Pub Date : 2022-12-05DOI: 10.1080/10485252.2022.2152813
Chen Zhong
ABSTRACT This paper extends the classical Glivenko–Cantelli theorem for the empirical cumulative distribution function based on the innovations in the ARCH model with a slowly time-varying trend. In this semiparametric time-varying model, strong consistency for the innovation density estimator via kernel smoothing method is established, given that the trend and ARCH parameter estimators meet some mild conditions. Besides, the strong consistency for the Gaussian quasi maximum likelihood estimator (QMLE) in the time-varying ARCH parameter is established as well. Moreover, in terms of the existence of the trend in the data, two major nonparametric trend estimators, B-spline and kernel estimators, are shown to be appropriate for the strong consistency results.
{"title":"Extended Glivenko–Cantelli theorem and L 1 strong consistency of innovation density estimator for time-varying semiparametric ARCH model","authors":"Chen Zhong","doi":"10.1080/10485252.2022.2152813","DOIUrl":"https://doi.org/10.1080/10485252.2022.2152813","url":null,"abstract":"ABSTRACT This paper extends the classical Glivenko–Cantelli theorem for the empirical cumulative distribution function based on the innovations in the ARCH model with a slowly time-varying trend. In this semiparametric time-varying model, strong consistency for the innovation density estimator via kernel smoothing method is established, given that the trend and ARCH parameter estimators meet some mild conditions. Besides, the strong consistency for the Gaussian quasi maximum likelihood estimator (QMLE) in the time-varying ARCH parameter is established as well. Moreover, in terms of the existence of the trend in the data, two major nonparametric trend estimators, B-spline and kernel estimators, are shown to be appropriate for the strong consistency results.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"99 1","pages":"373 - 396"},"PeriodicalIF":1.2,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83605981","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}
Pub Date : 2022-11-29DOI: 10.1080/10485252.2022.2150766
Gaku Igarashi
In regression discontinuity design (RDD), the continuity of the density of a running variable is required. Hence, a discontinuity test of density is used for RDD. In previous studies, tests using difference estimators between the left- and right-hand limits of a density at a (potential) discontinuity point were suggested. In the present paper, a new discontinuity test based on direct density ratio estimation using a beta kernel is proposed. By using the ratio estimator in the proposed test statistic, rather than a difference estimator, the characteristic form of the asymptotic variance of the test statistic is obtained. Consequently, the power of the proposed test is shown to increase when used as a one-tailed test. Simulation studies illustrate the larger power of the proposed test when used as a one-tailed test.
{"title":"A nonparametric discontinuity test of density using a beta kernel","authors":"Gaku Igarashi","doi":"10.1080/10485252.2022.2150766","DOIUrl":"https://doi.org/10.1080/10485252.2022.2150766","url":null,"abstract":"In regression discontinuity design (RDD), the continuity of the density of a running variable is required. Hence, a discontinuity test of density is used for RDD. In previous studies, tests using difference estimators between the left- and right-hand limits of a density at a (potential) discontinuity point were suggested. In the present paper, a new discontinuity test based on direct density ratio estimation using a beta kernel is proposed. By using the ratio estimator in the proposed test statistic, rather than a difference estimator, the characteristic form of the asymptotic variance of the test statistic is obtained. Consequently, the power of the proposed test is shown to increase when used as a one-tailed test. Simulation studies illustrate the larger power of the proposed test when used as a one-tailed test.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"11 1","pages":"323 - 354"},"PeriodicalIF":1.2,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88283660","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}
Pub Date : 2022-11-25DOI: 10.1080/10485252.2022.2149749
Tian Jiang
Nonparametric regression with missing at random (MAR) responses, univariate regression component of interest, and the scale function depending on both the predictor and auxiliary covariates, is considered. The asymptotic theory suggests that both heteroscedasticity and MAR mechanism affect the sharp constant of the minimax mean integrated squared error (MISE) convergence. Our sharp minimax procedure is based on the estimation of unknown nuisance scale function, design density and availability likelihood. The estimator is adaptive to the missing mechanism and unknown smoothness of the estimated regression function. Simulation studies and real examples also justify practical feasibility of the proposed method for this complex regression setting.
{"title":"Nonparametric regression with responses missing at random and the scale depending on auxiliary covariates","authors":"Tian Jiang","doi":"10.1080/10485252.2022.2149749","DOIUrl":"https://doi.org/10.1080/10485252.2022.2149749","url":null,"abstract":"Nonparametric regression with missing at random (MAR) responses, univariate regression component of interest, and the scale function depending on both the predictor and auxiliary covariates, is considered. The asymptotic theory suggests that both heteroscedasticity and MAR mechanism affect the sharp constant of the minimax mean integrated squared error (MISE) convergence. Our sharp minimax procedure is based on the estimation of unknown nuisance scale function, design density and availability likelihood. The estimator is adaptive to the missing mechanism and unknown smoothness of the estimated regression function. Simulation studies and real examples also justify practical feasibility of the proposed method for this complex regression setting.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"4 1","pages":"302 - 322"},"PeriodicalIF":1.2,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81655138","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}
Pub Date : 2022-11-24DOI: 10.1080/10485252.2022.2148667
Shuying Wang, Da Xu, Chunjie Wang, Jianguo Sun
Linear transformation models have been one type of models commonly used for regression analysis of failure time data partly due to their flexibility. More recently they have been generalised to the case where there may exist a cured subgroup or the censoring may be informative. In this paper, we consider a more complicated and general situation where both a cured subgroup and informative censoring, or more specifically informative interval censoring, exist. As pointed out in the literature, the analysis that fails to take into account either the cured subgroup or the informative censoring can yield biased estimation or misleading conclusions. For the problem, a three-component mixture cure model is presented and we develop a two-step estimation procedure with the use of B-splines to approximate unknown functions. The proposed approach is quite flexible and can be easily implemented. Also the proposed estimators of regression parameters are shown to be consistent and asymptotically normal. An extensive simulation study is conducted and suggests that the method works well for practical situations. Furthermore a real application is provided to illustrate the proposed methodology.
{"title":"Estimation of linear transformation cure models with informatively interval-censored failure time data","authors":"Shuying Wang, Da Xu, Chunjie Wang, Jianguo Sun","doi":"10.1080/10485252.2022.2148667","DOIUrl":"https://doi.org/10.1080/10485252.2022.2148667","url":null,"abstract":"Linear transformation models have been one type of models commonly used for regression analysis of failure time data partly due to their flexibility. More recently they have been generalised to the case where there may exist a cured subgroup or the censoring may be informative. In this paper, we consider a more complicated and general situation where both a cured subgroup and informative censoring, or more specifically informative interval censoring, exist. As pointed out in the literature, the analysis that fails to take into account either the cured subgroup or the informative censoring can yield biased estimation or misleading conclusions. For the problem, a three-component mixture cure model is presented and we develop a two-step estimation procedure with the use of B-splines to approximate unknown functions. The proposed approach is quite flexible and can be easily implemented. Also the proposed estimators of regression parameters are shown to be consistent and asymptotically normal. An extensive simulation study is conducted and suggests that the method works well for practical situations. Furthermore a real application is provided to illustrate the proposed methodology.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"21 1","pages":"283 - 301"},"PeriodicalIF":1.2,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88217244","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}
Pub Date : 2022-11-22DOI: 10.1080/10485252.2022.2147173
Qiqing Yu
The generalised maximum likelihood estimator (GMLE) of a survival function based on truncated interval-censored (TIC) data has been studied since 1990s (by Frydman, H. (1994), ‘A note on nonparametric estimation of the distribution function from interval censored and truncated data’, Journal of the Royal Statistical Society, Series B, 56, 71–74 among others). In the literature related to the GMLE based on TIC data, there are several issues that have not been properly settled in both methodology and theory including: (1) innermost intervals based on the TIC data are not correctly formulated and they lead to inconsistent estimators which are not the GMLE; and (2) the consistency of the GMLE has not been established. We settle these two issues in this paper. In particular, we specify the correct forms of innermost intervals and establish consistency results for the GMLE under a realistic model.
{"title":"The generalised MLE with truncated interval-censored data","authors":"Qiqing Yu","doi":"10.1080/10485252.2022.2147173","DOIUrl":"https://doi.org/10.1080/10485252.2022.2147173","url":null,"abstract":"The generalised maximum likelihood estimator (GMLE) of a survival function based on truncated interval-censored (TIC) data has been studied since 1990s (by Frydman, H. (1994), ‘A note on nonparametric estimation of the distribution function from interval censored and truncated data’, Journal of the Royal Statistical Society, Series B, 56, 71–74 among others). In the literature related to the GMLE based on TIC data, there are several issues that have not been properly settled in both methodology and theory including: (1) innermost intervals based on the TIC data are not correctly formulated and they lead to inconsistent estimators which are not the GMLE; and (2) the consistency of the GMLE has not been established. We settle these two issues in this paper. In particular, we specify the correct forms of innermost intervals and establish consistency results for the GMLE under a realistic model.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"73 1","pages":"266 - 282"},"PeriodicalIF":1.2,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82875942","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}
Pub Date : 2022-11-22DOI: 10.1080/10485252.2022.2146111
Yiming Wang, Sujit K. Ghosh
A nonparametric model using a sequence of Bernstein polynomials is constructed to approximate arbitrary isotropic covariance functions valid in and related approximation properties are investigated using the popular norm and norms. A computationally efficient sieve maximum likelihood (sML) estimation is then developed to nonparametrically estimate the unknown isotropic covariance function valid in . Consistency of the proposed sieve ML estimator is established under increasing domain regime. The proposed methodology is compared numerically with couple of existing nonparametric as well as with commonly used parametric methods. Numerical results based on simulated data show that our approach outperforms the parametric methods in reducing bias due to model misspecification and also the nonparametric methods in terms of having significantly lower values of expected and norms. Application to precipitation data is illustrated to showcase a real case study. Additional technical details and numerical illustrations are also made available.
{"title":"Nonparametric estimation of isotropic covariance function","authors":"Yiming Wang, Sujit K. Ghosh","doi":"10.1080/10485252.2022.2146111","DOIUrl":"https://doi.org/10.1080/10485252.2022.2146111","url":null,"abstract":"A nonparametric model using a sequence of Bernstein polynomials is constructed to approximate arbitrary isotropic covariance functions valid in and related approximation properties are investigated using the popular norm and norms. A computationally efficient sieve maximum likelihood (sML) estimation is then developed to nonparametrically estimate the unknown isotropic covariance function valid in . Consistency of the proposed sieve ML estimator is established under increasing domain regime. The proposed methodology is compared numerically with couple of existing nonparametric as well as with commonly used parametric methods. Numerical results based on simulated data show that our approach outperforms the parametric methods in reducing bias due to model misspecification and also the nonparametric methods in terms of having significantly lower values of expected and norms. Application to precipitation data is illustrated to showcase a real case study. Additional technical details and numerical illustrations are also made available.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"19 1","pages":"198 - 237"},"PeriodicalIF":1.2,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76341515","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}
Pub Date : 2022-11-21DOI: 10.1080/10485252.2022.2147172
Daojiang He, Huijun Shi, Kai Xu, M. Cao
In this paper, the problem of testing the equality of the mean vectors of k populations with possibly unknown and unequal covariance matrices is investigated in high-dimensional settings. The null distributions of most existing tests are asymptotically normal which inevitably imposes strong conditions on covariance matrices. However, we assume here only mild additional conditions on the proposed test, which offers much flexibility in practical applications. Additionally, the Welch–Satterthwaite -approximation we adopted can automatically mimic the shape of the null distribution of the proposed test statistic, while the normal approximation cannot achieve the adaptivity. Finally, an extensive simulation study shows that the proposed test has better performance on both size and power compared with existing methods.
{"title":"A high-dimensional test for the k-sample Behrens–Fisher problem","authors":"Daojiang He, Huijun Shi, Kai Xu, M. Cao","doi":"10.1080/10485252.2022.2147172","DOIUrl":"https://doi.org/10.1080/10485252.2022.2147172","url":null,"abstract":"In this paper, the problem of testing the equality of the mean vectors of k populations with possibly unknown and unequal covariance matrices is investigated in high-dimensional settings. The null distributions of most existing tests are asymptotically normal which inevitably imposes strong conditions on covariance matrices. However, we assume here only mild additional conditions on the proposed test, which offers much flexibility in practical applications. Additionally, the Welch–Satterthwaite -approximation we adopted can automatically mimic the shape of the null distribution of the proposed test statistic, while the normal approximation cannot achieve the adaptivity. Finally, an extensive simulation study shows that the proposed test has better performance on both size and power compared with existing methods.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"29 1","pages":"239 - 265"},"PeriodicalIF":1.2,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75115241","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}
Pub Date : 2022-11-18DOI: 10.1080/10485252.2023.2226779
E. Belitser, P. Serra, Alexandra G. J. Vegelien
A general many quantiles + noise model is studied in the robust formulation (allowing non-normal, non-independent observations), where the identifiability requirement for the noise is formulated in terms of quantiles rather than the traditional zero expectation assumption. We propose a penalization method based on the quantile loss function with appropriately chosen penalty function making inference on possibly sparse high-dimensional quantile vector. We apply a local approach to address the optimality by comparing procedures to the oracle sparsity structure. We establish that the proposed procedure mimics the oracle in the problems of estimation and uncertainty quantification (under the so called EBR condition). Adaptive minimax results over sparsity scale follow from our local results.
{"title":"Robust oracle estimation and uncertainty quantification for possibly sparse quantiles","authors":"E. Belitser, P. Serra, Alexandra G. J. Vegelien","doi":"10.1080/10485252.2023.2226779","DOIUrl":"https://doi.org/10.1080/10485252.2023.2226779","url":null,"abstract":"A general many quantiles + noise model is studied in the robust formulation (allowing non-normal, non-independent observations), where the identifiability requirement for the noise is formulated in terms of quantiles rather than the traditional zero expectation assumption. We propose a penalization method based on the quantile loss function with appropriately chosen penalty function making inference on possibly sparse high-dimensional quantile vector. We apply a local approach to address the optimality by comparing procedures to the oracle sparsity structure. We establish that the proposed procedure mimics the oracle in the problems of estimation and uncertainty quantification (under the so called EBR condition). Adaptive minimax results over sparsity scale follow from our local results.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"458 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78296795","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}
Pub Date : 2022-11-17DOI: 10.1080/10485252.2022.2146110
Kwan-Young Bak, J. Koo
This study investigates the minimaxity of a multi-task nonparametric regression problem. We formulate a simultaneous function estimation problem based on information pooling across multiple experiments under a low-dimensional structure. A nonparametric reduced rank regression estimator based on the nuclear norm penalisation scheme is proposed to incorporate the low-dimensional structure in the estimation process. A rank of a set of functions is defined in terms of their Fourier coefficients to formally characterise the dependence structure among functions. Minimax upper and lower bounds are established under various asymptotic scenarios to examine the role of the low-rank structure in determining optimal rates of convergence. The results confirm that exploiting the low-rank structure can significantly improve the convergence rate for the simultaneous estimation of multiple functions. The results also imply that the proposed estimator is rate optimal in the minimax sense for the rank-constraint Sobolev class of vector-valued functions.
{"title":"Minimax estimation in multi-task regression under low-rank structures","authors":"Kwan-Young Bak, J. Koo","doi":"10.1080/10485252.2022.2146110","DOIUrl":"https://doi.org/10.1080/10485252.2022.2146110","url":null,"abstract":"This study investigates the minimaxity of a multi-task nonparametric regression problem. We formulate a simultaneous function estimation problem based on information pooling across multiple experiments under a low-dimensional structure. A nonparametric reduced rank regression estimator based on the nuclear norm penalisation scheme is proposed to incorporate the low-dimensional structure in the estimation process. A rank of a set of functions is defined in terms of their Fourier coefficients to formally characterise the dependence structure among functions. Minimax upper and lower bounds are established under various asymptotic scenarios to examine the role of the low-rank structure in determining optimal rates of convergence. The results confirm that exploiting the low-rank structure can significantly improve the convergence rate for the simultaneous estimation of multiple functions. The results also imply that the proposed estimator is rate optimal in the minimax sense for the rank-constraint Sobolev class of vector-valued functions.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"57 1","pages":"122 - 144"},"PeriodicalIF":1.2,"publicationDate":"2022-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76856131","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}
Pub Date : 2022-11-11DOI: 10.1080/10485252.2022.2142222
Christophe Crambes, Chayma Daayeb, A. Gannoun, Yousri Henchiri
Dealing with missing values is an important issue in data observation or data recording process. In this paper, we consider a functional linear regression model with partially observed covariate and missing values in the response. We use a reconstruction operator that aims at recovering the missing parts of the explanatory curves, then we are interested in regression imputation method of missing data on the response variable, using functional principal component regression to estimate the functional coefficient of the model. We study the asymptotic behaviour of the prediction error when missing data are replaced by the imputed values in the original dataset. The practical behaviour of the method is also studied on simulated data and a real dataset.
{"title":"Functional linear model with partially observed covariate and missing values in the response","authors":"Christophe Crambes, Chayma Daayeb, A. Gannoun, Yousri Henchiri","doi":"10.1080/10485252.2022.2142222","DOIUrl":"https://doi.org/10.1080/10485252.2022.2142222","url":null,"abstract":"Dealing with missing values is an important issue in data observation or data recording process. In this paper, we consider a functional linear regression model with partially observed covariate and missing values in the response. We use a reconstruction operator that aims at recovering the missing parts of the explanatory curves, then we are interested in regression imputation method of missing data on the response variable, using functional principal component regression to estimate the functional coefficient of the model. We study the asymptotic behaviour of the prediction error when missing data are replaced by the imputed values in the original dataset. The practical behaviour of the method is also studied on simulated data and a real dataset.","PeriodicalId":50112,"journal":{"name":"Journal of Nonparametric Statistics","volume":"36 1","pages":"172 - 197"},"PeriodicalIF":1.2,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79794673","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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