This paper studies Graphical SLOPE for precision matrix estimation, with emphasis on its ability to recover both sparsity and clusters of edges with equal or similar strength. In a fixed-dimensional regime, we establish that the root-n scaled estimation error converges to the unique minimizer of a strictly convex optimization problem defined through the directional derivative of the SLOPE penalty. We also establish convergence of the induced SLOPE pattern, thereby obtaining an asymptotic characterization of the clustering structure selected by the estimator. A comparison with GLASSO shows that the grouping property of SLOPE can substantially improve estimation accuracy when the precision matrix exhibits structured edge patterns. To assess the effect of departures from Gaussianity, we then analyze Gaussian-loss precision matrix estimation under elliptical distributions. In this setting, we derive the limiting distribution and quantify the inflation in variability induced by heavy tails relative to the Gaussian benchmark. We also study TSLOPE, based on the multivariate t-loss, and derive its limiting distribution. The results show that TSLOPE offers clear advantages over GSLOPE under heavy-tailed data-generating mechanisms. Simulation evidence suggests that these qualitative conclusions persist in high-dimensional settings, and an empirical application shows that SLOPE-based estimators, especially TSLOPE, can uncover economically meaningful clustered dependence structures.

We develop a hierarchical Bayesian framework for covariance matrix estimation built on a key observation: while equivariance under the full general linear group GL(p) is well known, it is an extremely restrictive property – estimators equivariant to GL(p) are limited to scalar multiples of the sample covariance matrix and carry considerably larger risks than shrinkage estimators. By contrast, commonly used shrinkage estimators, including the Haff empirical Bayes estimator, and the Ledoit–Wolf estimators, are all equivariant under the smaller orthogonal group O(p). Exploiting this structure, we establish that the Haar measure Bayes rule in an oracle eigenvalue model is the minimum risk estimator within the class of O(p)-equivariant estimators, and derive oracle Bayes rules for the covariance and precision matrices under the squared Frobenius, Stein, and squared Stein loss functions. These oracle rules serve as theoretical benchmarks that dominate all commonly used estimators. To approximate them when the true eigenvalues are unknown, we introduce a hierarchical Bayes model that places a finite P'olya tree prior on the eigenvalue distribution and uses Gibbs sampling to generate posterior draws, yielding both shrinkage estimates for the eigenvalues and approximations to the oracle Bayes rules. Simulations suggest that the finite P'olya tree prior is able to recover the general form of the distribution of the eigenvalues, and confirm that the resulting estimators closely approach oracle performance, substantially outperforming classical competitors for both covariance and precision matrix estimation.

SLOPE is a popular method for dimensionality reduction in the high-dimensional regression. Indeed some regression coefficient estimates of SLOPE can be null (sparsity) or can be equal in absolute value (clustering). Consequently, SLOPE may eliminate irrelevant predictors and may identify groups of predictors having the same influence on the vector of responses. The notion of SLOPE pattern allows to derive theoretical properties on sparsity and clustering by SLOPE. Specifically, the SLOPE pattern of a vector provides: the sign of its components (positive, negative or null), the clusters (indices of components equal in absolute value) and clusters ranking. In this article we give a necessary and sufficient condition for SLOPE pattern recovery of an unknown vector of regression coefficients.

This article investigates the asymptotic distribution of penalized estimators with non-differentiable penalties designed to recover low-dimensional pattern structures. Patterns play a central role in estimation, as they reveal the underlying structure of the parameter – which coefficients are zero, which are equal, and how they are clustered. The main technical challenge stems from the discontinuous nature of these patterns (such as the sign function in the case of the Lasso penalty), a difficulty not previously addressed in the literature and only recently analyzed for the standard linear model. To overcome this, we extend classical results from empirical process theory for M-estimation by incorporating the distributional behavior of model patterns. We introduce a new mathematical framework for studying pattern convergence of regularized M-estimators. While classical approaches to distributional convergence rely on uniform conditions, our analysis employs a new local condition, stochastic Lipschitz differentiability (SLD), which controls fluctuations of the Taylor remainder. We demonstrate how this framework applies to a broad class of loss functions, covering generalized linear models (e.g., logistic and Poisson regression) and robust regression settings with non-smooth losses such as the Huber and quantile loss.

The Partial Correlation Graphical LASSO (PCGLASSO) offers a scale-invariant alternative to the standard GLASSO. This paper provides the first comprehensive treatment of the PCGLASSO estimator. We introduce a novel and highly efficient algorithm. Our central theoretical contribution is the first scale-invariant irrepresentability criterion for PCGLASSO, which guarantees consistent model selection. We prove this condition is significantly weaker than its GLASSO counterpart, providing the first theoretical justification for PCGLASSO's superior empirical performance, especially in recovering networks with hub structures. Furthermore, we deliver the first analysis of the estimator's non-convex solution landscape, establishing new conditions for global uniqueness and guaranteeing the consistency of all minimizers.

Gamma-ray bursts (GRBs) are among the most luminous explosions in the Universe and serve as powerful probes of the early cosmos. However, the rapid fading of their afterglows and the scarcity of spectroscopic measurements make photometric classification crucial for timely high-redshift identification. We present an ensemble machine learning framework for redshift classification of GRBs based solely on their optical plateau and prompt emission properties. Our dataset comprises 171 long GRBs observed by the Swift UVOT and more than 450 ground-based telescopes. The analysis pipeline integrates robust statistical techniques, including M-estimator outlier rejection, multivariate imputation using Multiple Imputation by Chained Equations, and Least Absolute Shrinkage and Selection Operator feature selection, followed by a SuperLearner ensemble combining parametric, semi-parametric, and non-parametric algorithms. The optimal model, trained on raw optical data with outlier removal at a redshift threshold of z equals 2.0, achieves a true positive rate of 74 percent and an area under the curve of 0.84, maintaining balanced generalization between training and test sets. At higher thresholds, such as z equals 3.0, the classifier sustains strong discriminative power with an area under the curve of 0.88. Validation on an independent GRB sample yields 97 percent overall accuracy, perfect specificity, and an ensemble area under the curve of 0.93. Compared to previous prompt- and X-ray-based classifiers, our optical framework offers enhanced sensitivity to high-redshift events, improved robustness against data incompleteness, and greater applicability to ground-based follow-up. We also publicly release a web application that enables real-time redshift classification, facilitating rapid identification of candidate high-redshift GRBs for cosmological studies.

Optimal minimum variance portfolios can be analytically computed and require only the estimate of the inverse of the covariance matrix, commonly referred to as the precision matrix. Graphical models, which have demonstrated exceptional performance in uncovering the conditional dependence structure among a given set of variables, can provide reliable estimates of the precision matrix. This paper introduces two novel graphical modeling techniques: Gslope and Tslope, which use the Sorted ℓ _1 -Penalized Estimator (Slope) to directly estimate the precision matrix. We develop ad hoc algorithms to efficiently solve the underlying optimization problems: the Alternating Direction Method of Multipliers for Gslope, and the Expectation-Maximization algorithm for Tslope. Our methods are suitable for both Gaussian and non-Gaussian distributed data and take into account the empirically observed distributional characteristics of asset returns. Through extensive simulation analysis, we demonstrate the superiority of our new methods over state-of-the-art estimation techniques, particularly regarding clustering and stability characteristics. The empirical results on real-world data support the validity of our new approaches, which often outperform state-of-the-art methods in terms of volatility, extreme risk, and risk-adjusted returns. Notably, they prove to be effective tools for dealing with high-dimensional problems and heavy-tailed distributions, two critical issues in the literature.

Context. Gamma-ray bursts (GRBs), which have been observed at redshifts as high as 9.4, could serve as valuable probes for investigating the distant Universe. However, using them in this manner necessitates an increase in the number of GRBs with determined redshifts, as currently only 12% of them have known redshifts due to observational biases. Aims. We aim to address the shortage of GRBs with measured redshifts to enable full realization of their potential as valuable cosmological probes. Methods. Following our previous approach, in this work we take a further step to overcome this issue by adding 30 more GRBs to our ensemble supervised machine learning training sample, representing an increase of 20%, which will help us obtain more accurate pseudo-redshifts. In addition, we have built a freely accessible and user-friendly web application that infers the redshift of long GRBs (LGRBs) with plateau emission using our machine learning model. The web app is the first of its kind for such a study and will allow the community to obtain pseudo-redshifts by entering the GRB parameters into the app. Results. Through our machine learning model, we successfully estimated redshifts for 276 LGRBs using X-ray afterglow parameters detected by the Neil Gehrels Swift Observatory and increased the sample of LGRBs with known redshifts by 110%. We also performed Monte Carlo simulations to demonstrate the future applicability of this research. Conclusions. The results presented in this work will enable the community to increase the sample of GRBs with known pseudoredshifts. This can help address many outstanding issues, such as GRB formation rate, luminosity function, and the true nature of low-luminosity GRBs, and it can enable the application of GRBs as standard candles.

Gamma-ray bursts (GRBs) are intense, short-lived bursts of gamma-ray radiation observed up to a high redshift ( z ∼ 10) due to their luminosities. Thus, they can serve as cosmological tools to probe the early Universe. However, we need a large sample of high- z GRBs, currently limited due to the difficulty in securing time at the large aperture telescopes. Thus, it is painstaking to determine quickly whether a GRB is high- z or low- z , which hampers the possibility of performing rapid follow-up observations. Previous efforts to distinguish between high- and low- z GRBs using GRB properties and machine learning (ML) have resulted in limited sensitivity. In this study, we aim to improve this classification by employing an ensemble ML method on 251 GRBs with measured redshifts and plateaus observed by the Neil Gehrels Swift Observatory. Incorporating the plateau phase with the prompt emission, we have employed an ensemble of classification methods to unprecedentedly enhance the sensitivity. Additionally, we investigate the effectiveness of various classification methods using different redshift thresholds, z _threshold = z _t at z _t = 2.0, 2.5, 3.0, and 3.5. We achieve a sensitivity of 87% and 89% with a balanced sampling for both z _t = 3.0 and z _t = 3.5, respectively, representing a 9% and 11% increase in the sensitivity over random forest used alone. Overall, the best results are at z _t = 3.5, where the difference between the sensitivity of the training set and the test set is the smallest. This enhancement of the proposed method paves the way for new and intriguing follow-up observations of high- z GRBs.

MOTIVATION:Bottom-up mass spectrometry-based proteomics studies changes in protein abundance and structure across conditions. Since the currency of these experiments are peptides, i.e. subsets of protein sequences that carry the quantitative information, conclusions at a different level must be computationally inferred. The inference is particularly challenging in situations where the peptides are shared by multiple proteins or post-translational modifications. While many approaches infer the underlying abundances from unique peptides, there is a need to distinguish the quantitative patterns when peptides are shared. RESULTS:We propose a statistical approach for estimating protein abundances, as well as site occupancies of post-translational modifications, based on quantitative information from shared peptides. The approach treats the quantitative patterns of shared peptides as convex combinations of abundances of individual proteins or modification sites, and estimates the abundance of each source in a sample together with the weights of the combination. In simulation-based evaluations, the proposed approach improved the precision of estimated fold changes between conditions. We further demonstrated the practical utility of the approach in experiments with diverse biological objectives, ranging from protein degradation and thermal proteome stability, to changes in protein post-translational modifications. AVAILABILITY AND IMPLEMENTATION:The approach is implemented in an open-source R package MSstatsWeightedSummary. The package is currently available at https://github.com/Vitek-Lab/MSstatsWeightedSummary (doi: 10.5281/zenodo.14662989). Code required to reproduce the results presented in this article can be found in a repository https://github.com/mstaniak/MWS_reproduction (doi: 10.5281/zenodo.14656053).

This paper explores the asymptotic distributions of low-dimensional patterns in linear regression with regularizers such as Lasso, Elastic Net, Generalized Lasso, and SLOPE, as the number of observations n grows and the penalty increases at rate √(n) . While the asymptotic distribution of rescaled estimation errors is well-understood, convergence of patterns lacks proof in the literature, even for Lasso. We provide a proof using the Hausdorff distance for subdifferentials. We also derive the limiting probability of recovering the true model pattern, which approaches 1 when the penalty scaling diverges and the regularizer-specific asymptotic irrepresentability condition is satisfied. We propose two-step procedures that asymptotically recover model patterns, regardless of the irrepresentability condition. Our theory shows that Fused Lasso cannot reliably recover its clustering pattern for independent regressors, but this can be resolved by concavifying its penalty coefficients. Simulation studies compare the asymptotic properties of Lasso, Fused Lasso, and SLOPE.

Polygenic risk scores (PRS) are widely used in post-GWAS analyses to predict complex traits across humans, animals, and plants. While significant progress has been made in developing new PRS methods, much less attention has been given to quantifying the uncertainty associated with these predictions. In this work, we propose a method for individualized uncertainty quantification based on quantile regression. When paired with conformal prediction, this approach enables the construction of prediction intervals with guaranteed coverage, offering lower and upper bounds within which the phenotype is likely to fall with high probability. We apply this framework to data from the UK Biobank and the ProgeNIA/SardiNIA studies, showing that the resulting prediction intervals: (1) maintain valid coverage under minimal model assumptions, (2) provide more realistic individualized estimates of uncertainty by allowing for asymmetry and individual-specific interval lengths, and (3) exhibit reduced uncertainty compared to existing methods. Overall, we present a novel framework for individualized uncertainty quantification in PRS analyses and highlight the importance of incorporating uncertainty into predictive modeling. ### Competing Interest Statement The authors have declared no competing interest.

We present a suite of packages in R, Python, Julia, and C++ that efficiently solve the Sorted L-One Penalized Estimation (SLOPE) problem. The packages feature a highly efficient hybrid coordinate descent algorithm that fits generalized linear models (GLMs) and supports a variety of loss functions, including Gaussian, binomial, Poisson, and multinomial logistic regression. Our implementation is designed to be fast, memory-efficient, and flexible. The packages support a variety of data structures (dense, sparse, and out-of-memory matrices) and are designed to efficiently fit the full SLOPE path as well as handle cross-validation of SLOPE models, including the relaxed SLOPE. We present examples of how to use the packages and benchmarks that demonstrate the performance of the packages on both real and simulated data and show that our packages outperform existing implementations of SLOPE in terms of speed.

The standard cosmological model, the $\Lambda$CDM model, is the most suitable description for our universe. This framework can explain the accelerated expansion phase of the universe but still is not immune to open problems when it comes to the comparison with observations. One of the most critical issues is the so-called Hubble constant ($H_0$) tension, namely, the difference of about $5\sigma$ as an average between the value of $H_0$ estimated locally and the cosmological value measured from the Last Scattering Surface. The value of this tension changes from 4 to 6 $\sigma$ according to the data used. The current analysis explores the $H_0$ tension in the \textit{Pantheon} sample (PS) of SNe Ia. Through the division of the PS in 3 and 4 bins, the value of $H_0$ is estimated for each bin and all the values are fitted with a decreasing function of the redshift ($z$). Remarkably, $H_0$ undergoes a slow decreasing evolution with $z$, having an evolutionary coefficient compatible with zero up to $5.8\sigma$. If this trend is not caused by hidden astrophysical biases or $z$-selection effects, then the $f(R)$ modified theories of gravity represent a valid model for explaining such a trend.

Context. Gamma-ray bursts (GRBs), observed at redshifts as high as 9.4, could serve as valuable probes for investigating the distant Universe. However, this necessitates an increase in the number of GRBs with determined redshifts, as currently, only 12 Aims. We aim to address the shortage of GRBs with measured redshifts, enabling us to fully realize their potential as valuable cosmological probes Methods. Following Dainotti et al. (2024c), we have taken a second step to overcome this issue by adding 30 more GRBs to our ensemble supervised machine learning training sample, an increase of 20 estimates. In addition, we have built a freely accessible and user-friendly web app that infers the redshift of long GRBs (LGRBs) with plateau emission using our machine learning model. The web app is the first of its kind for such a study and will allow the community to obtain redshift estimates by entering the GRB parameters in the app. Results. Through our machine learning model, we have successfully estimated redshifts for 276 LGRBs using X-ray afterglow parameters detected by the Neil Gehrels Swift Observatory and increased the sample of LGRBs with known redshifts by 110 demonstrate the future applicability of this research. Conclusions. The results presented in this research will enable the community to increase the sample of GRBs with known redshift estimates. This can help address many outstanding issues, such as GRB formation rate, luminosity function, and the true nature of low-luminosity GRBs, and enable the application of GRBs as standard candles

Gamma-Ray Bursts (GRBs), due to their high luminosities are detected up to redshift 10, and thus have the potential to be vital cosmological probes of early processes in the universe. Fulfilling this potential requires a large sample of GRBs with known redshifts, but due to observational limitations, only 11\% have known redshifts ($z$). There have been numerous attempts to estimate redshifts via correlation studies, most of which have led to inaccurate predictions. To overcome this, we estimated GRB redshift via an ensemble supervised machine learning model that uses X-ray afterglows of long-duration GRBs observed by the Neil Gehrels Swift Observatory. The estimated redshifts are strongly correlated (a Pearson coefficient of 0.93) and have a root mean square error, namely the square root of the average squared error $\langle\Delta z^2\rangle$, of 0.46 with the observed redshifts showing the reliability of this method. The addition of GRB afterglow parameters improves the predictions considerably by 63\% compared to previous results in peer-reviewed literature. Finally, we use our machine learning model to infer the redshifts of 154 GRBs, which increase the known redshifts of long GRBs with plateaus by 94\%, a significant milestone for enhancing GRB population studies that require large samples with redshift.

Type Ia Supernovae (SNe Ia) are considered the most reliable standard candles and they have played an invaluable role in cosmology since the discovery of the Universe's accelerated expansion. During the last decades, the SNe Ia samples have been improved in number, redshift coverage, calibration methodology, and systematics treatment. These efforts led to the most recent "Pantheon" (2018) and "Pantheon +" (2022) releases, which enable to constrain cosmological parameters more precisely than previous samples. In this era of precision cosmology, the community strives to find new ways to reduce uncertainties on cosmological parameters. To this end, we start our investigation even from the likelihood assumption of Gaussianity, implicitly used in this domain. Indeed, the usual practice involves constraining parameters through a Gaussian distance moduli likelihood. This method relies on the implicit assumption that the difference between the distance moduli measured and the ones expected from the cosmological model is Gaussianly distributed. In this work, we test this hypothesis for both the Pantheon and Pantheon + releases. We find that in both cases this requirement is not fulfilled and the actual underlying distributions are a logistic and a Student's t distribution for the Pantheon and Pantheon + data, respectively. When we apply these new likelihoods fitting a flat ΛCDM model, we significantly reduce the uncertainties on the matter density ΩM and the Hubble constant H0 of ∼40%. As a result, the Hubble tension is increased at >5σ level. This boosts the SNe Ia power in constraining cosmological parameters, thus representing a huge step forward to shed light on the current debated tensions in cosmology.

Gamma-ray bursts (GRBs) can be probes of the early Universe, but currently, only 26% of GRBs observed by the Neil Gehrels Swift Observatory have known redshifts ( z ) due to observational limitations. To address this, we estimated the GRB redshift (distance) via a supervised statistical learning model that uses optical afterglow observed by Swift and ground-based telescopes. The inferred redshifts are strongly correlated (a Pearson coefficient of 0.93) with the observed redshifts, thus proving the reliability of this method. The inferred and observed redshifts allow us to estimate the number of GRBs occurring at a given redshift (GRB rate) to be 8.47–9 yr ^−1 Gpc ^−1 for 1.9 < z < 2.3. Since GRBs come from the collapse of massive stars, we compared this rate with the star formation rate, highlighting a discrepancy of a factor of 3 at z < 1.

The LambdaCDM model is the most commonly accepted framework in modern cosmology. However, the local measurements of the Hubble constant, H0, via the Supernovae Type Ia (SNe Ia) calibrated on Cepheids provide a value which is in significant disagreement, from 4 to 6 sigma, with the value of H0 inferred from the Cosmic Microwave Background (CMB) observed by Planck. This disagreement is the so-called Hubble constant tension. To find out the reason for this discrepancy, we analyze the behaviour of the H0 in the Pantheon sample of SNe Ia through a binning approach: we divide the Pantheon into 3 and 4 bins ordered with redshift (z), and for each of them, we estimate the H0. After the H0 estimation, we fit the H0 values with a decreasing function of z, finding out that H0 undergoes a slow decreasing trend compatible with the evolution scenario in 2.0 sigma. [...] Together with SNe Ia, more astrophysical probes such as quasars (QSO) [...] and Gamma-Ray Bursts (GRBs) [...], are needed to tackle the H0 tension. In the realm of GRB-cosmology, one of the most promising correlations is the fundamental plane relation [...]. In the context of applying this relation as a cosmological tool, we also compute how many GRBs must be gathered to reach the same precision as the SNe Ia. Since we are about two decades away from reaching such precision, we also attempt to find additional correlations for the GRBs associated with SNe Ibc that could be exploited to standardize the class of GRB-SNe Ibc in the future. We find a hint of a correlation between the GRBs' end-of-plateau optical luminosity and the SNe's rest-frame peak time, suggesting that the GRBs with the most luminous optical plateau emission are associated with SNe with the most delayed peaks in their light curves. So far, it is the fundamental plane relation to be the most promising candle for exploring the high-z universe.

Cosmological models and their parameters are widely debated, especially about whether the current discrepancy between the values of the Hubble constant, H 0 , obtained by Type Ia supernovae (SNe Ia) and the Planck data from the cosmic microwave background radiation could be alleviated when alternative cosmological models are considered. Thus, combining high-redshift probes, such as gamma-ray bursts (GRBs) and quasi-stellar objects (QSOs, or quasars), together with baryon acoustic oscillations and SNe Ia is important to assess the viability of these alternative models and whether they can cast further light on the Hubble tension. In this work, for GRBs, we use a three-dimensional relation between the peak prompt luminosity, the rest-frame time at the end of the X-ray plateau, and its corresponding luminosity in X-rays: the 3D Dainotti fundamental plane relation. Regarding QSOs, we use the Risaliti−Lusso relation among the UV and X-ray luminosities for a sample of 2421 sources. We correct both the QSO and GRB relations by accounting for selection and evolutionary effects with a reliable statistical method. We here use both the traditional Gaussian likelihoods (   ) and the new best-fit likelihoods (   ) to infer cosmological parameters of nonflat Lambda cold dark matter (ΛCDM) and flat w CDM models. We obtain for all the parameters reduced uncertainties, up to 35% for H 0 , when applying the new   likelihoods in place of the Gaussian ones. Our results remain consistent with a flat ΛCDM model, although with a shift of the dark energy parameter w toward w < −1 and a curvature density parameter toward Ω k < 0.