We enhance the precision of predicting daily stock market price volatility using the maximum overlapping discrete wavelet transform (MODWT) spectral model and two learning approaches: the heuristic gradient descent (FS.HGD) and hybrid neural fuzzy inference system (HyFIS). The FS.HGD approach iteratively updates the model’s parameters based on the error function gradient, while the HyFIS approach combines the advantages of neural networks and fuzzy logic systems to create a more robust and accurate learning model. The MODWT uses five mathematical functions to form a discrete wavelet basis. The dataset used includes the daily closing prices of the Tadawul stock market from August 2011 to December 2019. Inputs were selected based on multiple regression, tolerance, and variance inflation factor tests, and the oil price (Loil) and repo rate (Repo) were identified as input variables. The output variable is represented by the logarithm of the Tadawul stock market price (LSCS). MODWT-LA8 (ARIMA(1,1,0) with drift) outperforms other WT functions on the 80% dataset, with an ME of (0.00000532), MAE of (0.003214182), and MAPE of (0.06449683). The addition of WT functions to the FS.HGD and HyFIS models increases their forecasting ability. Based on the reduced RMSE (0.048), MAE (0.038), and MAPE (0.538), the MODWT-LA8-FS.HGD outperforms traditional models in predicting the remaining 20% of datasets.

This paper investigates the estimation of the stress–strength reliability parameter $R=P(Y<X)$ when both stress and strength follow independent Garhy distributions under progressive Type-II censoring schemes. A closed-form expression for $\mathit{R}$ is explicitly derived, enabling effective and precise calculation without numerical integration. The Garhy distribution, a flexible one-parameter lifetime model with an increasing hazard function, is confirmed by full-scale goodness-of-fit diagnostics. A Bayesian estimation model is trained on non-informative priors (normal and extended Jeffreys priors) under squared error loss. The posterior expectations are analytically intractable; we adopt two complementary methods of computation: (i) Markov Chain Monte Carlo (MCMC) using the Metropolis–Hastings algorithm and (ii) the Tierney–Kadane (TK) approximation, which provides extremely precise analytical estimates with significantly reduced computational burden. Monte Carlo simulations are large-scale and compare the proposed estimators under different censoring schemes, sample sizes, and parameter configurations in terms of bias and mean squared error (MSE). The methodology is further applied to a real medical dataset comprising kidney dialysis patient survival times, demonstrating its practical relevance in clinical reliability assessment. Results consistently indicate that Bayesian methods, particularly with the extended Jeffreys prior, outperform classical MLEs in terms of stability and accuracy, especially under heavy censoring. Moreover, the TK approximation yields estimates virtually identical to MCMC while requiring only a fraction of the computational effort. We further extend the TK framework to approximate the posterior variance of R and the expected log-likelihood, providing a fully analytical alternative to MCMC for comprehensive Bayesian inference.

This paper investigates the stress–strength reliability parameter $R=P(Y<X)$ for the Garhy distribution under upper-, lower-, and mixed-record schemes. The Garhy distribution, a flexible one-parameter lifetime model, is shown to provide an excellent fit to precipitation and medical (kidney dialysis) data, with Kolmogorov–Smirnov p-values of 0.9904 and 0.9072, respectively. Maximum likelihood estimators (MLEs) of R are developed, alongside Bayesian estimators using Jeffreys and extended Jeffreys priors under squared error loss. Markov chain Monte Carlo (MCMC) methods are employed for posterior inference. An extensive Monte Carlo simulation study reveals that: (i) MLEs converge under all record schemes but exhibit larger bias and lower efficiency compared to Bayesian estimators; (ii) Bayesian estimators, in contrast, demonstrate superior stability, lower mean squared error, and better bias control, especially for pure upper and lower records; (iii) mixed records consistently yield the most balanced and reliable estimates, capturing distributional information more effectively than single-type records; and (iv) the extended Jeffreys prior provides effective bias correction, outperforming the standard Jeffreys prior in many scenarios. Analysis of real-world datasets confirms the simulation findings, with Bayesian estimators under mixed records producing stable and accurate reliability estimates, outperforming MLE in terms of precision and stability. The results strongly advocate for Bayesian methodology with non-informative priors when assessing stress–strength reliability from record-breaking data.

This paper addresses the estimation of the multi-component stress–strength reliability when both the strength variables and the stress variable follow the one-parameter Garhy distribution. Data are assumed to arise from a unified hybrid censoring scheme, which generalizes both Type-I and Type-II hybrid censoring. A closed-form expression for the reliability parameter $R_{m,k}=P(\mathrm{at}\mathrm{least}m\mathrm{of}(X_1,\dots ,X_k)>Y)$ is derived, enabling efficient computation. Three estimation procedures are developed: maximum likelihood estimation (MLE), Bayesian inference using Markov chain Monte Carlo (MCMC) with non-informative priors, and the Tierney–Kadane Laplace-type approximation for posterior moments. For each method, we provide complete mathematical derivations, including the likelihood function under unified hybrid censoring, the posterior conditionals, and the asymptotic distribution of the reliability via the Delta method. Furthermore, Bayesian estimation is extended to asymmetric loss functions, and posterior propriety is formally proven. To check the suitability of the proposed methods, a real data application on generator failure times in power systems is presented.

Recursive constructions derived from finite projective geometries generate scalable families of resolvable block designs exhibiting strong algebraic regularity and intrinsic symmetry. In this work, we investigate the structural optimization of recursion depth in such constructions and demonstrate that the combinatorial growth of recursive chains is governed by a quadratic scaling law originating from the asymptotic expansion of Gaussian binomial coefficients. We show that the resulting exponent is strictly concave, which guarantees the existence and uniqueness of an optimal recursion depth. This optimum occurs at the midpoint of the projective dimension and reflects the dual symmetry of the lattice of projective subspaces. To analyze this behavior, we introduce a normalized objective function that compares recursion depths and reveals a unique maximum corresponding to the midpoint of the projective dimension. Theoretical analysis is supported by exact enumeration and asymptotic validation, confirming that the optimal depth is robust to lower-order perturbations and remains invariant under the natural duality of projective geometry. The proposed framework establishes a direct connection between symmetry properties of discrete geometric structures and optimality in nonlinear discrete systems. These results provide a unified perspective on recursive design constructions, revealing that symmetry not only governs combinatorial structure but also induces a mathematically inevitable optimal configuration. The approach opens new directions for studying symmetry-induced optimality in combinatorial geometry, discrete optimization, and related nonlinear mathematical models.

This paper introduces a rigorous class of two-dimensional Markov-switching autoregressive moving-average (2D MS-ARMA) models for spatial lattice data exhibiting regime-dependent dynamics. The switching mechanism is governed by a latent causal Markov random field that drives spatial transitions between regime-specific autoregressive and moving-average structures. We provide sufficient conditions for the existence of a strictly stationary solution through the top Lyapunov exponent associated with a sequence of random matrices obtained from a state-space representation constructed along the lexicographic order. For the first-order bidirectional specification, we derive explicit spectral conditions linking stationarity to the regime-dependent spectral radii. Sufficient conditions ensuring the existence of finite second-order moments are also provided. Parameter estimation is carried out using a variational expectation–maximization (VEM) algorithm based on a mean-field approximation of the posterior distribution of the hidden regimes. The E-step yields closed-form coordinate ascent updates, while the M-step relies on gradient-based numerical optimization with derivatives computed via recursive differentiation. Under increasing-domain asymptotics, we discuss the consistency and asymptotic behavior of the variational estimator. The proposed framework fills a methodological gap between classical one-dimensional Markov-switching ARMA models and spatial autoregressive structures by extending regime-switching theory to multi-indexed processes with rigorous probabilistic foundations. It provides a comprehensive basis for statistical inference, model diagnostics, and prediction in spatially heterogeneous environments.