For learning about the causal effect of a treatment, a randomized controlled trial (RCT) is considered the gold standard. However, randomizing treatment is sometimes unethical or infeasible, and instead an observational study may be conducted. While some aspects of a well-designed RCT cannot be replicated in an observational study, one aspect that can is to have a protocol with prespecified hypotheses about prespecified outcomes and a prespecified analysis. We illustrate the value of protocols for observational studies in three applications—the effect of playing high school football on later life mental functioning, the effect of police seizing a gun when arresting a domestic violence suspect on future domestic violence and the effect of mountaintop mining on health. We then discuss methodologies for observational study protocols. We discuss considerations for protocols that are similar between observational studies and RCTs, and considerations that are different. The considerations that are different include (i) whether the protocol should be specified before treatment assignment is known or after; (ii) how multiple outcomes should be incorporated into the planned analysis and (iii) how subgroups should be incorporated into the planned analysis. We conclude with discussion of a few open problems in the methodology of observational study protocols.

In his 2022 IMS Medallion Lecture delivered at the Joint Statistical Meetings, Prof. Dylan S. Small eloquently advocated for the use of protocols in observational studies. We discuss his proposal and, inspired by his ideas, we develop a protocol for the regression discontinuity design.

Models such as the zero-inflated and zero-altered Poisson and zero-truncated binomial are now well established. We review important ideas behind the three operators involved (as well as deflation and modification) and other incremental extensions. We propose a super mixture model that unifies alteration, inflation, truncation and deflation for counts, given a 1- or 2-parameter parent or base distribution. Since all of the operators except for truncation have both parametric and nonparametric variants, it is necessary to consider a total of seven different scenarios. Highlights of this paper include the following: (i) the mixture distribution is exceeding flexible, accommodating up to seven modes; (ii) under-, equi- and over-dispersion can be handled using a negative binomial (NB) parent, with underdispersion handled by a novel Generally-Truncated-Expansion method; (iii) under- and over-dispersion are studied holistically in terms of the the four operators previously referred to; (iv) while generally-altered regression explains why observations are there, generally-inflated regression accounts for why they are there in excess, and generally-deflated regression explains why observations are not there. The important application of heaped and seeped data from retrospective self-reported surveys is briefly mentioned. The GAITD-NB has potential to become a Swiss army knife for analyzing count responses.

Large-sample Bayesian analogs exist for many frequentist methods, but are less well known for the widely-used “sandwich” or “robust” variance estimators. We review existing approaches to Bayesian analogs of sandwich variance estimators and propose a new analog, as the Bayes rule under a form of balanced loss function, that combines elements of standard parametric inference with fidelity of the data to the model. Our development is general, for essentially any regression setting with independent outcomes. Being the large-sample equivalent of its frequentist counterpart, we show by simulation that Bayesian robust standard error estimates can faithfully quantify the variability of parameter estimators even under model misspecification—thus retaining the major attraction of the original frequentist version. We demonstrate our Bayesian analog of standard error estimates when studying the association between age and systolic blood pressure in NHANES.

Consider a Bayesian setup in which we observe Y, whose distribution depends on a parameter θ, that is, $\mathit{Y}|\mathit{\theta }\sim {\mathit{\pi }}_{\mathit{Y}|\mathit{\theta }}$. The parameter θ is unknown and treated as random, and a prior distribution chosen from some parametric family $\{{\mathit{\pi }}_{\mathit{\theta }}(·;\mathit{h}),\mathit{h}\in \mathcal{H}\}$, is to be placed on it. For the subjective Bayesian there is a single prior in the family which represents his or her beliefs about θ, but determination of this prior is very often extremely difficult. In the empirical Bayes approach, the latent distribution on θ is estimated from the data. This is usually done by choosing the value of the hyperparameter h that maximizes some criterion. Arguably the most common way of doing this is to let $\mathit{m}(\mathit{h})$ be the marginal likelihood of h, that is, $\mathit{m}(\mathit{h})=\int {\mathit{\pi }}_{\mathit{Y}|\mathit{\theta }}{\mathit{\nu }}_{\mathit{h}}(\mathit{\theta })\mathit{d}\mathit{\theta }$, and choose the value of h that maximizes $\mathit{m}(·)$. Unfortunately, except for a handful of textbook examples, analytic evaluation of ${arg\hspace{0.17em}max}_{\mathit{h}}\mathit{m}(\mathit{h})$ is not feasible. The purpose of this paper is two-fold. First, we review the literature on estimating it and find that the most commonly used procedures are either potentially highly inaccurate or don’t scale well with the dimension of h, the dimension of θ, or both. Second, we present a method for estimating ${arg\hspace{0.17em}max}_{\mathit{h}}\mathit{m}(\mathit{h})$, based on Markov chain Monte Carlo, that applies very generally and scales well with dimension. Let g be a real-valued function of θ, and let $\mathit{I}(\mathit{h})$ be the posterior expectation of $\mathit{g}(\mathit{\theta })$ when the prior is ${\mathit{\nu }}_{\mathit{h}}$. As a byproduct of our approach, we show how to obtain point estimates and globally-valid confidence bands for the family $\mathit{I}(\mathit{h}),\mathit{h}\in \mathcal{H}$. To illustrate the scope of our methodology we provide three detailed examples, having different characters.

There is much interest lately in explainability in statistics and machine learning. One aspect of explainability is to quantify the importance of various features (or covariates). Two popular methods for defining variable importance are LOCO (Leave Out COvariates) and Shapley Values. We take a look at the properties of these methods and their advantages and disadvantages. We are particularly interested in the effect of correlation between features which can obscure interpretability. Contrary to some claims, Shapley values do not eliminate feature correlation. We critique the game theoretic axioms for Shapley values and we question their relevance for assessing feature importance. We propose new, more statistically oriented axioms for feature importance and some measures that satisfy these axioms. However, correcting for correlation is a Faustian bargain: removing the effect of correlation creates other forms of bias. Ultimately, we recommend a slightly modified version of LOCO. We briefly consider how to modify Shapley values to better address feature correlation.

We discuss the asymptotics of the nonparametric maximum likelihood estimator (NPMLE) in the normal mixture model. We then prove the convergence rate of the NPMLE decision in the empirical Bayes problem with normal observations. We point to (and heavily use) the connection between the NPMLE decision and Stein unbiased risk estimator (SURE). Next, we prove that the same solution is optimal in the compound decision problem where the unobserved parameters are not assumed to be random.

Similar results are usually claimed using an oracle-based argument. However, we contend that the standard oracle argument is not valid. It was only partially proved that it can be fixed, and the existing proofs of these partial results are tedious. Our approach, on the other hand, is straightforward and short.

In honor of the 100th birth anniversary of Lucien Le Cam (November 18, 1924–April 24, 2000), we work out a version of the van Trees inequality in a Hájek–Le Cam spirit, that is, under minimal assumptions that, in particular, involve no direct pointwise regularity assumptions on densities but rather almost-everywhere differentiability in quadratic mean of the model. Surprisingly, it suffices that the latter differentiability holds along canonical directions—not along all directions. Also, we identify a (slightly stronger) version of the van Trees inequality as a very instance of a Cramér–Rao bound, that is, the van Trees inequality is not just a Bayesian analog of the Cramér–Rao bound. We provide, as an illustration, an elementary proof of the local asymptotic minimax theorem for quadratic loss functions, again assuming differentiability in quadratic mean only along canonical directions.