F. De Bastiani, D. Stasinopoulos, R. Rigby, G. Heller, Lucas A. Silva
This study introduces the bucket plot, a visual tool to detect skewness and kurtosis in a continuously distributed random variable. The plot can be applied to both moment and centile skewness and kurtosis. The bucket plot is used to detect skewness and kurtosis either in a response variable, or in the residuals from a fitted model as a diagnostic tool by which to assess the adequacy of a fitted distribution to the response variable regarding skewness and kurtosis. We demonstrate the bucket plot in nine simulated skewness and kurtosis scenarios, and the usefulness of the plot is shown in a real-data situation.
{"title":"Bucket plot: A visual tool for skewness and kurtosis comparisons","authors":"F. De Bastiani, D. Stasinopoulos, R. Rigby, G. Heller, Lucas A. Silva","doi":"10.1214/22-bjps533","DOIUrl":"https://doi.org/10.1214/22-bjps533","url":null,"abstract":"This study introduces the bucket plot, a visual tool to detect skewness and kurtosis in a continuously distributed random variable. The plot can be applied to both moment and centile skewness and kurtosis. The bucket plot is used to detect skewness and kurtosis either in a response variable, or in the residuals from a fitted model as a diagnostic tool by which to assess the adequacy of a fitted distribution to the response variable regarding skewness and kurtosis. We demonstrate the bucket plot in nine simulated skewness and kurtosis scenarios, and the usefulness of the plot is shown in a real-data situation.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47499946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this short communication we show that basic tools from Malliavin calculus can be applied to derive the two-point function of the slope of the one-dimensional KPZ equation, starting from an arbitrary two-sided Brownian motion, in terms of the polymer end-point annealed distribution associated to the stochastic heat equation. We also prove that this distribution is given in terms of the derivative of the variance of the solution of the KPZ equation.
{"title":"On the two-point function of the one-dimensional KPZ equation","authors":"Sergio I. L'opez, Leandro P. R. Pimentel","doi":"10.1214/23-bjps576","DOIUrl":"https://doi.org/10.1214/23-bjps576","url":null,"abstract":"In this short communication we show that basic tools from Malliavin calculus can be applied to derive the two-point function of the slope of the one-dimensional KPZ equation, starting from an arbitrary two-sided Brownian motion, in terms of the polymer end-point annealed distribution associated to the stochastic heat equation. We also prove that this distribution is given in terms of the derivative of the variance of the solution of the KPZ equation.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41811139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we are concerned with a family of $N$-urn branching processes, where some particles are put into $N$ urns initially and then each particle gives birth to several new particles in some urn when dies. This model includes the $N$-urn Ehrenfest model and the $N$-urn branching random walk as special cases. We show that the scaling limit of the process is driven by a $C(mathbb{T})$-valued linear ordinary differential equation and the fluctuation of the process is driven by a generalized Ornstein-Uhlenbeck process in the dual of $C^infty(mathbb{T})$, where $mathbb{T}=(0, 1]$ is the one-dimensional torus. A crucial step for proofs of above main results is to show that numbers of particles in different urns are approximately independent. As applications of our main results, limit theorems of hitting times of the process are also discussed.
{"title":"Scaling limits and fluctuations of a family of N-urn branching processes","authors":"Xiaofeng Xue","doi":"10.1214/23-bjps567","DOIUrl":"https://doi.org/10.1214/23-bjps567","url":null,"abstract":"In this paper we are concerned with a family of $N$-urn branching processes, where some particles are put into $N$ urns initially and then each particle gives birth to several new particles in some urn when dies. This model includes the $N$-urn Ehrenfest model and the $N$-urn branching random walk as special cases. We show that the scaling limit of the process is driven by a $C(mathbb{T})$-valued linear ordinary differential equation and the fluctuation of the process is driven by a generalized Ornstein-Uhlenbeck process in the dual of $C^infty(mathbb{T})$, where $mathbb{T}=(0, 1]$ is the one-dimensional torus. A crucial step for proofs of above main results is to show that numbers of particles in different urns are approximately independent. As applications of our main results, limit theorems of hitting times of the process are also discussed.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46121517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The supremum of the standardized empirical process is a promising statistic for testing whether the distribution function $F$ of i.i.d. real random variables is either equal to a given distribution function $F_0$ (hypothesis) or $F ge F_0$ (one-sided alternative). Since cite{r5} it is well-known that an affine-linear transformation of the suprema converge in distribution to the Gumbel law as the sample size tends to infinity. This enables the construction of an asymptotic level-$alpha$ test. However, the rate of convergence is extremely slow. As a consequence the probability of the type I error is much larger than $alpha$ even for sample sizes beyond $10.000$. Now, the standardization consists of the weight-function $1/sqrt{F_0(x)(1-F_0(x))}$. Substituting the weight-function by a suitable random constant leads to a new test-statistic, for which we can derive the exact distribution (and the limit distribution) under the hypothesis. A comparison via a Monte-Carlo simulation shows that the new test is uniformly better than the Smirnov-test and an appropriately modified test due to cite{r20}. Our methodology also works for the two-sided alternative $F neq F_0$.
{"title":"Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process","authors":"D. Ferger","doi":"10.1214/23-bjps564","DOIUrl":"https://doi.org/10.1214/23-bjps564","url":null,"abstract":"The supremum of the standardized empirical process is a promising statistic for testing whether the distribution function $F$ of i.i.d. real random variables is either equal to a given distribution function $F_0$ (hypothesis) or $F ge F_0$ (one-sided alternative). Since cite{r5} it is well-known that an affine-linear transformation of the suprema converge in distribution to the Gumbel law as the sample size tends to infinity. This enables the construction of an asymptotic level-$alpha$ test. However, the rate of convergence is extremely slow. As a consequence the probability of the type I error is much larger than $alpha$ even for sample sizes beyond $10.000$. Now, the standardization consists of the weight-function $1/sqrt{F_0(x)(1-F_0(x))}$. Substituting the weight-function by a suitable random constant leads to a new test-statistic, for which we can derive the exact distribution (and the limit distribution) under the hypothesis. A comparison via a Monte-Carlo simulation shows that the new test is uniformly better than the Smirnov-test and an appropriately modified test due to cite{r20}. Our methodology also works for the two-sided alternative $F neq F_0$.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47487307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The presence of multicollinearity adversely affects the inferential properties of the maximum likelihood (ML) estimator in logistic regression model. It is a well established fact that the use of restrictions lowers the effect of multicollinearity. In this article, an alternative to the ML estimator has been introduced by combining the exact prior information into the logistic r − k class (Lrk) estimator. The estimator is named a logistic restricted r − k class estimator. Another estimator, logistic restricted PCR estimator, is also developed as a special case of the LRrk estimator. The asymptotic mean squared error (MSE) matrix properties of the estimators are studied and necessary and sufficient conditions are derived. Further, a Monte Carlo simulation study is performed to compare the performance of the estimators in terms of the scalar MSE and the prediction MSE. It is found that the proposed estimators perform better than the existing estimators in most of the cases considered. Moreover, a numerical example has also been presented for comparing the performance of the estimators.
{"title":"A general restricted estimator in binary logistic regression in the presence of multicollinearity","authors":"Gargi Tyagi, S. Chandra","doi":"10.1214/21-bjps527","DOIUrl":"https://doi.org/10.1214/21-bjps527","url":null,"abstract":"The presence of multicollinearity adversely affects the inferential properties of the maximum likelihood (ML) estimator in logistic regression model. It is a well established fact that the use of restrictions lowers the effect of multicollinearity. In this article, an alternative to the ML estimator has been introduced by combining the exact prior information into the logistic r − k class (Lrk) estimator. The estimator is named a logistic restricted r − k class estimator. Another estimator, logistic restricted PCR estimator, is also developed as a special case of the LRrk estimator. The asymptotic mean squared error (MSE) matrix properties of the estimators are studied and necessary and sufficient conditions are derived. Further, a Monte Carlo simulation study is performed to compare the performance of the estimators in terms of the scalar MSE and the prediction MSE. It is found that the proposed estimators perform better than the existing estimators in most of the cases considered. Moreover, a numerical example has also been presented for comparing the performance of the estimators.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47422069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we mainly study the consistency of the nearest neighbor estimator of the density function based on m-extended negatively dependent samples. The weak consistency, strong consistency, uniformly strong consistency and the convergence rate are established under some mild conditions. The results obtained in this paper extend and improve some existing ones in the literature.
{"title":"Consistency of nearest neighbor estimator of density function for m-END samples","authors":"Wei Wang, Yi Wu","doi":"10.1214/22-bjps530","DOIUrl":"https://doi.org/10.1214/22-bjps530","url":null,"abstract":"In this paper, we mainly study the consistency of the nearest neighbor estimator of the density function based on m-extended negatively dependent samples. The weak consistency, strong consistency, uniformly strong consistency and the convergence rate are established under some mild conditions. The results obtained in this paper extend and improve some existing ones in the literature.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45448990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One of the most important assumptions in multiple regression analysis is the independence of the explanatory variables, however, this assumption is violated in several situations. In this work, we investigate regression equations when this independence does not hold and the explanatory variables are connected by many of elliptical copulas. We apply the proposed regression equation to study its heteroscedasticity diagnostic and using simulated data we also assess our regression model. A cross-validation procedure is carried out to ensure the unbiasedness of the results. Also, a real data analysis is presented as an application.
{"title":"A heteroscedasticity diagnostic of a regression analysis with copula dependent random variables","authors":"A. Sheikhi, Fereshteh Arad, R. Mesiar","doi":"10.1214/22-bjps532","DOIUrl":"https://doi.org/10.1214/22-bjps532","url":null,"abstract":"One of the most important assumptions in multiple regression analysis is the independence of the explanatory variables, however, this assumption is violated in several situations. In this work, we investigate regression equations when this independence does not hold and the explanatory variables are connected by many of elliptical copulas. We apply the proposed regression equation to study its heteroscedasticity diagnostic and using simulated data we also assess our regression model. A cross-validation procedure is carried out to ensure the unbiasedness of the results. Also, a real data analysis is presented as an application.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43736550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Model selection for functional linear regression with hierarchical structure","authors":"S. Feng, Xinyu Zhang, Hui Liang, Lifang Pei","doi":"10.1214/21-bjps525","DOIUrl":"https://doi.org/10.1214/21-bjps525","url":null,"abstract":"","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42795898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Identifying key nodes, estimating the probability of connection between them, and distinguishing latent groups are some of the main objectives of social network analysis. In this paper, we propose a class of blockmodels to model stochastic equivalence and visualize groups in an unobservable space. In this setting, the proposed method is based on two approaches: latent distances and latent dissimilarities at the group level. The projection proposed in the paper is performed without needing to project individuals, unlike the main approaches in the literature. Our approach can be used in undirected or directed graphs and is flexible enough to cluster and quantify between and within-group tie probabilities in social networks. The effectiveness of the methodology in representing groups in latent spaces was analyzed under artificial datasets and in two case studies.
{"title":"An alternative class of models to position social network groups in latent spaces","authors":"Izabel Nolau, G. S. Ferreira","doi":"10.1214/21-bjps526","DOIUrl":"https://doi.org/10.1214/21-bjps526","url":null,"abstract":"Identifying key nodes, estimating the probability of connection between them, and distinguishing latent groups are some of the main objectives of social network analysis. In this paper, we propose a class of blockmodels to model stochastic equivalence and visualize groups in an unobservable space. In this setting, the proposed method is based on two approaches: latent distances and latent dissimilarities at the group level. The projection proposed in the paper is performed without needing to project individuals, unlike the main approaches in the literature. Our approach can be used in undirected or directed graphs and is flexible enough to cluster and quantify between and within-group tie probabilities in social networks. The effectiveness of the methodology in representing groups in latent spaces was analyzed under artificial datasets and in two case studies.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46862529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Time series analysis is widely used in the fields of economics, ecology and medicine. Robust variable selection procedures through penalized regression have been gaining increased attention. In our work, a robust penalized regression estimator based on exponential squared loss for autoregressive (AR) models is proposed and discussed. The objective model with adaptive Lasso penalty realizes variable selection and parameter estimation simultaneously. Under some regular conditions, we establish the asymptotic and “Oracle” properties of the proposed estimator. In particular, the induced non-convex and non-differentiable mathematical programming problem offers challenges for solving algorithms. To solve this problem efficiently, we specially design a block coordinate descent (BCD) algorithm equipped with concave-convex process (CCCP) and provide a convergence guarantee. Numerical simulation studies are carried out to show that the proposed method is particularly robust and applicable compared with some recent methods when there are different types of noise or different intensity of noise. Furthermore, an application on a dataset of daily minimum temperature in Melbourne over 1981-1990 is performed.
{"title":"Exponential squared loss based robust variable selection of AR models","authors":"Yaxin Wu, Yunquan Song, Xijun Liang, Yujie Gai","doi":"10.1214/21-bjps524","DOIUrl":"https://doi.org/10.1214/21-bjps524","url":null,"abstract":"Time series analysis is widely used in the fields of economics, ecology and medicine. Robust variable selection procedures through penalized regression have been gaining increased attention. In our work, a robust penalized regression estimator based on exponential squared loss for autoregressive (AR) models is proposed and discussed. The objective model with adaptive Lasso penalty realizes variable selection and parameter estimation simultaneously. Under some regular conditions, we establish the asymptotic and “Oracle” properties of the proposed estimator. In particular, the induced non-convex and non-differentiable mathematical programming problem offers challenges for solving algorithms. To solve this problem efficiently, we specially design a block coordinate descent (BCD) algorithm equipped with concave-convex process (CCCP) and provide a convergence guarantee. Numerical simulation studies are carried out to show that the proposed method is particularly robust and applicable compared with some recent methods when there are different types of noise or different intensity of noise. Furthermore, an application on a dataset of daily minimum temperature in Melbourne over 1981-1990 is performed.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46292382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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