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We investigate the fairness issue in classification, where automated decisions are made for individuals from different protected groups. In high-consequence scenarios, decision errors can disproportionately affect certain protected groups, leading to unfair outcomes. To address this issue, we propose a fairness-adjusted selective inference (FASI) framework and develop data-driven algorithms that achieve statistical parity by controlling the false selection rate (FSR) among protected groups. Our FASI algorithm operates by converting the outputs of black-box classifiers into R-values, which are both intuitive and computationally efficient. These R-values serve as the basis for selection rules that are provably valid for FSR control in finite samples for protected groups, effectively mitigating the unfairness in group-wise error rates. We demonstrate the numerical performance of our approach using both simulated and real data.

The simultaneous estimation of many parameters based on data collected from corresponding studies is a key research problem that has received renewed attention in the high-dimensional setting. Many practical situations involve heterogeneous data where heterogeneity is captured by a nuisance parameter. Effectively pooling information across samples while correctly accounting for heterogeneity presents a significant challenge in large-scale estimation problems. We address this issue by introducing the ``Nonparametric Empirical Bayes Structural Tweedie" (NEST) estimator, which efficiently estimates the unknown effect sizes and properly adjusts for heterogeneity via a generalized version of Tweedie's formula. For the normal means problem, NEST simultaneously handles the two main selection biases introduced by heterogeneity: one, the selection bias in the mean, which cannot be effectively corrected without also correcting for, two, selection bias in the variance. We develop theory to show that NEST is asymptotically as good as the optimal Bayes rule that uniquely minimizes a weighted squared error loss. In our simulation studies NEST outperforms competing methods, with much efficiency gains in many settings. The proposed method is demonstrated on estimating the batting averages of baseball players and Sharpe ratios of mutual fund returns. Extensions to other members of the two-parameter exponential family are discussed., Comment: Citations corrected and a new author added. No change in content!

We consider the common setting where one observes probability estimates for a large number of events, such as default risks for numerous bonds. Unfortunately, even with unbiased estimates, selecting events corresponding to the most extreme probabilities can result in systematically underestimating the true level of uncertainty. We develop an empirical Bayes approach "Excess Certainty Adjusted Probabilities" (ECAP), using a variant of Tweedie's formula, which updates probability estimates to correct for selection bias. ECAP is a flexible non-parametric method, which directly estimates the score function associated with the probability estimates, so it does not need to make any restrictive assumptions about the prior on the true probabilities. ECAP also works well in settings where the probability estimates are biased. We demonstrate through theoretical results, simulations, and an analysis of two real world data sets, that ECAP can provide significant improvements over the original probability estimates.

Standardization has been a widely adopted practice in multiple testing, for it takes into account the variability in sampling and makes the test statistics comparable across different study units. However, despite conventional wisdom to the contrary, we show that there can be a significant loss in information from basing hypothesis tests on standardized statistics rather than the full data. We develop a new class of heteroscedasticity--adjusted ranking and thresholding (HART) rules that aim to improve existing methods by simultaneously exploiting commonalities and adjusting heterogeneities among the study units. The main idea of HART is to bypass standardization by directly incorporating both the summary statistic and its variance into the testing procedure. A key message is that the variance structure of the alternative distribution, which is subsumed under standardized statistics, is highly informative and can be exploited to achieve higher power. The proposed HART procedure is shown to be asymptotically valid and optimal for false discovery rate (FDR) control. Our simulation results demonstrate that HART achieves substantial power gain over existing methods at the same FDR level. We illustrate the implementation through a microarray analysis of myeloma., Comment: 55 pages, 13 figures

We suggest a new method, called Functional Additive Regression, or FAR, for efficiently performing high-dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, $X(t)$, and a scalar response, $Y$, in two key respects. First, FAR uses a penalized least squares optimization approach to efficiently deal with high-dimensional problems involving a large number of functional predictors. Second, FAR extends beyond the standard linear regression setting to fit general nonlinear additive models. We demonstrate that FAR can be implemented with a wide range of penalty functions using a highly efficient coordinate descent algorithm. Theoretical results are developed which provide motivation for the FAR optimization criterion. Finally, we show through simulations and two real data sets that FAR can significantly outperform competing methods., Comment: Published at http://dx.doi.org/10.1214/15-AOS1346 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

While functional regression models have received increasing attention recently, most existing approaches assume both a linear relationship and a scalar response variable. We suggest a new method, "Functional Response Additive Model Estimation" (FRAME), which extends the usual linear regression model to situations involving both functional predictors, $X_j(t)$, scalar predictors, $Z_k$, and functional responses, $Y(s)$. Our approach uses a penalized least squares optimization criterion to automatically perform variable selection in situations involving multiple functional and scalar predictors. In addition, our method uses an efficient coordinate descent algorithm to fit general nonlinear additive relationships between the predictors and response. We develop our model for novel forecasting challenges in the entertainment industry. In particular, we set out to model the decay rate of demand for Hollywood movies using the predictive power of online virtual stock markets (VSMs). VSMs are online communities that, in a market-like fashion, gather the crowds' prediction about demand for a particular product. Our fully functional model captures the pattern of pre-release VSM trading prices and provides superior predictive accuracy of a movie's post-release demand in comparison to traditional methods. In addition, we propose graphical tools which give a glimpse into the causal relationship between market behavior and box office revenue patterns, and hence provide valuable insight to movie decision makers., Comment: Published in at http://dx.doi.org/10.1214/14-AOAS781 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Recently, considerable interest has focused on variable selection methods in regression situations where the number of predictors, $p$, is large relative to the number of observations, $n$. Two commonly applied variable selection approaches are the Lasso, which computes highly shrunk regression coefficients, and Forward Selection, which uses no shrinkage. We propose a new approach, "Forward-Lasso Adaptive SHrinkage" (FLASH), which includes the Lasso and Forward Selection as special cases, and can be used in both the linear regression and the Generalized Linear Model domains. As with the Lasso and Forward Selection, FLASH iteratively adds one variable to the model in a hierarchical fashion but, unlike these methods, at each step adjusts the level of shrinkage so as to optimize the selection of the next variable. We first present FLASH in the linear regression setting and show that it can be fitted using a variant of the computationally efficient LARS algorithm. Then, we extend FLASH to the GLM domain and demonstrate, through numerous simulations and real world data sets, as well as some theoretical analysis, that FLASH generally outperforms many competing approaches., Comment: Published in at http://dx.doi.org/10.1214/10-AOAS375 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In many organisms the expression levels of each gene are controlled by the activation levels of known "Transcription Factors" (TF). A problem of considerable interest is that of estimating the "Transcription Regulation Networks" (TRN) relating the TFs and genes. While the expression levels of genes can be observed, the activation levels of the corresponding TFs are usually unknown, greatly increasing the difficulty of the problem. Based on previous experimental work, it is often the case that partial information about the TRN is available. For example, certain TFs may be known to regulate a given gene or in other cases a connection may be predicted with a certain probability. In general, the biology of the problem indicates there will be very few connections between TFs and genes. Several methods have been proposed for estimating TRNs. However, they all suffer from problems such as unrealistic assumptions about prior knowledge of the network structure or computational limitations. We propose a new approach that can directly utilize prior information about the network structure in conjunction with observed gene expression data to estimate the TRN. Our approach uses $L_1$ penalties on the network to ensure a sparse structure. This has the advantage of being computationally efficient as well as making many fewer assumptions about the network structure. We use our methodology to construct the TRN for E. coli and show that the estimate is biologically sensible and compares favorably with previous estimates., Comment: Published in at http://dx.doi.org/10.1214/10-AOAS350 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

High-dimensional classification has become an increasingly important problem. In this paper we propose a "Multivariate Adaptive Stochastic Search" (MASS) approach which first reduces the dimension of the data space and then applies a standard classification method to the reduced space. One key advantage of MASS is that it automatically adjusts to mimic variable selection type methods, such as the Lasso, variable combination methods, such as PCA, or methods that combine these two approaches. The adaptivity of MASS allows it to perform well in situations where pure variable selection or variable combination methods fail. Another major advantage of our approach is that MASS can accurately project the data into very low-dimensional non-linear, as well as linear, spaces. MASS uses a stochastic search algorithm to select a handful of optimal projection directions from a large number of random directions in each iteration. We provide some theoretical justification for MASS and demonstrate its strengths on an extensive range of simulation studies and real world data sets by comparing it to many classical and modern classification methods., Comment: Published in at http://dx.doi.org/10.1214/09-AOAS284 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Regression models to relate a scalar $Y$ to a functional predictor $X(t)$ are becoming increasingly common. Work in this area has concentrated on estimating a coefficient function, $\beta(t)$, with $Y$ related to $X(t)$ through $\int\beta(t)X(t) dt$. Regions where $\beta(t)\ne0$ correspond to places where there is a relationship between $X(t)$ and $Y$. Alternatively, points where $\beta(t)=0$ indicate no relationship. Hence, for interpretation purposes, it is desirable for a regression procedure to be capable of producing estimates of $\beta(t)$ that are exactly zero over regions with no apparent relationship and have simple structures over the remaining regions. Unfortunately, most fitting procedures result in an estimate for $\beta(t)$ that is rarely exactly zero and has unnatural wiggles making the curve hard to interpret. In this article we introduce a new approach which uses variable selection ideas, applied to various derivatives of $\beta(t)$, to produce estimates that are both interpretable, flexible and accurate. We call our method "Functional Linear Regression That's Interpretable" (FLiRTI) and demonstrate it on simulated and real-world data sets. In addition, non-asymptotic theoretical bounds on the estimation error are presented. The bounds provide strong theoretical motivation for our approach., Comment: Published in at http://dx.doi.org/10.1214/08-AOS641 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A significant problem with most functional data analyses is that of misaligned curves. Without adjustment, even an analysis as simple as estimation of the mean will fail. One common method to synchronize a set of curves involves equating ``landmarks'' such as peaks or troughs. The landmarks method can work well but will fail if marker events can not be identified or are missing from some curves. An alternative approach, the ``continuous monotone registration'' method, works by transforming the curves so that they are as close as possible to a target function. This method can also perform well but is highly dependent on identifying an accurate target function. We develop an alignment method based on equating the ``moments'' of a given set of curves. These moments are intended to capture the locations of important features which may represent local behavior, such as maximums and minimums, or more global characteristics, such as the slope of the curve averaged over time. Our method works by equating the moments of the curves while also shrinking toward a common shape. This allows us to capture the advantages of both the landmark and continuous monotone registration approaches. The method is illustrated on several data sets and a simulation study is performed., Comment: Published in at http://dx.doi.org/10.1214/07-AOAS127 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)