Mostrando recursos 1 - 20 de 581

  1. Recursive construction of confidence regions

    Bielecki, Tomasz; Chen, Tao; Cialenco, Igor
    Assuming that one-step transition kernel of a discrete time, time-homogenous Markov chain model is parameterized by a parameter $\theta\in\boldsymbol{\Theta}$, we derive a recursive (in time) construction of confidence regions for the unknown parameter of interest, say $\theta^{*}\in\boldsymbol{\Theta}$. It is supposed that the observed data used in the construction of the confidence regions is generated by a Markov chain whose transition kernel corresponds to $\theta^{*}$. The key step in our construction is the derivation of a recursive scheme for an appropriate point estimator of $\theta^{*}$. To achieve this, we start by what we call the base recursive point estimator, using which...

  2. The control of the false discovery rate in fixed sequence multiple testing

    Lynch, Gavin; Guo, Wenge; Sarkar, Sanat K.; Finner, Helmut
    Controlling false discovery rate (FDR) is a powerful approach to multiple testing. In many applications, the tested hypotheses have an inherent hierarchical structure. In this paper, we focus on the fixed sequence structure where the testing order of the hypotheses has been strictly specified in advance. We are motivated to study such a structure, since it is the most basic of hierarchical structure, yet it is often seen in real applications such as statistical process control and streaming data analysis. We first consider a conventional fixed sequence method that stops testing once an acceptance occurs, and develop such a method...

  3. Geometric ergodicity of Rao and Teh’s algorithm for Markov jump processes and CTBNs

    Miasojedow, Błażej; Niemiro, Wojciech
    Rao and Teh (2012, 2013) introduced an efficient MCMC algorithm for sampling from the posterior distribution of a hidden Markov jump process. The algorithm is based on the idea of sampling virtual jumps. In the present paper we show that the Markov chain generated by Rao and Teh’s algorithm is geometrically ergodic. To this end we establish a geometric drift condition towards a small set. A similar result is also proved for a special version of the algorithm, used for probabilistic inference in Continuous Time Bayesian Networks.

  4. Detection of low dimensionality and data denoising via set estimation techniques

    Aaron, Catherine; Cholaquidis, Alejandro; Cuevas, Antonio
    This work is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set $S\subset{\mathbb{R}}^{d}$. The general aim is to identify (via stochastic procedures) some qualitative or quantitative features of $S$, of geometric or topological character. The available information is just a random sample of points drawn on $S$. The term “to identify” means here to achieve a correct answer almost surely (a.s.) when the sample size tends to infinity. More specifically the paper aims at giving some partial answers to the following questions: is $S$ full dimensional? Is $S$...

  5. Revisiting the Hodges-Lehmann estimator in a location mixture model: Is asymptotic normality good enough?

    Balabdaoui, Fadoua
    Location mixture models, resulting in shifting a common distribution with some probability, have been widely used to account for existence of clusters in the data. Assuming only symmetry of this common distribution allows for great flexibility, especially when the traditional normality assumption is violated. This semi-parametric model has been studied in several papers, where the mixture parameters are first estimated before constructing an estimator for the non-parametric component. The plug-in method suggested by Hunter et al. (2007) has the merit to be easily implementable and fast to compute. However, no result is available on the limit distribution of the obtained...

  6. Revisiting the Hodges-Lehmann estimator in a location mixture model: Is asymptotic normality good enough?

    Balabdaoui, Fadoua
    Location mixture models, resulting in shifting a common distribution with some probability, have been widely used to account for existence of clusters in the data. Assuming only symmetry of this common distribution allows for great flexibility, especially when the traditional normality assumption is violated. This semi-parametric model has been studied in several papers, where the mixture parameters are first estimated before constructing an estimator for the non-parametric component. The plug-in method suggested by Hunter et al. (2007) has the merit to be easily implementable and fast to compute. However, no result is available on the limit distribution of the obtained...

  7. Sharp minimax adaptation over Sobolev ellipsoids in nonparametric testing

    Ji, Pengsheng; Nussbaum, Michael
    In the problem of testing for signal in Gaussian white noise, over a smoothness class with an $L_{2}$-ball removed, minimax rates of convergences (separation rates) are well known (Ingster [24]); they are expressed in the rate of the ball radius tending to zero along with noise intensity, such that a nontrivial asymptotic power is possible. It is also known that, if the smoothness class is a Sobolev type ellipsoid of degree $\beta$ and size $M$, the optimal rate result can be sharpened towards a Pinsker type asymptotics for the critical radius (Ermakov [9]). The minimax optimal tests in that setting...

  8. Sharp minimax adaptation over Sobolev ellipsoids in nonparametric testing

    Ji, Pengsheng; Nussbaum, Michael
    In the problem of testing for signal in Gaussian white noise, over a smoothness class with an $L_{2}$-ball removed, minimax rates of convergences (separation rates) are well known (Ingster [24]); they are expressed in the rate of the ball radius tending to zero along with noise intensity, such that a nontrivial asymptotic power is possible. It is also known that, if the smoothness class is a Sobolev type ellipsoid of degree $\beta$ and size $M$, the optimal rate result can be sharpened towards a Pinsker type asymptotics for the critical radius (Ermakov [9]). The minimax optimal tests in that setting...

  9. Bounded isotonic regression

    Luss, Ronny; Rosset, Saharon
    Isotonic regression offers a flexible modeling approach under monotonicity assumptions, which are natural in many applications. Despite this attractive setting and extensive theoretical research, isotonic regression has enjoyed limited interest in practical modeling primarily due to its tendency to suffer significant overfitting, even in moderate dimension, as the monotonicity constraints do not offer sufficient complexity control. Here we propose to regularize isotonic regression by penalizing or constraining the range of the fitted model (i.e., the difference between the maximal and minimal predictions). We show that the optimal solution to this problem is obtained by constraining the non-penalized isotonic regression model...

  10. Bounded isotonic regression

    Luss, Ronny; Rosset, Saharon
    Isotonic regression offers a flexible modeling approach under monotonicity assumptions, which are natural in many applications. Despite this attractive setting and extensive theoretical research, isotonic regression has enjoyed limited interest in practical modeling primarily due to its tendency to suffer significant overfitting, even in moderate dimension, as the monotonicity constraints do not offer sufficient complexity control. Here we propose to regularize isotonic regression by penalizing or constraining the range of the fitted model (i.e., the difference between the maximal and minimal predictions). We show that the optimal solution to this problem is obtained by constraining the non-penalized isotonic regression model...

  11. Partition structure and the $A$-hypergeometric distribution associated with the rational normal curve

    Mano, Shuhei
    A distribution whose normalization constant is an $A$-hypergeometric polynomial is called an $A$-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an $A$-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associated with the rational normal curve, appears in inferences involving exchangeable partition structures. An exact sampling algorithm is presented for the general (any number of rows) $A$-hypergeometric distributions. Then, the maximum likelihood estimation of the $A$-hypergeometric distribution associated with the rational normal curve, which is an...

  12. Partition structure and the $A$-hypergeometric distribution associated with the rational normal curve

    Mano, Shuhei
    A distribution whose normalization constant is an $A$-hypergeometric polynomial is called an $A$-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an $A$-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associated with the rational normal curve, appears in inferences involving exchangeable partition structures. An exact sampling algorithm is presented for the general (any number of rows) $A$-hypergeometric distributions. Then, the maximum likelihood estimation of the $A$-hypergeometric distribution associated with the rational normal curve, which is an...

  13. Gradient angle-based analysis for spatiotemporal point processes

    Zhang, Tonglin; Huang, Yen-Ning
    Spatiotemporal point processes (STPPs) are important in modeling randomly appeared events developed in space and time. Statistical methods of STPPs have been widely used in applications. In all of these methods, evaluations and inferences of intensity functions are the primary issues. The present article proposes a new method, which attempts to evaluate angles of gradient vectors of intensity functions rather than the intensity functions themselves. According to the nature of many natural and human phenomena, the evaluation of angle patterns of the gradient vectors is more important than the evaluation of their magnitude patterns because changes of angle patterns often...

  14. Gradient angle-based analysis for spatiotemporal point processes

    Zhang, Tonglin; Huang, Yen-Ning
    Spatiotemporal point processes (STPPs) are important in modeling randomly appeared events developed in space and time. Statistical methods of STPPs have been widely used in applications. In all of these methods, evaluations and inferences of intensity functions are the primary issues. The present article proposes a new method, which attempts to evaluate angles of gradient vectors of intensity functions rather than the intensity functions themselves. According to the nature of many natural and human phenomena, the evaluation of angle patterns of the gradient vectors is more important than the evaluation of their magnitude patterns because changes of angle patterns often...

  15. Approximate likelihood inference in generalized linear latent variable models based on the dimension-wise quadrature

    Bianconcini, Silvia; Cagnone, Silvia; Rizopoulos, Dimitris
    We propose a new method to perform approximate likelihood inference in latent variable models. Our approach provides an approximation of the integrals involved in the likelihood function through a reduction of their dimension that makes the computation feasible in situations in which classical and adaptive quadrature based methods are not applicable. We derive new theoretical results on the accuracy of the obtained estimators. We show that the proposed approximation outperforms several existing methods in simulations, and it can be successfully applied in presence of multidimensional longitudinal data when standard techniques are not applicable or feasible.

  16. Approximate likelihood inference in generalized linear latent variable models based on the dimension-wise quadrature

    Bianconcini, Silvia; Cagnone, Silvia; Rizopoulos, Dimitris
    We propose a new method to perform approximate likelihood inference in latent variable models. Our approach provides an approximation of the integrals involved in the likelihood function through a reduction of their dimension that makes the computation feasible in situations in which classical and adaptive quadrature based methods are not applicable. We derive new theoretical results on the accuracy of the obtained estimators. We show that the proposed approximation outperforms several existing methods in simulations, and it can be successfully applied in presence of multidimensional longitudinal data when standard techniques are not applicable or feasible.

  17. A provable smoothing approach for high dimensional generalized regression with applications in genomics

    Han, Fang; Ji, Hongkai; Ji, Zhicheng; Wang, Honglang
    In many applications, linear models fit the data poorly. This article studies an appealing alternative, the generalized regression model. This model only assumes that there exists an unknown monotonically increasing link function connecting the response $Y$ to a single index $\boldsymbol{X} ^{\mathsf{T}}\boldsymbol{\beta } ^{*}$ of explanatory variables $\boldsymbol{X} \in{\mathbb{R}} ^{d}$. The generalized regression model is flexible and covers many widely used statistical models. It fits the data generating mechanisms well in many real problems, which makes it useful in a variety of applications where regression models are regularly employed. In low dimensions, rank-based M-estimators are recommended to deal with the...

  18. A provable smoothing approach for high dimensional generalized regression with applications in genomics

    Han, Fang; Ji, Hongkai; Ji, Zhicheng; Wang, Honglang
    In many applications, linear models fit the data poorly. This article studies an appealing alternative, the generalized regression model. This model only assumes that there exists an unknown monotonically increasing link function connecting the response $Y$ to a single index $\boldsymbol{X} ^{\mathsf{T}}\boldsymbol{\beta } ^{*}$ of explanatory variables $\boldsymbol{X} \in{\mathbb{R}} ^{d}$. The generalized regression model is flexible and covers many widely used statistical models. It fits the data generating mechanisms well in many real problems, which makes it useful in a variety of applications where regression models are regularly employed. In low dimensions, rank-based M-estimators are recommended to deal with the...

  19. Nonparametric estimating equations for circular probability density functions and their derivatives

    Di Marzio, Marco; Fensore, Stefania; Panzera, Agnese; Taylor, Charles C.
    We propose estimating equations whose unknown parameters are the values taken by a circular density and its derivatives at a point. Specifically, we solve equations which relate local versions of population trigonometric moments with their sample counterparts. Major advantages of our approach are: higher order bias without asymptotic variance inflation, closed form for the estimators, and absence of numerical tasks. We also investigate situations where the observed data are dependent. Theoretical results along with simulation experiments are provided.

  20. Nonparametric estimating equations for circular probability density functions and their derivatives

    Di Marzio, Marco; Fensore, Stefania; Panzera, Agnese; Taylor, Charles C.
    We propose estimating equations whose unknown parameters are the values taken by a circular density and its derivatives at a point. Specifically, we solve equations which relate local versions of population trigonometric moments with their sample counterparts. Major advantages of our approach are: higher order bias without asymptotic variance inflation, closed form for the estimators, and absence of numerical tasks. We also investigate situations where the observed data are dependent. Theoretical results along with simulation experiments are provided.

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