arXiv subjects

Haojin Zhou

Publications and source records attributed to Haojin Zhou.

4 recordsLinked to original sources

arXiv2026

The Equivariance Criterion in a Linear Model for Random-$X$ Cases

Equivariance is increasingly used in machine learning and statistics, often without systematic justification. In a companion article, the equivariance criterion was applied to the normal linear model with a fixed design matrix (fixed-$X$), yielding the minimum risk equivariant (MRE) estimators of the coefficient vector and of the condensed diagonal covariance matrix under a multivariate invariant location--scale group. We extend these results to the random-$X$ case, with covariates sampled from a population. The extension hinges on a distinction vacuous for fixed-$X$ but fundamental for random-$X$: whether risk and unbiasedness are evaluated conditionally on the realized design or after averaging over the design distribution. Under conditional evaluation, the fixed-$X$ group applies given $X$: least squares remains the best equivariant estimator of the coefficient vector, and the MRE estimators of the population variances keep their fixed-$X$ forms with population sizes at the realized design. Under absolute evaluation with an i.i.d.\ design, the picture changes qualitatively: the natural scale group acting jointly on $(Y,X)$ fixes the coefficient vector, the induced parameter-space action is intransitive, equivariant risks are constant only along orbits indexed by the signal-to-noise ratio $ρ=\|β\|^2/σ^2$, and no uniformly minimum risk equivariant estimator exists. In the scalar case the optimal equivariant weight is the oracle shrinkage factor $w^*(ρ)=ρ/(ρ+E[T^{-1}])$, with least squares recovered as the infinite-signal limit $ρ\to\infty$---explaining and refining the known failure of the Gauss--Markov theorem with random regressors. For a centered design under location--scale transformations, least squares remains optimal within the natural invariant-contrast class, and the MRE estimator $S^2/(n-p+2)$ of the error variance is valid under both modes.

math.ST↗

arXiv2026

Pitman closest equivariant estimators under multivariate scale and location--scale models

For multivariate scale and location--scale models with independent components, we extend the univariate results of Zhou and Nayak (2012) and derive optimum equivariant estimators under the generalized Pitman closeness criterion. We first show, by a counterexample, that in the multivariate case the Pitman closeness comparison within the class of equivariant estimators is not transitive, so that a Pitman closest equivariant estimator does not exist in general. We then enlarge the transformation group by the coordinate permutations---equivalently, impose the formal equivariance principle of Berger (1985) across isomorphic component problems, in the spirit of the separable rules of Robbins' (1951) compound decision theory---and show that within the resulting restricted class an optimum is restored. A multivariate median lemma based on strictly convex losses then yields explicit Pitman closest equivariant estimators of the scale parameters, powers of the scale parameters, and the location parameters, given by median-adjusted versions of any given equivariant estimator. Applications to the multivariate uniform and multivariate normal distributions are worked out in detail. Monte Carlo experiments for the Rayleigh distribution and for a competing risks model with Rayleigh component lifetimes confirm that the proposed estimators dominate the maximum likelihood and Bayes estimators under the Pitman closeness criterion, and a real industrial data set on ball bearing failure times illustrates the feasibility of the method in practice.

math.ST↗

arXiv2025

The Equivariance Criterion in a Linear Model for Fixed-X Cases

The field of machine have seen rising applications of equivariance criterion. However, there is no systematic way to justify its usage, including why it works, whether there is an optimal solution and if so, what form it carries. In this article, we explored the usage of equivariance criterion in a normal linear model with fixed-$X$ and extended the model to allow multiple populations, which, in turn, leads to a multivariate invariant location-scale transformation group, compared than the commonly used univariate one. The minimum risk equivariant estimators of the coefficient vector and the diagonal covariance matrix were derived, which were consistent with literature works. This work serves as an early exploration of the usage of equivariance criterion in machine learning, where we confirmed that the least square approach widely used in machine learning indeed carries optimality in some sense at least in the framework of estimation. Meanwhile, the problems can be shown to be equivalent to a mixture from $p$ independent normal samples and via the principle of functional equivariance, an alternative proof can be derived. However, such an approach carries its own limitation with a strong tie to equivariance criterion.

math.ST↗

arXiv2023

A Dual Cox Model Theory And Its Applications In Oncology

Given the prominence of targeted therapy and immunotherapy in cancer treatment, it becomes imperative to consider heterogeneity in patients' responses to treatments, which contributes greatly to the widely used proportional hazard assumption invalidated as in several clinical trials. To address the challenge, we develop a Dual Cox model theory including a Dual Cox model and a fitting algorithm. As one of the finite mixture models, the proposed Dual Cox model consists of two independent Cox models based on patients' responses to one designated treatment (usually the experimental one) in the clinical trial. Responses of patients in the designated treatment arm can be observed and hence those patients are known responders or non-responders. From the perspective of subgroup classification, such a phenomenon renders the proposed model as a semi-supervised problem, compared to the typical finite mixture model where the subgroup classification is usually unsupervised. A specialized expectation-maximization algorithm is utilized for model fitting, where the initial parameter values are estimated from the patients in the designated treatment arm and then the iteratively reweighted least squares (IRLS) is applied. Under mild assumptions, the consistency and asymptotic normality of its estimators of effect parameters in each Cox model are established. In addition to strong theoretical properties, simulations demonstrate that our theory can provide a good approximation to a wide variety of survival models, is relatively robust to the change of censoring rate and response rate, and has a high prediction accuracy and stability in subgroup classification while it has a fast convergence rate. Finally, we apply our theory to two clinical trials with cross-overed KM plots and identify the subgroups where the subjects benefit from the treatment or not.

stat.ME↗