Unconditional Optimality of Conditional Tests in the Presence of Nuisance Parameters

1997 ◽  
Vol 47 (1-2) ◽  
pp. 43-58
Author(s):  
Gaurangadeb Chattopadhyay
Keyword(s):  
Nuisance Parameter ◽  
Loss Functions ◽  
Nuisance Parameters ◽  
Conditional Test ◽  
Ancillary Statistic ◽  
Conditional Tests

The results of Chatterjee and Chanopadhyay (1990) are extended here to the case where nuisance parameters are present. The extension involves the inclusion of the nuisance parameter alongwith the partial ancillary statistic in the loss functions. It is shown that the usual best conditional test is actually unconditionally optimum irrespective of any knowledge about the nuisance parameter. Some standard examples are considered where loss functions can be so chosen as to realise certain intuitively reasonable properties.

scholarly journals Nuisance hardened data compression for fast likelihood-free inference

2019 ◽  
Vol 488 (4) ◽  
pp. 5093-5103 ◽  
Author(s):  
Justin Alsing ◽  
Benjamin Wandelt
Keyword(s):  
Latent Variables ◽  
Cosmological Parameters ◽  
Nuisance Parameter ◽  
Nuisance Parameters ◽  
Weak Lensing ◽  
Posterior Inference ◽  
Higher Dimensional ◽  
Set Up ◽  
Data Summaries ◽  
Inference Task

ABSTRACT We show how nuisance parameter marginalized posteriors can be inferred directly from simulations in a likelihood-free setting, without having to jointly infer the higher dimensional interesting and nuisance parameter posterior first and marginalize a posteriori. The result is that for an inference task with a given number of interesting parameters, the number of simulations required to perform likelihood-free inference can be kept (roughly) the same irrespective of the number of additional nuisances to be marginalized over. To achieve this, we introduce two extensions to the standard likelihood-free inference set-up. First, we show how nuisance parameters can be recast as latent variables and hence automatically marginalized over in the likelihood-free framework. Secondly, we derive an asymptotically optimal compression from N data to n summaries – one per interesting parameter - such that the Fisher information is (asymptotically) preserved, but the summaries are insensitive to the nuisance parameters. This means that the nuisance marginalized inference task involves learning n interesting parameters from n ‘nuisance hardened’ data summaries, regardless of the presence or number of additional nuisance parameters to be marginalized over. We validate our approach on two examples from cosmology: supernovae and weak-lensing data analyses with nuisance parametrized systematics. For the supernova problem, high-fidelity posterior inference of Ωm and w0 (marginalized over systematics) can be obtained from just a few hundred data simulations. For the weak-lensing problem, six cosmological parameters can be inferred from just $\mathcal {O}(10^3)$ simulations, irrespective of whether 10 additional nuisance parameters are included in the problem or not.

scholarly journals Data-Adaptive Bias-Reduced Doubly Robust Estimation

2016 ◽  
Vol 12 (1) ◽  
pp. 253-282 ◽  
Author(s):  
Karel Vermeulen ◽  
Stijn Vansteelandt
Keyword(s):  
Nuisance Parameter ◽  
Nuisance Parameters ◽  
Robust Estimator ◽  
Robust Estimators ◽  
Infinite Dimensional ◽  
Working Models ◽  
Finite Dimensional ◽  
Doubly Robust Estimation ◽  
Data Adaptive ◽  
Doubly Robust

Abstract Doubly robust estimators have now been proposed for a variety of target parameters in the causal inference and missing data literature. These consistently estimate the parameter of interest under a semiparametric model when one of two nuisance working models is correctly specified, regardless of which. The recently proposed bias-reduced doubly robust estimation procedure aims to partially retain this robustness in more realistic settings where both working models are misspecified. These so-called bias-reduced doubly robust estimators make use of special (finite-dimensional) nuisance parameter estimators that are designed to locally minimize the squared asymptotic bias of the doubly robust estimator in certain directions of these finite-dimensional nuisance parameters under misspecification of both parametric working models. In this article, we extend this idea to incorporate the use of data-adaptive estimators (infinite-dimensional nuisance parameters), by exploiting the bias reduction estimation principle in the direction of only one nuisance parameter. We additionally provide an asymptotic linearity theorem which gives the influence function of the proposed doubly robust estimator under correct specification of a parametric nuisance working model for the missingness mechanism/propensity score but a possibly misspecified (finite- or infinite-dimensional) outcome working model. Simulation studies confirm the desirable finite-sample performance of the proposed estimators relative to a variety of other doubly robust estimators.

CONSISTENT SPECIFICATION TESTING WITH NUISANCE PARAMETERS PRESENT ONLY UNDER THE ALTERNATIVE

1998 ◽  
Vol 14 (3) ◽  
pp. 295-325 ◽  
Author(s):  
Maxwell B. Stinchcombe ◽  
Halbert White
Keyword(s):  
Statistical Models ◽  
Nuisance Parameter ◽  
Nuisance Parameters ◽  
Law Of Iterated Logarithm ◽  
Specification Testing ◽  
Specification Tests ◽  
Recent Developments ◽  
Topological Measures ◽  
Parameter Approach ◽  
Tests Of Hypotheses

The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. How and why the nuisance parameter approach works and how it can be extended bear closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banach-valued random elements and applying the Banach central limit theorem and law of iterated logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted.

scholarly journals High dimensional nuisance parameters: an example from parametric survival analysis

2020 ◽  
Vol 3 (2) ◽  
pp. 119-148
Author(s):  
H. S. Battey ◽  
D. R. Cox
Keyword(s):  
Survival Analysis ◽  
Random Effects ◽  
Nuisance Parameter ◽  
Nuisance Parameters ◽  
High Dimensional ◽  
Open Problems ◽  
Complex Dependence ◽  
Alternative Approaches ◽  
Parametric Survival ◽  
Parametric Survival Analysis

AbstractParametric statistical problems involving both large amounts of data and models with many parameters raise issues that are explicitly or implicitly differential geometric. When the number of nuisance parameters is comparable to the sample size, alternative approaches to inference on interest parameters treat the nuisance parameters either as random variables or as arbitrary constants. The two approaches are compared in the context of parametric survival analysis, with emphasis on the effects of misspecification of the random effects distribution. Notably, we derive a detailed expression for the precision of the maximum likelihood estimator of an interest parameter when the assumed random effects model is erroneous, recovering simply derived results based on the Fisher information in the correctly specified situation but otherwise illustrating complex dependence on other aspects. Methods of assessing model adequacy are given. The results are both directly applicable and illustrate general principles of inference when there is a high-dimensional nuisance parameter. Open problems with an information geometrical bearing are outlined.

scholarly journals Targeted Estimation of Nuisance Parameters to Obtain Valid Statistical Inference

2014 ◽  
Vol 10 (1) ◽  
Author(s):  
Mark J. van der Laan
Keyword(s):  
Propensity Score ◽  
Statistical Model ◽  
Statistical Inference ◽  
Ensemble Learning ◽  
Nuisance Parameter ◽  
Bias Reduction ◽  
Nuisance Parameters ◽  
Regularity Conditions ◽  
Trade Off ◽  
Baseline Covariates

AbstractIn order to obtain concrete results, we focus on estimation of the treatment specific mean, controlling for all measured baseline covariates, based on observing independent and identically distributed copies of a random variable consisting of baseline covariates, a subsequently assigned binary treatment, and a final outcome. The statistical model only assumes possible restrictions on the conditional distribution of treatment, given the covariates, the so-called propensity score. Estimators of the treatment specific mean involve estimation of the propensity score and/or estimation of the conditional mean of the outcome, given the treatment and covariates. In order to make these estimators asymptotically unbiased at any data distribution in the statistical model, it is essential to use data-adaptive estimators of these nuisance parameters such as ensemble learning, and specifically super-learning. Because such estimators involve optimal trade-off of bias and variance w.r.t. the infinite dimensional nuisance parameter itself, they result in a sub-optimal bias/variance trade-off for the resulting real-valued estimator of the estimand. We demonstrate that additional targeting of the estimators of these nuisance parameters guarantees that this bias for the estimand is second order and thereby allows us to prove theorems that establish asymptotic linearity of the estimator of the treatment specific mean under regularity conditions. These insights result in novel targeted minimum loss-based estimators (TMLEs) that use ensemble learning with additional targeted bias reduction to construct estimators of the nuisance parameters. In particular, we construct collaborative TMLEs (C-TMLEs) with known influence curve allowing for statistical inference, even though these C-TMLEs involve variable selection for the propensity score based on a criterion that measures how effective the resulting fit of the propensity score is in removing bias for the estimand. As a particular special case, we also demonstrate the required targeting of the propensity score for the inverse probability of treatment weighted estimator using super-learning to fit the propensity score.

scholarly journals Approximating posteriors with high-dimensional nuisance parameters via integrated rotated Gaussian approximation

Biometrika ◽  
2020 ◽  
Author(s):  
W Van Den Boom ◽  
G Reeves ◽  
D B Dunson
Keyword(s):  
Gaussian Approximation ◽  
Broad Class ◽  
Nuisance Parameter ◽  
Real Data ◽  
Nuisance Parameters ◽  
High Dimensional ◽  
Data Sets ◽  
Posterior Distributions ◽  
Approximation Accuracy ◽  
Dimensional Parameter

Abstract Posterior computation for high-dimensional data with many parameters can be challenging. This article focuses on a new method for approximating posterior distributions of a low- to moderate-dimensional parameter in the presence of a high-dimensional or otherwise computationally challenging nuisance parameter. The focus is on regression models and the key idea is to separate the likelihood into two components through a rotation. One component involves only the nuisance parameters, which can then be integrated out using a novel type of Gaussian approximation. We provide theory on approximation accuracy that holds for a broad class of forms of the nuisance component and priors. Applying our method to simulated and real data sets shows that it can outperform state-of-the-art posterior approximation approaches.

On Invariance Principles for LMP Conditional test Statistics*

1981 ◽  
Vol 30 (1-2) ◽  
pp. 41-56 ◽  
Author(s):  
Pranab Kumar Sen
Keyword(s):  
Random Variables ◽  
Rank Test ◽  
Test Statistics ◽  
Invariance Principles ◽  
Conditional Test ◽  
Special Cases ◽  
Conditional Tests

For a set of random variables, not necessarily independent, along with a characterization of locally most powerful (LMP) conditional tests, some weak as well as strong invariancc principles for tbe allied test statistics arc considered. In the light of asymptotic sufficiency of sub-sigma fields (generating tbe conditional tests). the closeness of LMP and LMP conditional test statistics is examined with reference to the invariancc principles for these statistics. these results include, as special cases, parallel results on various LMP rank test statistica. scattered in the literature.

scholarly journals A Generally Efficient Targeted Minimum Loss Based Estimator based on the Highly Adaptive Lasso

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Mark van der Laan
Keyword(s):  
Nuisance Parameter ◽  
Adaptive Lasso ◽  
Nuisance Parameters ◽  
Minimum Loss ◽  
Target Parameter ◽  
Empirical Risk ◽  
Super Learner ◽  
One Step ◽  
Variation Norm ◽  
Asymptotically Efficient

Abstract Suppose we observe $n$ independent and identically distributed observations of a finite dimensional bounded random variable. This article is concerned with the construction of an efficient targeted minimum loss-based estimator (TMLE) of a pathwise differentiable target parameter of the data distribution based on a realistic statistical model. The only smoothness condition we will enforce on the statistical model is that the nuisance parameters of the data distribution that are needed to evaluate the canonical gradient of the pathwise derivative of the target parameter are multivariate real valued cadlag functions (right-continuous and left-hand limits, (G. Neuhaus. On weak convergence of stochastic processes with multidimensional time parameter. Ann Stat 1971;42:1285–1295.) and have a finite supremum and (sectional) variation norm. Each nuisance parameter is defined as a minimizer of the expectation of a loss function over over all functions it its parameter space. For each nuisance parameter, we propose a new minimum loss based estimator that minimizes the loss-specific empirical risk over the functions in its parameter space under the additional constraint that the variation norm of the function is bounded by a set constant. The constant is selected with cross-validation. We show such an MLE can be represented as the minimizer of the empirical risk over linear combinations of indicator basis functions under the constraint that the sum of the absolute value of the coefficients is bounded by the constant: i.e., the variation norm corresponds with this $L_1$-norm of the vector of coefficients. We will refer to this estimator as the highly adaptive Lasso (HAL)-estimator. We prove that for all models the HAL-estimator converges to the true nuisance parameter value at a rate that is faster than $n^{-1/4}$ w.r.t. square-root of the loss-based dissimilarity. We also show that if this HAL-estimator is included in the library of an ensemble super-learner, then the super-learner will at minimal achieve the rate of convergence of the HAL, but, by previous results, it will actually be asymptotically equivalent with the oracle (i.e., in some sense best) estimator in the library. Subsequently, we establish that a one-step TMLE using such a super-learner as initial estimator for each of the nuisance parameters is asymptotically efficient at any data generating distribution in the model, under weak structural conditions on the target parameter mapping and model and a strong positivity assumption (e.g., the canonical gradient is uniformly bounded). We demonstrate our general theorem by constructing such a one-step TMLE of the average causal effect in a nonparametric model, and establishing that it is asymptotically efficient.

Valence electron spectroscopy of inhomogeneous media

1992 ◽  
Vol 50 (2) ◽  
pp. 1150-1151
Author(s):  
A. Howie ◽  
D.W. McComb
Keyword(s):  
Specific Gravity ◽  
Loss Function ◽  
Electron Spectroscopy ◽  
Valence Electron ◽  
Peak Height ◽  
Loss Functions ◽  
Loss Peak ◽  
High Energies ◽  
Valence Electron Density ◽  
Electron Microscopist

The bulk loss function Im(-l/ε (ω)), a well established tool for the interpretation of valence loss spectra, is being progressively adapted to the wide variety of inhomogeneous samples of interest to the electron microscopist. Proportionality between n, the local valence electron density, and ε-1 (Sellmeyer's equation) has sometimes been assumed but may not be valid even in homogeneous samples. Figs. 1 and 2 show the experimentally measured bulk loss functions for three pure silicates of different specific gravity ρ - quartz (ρ = 2.66), coesite (ρ = 2.93) and a zeolite (ρ = 1.79). Clearly, despite the substantial differences in density, the shift of the prominent loss peak is very small and far less than that predicted by scaling e for quartz with Sellmeyer's equation or even the somewhat smaller shift given by the Clausius-Mossotti (CM) relation which assumes proportionality between n (or ρ in this case) and (ε - 1)/(ε + 2). Both theories overestimate the rise in the peak height for coesite and underestimate the increase at high energies.

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