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We consider the problem of uncertainty assessment for low dimensional components in high dimensional models. Specifically, we propose a novel 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 method provides a general framework for high dimensional inference and is applicable to a wide variety of applications. In particular, we apply this general framework to study five illustrative examples: linear regression, logistic regression, Poisson regression, Gaussian graphical model and additive hazards model. For hypothesis testing, 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. For confidence region construction, we show that the decorrelated score function can be used to construct point estimators that are asymptotically normal and semiparametrically efficient. We further generalize this framework to handle the settings of misspecified models. Thorough numerical results are provided to back up the developed theory.

Over the past decade, doubly robust estimators have been proposed for a variety of target parameters in causal inference and missing data models. These are asymptotically unbiased when at least one of two nuisance working models is correctly specified, regardless of which. While their asymptotic distribution is not affected by the choice of root- n consistent estimators of the nuisance parameters indexing these working models when all working models are correctly specified, this choice of estimators can have a dramatic impact under misspecification of at least one working model. In this article, we will therefore propose a simple and generic estimation principle for the nuisance parameters indexing each of the working models, which is designed to improve the performance of the doubly robust estimator of interest, relative to the default use of maximum likelihood estimators for the nuisance parameters. The proposed approach locally minimizes the squared first-order asymptotic bias of the doubly robust estimator under misspecification of both working models and results in doubly robust estimators with easy-to-calculate asymptotic variance. It moreover improves the stability of the weights in those doubly robust estimators which invoke inverse probability weighting. Simulation studies confirm the desirable finite-sample performance of the proposed estimators. Supplementary materials for this article are available online.

We develop a hybrid framework for quantum parameter estimation in the presence of nuisance parameters. In this scheme, the parameters of interest are treated as fixed non-random parameters while nuisance parameters are integrated out with respect to a prior (random parameters). Within this setting, we introduce the hybrid partial quantum Fisher information matrix (hpQFIM), defined by prior-averaging the nuisance block of the QFIM and taking a Schur complement, and derive a corresponding Cramér–Rao-type lower bound on the hybrid risk. We establish the structural properties of the hpQFIM, including inequalities that bracket it between computationally tractable approximations, as well as limiting behaviors under extreme priors. Operationally, the hybrid approach improves over pure point estimation since the optimal measurement for the parameters of interest depends only on the prior distribution of the nuisance, rather than on its unknown value. We illustrate the framework with analytically solvable qubit models and numerical examples, clarifying how partial prior information on nuisance variables can be systematically exploited in quantum metrology.

An open problem of Markov switching models is identifying the number of states, generally fixed a priori; it is impossible to apply classical tests due to the issue of the nuisance parameters present only under the alternative hypothesis. In this work, we show, by Monte Carlo simulations, that fuzzy clustering is able to reproduce the parametric state inference derived from the Hamilton filter and that the typical indices used in clustering to determine the number of groups can be used to identify the number of states in this framework. The procedure is very simple to apply, considering that it is performed independently of the data generating process and that the indicators we use are available in most statistical packages. Furthermore, the proposed approach appears to be sufficiently robust to perturbations in the data generating processes. A final application of real data completes the analysis.

Seafloor transponder coordinates are determined by measurements between a ship-borne GNSS-acoustic transducer and the transponder. Differencing techniques can be applied to eliminate the impact of measurement biases for precise positioning effectively. The problem is that the correlations between differenced measurements must be adequately considered in the covariance matrix, which might cause a great number of calculations. This paper presents a set of conversion formulae to derive the differenced solution from the undifferenced model without nuisance parameters, and then we propose a dimension-reduction algorithm to fast solve the Gauss–Markov model augmented with nuisance parameters. The equivalence of the differenced and undifferenced solution is discussed within a wider range. It shows that: (1) the undifferenced solution can be converted into the differenced solution with only a few additional calculations; (2) there are a class of differencing schemes which are completely equivalent to each other having unique differencing equivalence weight (DEW) matrix; (3) the proposed algorithm is more efficient and has a good numerical stability relative to the blocking–stacking algorithm and the one-by-one elimination. The simulation and the real trial performed in a 3000-m depth sea area verified the proposed results.

A statistical manifold M can be defined as a Riemannian manifold each of whose points is a probability distribution on the same support. In fact, statistical manifolds possess a richer geometric structure beyond the Fisher information metric defined on the tangent bundle TM. Recognizing that points in M are distributions and not just generic points in a manifold, TM can be extended to a Hilbert bundle HM. This extension proves fundamental when we generalize the classical notion of a point estimate—a single point in M—to a function on M that characterizes the relationship between observed data and each distribution in M. The log likelihood and score functions are important examples of generalized estimators. In terms of a parameterization θ:M→Θ⊂Rk, θ^ is a distribution on Θ while its generalization gθ^=θ^−Eθ^ as an estimate is a function over Θ that indicates inconsistency between the model and data. As an estimator, gθ^ is a distribution of functions. Geometric properties of these functions describe statistical properties of gθ^. In particular, the expected slopes of gθ^ are used to define Λ(gθ^), the Λ-information of gθ^. The Fisher information I is an upper bound for the Λ-information: for all g, Λ(g)≤I. We demonstrate the utility of this geometric perspective using the two-sample problem.

With very large amounts of data, important aspects of statistical analysis may appear largely descriptive in that the role of probability sometimes seems limited or totally absent. The main emphasis of the present paper lies on contexts where formulation in terms of a probabilistic model is feasible and fruitful but to be at all realistic large numbers of unknown parameters need consideration. Then many of the standard approaches to statistical analysis, for instance direct application of the method of maximum likelihood, or the use of flat priors, often encounter difficulties. After a brief discussion of broad conceptual issues, we provide some new perspectives on aspects of high-dimensional statistical theory, emphasizing a number of open problems.

We consider testing regression coefficients in high dimensional generalized linear models. By modifying the test statistic of Goeman and his colleagues for large but fixed dimensional settings, we propose a new test, based on an asymptotic analysis, that is applicable for diverging dimensions and is robust to accommodate a wide range of link functions. The power properties of the tests are evaluated asymptotically under two families of alternative hypotheses. In addition, a test in the presence of nuisance parameters is also proposed. The tests can provide p-values for testing significance of multiple gene sets, whose application is demonstrated in a case-study on lung cancer.

Higher-order asymptotic methods for nonlinear models with nuisance parameters are developed. We allow for both one-step estimators, in which the nuisance and parameters of interest are jointly estimated; and also two-step (or iterated) estimators, in which the nuisance parameters are first estimated. The properties of the former, although in principle simpler to conceptualize, are more difficult to establish explicitly. The iterated estimators allow for a variety of scenarios. The results indicate when second-order considerations should be taken into account when conducting inferences with two-step estimators. The results in the paper accomplish three objectives: (i) provide simpler methods for deriving higher-order moments when nuisance parameters are present; (ii) indicate more explicitly the sources of deviations of estimators’ sampling distributions from that given by standard first-order asymptotic theory; and, in turn, (iii) indicate in which situations the corrections (either analytically or by a resampling method such as bootstrap or jackknife) should be made when making inferences. We illustrate using several popular examples in econometrics. We also provide a numerical example which highlights how a simple analytical bias correction can improve inferences.

The inferential models (IM) framework provides prior-free, frequency-calibrated, and posterior probabilistic inference. The key is the use of random sets to predict unobservable auxiliary variables connected to the observable data and unknown parameters. When nuisance parameters are present, a marginalization step can reduce the dimension of the auxiliary variable which, in turn, leads to more efficient inference. For regular problems, exact marginalization can be achieved, and we give conditions for marginal IM validity. We show that our approach provides exact and efficient marginal inference in several challenging problems, including a many-normal-means problem. In nonregular problems, we propose a generalized marginalization technique and prove its validity. Details are given for two benchmark examples, namely, the Behrens-Fisher and gamma mean problems.