Optimal asymptotic tests of composite hypotheses for continuous time stochastic processes
01 Feb 1996-
TL;DR: In this paper, a locally asymptotically most powerful test for testing the composite hypothesis H 0 : γ = γ 0 against H 1: γ ≠ γ 1 in the presence of a nuisance parameter θ is developed following the concept of C(α)-tests introduced by Neyman.
Abstract: Consider a stochastic process {X t , t ≥ O} whose distributions depend on an unknown parameter (γ,θ). A locally asymptotically most powerful test, for testing the composite hypothesis H 0 : γ = γ 0 against H 1 : γ ≠ γ 0 in the presence of a nuisance parameter θ is developed following the concept of C(α)-tests introduced by Neyman. Results are illustrated by means of example of process {X(t),t ≥ 0} satisfying the linear stochastic differential equation dX(t) = (γX(t) + θ)dt+dW(t),t ≥ 0.
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TL;DR: By focusing on excess risk rather than parameter estimation, this work can give guarantees under weaker assumptions than in previous works and accommodate the case where the target parameter belongs to a complex nonparametric class.
Abstract: We provide non-asymptotic excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate the target parameter depends on an unknown nuisance parameter that must be estimated from data. We analyze a two-stage sample splitting meta-algorithm that takes as input two arbitrary estimation algorithms: one for the target parameter and one for the nuisance parameter. We show that if the population risk satisfies a condition called Neyman orthogonality, the impact of the nuisance estimation error on the excess risk bound achieved by the meta-algorithm is of second order. Our theorem is agnostic to the particular algorithms used for the target and nuisance and only makes an assumption on their individual performance. This enables the use of a plethora of existing results from statistical learning and machine learning to give new guarantees for learning with a nuisance component. Moreover, by focusing on excess risk rather than parameter estimation, we can give guarantees under weaker assumptions than in previous works and accommodate settings in which the target parameter belongs to a complex nonparametric class. We provide conditions on the metric entropy of the nuisance and target classes such that oracle rates---rates of the same order as if we knew the nuisance parameter---are achieved. We also derive new rates for specific estimation algorithms such as variance-penalized empirical risk minimization, neural network estimation and sparse high-dimensional linear model estimation. We highlight the applicability of our results in four settings of central importance: 1) heterogeneous treatment effect estimation, 2) offline policy optimization, 3) domain adaptation, and 4) learning with missing data.
122 citations
Cites methods from "Optimal asymptotic tests of composi..."
...We show that Neyman orthogonality, which has been used to prove oracle rates for inference in semiparametric models (Neyman, 1959, 1979; Chernozhukov et al., 2018a,b), is key to providing oracle rates for statistical learning with a nuisance component....
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01 Jan 2003
TL;DR: For any generalized linear model, the Pearson goodness of fit statistic is the score test statistic for testing the current model against the saturated model, and the relationship between the Pearson statistic and the residual deviance is therefore explained in this paper.
Abstract: For any generalized linear model, the Pearson goodness of fit statistic is the score test statistic for testing the current model against the saturated model. The relationship between the Pearson statistic and the residual deviance is therefore the relationship between the score test and the likelihood ratio test statistics, and this clarifies the role of the Pearson statistic in generalized linear models. The result is extended to cases in which there are multiple reponse observations for the same combination of explanatory variables.
62 citations
Cites background from "Optimal asymptotic tests of composi..."
...In that case, ί\ and £2 are independent and /2.1 = In, meaning that the information matrix /22 does not need to be adjusted for estimation of θ 1, Neyman [15] and Neyman and Scott [16] show that the asymptotic distribution and efficiency of the score statistic S is unchanged if an estimator other than the maximum likelihood estimator is used for the nuisance parameters, provided that the estimator is consistent with convergence rate at least O(n~1/2), where n is the number of observations....
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...1 = In, meaning that the information matrix /22 does not need to be adjusted for estimation of θ 1, Neyman [15] and Neyman and Scott [16] show that the asymptotic distribution and efficiency of the score statistic S is unchanged if an estimator other than the maximum likelihood estimator is used for the nuisance parameters, provided that the estimator is consistent with convergence rate at least O(n~(1)/(2)), where n is the number of observations....
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TL;DR: In this paper, the known estimation and simulation methods for multivariate t distributions are reviewed and a review of selected applications is also provided, which will serve as an important reference and encourage further research activities in the area.
Abstract: The known estimation and simulation methods for multivariate t distributions are reviewed. A review of selected applications is also provided. We believe that this review will serve as an important reference and encourage further research activities in the area.
53 citations
Cites background from "Optimal asymptotic tests of composi..."
...Sutradhar [53] proposed Neyman’s [42] score test for this test for large n....
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...Sutradhar [53] proposed Neyman’s [42] score test for this test for large n. Le r = (r11, . . . , rhl, . . . , rpp)T be the p(p + 1)/2 × 1 vector formed by stacking the distinct elements of R, with rhl being the (h, l)th element of the p × p matrix R....
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...Then, Neyman’s partial score test statistic is given by W(μ̂, ν̂) = TT [ M̂11 − ( M̂12M̂13 )(M̂22 M̂23 M̂33 )−1 (M̂T12 M̂T13 )]−1 T, (13) where T ≡ [T1(r0, μ̂, ν̂), . . . , Tp(p+1)/2(r0, μ̂, ν̂)]T for i, r = 1,2,3; M̂ir are obtained from Mir = E(−Dir) by replacing μ and ν with their consistent estimates; and Dir for i, r = 1,2,3 are the derivatives given by D11 = ∂ 2G ∂r∂r′ , D12 = ∂ 2G ∂r∂μ′ , D13 = ∂ 2G ∂r∂ν , D22 = ∂ 2G ∂μ∂μ′ , D23 = ∂ 2G ∂μ∂ν , and D33 = ∂ 2G ∂ν2 ....
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01 Jan 2005
TL;DR: In this article, the class of zero-inated over-dispersed generalized linear models and score tests for selecting a model that can handle such data were proposed, and the power properties of the tests were examined by a limited simulation study.
Abstract: Discrete data in the form of counts often exhibit extra variation that cannot be explained by a simple model, such as the binomial or the Poisson. Also, these data sometimes show more zero counts than what can be predicted by a simple model. Therefore, a discrete generalized linear model (Poisson or binomial) may fail to t a set of discrete data either because of zero-ination, because of over- dispersion, or because there is zero-ination as well as over-dispersion in the data. Previous published work deals with goodness of t tests of the generalized linear model against zero-ination and against over-dispersion separately. In this paper we deal with the class of zero-inated over-dispersed generalized linear models and propose procedures based on score tests for selecting a model that ts such data. For over-dispersion we consider a general over-dispersion model and specic over- dispersion models. We show that in certain cases and under certain conditions, the score tests derived using the general over-dispersion model and those devel- oped under specic over-dispersion models are identical. Empirical level and power properties of the tests are examined by a limited simulation study. Simulations show that the score tests, in general, hold nominal levels well and have good power properties. Two illustrative examples and a discussion are presented.
46 citations
Cites background from "Optimal asymptotic tests of composi..."
...The score test (Rao (1947)) is a special case of the more general C(α) test (Neyman (1959)) in which the nuisance parameters are replaced by maximum likelihood estimates which are √ N (N=number of observations used in estimat- ing the parameters) consistent estimates....
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TL;DR: The relatively simple Wald's test (WT) is as powerful as the likelihood ratio test (LRT) and that both have consistently greater power than the score test, and the regression test holds its empirical levels, and in some occasions is aspowerful as the WT and the LRT.
Abstract: The within-subject coefficient of variation and intra-class correlation coefficient are commonly used to assess the reliability or reproducibility of interval-scale measurements. Comparison of reproducibility or reliability of measurement devices or methods on the same set of subjects comes down to comparison of dependent reliability or reproducibility parameters. In this paper, we develop several procedures for testing the equality of two dependent within-subject coefficients of variation computed from the same sample of subjects, which is, to the best of our knowledge, has not yet been dealt with in the statistical literature. The Wald test, the likelihood ratio, and the score tests are developed. A simple regression procedure based on results due to Pitman and Morgan is constructed. Furthermore we evaluate the statistical properties of these methods via extensive Monte Carlo simulations. The methodologies are illustrated on two data sets; the first are the microarray gene expressions measured by two plat- forms; the Affymetrix and the Amersham. Because microarray experiments produce expressions for a large number of genes, one would expect that the statistical tests to be asymptotically equivalent. To explore the behaviour of the tests in small or moderate sample sizes, we illustrated the methodologies on data from computer-aided tomographic scans of 50 patients. It is shown that the relatively simple Wald's test (WT) is as powerful as the likelihood ratio test (LRT) and that both have consistently greater power than the score test. The regression test holds its empirical levels, and in some occasions is as powerful as the WT and the LRT. A comparison between the reproducibility of two measuring instruments using the same set of subjects leads naturally to a comparison of two correlated indices. The presented methodology overcomes the difficulty noted by data analysts that dependence between datasets would confound any inferences one could make about the differences in measures of reliability and reproducibility. The statistical tests presented in this paper have good properties in terms of statistical power.
39 citations
References
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TL;DR: In this article, a locally asymptotically most powerful test for a composite hypothesis $H:\xi = \xi_0$ has been developed for the case where the observable random variables $\{X_{nk}, k = 1, 2, \cdots, n\}$ are independently but not necessarily identically distributed.
Abstract: A locally asymptotically most powerful test for a composite hypothesis $H:\xi = \xi_0$ has been developed for the case where the observable random variables $\{X_{nk}, k = 1, 2, \cdots, n\}$ are independently but not necessarily identically distributed. However, their distributions depend on $s + 1$ parameters, one being $\xi$ under test and the other being a vector $\theta = (\theta_1, \cdots, \theta_s)$ of nuisance parameters. The theory is illustrated with an example from the field of astronomy.
309 citations
01 Jan 1982
TL;DR: In this paper, a survey of the central limit theorems for discrete time martingales with continuous time is presented, and several related sets of conditions for convergence are formulated, where conditions are given in terms of conditional moments of truncated variables.
Abstract: This survey paper consists of two parts. In the first part (up to and including setion 3) we review the central limit theorems for discrete time martingales, and show that many different sets of conditions for convergence may be reduced to one basic set, where conditions are given in terms of conditional moments of truncated variables, given the past. In the second part (sections 4 and 5) we first recall some basic facts from the modern "French" theory of stochastic processes, then show that Rebolledo's recent functional limit theorems for martingales with continuous time can be deduced from the limit theorems for discrete time martingales. Again, several related sets of conditions for convergence are formulated.
229 citations
TL;DR: In this article, the asymptotic theory of estimates of an unknown parameter in continuous-time Markov processes, which are described by non-linear stochastic differential equations, is investigated.
Abstract: This paper is concerned with the asymptotic theory of estimates of an unknown parameter in continuous-time Markov processes, which are described by non-linear stochastic differential equations. The maximum likelihood estimate and the minimum contrast estimate are investigated. For these estimates strong consistency and asymptotic normality are proved. The unknown parameter is assumed to take its values either in an open or in a compact set of real numbers.
54 citations