The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener‐Itô integrals.
{"title":"The basic distributional theory for the product of zero mean correlated normal random variables","authors":"Robert E. Gaunt","doi":"10.1111/stan.12267","DOIUrl":"https://doi.org/10.1111/stan.12267","url":null,"abstract":"The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener‐Itô integrals.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"32 1","pages":"450 - 470"},"PeriodicalIF":1.5,"publicationDate":"2021-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76103266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The beta family owes its privileged status within unit interval distributions to several relevant features such as, for example, easiness of interpretation and versatility in modeling different types of data. However, the flexibility of its density at the endpoints of the support is poor enough to prevent from properly modeling the data portions having values next to zero and one. Such a drawback can be overcome by resorting to the class of the noncentral beta distributions. Indeed, the latter allows the density to take on arbitrary positive and finite limits which have a really simple form. Nevertheless, the analytical and mathematical complexity of this distribution poses strong limitations on its use as a model for data on the real interval (0, 1). That said, an in‐depth study of a newly found analogue of the noncentral beta distribution is carried out in this article. The latter preserves the applicative potential of the standard noncentral beta class but with the advantage of showing a more straightforward and easily handleable density.
{"title":"On the conditional noncentral beta distribution","authors":"C. Orsi","doi":"10.1111/stan.12249","DOIUrl":"https://doi.org/10.1111/stan.12249","url":null,"abstract":"The beta family owes its privileged status within unit interval distributions to several relevant features such as, for example, easiness of interpretation and versatility in modeling different types of data. However, the flexibility of its density at the endpoints of the support is poor enough to prevent from properly modeling the data portions having values next to zero and one. Such a drawback can be overcome by resorting to the class of the noncentral beta distributions. Indeed, the latter allows the density to take on arbitrary positive and finite limits which have a really simple form. Nevertheless, the analytical and mathematical complexity of this distribution poses strong limitations on its use as a model for data on the real interval (0, 1). That said, an in‐depth study of a newly found analogue of the noncentral beta distribution is carried out in this article. The latter preserves the applicative potential of the standard noncentral beta class but with the advantage of showing a more straightforward and easily handleable density.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"68 1","pages":"164 - 189"},"PeriodicalIF":1.5,"publicationDate":"2021-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80063061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Random walks, intrinsic autoregression, state‐space models, smoothing splines, and so on have been widely used in various areas of statistics. However, practitioners wanting to fit these models using existing packages for random‐effects models are often faced with the difficulty that their covariance matrices are not uniquely determined. Unfortunately, different specifications of the model lead to different covariance structures, giving different analyses. Even if we make a decision on specification it is not immediately obvious how to make inferences from these models. There have been various suggestions on how to overcome such difficulties. However, they differ, implying that there is as yet no agreed remedy. In this article we provide a unified view on these alternatives and show how the analysis can be made invariant with respect to the choice of covariance by inclusion of a suitable set of covariates. Several examples are used to illustrate the approach.
{"title":"Resolving the ambiguity of random‐effects models with singular precision matrix","authors":"Woojoo Lee, H. Piepho, Youngjo Lee","doi":"10.1111/stan.12244","DOIUrl":"https://doi.org/10.1111/stan.12244","url":null,"abstract":"Random walks, intrinsic autoregression, state‐space models, smoothing splines, and so on have been widely used in various areas of statistics. However, practitioners wanting to fit these models using existing packages for random‐effects models are often faced with the difficulty that their covariance matrices are not uniquely determined. Unfortunately, different specifications of the model lead to different covariance structures, giving different analyses. Even if we make a decision on specification it is not immediately obvious how to make inferences from these models. There have been various suggestions on how to overcome such difficulties. However, they differ, implying that there is as yet no agreed remedy. In this article we provide a unified view on these alternatives and show how the analysis can be made invariant with respect to the choice of covariance by inclusion of a suitable set of covariates. Several examples are used to illustrate the approach.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"6 1","pages":"482 - 499"},"PeriodicalIF":1.5,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82348589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. M. Bayer, Francisco Cribari‐Neto, Jéssica Santos
Models based on the Kumaraswamy law are used with variables that assume values in (0, 1). In some cases, however, the data contain zeros and/or ones, that is, there is data inflation. We introduce a class of regression models that can be used with such inflated data, namely: the class of inflated Kumaraswamy regression models. We consider inflation at zero, at one, and at both zero and one. We introduce the model and provide closed‐form expressions for its score vector and Fisher's information matrix. The proposed model is used to evaluate the impacts of different conditioning variables on the proportion of people who live in households with inadequate water supply and sewage in Brazilian municipalities. Our results reveal that policies directed to increasing the population share with college education in places where it is low are particularly effective in reducing the prevalence of people who live under inadequate sanitation conditions.
{"title":"Inflated Kumaraswamy regressions with application to water supply and sanitation in Brazil","authors":"F. M. Bayer, Francisco Cribari‐Neto, Jéssica Santos","doi":"10.1111/stan.12242","DOIUrl":"https://doi.org/10.1111/stan.12242","url":null,"abstract":"Models based on the Kumaraswamy law are used with variables that assume values in (0, 1). In some cases, however, the data contain zeros and/or ones, that is, there is data inflation. We introduce a class of regression models that can be used with such inflated data, namely: the class of inflated Kumaraswamy regression models. We consider inflation at zero, at one, and at both zero and one. We introduce the model and provide closed‐form expressions for its score vector and Fisher's information matrix. The proposed model is used to evaluate the impacts of different conditioning variables on the proportion of people who live in households with inadequate water supply and sewage in Brazilian municipalities. Our results reveal that policies directed to increasing the population share with college education in places where it is low are particularly effective in reducing the prevalence of people who live under inadequate sanitation conditions.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"30 1","pages":"453 - 481"},"PeriodicalIF":1.5,"publicationDate":"2021-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79501911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a system of dependent Poisson variables, where each variable is the sum of an independent variate and a common variate. It is the common variate that creates the dependence. Within this system, a test of independence may be constructed where the null hypothesis is that the common variate is identically zero. In the present paper, we consider the maximum log likelihood ratio test. For this test, it is well‐known that the asymptotic distribution of the test statistic is an equal mixture of zero and a chi‐square distribution with one degree of freedom. We examine a Bartlett correction of the test, in the hope that we will get better approximation of the nominal size for moderately large sample sizes. A correction of this type is explicitly derived, and its usefulness is explored in a simulation study. For practical purposes, the correction is found to be useful in dimension two, but not in higher dimensions.
{"title":"Bartlett correction of an independence test in a multivariate Poisson model","authors":"Rolf Larsson","doi":"10.1111/stan.12265","DOIUrl":"https://doi.org/10.1111/stan.12265","url":null,"abstract":"We consider a system of dependent Poisson variables, where each variable is the sum of an independent variate and a common variate. It is the common variate that creates the dependence. Within this system, a test of independence may be constructed where the null hypothesis is that the common variate is identically zero. In the present paper, we consider the maximum log likelihood ratio test. For this test, it is well‐known that the asymptotic distribution of the test statistic is an equal mixture of zero and a chi‐square distribution with one degree of freedom. We examine a Bartlett correction of the test, in the hope that we will get better approximation of the nominal size for moderately large sample sizes. A correction of this type is explicitly derived, and its usefulness is explored in a simulation study. For practical purposes, the correction is found to be useful in dimension two, but not in higher dimensions.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"27 1","pages":"391 - 417"},"PeriodicalIF":1.5,"publicationDate":"2021-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87499206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 introduce the mixed estimators based on product least relative error estimation and least squares estimation in a multiplicative linear regression model. The asymptotic properties for the mixed estimators are established. We present some explicit expressions of the optimal estimator of the mixed estimators, and we also suggest some numerical solutions in the simulation studies and real data analysis. Studying model checking problems for multiplicative linear regression models, we propose four test statistics. One is the score‐type test statistic, the second one is the residual‐based empirical process test statistic marked by proper functions of the covariates. The third one is the integrated conditional moment test statistic by using linear projection weighting function, and the fourth one is the adaptive model test statistic. These test statistics are all related to the mixed estimators. The asymptotic properties of these test statistics are established, and some bootstrap procedures for calculating the critical values are also proposed. Simulation studies are conducted to demonstrate the performance of the proposed estimation procedures, and a real example is analyzed to illustrate its practical usage.
{"title":"Model checking for multiplicative linear regression models with mixed estimators","authors":"Jun Zhang","doi":"10.1111/stan.12239","DOIUrl":"https://doi.org/10.1111/stan.12239","url":null,"abstract":"In this paper, we introduce the mixed estimators based on product least relative error estimation and least squares estimation in a multiplicative linear regression model. The asymptotic properties for the mixed estimators are established. We present some explicit expressions of the optimal estimator of the mixed estimators, and we also suggest some numerical solutions in the simulation studies and real data analysis. Studying model checking problems for multiplicative linear regression models, we propose four test statistics. One is the score‐type test statistic, the second one is the residual‐based empirical process test statistic marked by proper functions of the covariates. The third one is the integrated conditional moment test statistic by using linear projection weighting function, and the fourth one is the adaptive model test statistic. These test statistics are all related to the mixed estimators. The asymptotic properties of these test statistics are established, and some bootstrap procedures for calculating the critical values are also proposed. Simulation studies are conducted to demonstrate the performance of the proposed estimation procedures, and a real example is analyzed to illustrate its practical usage.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"1 1","pages":"364 - 403"},"PeriodicalIF":1.5,"publicationDate":"2021-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87409760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Monotone single index models have gained increasing popularity over the past decades due to their flexibility and versatile use in diverse areas. Semi‐parametric estimators such as the least squares and maximum likelihood estimators of the unknown index and monotone ridge function were considered to make inference in such models without having to choose some tuning parameter. Description of the asymptotic behavior of those estimators crucially depends on acquiring a good understanding of the optimization problems associated with the corresponding population criteria. In this paper, we give several insights into these criteria by proving existence of minimizers thereof over general classes of parameters. In order to describe these minimizers, we prove different results which give the direction of variation of the population criteria in general and in the special case where the common distribution of the covariates is Gaussian. A complementary simulation study was performed and whose results give support to our main theorems.
{"title":"On the population least‐squares criterion in the monotone single index model","authors":"F. Balabdaoui, C. Durot, Christopher Fragneau","doi":"10.1111/stan.12240","DOIUrl":"https://doi.org/10.1111/stan.12240","url":null,"abstract":"Monotone single index models have gained increasing popularity over the past decades due to their flexibility and versatile use in diverse areas. Semi‐parametric estimators such as the least squares and maximum likelihood estimators of the unknown index and monotone ridge function were considered to make inference in such models without having to choose some tuning parameter. Description of the asymptotic behavior of those estimators crucially depends on acquiring a good understanding of the optimization problems associated with the corresponding population criteria. In this paper, we give several insights into these criteria by proving existence of minimizers thereof over general classes of parameters. In order to describe these minimizers, we prove different results which give the direction of variation of the population criteria in general and in the special case where the common distribution of the covariates is Gaussian. A complementary simulation study was performed and whose results give support to our main theorems.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"1 1","pages":"408 - 436"},"PeriodicalIF":1.5,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86521625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article focuses on the efficient estimation problem of an arbitrary‐order periodic integer‐valued autoregressive (PINAR(p)) model. Both the local asymptotic normality (LAN) property and the local asymptotic linearity property satisfied by the central sequence of the underlying model are established. Using these results, we construct efficient estimators for the parameters in a parametric framework. The consistency property of these efficient estimations is evaluated via an intensive simulation study. Moreover, the performances of these efficient estimations, over the conditional maximum likelihood (CML) and the conditional least squares (CLS) estimations, are also illustrated via an intensive simulation study and an application on real data set.
{"title":"Locally asymptotically efficient estimation for parametric PINAR(p) models","authors":"Mohamed Sadoun, M. Bentarzi","doi":"10.1111/stan.12234","DOIUrl":"https://doi.org/10.1111/stan.12234","url":null,"abstract":"This article focuses on the efficient estimation problem of an arbitrary‐order periodic integer‐valued autoregressive (PINAR(p)) model. Both the local asymptotic normality (LAN) property and the local asymptotic linearity property satisfied by the central sequence of the underlying model are established. Using these results, we construct efficient estimators for the parameters in a parametric framework. The consistency property of these efficient estimations is evaluated via an intensive simulation study. Moreover, the performances of these efficient estimations, over the conditional maximum likelihood (CML) and the conditional least squares (CLS) estimations, are also illustrated via an intensive simulation study and an application on real data set.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"16 1","pages":"257 - 289"},"PeriodicalIF":1.5,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80321948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new estimation method is proposed founded upon a nonlinear conjugate gradient‐type algorithm having an efficient line search technique for cure rate models with competing risks, which are subject to elimination. An extensive simulation study is carried out to compare the performance of the proposed algorithm with some existing algorithms, including other conjugate gradient‐type algorithms and the expectation maximization algorithm. For this purpose, it is assumed that the initial competing risks follow an exponentially weighted Poisson distribution. In particular, it is shown that that the proposed algorithm produces estimates that are more accurate and efficient (i.e., the bias and root mean square errors are smaller), specifically with respect to the parameters related to the cure rate. Although for the purpose of simulation study an exponentially weighted Poisson competing risks distribution is assumed, the proposed algorithm incorporates a generic framework that can accommodate any competing risks distribution. Finally, a real data application is provided.
{"title":"On the estimation of destructive cure rate model: A new study with exponentially weighted Poisson competing risks","authors":"S. Pal, Souvik Roy","doi":"10.1111/stan.12237","DOIUrl":"https://doi.org/10.1111/stan.12237","url":null,"abstract":"A new estimation method is proposed founded upon a nonlinear conjugate gradient‐type algorithm having an efficient line search technique for cure rate models with competing risks, which are subject to elimination. An extensive simulation study is carried out to compare the performance of the proposed algorithm with some existing algorithms, including other conjugate gradient‐type algorithms and the expectation maximization algorithm. For this purpose, it is assumed that the initial competing risks follow an exponentially weighted Poisson distribution. In particular, it is shown that that the proposed algorithm produces estimates that are more accurate and efficient (i.e., the bias and root mean square errors are smaller), specifically with respect to the parameters related to the cure rate. Although for the purpose of simulation study an exponentially weighted Poisson competing risks distribution is assumed, the proposed algorithm incorporates a generic framework that can accommodate any competing risks distribution. Finally, a real data application is provided.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"22 1","pages":"324 - 342"},"PeriodicalIF":1.5,"publicationDate":"2021-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90366481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Cancho, Gladys D. C. Barriga, G. Cordeiro, E. Ortega, A. K. Suzuki
It is introduced the proportional hazards frailty model to allow a discrete distribution for the frailty variable. Frailty zero can be interpreted as being immune or cured. It is defined a class of survival models induced by a discrete frailty having a mixed Poisson distribution, which can account for unobserved dispersion. Further, a new regression to evaluate the effects of covariates in the cure fraction is constructed. Several former cure survival models are special cases of the proposed modeling framework. The inferential approach is based on Bayesian methods. Some simulation results are provided to assess the performance of the new regression. Its importance is illustrated by means of an application to colorectal cancer data.
{"title":"Bayesian survival model induced by frailty for lifetime with long‐term survivors","authors":"V. Cancho, Gladys D. C. Barriga, G. Cordeiro, E. Ortega, A. K. Suzuki","doi":"10.1111/stan.12236","DOIUrl":"https://doi.org/10.1111/stan.12236","url":null,"abstract":"It is introduced the proportional hazards frailty model to allow a discrete distribution for the frailty variable. Frailty zero can be interpreted as being immune or cured. It is defined a class of survival models induced by a discrete frailty having a mixed Poisson distribution, which can account for unobserved dispersion. Further, a new regression to evaluate the effects of covariates in the cure fraction is constructed. Several former cure survival models are special cases of the proposed modeling framework. The inferential approach is based on Bayesian methods. Some simulation results are provided to assess the performance of the new regression. Its importance is illustrated by means of an application to colorectal cancer data.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"4 1","pages":"299 - 323"},"PeriodicalIF":1.5,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89128639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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