Hilbert Space Methods in Probability and Statistical Inference

Small, Christopher G.; McLeish, D. L.

doi:10.1002/9781118165522

Cited by 79 publications

(55 citation statements)

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“…As suggested by the notation, a β and b β can depend on β. It is shown by Small and McLeish (2011) that, the space T is a Hilbert space with inner product given by g 1 (U ; β), g 2 (U ; β) = E β (g 1 (U ; β)g 2 (U ; β)), for any g 1 (U ; β), g 2 (U ; β) ∈ T . Similarly, consider the linear space spanned by the nuisance score functions…”

Section: Geometric Interpretation and Further Discussionmentioning

confidence: 99%

A general theory of hypothesis tests and confidence regions for sparse high dimensional models

Ning

Liu²

2017

Ann. Statist.

246

327

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We consider the problem of uncertainty assessment for low dimensional components in high dimensional models. Specifically, we propose a decorrelated score function to handle the impact of high dimensional nuisance parameters. We consider both hypothesis tests and confidence regions for generic penalized M-estimators. Unlike most existing inferential methods which are tailored for individual models, our approach provides a general framework for high dimensional inference and is applicable to a wide range of applications. From the testing perspective, we develop general theorems to characterize the limiting distributions of the decorrelated score test statistic under both null hypothesis and local alternatives. These results provide asymptotic guarantees on the type I errors and local powers of the proposed test. Furthermore, we show that the decorrelated score function can be used to construct point estimators that are semiparametrically efficient and the optimal confidence regions. We also generalize this framework to broaden its applications. First, we extend it to handle high dimensional null hypothesis, where the number of parameters of interest can increase exponentially fast with the sample size. Second, we establish the theory for model misspecification. Third, we go beyond the likelihood framework, by introducing the generalized score test based on general loss functions. Thorough numerical studies are conducted to back up the developed theoretical results.

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Section: Geometric Interpretation and Further Discussionmentioning

confidence: 99%

A general theory of hypothesis tests and confidence regions for sparse high dimensional models

Ning

Liu²

2017

Ann. Statist.

246

327

View full text Add to dashboard Cite

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“…This leads to the matching quantiles estimation (MQE) for the purpose of matching a target distribution. To the best of our knowledge, MQE has not been used in this particular context, though the idea of matching quantiles has been explored in other contexts; see, for example, Karian and Dudewicz (1999), Small and McLeish (1994), and Dominicy and Veredas (2013). Furthermore, our inference procedure is different from those in the aforementioned papers due to the different nature of our problem.…”

Section: Introductionmentioning

confidence: 84%

Matching a Distribution by Matching Quantiles Estimation

Sgouropoulos

Yao

Yastremiz

2015

Journal of the American Statistical Association

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Motivated by the problem of selecting representative portfolios for backtesting counterparty credit risks, we propose a matching quantiles estimation (MQE) method for matching a target distribution by that of a linear combination of a set of random variables. An iterative procedure based on the ordinary least-squares estimation (OLS) is proposed to compute MQE. MQE can be easily modified by adding a LASSO penalty term if a sparse representation is desired, or by restricting the matching within certain range of quantiles to match a part of the target distribution. The convergence of the algorithm and the asymptotic properties of the estimation, both with or without LASSO, are established. A measure and an associated statistical test are proposed to assess the goodness-of-match. The finite sample properties are illustrated by simulation. An application in selecting a counterparty representative portfolio with a real dataset is reported. The proposed MQE also finds applications in portfolio tracking, which demonstrates the usefulness of combining MQE with LASSO.

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“…Small and McLeish (1994) were the first to use estimating functions based on the characteristic function to study inference for the stable distributions based on independent observations. For models with heavy tailed distributions, Thavaneswaran and Heyde (1999) discussed the superiority of the LAD estimating function over the least squares estimating function.…”

Section: Estimating Function Based On Characteristic Functionmentioning

confidence: 99%

Estimation for Wrapped Zero Inflated Poisson and Wrapped Poisson Distributions

Thavaneswaran

Mandal

Pathmanathan

2016

IJSP

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There has been a growing interest in discrete circular models such as wrapped zero inflated Poisson and wrapped Poisson distributions and the trigonometric moments (see Brobbey et al., 2016 andGirija et al., 2014). Also, characteristic functions of stable processes have been used to study the estimation of the model parameters using estimating function approach (see Thavaneswaran et al., 2013). One difficulty in estimating the circular mean and the resultant mean length parameter of wrapped Poisson (WP) or wrapped zero inflated Poisson (WZIP) is that neither the likelihood of WP/WZIP random variable nor the score function is available in closed form, which leads one to use either trigonometric method of moment estimation (TMME) or an estimating function approach. In this paper, we study the estimation of WZIP distribution and WP distribution using estimating functions and obtain the closed form expression of the information matrix. We also derive the asymptotic distribution of the tangent of the mean direction for both the WZIP and WP distributions.

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Hilbert Space Methods in Probability and Statistical Inference

Cited by 79 publications

References 0 publications

A general theory of hypothesis tests and confidence regions for sparse high dimensional models

A general theory of hypothesis tests and confidence regions for sparse high dimensional models

Matching a Distribution by Matching Quantiles Estimation

Estimation for Wrapped Zero Inflated Poisson and Wrapped Poisson Distributions

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