Logistic regression is a commonly used classification algorithm in machine learning. It allows categorizing data into discrete classes by learning the relationship from a given set of labeled data. It learns a linear relationship from the given dataset and then introduces nonlinearity through an activation function to determine a hyperplane that separates the learning points into two subclasses. In the case of logistic regression, the logistic function is the most used activation function to perform binary classification. The choice of logistic function for binary classifications is justified by its ability to transform any real number into a probability between 0 and 1. This study provides, through two different approaches, a rigorous statistical answer to the crucial question that torments us, namely where does this logistic function on which most neural network algorithms are based come from? Moreover, it determines the computational cost of logistic regression, using theoretical and experimental approaches.

A smoothed bootstrap method is introduced for right-censored data based on the right-censoring-A_(n) assumption introduced by Coolen and Yan, which is a generalization of Hill’s A_(n) assumption for right-censored data. The smoothed bootstrap method is compared to Efron’s method for right-censored data through simulations. The comparison is conducted in terms of the coverage of percentile confidence intervals for the quartiles. From the study, it is found that the smoothed bootstrap method mostly performs better than Efron’s method, in particular for small data sets. We also illustrate the use of the method for survival function inference and compare it to a smoothed Kaplan-Meier bootstrap method through simulations.

Nonresponse is a major problem encountered by surveyors when conducting sampling surveys. The present study suggested a naïve modified Searls method for the elevated estimation of the population mean of the primary variable under investigation by utilizing the known auxiliary parameters. The bias along with the mean squared error (MSE) of the introduced estimator is calculated up to the approximation of the first order. We compared the presented estimator with a competing usual unbiased estimator and other competing population mean estimators regarding the issue of nonresponse. The efficiency criteria of the introduced estimator outperforming the other estimators in the competition are determined and verified using five real data sets. The MSEs for the introduced estimator and the other estimators in the competition are calculated for the five considered populations. The estimator with the least MSE or highest percentage relative efficiency (PRE) is recommended for practical exercise in different areas of applications.

This paper demonstrates the application of smoothed bootstrap methods and Efron’s methods for hypothesis testing on real-valued data, right-censored data and bivariate data. The tests include quartile hypothesis tests, two sample medians and Pearson and Kendall correlation tests. Simulation studies indicate that the smoothed bootstrap methods outperform Efron’s methods in most scenarios, particularly for small datasets. The smoothed bootstrap methods provide smaller discrepancies between the actual and nominal error rates, which makes them more reliable for testing hypotheses.

Early and accurate diagnosis of lung cancer remains challenging due to the heterogeneity of tumor morphology and the variability across imaging modalities. This study proposed a deep learning framework that integrated computed tomography (CT), positron emission tomography/computed tomography (PET/CT), and chest X-ray (CXR) within a unified multi-modal transformer architecture for early lung cancer detection. The framework employed modality-specific encoders combining convolutional and state-space blocks to extract spatial-frequency representations, followed by a gated cross-modal fusion transformer designed to align heterogeneous features and handle missing modalities through mixture-of-experts routing and low-rank imputation. Multi-task heads were jointly optimized for nodule detection, segmentation, malignancy classification, and survival risk prediction. Explainability was embedded through concept bottlenecks, prototype reasoning, gradient-based attribution, and counterfactual concept editing, offering case-level interpretability and clinically meaningful evidence maps. Uncertainty was estimated via Monte-Carlo dropout, deep ensembles, and temperature scaling to ensure calibrated confidence estimates and defer-to-expert safety decisions. Lung image database consortium and image database resource initiative (LIDC-IDRI) (CT), the cancer imaging archive (TCIA) (PET/CT), and national lung screening trial (NLST) (CXR) benchmark datasets revealed that our methods work better than the best methods available. The proposed technique yielded Dice scores of 0.879, 0.872, and 0.876, together with AUC values of 0.944, 0.952, and 0.938, and an expected calibration error (ECE) of 0.02 across all modalities. Under domain shift, cross-dataset analysis showed substantial generalization ($ AUC > 0.92 $). A generalizable framework for multi-modal diagnostics made it possible to use AI to help with lung cancer screening in a way that was clear, trustworthy, and scalable.