Figure 1

Open AccessArticle

Recovering Yield Curves from Dynamic Term Structure Models with Time-Varying Factors

Hiroyuki Kawakatsu

Stats 2020, 3(3), 284-329; https://doi.org/10.3390/stats3030020 - 22 Aug 2020

Abstract

A dynamic version of the Nelson-Siegel-Svensson term structure model with time-varying factors is considered for predicting out-of-sample maturity yields. Simple linear interpolation cannot be applied to recover yields at the very short- and long- end of the term structure where data are often missing. This motivates the use of dynamic parametric term structure models that exploit both time series and cross-sectional variation in yield data to predict missing data at the extreme ends of the term structure. Although the dynamic Nelson–Siegel–Svensson model is weakly identified when the two decay factors become close to each other, their predictions may be more accurate than those from more restricted models depending on data and maturity. Full article

(This article belongs to the Special Issue Time Series Analysis and Forecasting)

► Show Figures

Figure 1

Open AccessArticle

L_p Loss Functions in Invariance Alignment and Haberman Linking with Few or Many Groups

Alexander Robitzsch

Stats 2020, 3(3), 246-283; https://doi.org/10.3390/stats3030019 - 05 Aug 2020

Abstract

The comparison of group means in latent variable models plays a vital role in empirical research in the social sciences. The present article discusses an extension of invariance alignment and Haberman linking by choosing the robust power loss function

ρ (x)

[...] Read more.

ρ (x) = {| x |}^{p}

(p > 0)

. This power loss function with power values p smaller than one is particularly suited for item responses that are generated under partial invariance. For a general class of linking functions, asymptotic normality of estimates is shown. Moreover, the theory of M-estimation is applied for obtaining linking errors (i.e., inference with respect to a population of items) for this class of linking functions. In a simulation study, it is shown that invariance alignment and Haberman linking have comparable performance, and in some conditions, the newly proposed robust Haberman linking outperforms invariance alignment. In three examples, the influence of the choice of a particular linking function on the estimation of group means is demonstrated. It is concluded that the choice of the loss function in linking is related to structural assumptions about the pattern of noninvariance in item parameters. Full article

(This article belongs to the Section Multivariate Analysis)

► Show Figures

Figure 1

Open AccessEditorial

Learning from Data to Optimize Control in Precision Farming

Alexander Kocian and

Luca Incrocci

Stats 2020, 3(3), 239-245; https://doi.org/10.3390/stats3030018 - 24 Jul 2020

Abstract

Precision farming is one way of many to meet a 55 percent increase in global demand for agricultural products on current agricultural land by 2050 at reduced need of fertilizers and efficient use of water resources. The catalyst for the emergence of precision farming has been satellite positioning and navigation followed by Internet-of-Things, generating vast information that can be used to optimize farming processes in real-time. Statistical tools from data mining, predictive modeling, and machine learning analyze patterns in historical data, to make predictions about future events as well as intelligent actions. This special issue presents the latest development in statistical inference, machine learning, and optimum control for precision farming. Full article

(This article belongs to the Special Issue Statistical Tools in Precision Farming)

Open AccessArticle

A Bayesian Adaptive Design in Cancer Phase I Trials Using Dose Combinations with Ordinal Toxicity Grades

Márcio A. Diniz,

Sungjin Kim and

Mourad Tighiouart

Stats 2020, 3(3), 221-238; https://doi.org/10.3390/stats3030017 - 17 Jul 2020

Abstract

We propose a Bayesian adaptive design for early phase drug combination cancer trials incorporating ordinal grade of toxicities. Parametric models are used to describe the relationship between the dose combinations and the probabilities of the ordinal toxicities under the proportional odds assumption. Trial design proceeds by treating cohorts of two patients simultaneously receiving different dose combinations. Specifically, at each stage of the trial, we seek the dose of one agent by minimizing the Bayes risk with respect to a loss function given the current dose of the other agent. We consider two types of loss functions corresponding to the Continual Reassessment Method (CRM) and Escalation with Overdose Control (EWOC). At the end of the trial, we estimate the MTD curve as a function of Bayes estimates of the model parameters. We evaluate design operating characteristics in terms of safety of the trial and percent of dose recommendation at dose combination neighborhoods around the true MTD by comparing this design to the one that uses a binary indicator of DLT. The methodology is further adapted to the case of a pre-specified discrete set of dose combinations. Full article

(This article belongs to the Section Biostatistics)

► Show Figures

Figure 1

Open AccessArticle

Multivariate Mixed Response Model with Pairwise Composite-Likelihood Method

Hao Bai,

Yuan Zhong,

Xin Gao and

Wei Xu

Stats 2020, 3(3), 203-220; https://doi.org/10.3390/stats3030016 - 15 Jul 2020

Cited by 1

Abstract

In clinical research, study outcomes usually consist of various patients’ information corresponding to the treatment. To have a better understanding of the effects of different treatments, one often needs to analyze multiple clinical outcomes simultaneously, while the data are usually mixed with both continuous and discrete variables. We propose the multivariate mixed response model to implement statistical inference based on the conditional grouped continuous model through a pairwise composite-likelihood approach. It can simplify the multivariate model by dealing with three types of bivariate models and incorporating the asymptotical properties of the composite likelihood via the Godambe information. We demonstrate the validity and the statistic power of the multivariate mixed response model through simulation studies and clinical applications. This composite-likelihood method is advantageous for statistical inference on correlated multivariate mixed outcomes. Full article

► Show Figures

Figure 1

Open AccessArticle

Non-Negativity of a Quadratic form with Applications to Panel Data Estimation, Forecasting and Optimization

Bhimasankaram Pochiraju,

Sridhar Seshadri,

Dimitrios D. Thomakos and

Konstantinos Nikolopoulos

Stats 2020, 3(3), 185-202; https://doi.org/10.3390/stats3030015 - 06 Jul 2020

Abstract

For a symmetric matrix B, we determine the class of Q such that

Q^{t} BQ

is non-negative definite and apply it to panel data estimation and forecasting: the Hausman test for testing the endogeneity of the random effects in panel data [...] Read more.

For a symmetric matrix B, we determine the class of Q such that

Q^{t} BQ

is non-negative definite and apply it to panel data estimation and forecasting: the Hausman test for testing the endogeneity of the random effects in panel data models. We show that the test can be performed if the estimated error variances in the fixed and random effects models satisfy a specific inequality. If it fails, we discuss the restrictions under which the test can be performed. We show that estimators satisfying the inequality exist. Furthermore, we discuss an application to a constrained quadratic minimization problem with an indefinite objective function. Full article

Open AccessArticle

A Family of Correlated Observations: From Independent to Strongly Interrelated Ones

Daniel A. Griffith

Stats 2020, 3(3), 166-184; https://doi.org/10.3390/stats3030014 - 30 Jun 2020

Abstract

This paper proposes a new classification of correlated data types based upon the relative number of direct connections among observations, producing a family of correlated observations embracing seven categories, one whose empirical counterpart currently is unknown, and ranging from independent (i.e., no links) to approaching near-complete linkage (i.e., n(n − 1)/2 links). Analysis of specimen datasets from publicly available data sources furnishes empirical illustrations for these various categories. Their descriptions also include their historical context and calculation of their effective sample sizes (i.e., an equivalent number of independent observations). Concluding comments contain some state-of-the-art future research topics. Full article

(This article belongs to the Section Statistical Methods)

► Show Figures

Figure 1

Open AccessArticle

Generalized Mutual Information

Zhiyi Zhang

Stats 2020, 3(2), 158-165; https://doi.org/10.3390/stats3020013 - 10 Jun 2020

Abstract

Mutual information is one of the essential building blocks of information theory. It is however only finitely defined for distributions in a subclass of the general class of all distributions on a joint alphabet. The unboundedness of mutual information prevents its potential utility from being extended to the general class. This is in fact a void in the foundation of information theory that needs to be filled. This article proposes a family of generalized mutual information whose members are indexed by a positive integer n, with the nth member being the mutual information of nth order. The mutual information of the first order coincides with Shannon’s, which may or may not be finite. It is however established (a) that each mutual information of an order greater than 1 is finitely defined for all distributions of two random elements on a joint countable alphabet, and (b) that each and every member of the family enjoys all the utilities of a finite Shannon’s mutual information. Full article

Open AccessArticle

Modelling the Behaviour of Currency Exchange Rates with Singular Spectrum Analysis and Artificial Neural Networks

Paulo Canas Rodrigues,

Olushina Olawale Awe,

Jonatha Sousa Pimentel and

Rahim Mahmoudvand

Stats 2020, 3(2), 137-157; https://doi.org/10.3390/stats3020012 - 01 Jun 2020

Abstract

A proper understanding and analysis of suitable models involved in forecasting currency exchange rates dynamics is essential to provide reliable information about the economy. This paper deals with model fit and model forecasting of eight time series of historical data about currency exchange rate considering the United States dollar as reference. The time series techniques: classical autoregressive integrated moving average model, the non-parametric univariate and multivariate singular spectrum analysis (SSA), artificial neural network (ANN) algorithms, and a recent prominent hybrid method that combines SSA and ANN, are considered and their performance compared in terms of model fit and model forecasting. Moreover, specific methodological and computational adaptations were conducted to allow for these analyses and comparisons. Full article

(This article belongs to the Special Issue Time Series Analysis and Forecasting)

► Show Figures

Figure 1

Open AccessArticle

A-Spline Regression for Fitting a Nonparametric Regression Function with Censored Data

Ersin Yılmaz,

Syed Ejaz Ahmed and

Dursun Aydın

Stats 2020, 3(2), 120-136; https://doi.org/10.3390/stats3020011 - 29 May 2020

Abstract

This paper aims to solve the problem of fitting a nonparametric regression function with right-censored data. In general, issues of censorship in the response variable are solved by synthetic data transformation based on the Kaplan–Meier estimator in the literature. In the context of synthetic data, there have been different studies on the estimation of right-censored nonparametric regression models based on smoothing splines, regression splines, kernel smoothing, local polynomials, and so on. It should be emphasized that synthetic data transformation manipulates the observations because it assigns zero values to censored data points and increases the size of the observations. Thus, an irregularly distributed dataset is obtained. We claim that adaptive spline (A-spline) regression has the potential to deal with this irregular dataset more easily than the smoothing techniques mentioned here, due to the freedom to determine the degree of the spline, as well as the number and location of the knots. The theoretical properties of A-splines with synthetic data are detailed in this paper. Additionally, we support our claim with numerical studies, including a simulation study and a real-world data example. Full article

► Show Figures

Figure 1

Open AccessArticle

A Brief Overview of Restricted Mean Survival Time Estimators and Associated Variances

Szilárd Nemes,

Erik Bülow and

Andreas Gustavsson

Stats 2020, 3(2), 107-119; https://doi.org/10.3390/stats3020010 - 26 May 2020

Abstract

Restricted Mean Survival Time (

R M S T

) experiences a renaissance and is advocated as a model-free, easy to interpret alternative to proportional hazards regression and hazard rates with implication in causal inference. Estimation of

R M S T

and associated [...] Read more.

Restricted Mean Survival Time (

R M S T

) experiences a renaissance and is advocated as a model-free, easy to interpret alternative to proportional hazards regression and hazard rates with implication in causal inference. Estimation of

R M S T

and associated variance is mainly done by numerical integration of Kaplan–Meier curves. In this paper we briefly review the two main alternatives to the Kaplan–Meier method; analysis based on pseudo-observations, and the flexible parametric survival method. Using computer simulations, we assess the efficacy of the three methods compared to a fully parametric approach where the distribution of survival times is known. Thereafter, the three methods are directly compared without any distributional assumption for the survival data. Generally, flexible parametric survival methods outperform both competitors, however the differences are small. Full article

(This article belongs to the Section Statistical Methods)

► Show Figures

Figure 1

Open AccessArticle

Depth Induced Regression Medians and Uniqueness

Yijun Zuo

Stats 2020, 3(2), 94-106; https://doi.org/10.3390/stats3020009 - 10 Apr 2020

Abstract

The notion of median in one dimension is a foundational element in nonparametric statistics. It has been extended to multi-dimensional cases both in location and in regression via notions of data depth. Regression depth (RD) and projection regression depth (PRD) represent the two [...] Read more.

D_{C}

is another depth notion in regression. Depth-induced regression medians (maximum depth estimators) serve as robust alternatives to the classical least squares estimator. The uniqueness of regression medians is indispensable in the discussion of their properties and the asymptotics (consistency and limiting distribution) of sample regression medians. Are the regression medians induced from RD, PRD, and

D_{C}

unique? Answering this question is the main goal of this article. It is found that only the regression median induced from PRD possesses the desired uniqueness property. The conventional remedy measure for non-uniqueness, taking average of all medians, might yield an estimator that no longer possesses the maximum depth in both RD and