Scalar Parameter Of Interest Research Articles

Optimal HAR inference

This paper considers the problem of deriving heteroskedasticity and autocorrelation robust (HAR) inference about a scalar parameter of interest. The main assumption is that there is a known upper bound on the degree of persistence in data. I derive finite‐sample optimal tests in the Gaussian location model and show that the robustness‐efficiency tradeoffs embedded in the optimal tests are essentially determined by the maximal persistence. I find that with an appropriate adjustment to the critical value, it is nearly optimal to use the so‐called equal‐weighted cosine (EWC) test, where the long‐run variance is estimated by projections onto q type II cosines. The practical implications are an explicit link between the choice of q and assumptions on the underlying persistence, as well as a corresponding adjustment to the usual Student‐ t critical value. I illustrate the results in two empirical examples.

Confidence intervals in general regression models that utilize uncertain prior information

– We consider a general regression model, either without a scale parameter or with a known scale parameter. We construct a confidence interval for a scalar parameter of interest that utilizes the uncertain prior information that a distinct scalar parameter takes a specified value. This confidence interval has excellent coverage properties. It also has local scaled expected length, where the scaling is with respect to the usual confidence interval, that (a) is substantially less than 1 when the prior information is correct, (b) has a maximum value that is not too large, and (c) is close to 1 when the data and prior information are highly discordant. An important practical application of this confidence interval is to a quantal bioassay carried out to compare two similar compounds. In this context, the uncertain prior information is that the hypothesis of “parallelism” holds. We provide extensive numerical results that illustrate the attractive properties of this confidence interval in this context.

Quantum-Inspired Multi-Parameter Adaptive Bayesian Estimation for Sensing and Imaging

It is well known in Bayesian estimation theory that the conditional estimator attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest. In quantum, e.g., optical and atomic, imaging and sensing tasks the user has access to the quantum state that encodes the parameter. The choice of a measurement operator, i.e. a positive-operator valued measure (POVM), leads to a measurement outcome on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM that attains the MMSE= over all possible physically allowable measurements and the resulting MMSE [1]. This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements without considering any prior probability. For multi-parameter estimation, in quantum Fisher estimation theory the inverse of the QFI matrix provides an operator lower bound on the covariance of an unbiased estimator, and this bound is understood in the positive semidefinite sense. However, there has been little work on quantifying the quantum limits and measurement designs, for multi-parameter quantum estimation in a Bayesian setting. In this work, we build upon Personick's result to construct a Bayesian adaptive (greedy) measurement scheme for multiparameter estimation. We illustrate our proposed measurement scheme with the application of localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs superior to quantum-noise-limited focal-plane direct imaging.

Parametric Bootstrap Inference for Stratified Models With High-Dimensional Nuisance Specifications

Inference about a scalar parameter of interest typically relies on the asymptotic normality of common likelihood pivots, such as the signed likelihood root, the score and Wald statistics. Nevertheless, the resulting inferential procedures have been known to perform poorly when the dimension of the nuisance parameter is large relative to the sample size and when the information about the parameters is limited. In such cases, the use of asymptotic normality of analytical modifications of the signed likelihood root is known to recover inferential performance. It is proved here that parametric bootstrap of standard likelihood pivots results in as accurate inferences as analytical modifications of the signed likelihood root do in stratified models with stratum specific nuisance parameters. We focus on the challenging case where the number of strata increases as fast or faster than the stratum samples size. It is also shown that this equivalence holds regardless of whether constrained or unconstrained bootstrap is used. This is in contrast to when the dimension of the parameter space is fixed relative to the sample size, where constrained bootstrap is known to correct inference to higher-order than unconstrained bootstrap does. Large scale simulation experiments support the theoretical findings and demonstrate the excellent performance of bootstrap in some extreme modelling scenarios.

Confidence sequences with composite likelihoods

AbstractIn dominated parametric statistical models, confidence sequences provide conservatively valid frequentist inference directly from a likelihood ratio. They ensure a specific mode of replicability when inference is performed on accumulating data: inferential conclusions that are compatible with a guaranteed probability when the sample is enlarged, in the form of overlapping confidence regions. Here we consider both Robbins' mixture confidence sequences and running maximum likelihood confidence sequences recently considered by Wasserman, Ramdas, and Balakrishnan. We compare through simulation the replicability properties of the two kinds of confidence sequences, evaluating, along a prospected enlargement of the sample, the frequency of incompatible estimation intervals and the frequency of failure of simultaneous coverage of the true parameter value. Moreover, we propose a shortcut to extend the application of mixture confidence sequences to pseudo‐likelihoods, in particular to composite likelihood. The main assumption required is that normal asymptotic theory offers a good approximation to the density of the maximizer of the pseudo‐likelihood. When inference is about a scalar parameter of interest, the computation of the proposed sequence of confidence intervals is straightforward. The method is illustrated by an example with replicability properties evaluated through simulation.

Can Bayesian, confidence distribution and frequentist inference agree?

We discuss and characterise connections between frequentist, confidence distribution and objective Bayesian inference, when considering higher-order asymptotics, matching priors, and confidence distributions based on pivotal quantities. The focus is on testing precise or sharp null hypotheses on a scalar parameter of interest. Moreover, we illustrate that the application of these procedures requires little additional effort compared to the application of standard first-order theory. In this respect, using the R software, we indicate how to perform in practice the computation with three examples in the context of data from inter-laboratory studies, of the stress–strength reliability, and of a growth curve from dose–response data.

Simple and honest confidence intervals in nonparametric regression

We consider the problem of constructing honest confidence intervals (CIs) for a scalar parameter of interest, such as the regression discontinuity parameter, in nonparametric regression based on kernel or local polynomial estimators. To ensure that our CIs are honest, we use critical values that take into account the possible bias of the estimator upon which the CIs are based. We show that this approach leads to CIs that are more efficient than conventional CIs that achieve coverage by undersmoothing or subtracting an estimate of the bias. We give sharp efficiency bounds of using different kernels, and derive the optimal bandwidth for constructing honest CIs. We show that using the bandwidth that minimizes the maximum mean‐squared error results in CIs that are nearly efficient and that in this case, the critical value depends only on the rate of convergence. For the common case in which the rate of convergence is n −2/5, the appropriate critical value for 95% CIs is 2.18, rather than the usual 1.96 critical value. We illustrate our results in a Monte Carlo analysis and an empirical application.

Location-adjusted Wald statistics for scalar parameters

Inference about a scalar parameter of interest is a core statistical task that has attracted immense research in statistics. The Wald statistic is a prime candidate for the task, on the grounds of the asymptotic validity of the standard normal approximation to its finite-sample distribution, simplicity and low computational cost. It is well known, though, that this normal approximation can be inadequate, especially when the sample size is small or moderate relative to the number of parameters. A novel, algebraic adjustment to the Wald statistic is proposed, delivering significant improvements in inferential performance with only small implementation and computational overhead, predominantly due to additional matrix multiplications. The Wald statistic is viewed as an estimate of a transformation of the model parameters and is appropriately adjusted, using either maximum likelihood or reduced-bias estimators, bringing its expectation asymptotically closer to zero. The location adjustment depends on the expected information, an approximation to the bias of the estimator, and the derivatives of the transformation, which are all either readily available or easily obtainable in standard software for a wealth of models. An algorithm for the implementation of the location-adjusted Wald statistics in general models is provided, as well as a bootstrap scheme for the further scale correction of the location-adjusted statistic. Ample analytical and numerical evidence is presented for the adoption of the location-adjusted statistic in prominent modelling settings, including inference about log-odds and binomial proportions, logistic regression in the presence of nuisance parameters, beta regression, and gamma regression. The location-adjusted Wald statistics are used for the construction of significance maps for the analysis of multiple sclerosis lesions from MRI data.

The p-value Function and Statistical Inference

ABSTRACT This article has two objectives. The first and narrower is to formalize the p-value function, which records all possible p-values, each corresponding to a value for whatever the scalar parameter of interest is for the problem at hand, and to show how this p-value function directly provides full inference information for any corresponding user or scientist. The p-value function provides familiar inference objects: significance levels, confidence intervals, critical values for fixed-level tests, and the power function at all values of the parameter of interest. It thus gives an immediate accurate and visual summary of inference information for the parameter of interest. We show that the p-value function of the key scalar interest parameter records the statistical position of the observed data relative to that parameter, and we then describe an accurate approximation to that p-value function which is readily constructed.

The large sample coverage probability of confidence intervals in general regression models after a preliminary hypothesis test

AbstractWe derive a computationally convenient formula for the large sample coverage probability of a confidence interval for a scalar parameter of interest following a preliminary hypothesis test that a specified vector parameter takes a given value in a general regression model. Previously, this large sample coverage probability could only be estimated by simulation. Our formula only requires the evaluation, by numerical integration, of either a double or a triple integral, irrespective of the dimension of this specified vector parameter. We illustrate the application of this formula to a confidence interval for the odds ratio of myocardial infarction when the exposure is recent oral contraceptive use, following a preliminary test where two specified interactions in a logistic regression model are zero. For this real‐life data, we compare this large sample coverage probability with the actual coverage probability of this confidence interval, obtained by simulation.

Simple and Honest Confidence Intervals in Nonparametric Regression

We consider the problem of constructing honest confidence intervals (CIs) for a scalar parameter of interest, such as the regression discontinuity parameter, in nonparametric regression based on kernel or local polynomial estimators. To ensure that our CIs are honest, we derive and tabulate novel critical values that take into account the possible bias of the estimator upon which the CIs are based. We give sharp efficiency bounds of using different kernels, and derive the optimal bandwidth for constructing honest CIs. We show that using the bandwidth that minimizes the maximum meansquared error results in CIs that are nearly efficient and that in this case, the critical value depends only on the rate of convergence. For the common case in which the rate of convergence is n^{-4/5}, the appropriate critical value for 95% CIs is 2.18, rather than the usual 1.96 critical value. We illustrate our results in an empirical application.

Median bias reduction of maximum likelihood estimates

For regular parametric problems, we show how median centring of the maximum likelihood estimate can be achieved by a simple modification of the score equation. For a scalar parameter of interest, the estimator is equivariant under interest-respecting reparameterizations and is third-order median unbiased. With a vector parameter of interest, componentwise equivariance and third-order median centring are obtained. Like the implicit method of Firth (1993) for bias reduction, the new method does not require finiteness of the maximum likelihood estimate and is effective in preventing infinite estimates. Simulation results for continuous and discrete models, including binary and beta regression, confirm that the method succeeds in achieving componentwise median centring and in solving the boundary estimate problem, while keeping comparable dispersion and the same approximate distribution as its main competitors.

Bootstrap adjustments of signed scoring rule root statistics

ABSTRACTScoring rules give rise to methods for statistical inference and are useful tools to achieve robustness or reduce computations. Scoring rule inference is generally performed through first-order approximations to the distribution of the scoring rule estimator or of the ratio-type statistic. In order to improve the accuracy of first-order methods even in simple models, we propose bootstrap adjustments of signed scoring rule root statistics for a scalar parameter of interest in presence of nuisance parameters. The method relies on the parametric bootstrap approach that avoids onerous calculations specific of analytical adjustments. Numerical examples illustrate the accuracy of the proposed method.

The formal relationship between analytic and bootstrap approaches to parametric inference

Two routes most commonly proposed for accurate inference on a scalar interest parameter in the presence of a (possibly high-dimensional) nuisance parameter are parametric simulation (‘bootstrap’) methods, and analytic procedures based on normal approximation to adjusted forms of the signed root likelihood ratio statistic. Under some null hypothesis of interest, both methods yield p-values which are uniformly distributed to error of third-order in the available sample size. But, given a specific dataset, what is the formal relationship between p-values calculated by the two approaches? We show that the two methodologies give the same inference to second order in general: the analytic p-value calculated from a dataset will agree with the bootstrap p-value constructed from that same dataset to O(n−1), where n is the sample size. In practice, the agreement is often startling.

P-Values: The Insight to Modern Statistical Inference

I introduce a p-value function that derives from the continuity inherent in a wide range of regular statistical models. This provides confidence bounds and confidence sets, tests, and estimates that all reflect model continuity. The development starts with the scalar-variable scalar-parameter exponential model and extends to the vector-parameter model with scalar interest parameter, then to general regular models, and then references for testing vector interest parameters are available. The procedure does not use sufficiency but applies directly to general models, although it reproduces sufficiency-based results when sufficiency is present. The emphasis is on the coherence of the full procedure, and technical details are not emphasized.

Bayes, Reproducibility and the Quest for Truth

We consider the use of default priors in the Bayes methodology for seeking information concerning the true value of a parameter. By default prior, we mean the mathematical prior as initiated by Bayes [Philos. Trans. R. Soc. Lond. 53 (1763) 370–418] and pursued by Laplace [Theorie Analytique des Probabilites (1812) Courcier], Jeffreys [Theory of Probability (1961) Clarendon Press], Bernardo [J. Roy. Statist. Soc. Ser. B 41 (1979) 113–147] and many more, and then recently viewed as “potentially dangerous” [Science 340 (2013) 1177–1178] and “potentially useful” [Science 341 (2013) 1452]. We do not mean, however, the genuine prior [Science 340 (2013) 1177–1178] that has an empirical reference and would invoke standard frequency modelling. And we do not mean the subjective or opinion prior that an individual might have and would be viewed as specific to that individual. A mathematical prior has no referenced frequency information, but on occasion is known otherwise to lead to repetition properties called confidence. We investigate the presence of such supportive property, and ask can Bayes give reliability for other than the particular parameter weightings chosen for the conditional calculation. Thus, does the methodology have reproducibility? Or is it a leap of faith. For sample-space analysis, recent higher-order likelihood methods with regular models show that third-order accuracy is widely available using profile contours [In Past, Present and Future of Statistical Science (2014) 237– 252 CRC Press]. But for parameter-space analysis, accuracy is widely limited to first order. An exception arises with a scalar full parameter and the use of the scalar Jeffreys [J. Roy. Statist. Soc. Ser. B 25 (1963) 318–329]. But for vector full parameter even with a scalar interest parameter, difficulties have long been known [J. Roy. Statist. Soc. Ser. B 35 (1973) 189–233] and with parameter curvature, accuracy beyond first order can be unavailable [Statist. Sci. 26 (2011) 299–316].We show, however, that calculations on the parameter space can give full second-order information for a chosen scalar interest parameter; these calculations, however, require a Jeffreys prior that is used fully restricted to the one-dimensional profile for that interest parameter. Such a prior is effectively data-dependent and parameter-dependent and is focally restricted to the one-dimensional contour; these priors fall outside the usual Bayes approach and yet with substantial calculations can still give less than frequency analysis. We provide simple examples using discrete extensions of Jeffreys prior. These serve as counter-examples to general claims that Bayes can offer accuracy for statistical inference. To obtain this accuracy with Bayes, more effort is required compared to recent likelihood methods, which still remain more accurate. And with vector full parameters, accuracy beyond first order is routinely not available, as a change in parameter curvature causes Bayes and frequentist values to change in opposite direction, yet frequentist has full reproducibility. An alternative is to view default Bayes as an exploratory technique and then ask does it do as it overtly claims? Is it reproducible as understood in contemporary science? The posterior gives a distribution for an interest parameter and, thereby, a quantile for the interest parameter; an oracle could record whether it was left or right of the true value. If the average split in evaluative repetitions is in accord with the nominal level, then the approach is providing accuracy. And if not, then what is up, other than performance specific to the parameter frequencies in the prior. No one has answers although speculative claims abound.

Distribution of likelihood-basedp-values under a local alternative hypothesis

We consider inference on a scalar parameter of interest in the presence of a nuisance parameter, using a likelihood-based statistic which is asymptotically normally distributed under the null hypothesis. Higher-order expansions are used to compare the repeated sampling distribution, under a general contiguous alternative hypothesis, of $p$-values calculated from the asymptotic normal approximation to the null sampling distribution of the statistic with the distribution of $p$-values calculated by bootstrap approximations. The results of comparisons in terms of power of different testing procedures under an alternative hypothesis are closely related to differences under the null hypothesis, specifically the extent to which testing procedures are conservative or liberal under the null. Empirical examples are given which demonstrate that higher-order asymptotic effects may be seen clearly in small-sample contexts.

Simple and Honest Confidence Intervals in Nonparametric Regression

We consider the problem of constructing honest confidence intervals (CIs) for a scalar parameter of interest, such as the regression discontinuity parameter, in nonparametric regression based on kernel or local polynomial estimators. To ensure that our CIs are honest, we derive and tabulate novel critical values that take into account the possible bias of the estimator upon which the CIs are based. We give sharp efficiency bounds of using different kernels, and derive the optimal bandwidth for constructing honest CIs. We show that using the bandwidth that minimizes the maximum mean-squared error results in CIs that are nearly efficient and that in this case, the critical value depends only on the rate of convergence. For the common case in which the rate of convergence is n^{−4/5}, the appropriate critical value for 95% CIs is 2.18, rather than the usual 1.96 critical value. We illustrate our results in an empirical application.

Inference with Few Heterogeneous Clusters

Abstract Suppose estimating a model on each of a small number of potentially heterogeneous clusters yields approximately independent, unbiased, and Gaussian parameter estimators. We make two contributions in this setup. First, we show how to compare a scalar parameter of interest between treatment and control units using a two-sample t-statistic, extending previous results for the one-sample t-statistic. Second, we develop a test for the appropriate level of clustering; it tests the null hypothesis that clustered standard errors from a much finer partition are correct. We illustrate the approach by revisiting empirical studies involving clustered, time series, and spatially correlated data.

Quantifying nuisance parameter effects via decompositions of asymptotic refinements for likelihood-based statistics

Accurate inference on a scalar interest parameter in the presence of a nuisance parameter may be obtained using an adjusted version of the signed root likelihood ratio statistic, in particular Barndorff-Nielsen’s R∗ statistic. The adjustment made by this statistic may be decomposed into a sum of two terms, interpreted as correcting respectively for the possible effect of nuisance parameters and the deviation from standard normality of the signed root likelihood ratio statistic itself. We show that the adjustment terms are determined to second-order in the sample size by their means. Explicit expressions are obtained for the leading terms in asymptotic expansions of these means. These are easily calculated, allowing a simple way of quantifying and interpreting the respective effects of the two adjustments, in particular of the effect of a high dimensional nuisance parameter. Illustrations are given for a number of examples, which provide theoretical insight to the effect of nuisance parameters on parametric inference. The analysis provides a decomposition of the mean of the signed root statistic involving two terms: the first has the property of taking the same value whether there are no nuisance parameters or whether there is an orthogonal nuisance parameter, while the second is zero when there are no nuisance parameters. Similar decompositions are discussed for the Bartlett correction factor of the likelihood ratio statistic, and for other asymptotically standard normal pivots.