Pub Date : 2019-08-05DOI: 10.3103/s1066530719020054
G. Golubev, M. Safarian
Let X1, X2,... be independent random variables observed sequentially and such that X1,..., Xθ−1 have a common probability density p0, while Xθ, Xθ+1,... are all distributed according to p1 ≠ p0. It is assumed that p0 and p1 are known, but the time change θ ∈ ℤ+ is unknown and the goal is to construct a stopping time τ that detects the change-point θ as soon as possible. The standard approaches to this problem rely essentially on some prior information about θ. For instance, in the Bayes approach, it is assumed that θ is a random variable with a known probability distribution. In the methods related to hypothesis testing, this a priori information is hidden in the so-called average run length. The main goal in this paper is to construct stopping times that are free from a priori information about θ. More formally, we propose an approach to solving approximately the following minimization problem:$$Delta(theta;{tau^alpha})rightarrowmin_{tau^alpha};;text{subject};text{to};;alpha(theta;{tau^alpha})leqalpha;text{for};text{any};thetageq1,$$where α(θ; τ) = Pθ{τ < θ} is the false alarm probability and Δ(θ; τ) = Eθ(τ − θ)+ is the average detection delay computed for a given stopping time τ. In contrast to the standard CUSUM algorithm based on the sequential maximum likelihood test, our approach is related to a multiple hypothesis testing methods and permits, in particular, to construct universal stopping times with nearly Bayes detection delays.
{"title":"A Multiple Hypothesis Testing Approach to Detection Changes in Distribution","authors":"G. Golubev, M. Safarian","doi":"10.3103/s1066530719020054","DOIUrl":"https://doi.org/10.3103/s1066530719020054","url":null,"abstract":"Let <i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>,... be independent random variables observed sequentially and such that <i>X</i><sub>1</sub>,..., <i>X</i><sub><i>θ</i>−1</sub> have a common probability density <i>p</i><sub><i>0</i></sub>, while <i>X</i><sub><i>θ</i></sub>, <i>X</i><sub><i>θ</i>+1</sub>,... are all distributed according to <i>p</i><sub>1</sub> ≠ <i>p</i><sub>0</sub>. It is assumed that <i>p</i><sub>0</sub> and <i>p</i><sub>1</sub> are known, but the time change <i>θ</i> ∈ ℤ<sup>+</sup> is unknown and the goal is to construct a stopping time <i>τ</i> that detects the change-point <i>θ</i> as soon as possible. The standard approaches to this problem rely essentially on some prior information about <i>θ</i>. For instance, in the Bayes approach, it is assumed that <i>θ</i> is a random variable with a known probability distribution. In the methods related to hypothesis testing, this a priori information is hidden in the so-called average run length. The main goal in this paper is to construct stopping times that are free from a priori information about <i>θ.</i> More formally, we propose an approach to solving approximately the following minimization problem:<span>$$Delta(theta;{tau^alpha})rightarrowmin_{tau^alpha};;text{subject};text{to};;alpha(theta;{tau^alpha})leqalpha;text{for};text{any};thetageq1,$$</span>where <i>α</i>(<i>θ; τ</i>) = P<sub><i>θ</i></sub>{<i>τ < θ</i>} is <i>the false alarm probability</i> and <i>Δ</i>(<i>θ</i>; <i>τ</i>) = E<sub><i>θ</i></sub>(<i>τ − θ</i>)<sub>+</sub> is <i>the average detection delay</i> computed for a given stopping time <i>τ</i>. In contrast to the standard CUSUM algorithm based on the sequential maximum likelihood test, our approach is related to a multiple hypothesis testing methods and permits, in particular, to construct universal stopping times with nearly Bayes detection delays.","PeriodicalId":46039,"journal":{"name":"Mathematical Methods of Statistics","volume":"52 1","pages":"155-167"},"PeriodicalIF":0.5,"publicationDate":"2019-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-07-01DOI: 10.3103/S1066530719030013
S. Bouzebda, B. Nemouchi
{"title":"Central Limit Theorems for Conditional Empirical and Conditional U-Processes of Stationary Mixing Sequences","authors":"S. Bouzebda, B. Nemouchi","doi":"10.3103/S1066530719030013","DOIUrl":"https://doi.org/10.3103/S1066530719030013","url":null,"abstract":"","PeriodicalId":46039,"journal":{"name":"Mathematical Methods of Statistics","volume":"28 1","pages":"169 - 207"},"PeriodicalIF":0.5,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3103/S1066530719030013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43823131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-03DOI: 10.3103/s1066530719010022
A. Havet, M. Lerasle, É. Moulines
We consider the problem of nonparametric density estimation of a random environment from the observation of a single trajectory of a random walk in this environment. We build several density estimators using the beta-moments of this distribution. Then we apply the Goldenschluger-Lepski method to select an estimator satisfying an oracle type inequality. We obtain non-asymptotic bounds for the supremum norm of these estimators that hold when the RWRE is recurrent or transient to the right. A simulation study supports our theoretical findings.
{"title":"Density Estimation for RWRE","authors":"A. Havet, M. Lerasle, É. Moulines","doi":"10.3103/s1066530719010022","DOIUrl":"https://doi.org/10.3103/s1066530719010022","url":null,"abstract":"We consider the problem of nonparametric density estimation of a random environment from the observation of a single trajectory of a random walk in this environment. We build several density estimators using the beta-moments of this distribution. Then we apply the Goldenschluger-Lepski method to select an estimator satisfying an oracle type inequality. We obtain non-asymptotic bounds for the supremum norm of these estimators that hold when the RWRE is recurrent or transient to the right. A simulation study supports our theoretical findings.","PeriodicalId":46039,"journal":{"name":"Mathematical Methods of Statistics","volume":"289 ","pages":"18-38"},"PeriodicalIF":0.5,"publicationDate":"2019-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-03DOI: 10.3103/s1066530719010046
M. V. Boldin
We consider a stationary linear AR(p) model with contamination (gross errors in the observations). The autoregression parameters are unknown, as well as the distribution of innovations. Based on the residuals from the parameter estimates, an analog of the empirical distribution function is defined and a test of Pearson’s chi-square type is constructed for testing hypotheses on the distribution of innovations. We obtain the asymptotic power of this test under local alternatives and establish its qualitative robustness under the hypothesis and alternatives.
{"title":"On the Power of Pearson’s Test under Local Alternatives in Autoregression with Outliers","authors":"M. V. Boldin","doi":"10.3103/s1066530719010046","DOIUrl":"https://doi.org/10.3103/s1066530719010046","url":null,"abstract":"We consider a stationary linear <i>AR</i>(<i>p</i>) model with contamination (gross errors in the observations). The autoregression parameters are unknown, as well as the distribution of innovations. Based on the residuals from the parameter estimates, an analog of the empirical distribution function is defined and a test of Pearson’s chi-square type is constructed for testing hypotheses on the distribution of innovations. We obtain the asymptotic power of this test under local alternatives and establish its qualitative robustness under the hypothesis and alternatives.","PeriodicalId":46039,"journal":{"name":"Mathematical Methods of Statistics","volume":"104 ","pages":"57-65"},"PeriodicalIF":0.5,"publicationDate":"2019-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-03DOI: 10.3103/s1066530719010058
C. Joutard
We establish a large deviation approximation for the density of an arbitrary sequence of random vectors, by assuming several assumptions on the normalized cumulant generating function and its derivatives. We give two statistical applications to illustrate the result, the first one dealing with a vector of independent sample variances and the second one with a Gaussian multiple linear regression model. Numerical comparisons are eventually provided for these two examples.
{"title":"A Large Deviation Approximation for Multivariate Density Functions","authors":"C. Joutard","doi":"10.3103/s1066530719010058","DOIUrl":"https://doi.org/10.3103/s1066530719010058","url":null,"abstract":"We establish a large deviation approximation for the density of an arbitrary sequence of random vectors, by assuming several assumptions on the normalized cumulant generating function and its derivatives. We give two statistical applications to illustrate the result, the first one dealing with a vector of independent sample variances and the second one with a Gaussian multiple linear regression model. Numerical comparisons are eventually provided for these two examples.","PeriodicalId":46039,"journal":{"name":"Mathematical Methods of Statistics","volume":"5 8 1","pages":"66-73"},"PeriodicalIF":0.5,"publicationDate":"2019-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-03DOI: 10.3103/s1066530719010034
S. Khardani
In this work we suppose that the random vector (X, Y) satisfies the regression model Y = m(X) + ϵ, where m(·) belongs to some parametric class {({m_beta}(cdot):beta in mathbb{K})} and the error ϵ is independent of the covariate X. The response Y is subject to random right censoring. Using a nonlinear mode regression, a new estimation procedure for the true unknown parameter vector β0is proposed that extends the classical least squares procedure for nonlinear regression. We also establish asymptotic properties for the proposed estimator under assumptions of the error density. We investigate the performance through a simulation study.
在这项工作中，我们假设随机向量(X, Y)满足回归模型Y = m(X) + λ，其中m(·)属于某个参数类{({m_beta}(cdot):beta in mathbb{K})}，并且误差λ独立于协变量X。响应Y受到随机右删减。利用非线性模态回归，提出了一种新的真未知参数向量β0的估计方法，扩展了经典的非线性回归最小二乘估计方法。在误差密度的假设下，我们还建立了所提估计量的渐近性质。我们通过仿真研究来考察其性能。
{"title":"A Semi-Parametric Mode Regression with Censored Data","authors":"S. Khardani","doi":"10.3103/s1066530719010034","DOIUrl":"https://doi.org/10.3103/s1066530719010034","url":null,"abstract":"In this work we suppose that the random vector (<i>X</i>, <i>Y</i>) satisfies the regression model <i>Y</i> = <i>m</i>(<i>X</i>) + <i>ϵ</i>, where <i>m</i>(·) belongs to some parametric class {<span>({m_beta}(cdot):beta in mathbb{K})</span>} and the error <i>ϵ</i> is independent of the covariate <i>X</i>. The response <i>Y</i> is subject to random right censoring. Using a nonlinear mode regression, a new estimation procedure for the true unknown parameter vector <i>β</i><sub>0</sub>is proposed that extends the classical least squares procedure for nonlinear regression. We also establish asymptotic properties for the proposed estimator under assumptions of the error density. We investigate the performance through a simulation study.","PeriodicalId":46039,"journal":{"name":"Mathematical Methods of Statistics","volume":"293 ","pages":"39-56"},"PeriodicalIF":0.5,"publicationDate":"2019-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-01DOI: 10.3103/S1066530719020042
M. Boldin
{"title":"On the Asymptotic Power of Tests of Fit under Local Alternatives in Autoregression","authors":"M. Boldin","doi":"10.3103/S1066530719020042","DOIUrl":"https://doi.org/10.3103/S1066530719020042","url":null,"abstract":"","PeriodicalId":46039,"journal":{"name":"Mathematical Methods of Statistics","volume":"28 1","pages":"144 - 154"},"PeriodicalIF":0.5,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3103/S1066530719020042","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69418834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：