Outcomes are regressed on a calibrated probability vector for unobserved class membership. Under a structural conditional mean excluding the score and conditional calibration, the observed-data model reduces to a partially linear regression. The probability vector is a Berkson-type surrogate for membership, so the effect vector $\tau$ is identified without attenuation. In practice the vector is often coarsened to a hard label - an argmax, a confidence threshold - and that label need not retain the Berkson property. For any coarsening the plug-in estimator converges to $\mathcal{A}\tau$, where $\mathcal{A}-I$ is determined by the regression of the discarded part of the score on the retained part. Coarsening therefore leaves $\tau$ undistorted exactly when that regression vanishes, and otherwise distorts some contrasts far more than others. The same operator determines the bias that drives coverage loss. Where that bias is of the order of the standard error, the Wald interval has limiting coverage $\Phi(z-\nu)-\Phi(-z-\nu)$, with $\nu$ their ratio. A fixed bias sends coverage to zero. Operator, standard error, and - through the uncoarsened estimator - the bias are estimable from observed data, so the implied coverage can be approximated before the interval is reported. Simulations show severe coverage loss after argmax coarsening. Three real-data audits exhibit the direction-specific distortion.
A novel perturbation test based on a nonsmooth max-difference revenue statistic comparing the best null assortment with the best alternative assortment and asymptotic validity of the proposed p-value under adaptive assortment selection is proposed.
The power prior of Ibrahim and Chen incorporates historical data into a Bayesian analysis by raising the historical likelihood to a power $a_0 \in [0, 1]$. The choice of the exponent has remained an open question. This paper gives a closed-form answer under the predictive log-loss. For a model with $d$ parameters, a historical sample of size $N_0$, and average Kullback--Leibler divergence $\bar{D}_0$ between the historical and current data-generating distributions, the optimal exponent is $a_0^{*} = d/(2 N_0 \bar{D}_0 + d)$. Equivalently, the optimally borrowed effective sample size obeys the harmonic law $1/E^{*} = 1/N_0 + 2\bar{D}_0/d$: compatible data are pooled in full, and any difference caps the borrowed information at $d/(2\bar{D}_0)$ observations. The result is exact for multinomial data and extends to smooth parametric families. The law benchmarks adaptive borrowing, explains the reported degeneracy of the normalized power prior, and shows that neither subsetting the data nor decaying the exponent improves on the correctly discounted constant.
Invertible Logits Transformation (InvLT), which applies a learned scalar MLP element-wise to the pre-softmax logits, consistently outperforms a broad set of post-hoc baselines on standard calibration metrics.
Rolling covariance estimates feed two objects that are routinely treated as market structure. The first is the dominant eigenspace, monitored through the projector movement $\widehat D_{K,t}=\|\widehat P_{K,t}-\widehat P_{K,t-1}\|_F$; the second comprises scalar spectral functionals such as the absorption ratio and the leading-eigenvalue share. Both fluctuate under estimation noise, and shrinkage changes the law of that noise, so reading their movements as structural change requires calibration. For the eigenspace, we derive a first-order null law for $\widehat D_{K,t}$ between overlapping windows that share most of their data and show that it transfers without change to rotation-equivariant shrinkage estimators. A distribution-free Davis-Kahan band gauges whether the eigenspace is identified, an estimator-aware bootstrap provides the calibrated test, and a companion power analysis gives an approximate design rule for the smallest detectable rotation. For the scalar functionals, we show that first-order immunity to elliptical kurtosis holds for scale-invariant functionals and only for them, so that one estimated scalar calibrates the projector null and the absorption-ratio and leading-share intervals across the elliptical family. In high dimensions, where shrinkage cleaning biases the absorption ratio, we give a trace-preserving spike-debiased estimator that removes the bias. The results are verified by simulation under a known population covariance; an equity-panel appendix shows the procedures as diagnostics when the population is unknown.
We study finite-sample linear regression in the presence of varied and unknown label noise, focusing on the heteroskedastic and adaptive linear regression models. Heteroskedastic linear regression models settings where the labels are of varying quality. We receive $n$ pairs $(X_i,Y_i)$ with labels $Y_i=X_i^\top\beta+\varepsilon_i$, where $\varepsilon_i\sim N(0,\sigma_i^2)$ and the variances are unknown to the estimator. One natural measurement of the difficulty of this problem is the number of samples $m$ for which $\sigma_i^2\le1$ (larger $m$ is easier). We obtain a polynomial-time estimator with rate $\tilde{O}((nd^3/m^4)^{1/6})$ when $m\gg d^{3/4}n^{1/4}$, as well as nearly-matching lower bounds. For $d=O(1)$, our estimator achieves error $o(1)$ when $m\gg n^{1/4}$, whereas $L_1$ regression and other traditional approaches require $m\gg n^{1/2}$. In adaptive linear regression, the errors are drawn i.i.d. from an unknown distribution $p$, and our goal is to design a generic estimator that performs nearly as well as the best custom estimator that knows $p$. We introduce a (computationally inefficient) adaptive estimator that, so long as $p$ is a mixture of $k$ symmetric log-concave densities, achieves error comparable with the optimal estimator that knows $p$ and has $\tilde\Theta(n/k)$ samples. For $k=1$, we show that $L_q$ regression (with data-dependent $q$) gives a polynomial-time estimator. Finally, to study the computational limits of both problems, we introduce the planted linear regression problem, where $X_i\sim N(0,I_d)$, $m$ unknown samples are noiseless, and the rest have error $\varepsilon_i\sim N(0,1)$. We conjecture that recovering $\beta$ up to error $\ll\sqrt{d/n}$ (or exactly) may have an information-computation gap between $m=d+1$ and $m\sim d^{3/4}n^{1/4}$, as is suggested by our near-matching polynomial-time estimator and statistical query (SQ) lower bound.
Goodness-of-fit assessment for the binary logistic regression model is difficult when covariates are continuous: the data are effectively sparse, the classical Pearson and deviance tests fail, and practitioners rely on partition-based tests, such as the Hosmer-Lemeshow test, that group observations before comparing observed and expected counts. We study a partition test that modifies the Hosmer-Lemeshow statistic with a single directional correction term, weighted by $(1-2\bar\pi_g)$ and referred to a $\chi^2_{G-2}$ distribution. The correction is the grouped form of the Osius-Rojek/Farrington standardization; grouping makes it well defined in the sparse regime, and it targets the asymmetric over- and under-prediction that a misspecified link induces. A single alignment functional captures its effect, predicting where the test gains power (asymmetric-link misspecification) and where it does not (symmetric departures, and covariate-space structure that no probability-grouping test can see). In simulations the test holds its size; no well-calibrated partition test is more sensitive to asymmetric-link misfit, and it clearly exceeds Hosmer-Lemeshow there, most so for the complementary log-log link -- a modest gain that fades as $n$ grows; it ties Hosmer-Lemeshow on an omitted interaction and is less powerful on an omitted quadratic (by about ten percentage points at $n=1000$). A real-data application illustrates its use, and the test is implemented in the R package ebrahim.gof.
Ebrahim Khaled Ebrahim, Ahmed El-Kotory· 1 citation