Images are often degraded during the data acquisition process. The degradation may involve blurring, information loss due to sampling, and various sources of noise. The purpose of image restoration is to estimate the original image from the degraded data. The present work sets forward a restoration technique for exponential dispersion noise based on Particle filtering (PF) using Hidden Markov Model. In order not to take observation information into account in general, the PF algorithm produced an incorrect sample from a discrete approximation distribution. To resolve this problem, we propose in the resampling stage of PF, samples which are generated from a continuous distribution rather than a discrete one based on Exponential Dispersion Models (EDM). An iterative approach, called the Expectation-Maximization (EM) algorithm, is used to find the maximum likelihood estimates of the relevant unknown parameters of the EDM. Moreover, under some conditions, the concavity of the conditional expected log-likelihood function is established in the maximization step of the EM algorithm. The proposed approach is rooted in ideas from statistics, control theory and signal processing. Experimental results are eventually displayed with simulation and satellite images, which demonstrate the good performance of the proposed approach.

The estimation of parameters for a distribution function is a significant and prominent field within statistical inference. This particular problem holds great relevance in various domains, including industries, stock markets, image processing, and reliability studies. There are two recognized approaches to estimation: point estimation and interval estimation, also known as confidence intervals. In this study, our primary focus lies in the point estimation of parameters associated with an exponential dispersion distribution function. In this process, we consider one of the parameters as a random variable that requires estimation. To tackle this, we adopt a Bayesian inference approach utilizing a one-parameter dispersion distribution. We explore non-informative priors, such as uniform and Jeffrey’s priors, and provide evidence of the effectiveness of our method through simulation studies.

The basic objective of this paper is to introduce a new parametrization and extension of stochastic volatility models where the diffusion of the stock return is generated by the shifted compound Poisson distribution. This new class generalizes the affine standard process and can be better adapted to several empirical characteristics of return data. The shifted compound Poisson Model is more flexible and is usually implemented to modeling negative and non negative continuous data with a discrete probability mass at zero. The proposed model parameters, based on the Bayesian analysis, have been estimated. Furthermore, the dispersion stochastic volatility ${\varphi }_t$ which follows an infinite mixture inverse Gamma distribution is elaborated. The model’s performance has been analyzed with real data for its selection by varying the power parameter p in $\left\{0\right\}\cup (1,2).$ The paper also develops formal tools for comparing the basic standard stochastic volatility and the proposed model. Moreover, the proposed methodology is illustrated by analyzing four series of real indexes returns.

Lifetime data arising in engineering reliability, survival analysis, actuarial science, and environmental studies often exhibit substantial right-skewness, extreme observations, and heterogeneous hazard-rate structures. Classical lifetime distributions may not adequately capture these characteristics, thereby affecting risk assessment, reliability evaluation, and predictive performance. In this paper, we introduce the Heavy-Tailed QLindley (HTQL) distribution, a new two-parameter heavy-tailed extension of the QLindley model obtained through the New Family of Heavy-Tailed (NFHT) transformation. The proposed distribution provides greater flexibility for modelling positively skewed and heavy-tailed data while preserving analytical tractability. The HTQL model accommodates increasing, decreasing, bathtub-shaped, unimodal, and nearly constant hazard rate functions, making it suitable for applications in reliability analysis, survival studies, actuarial science, and environmental modelling. Several mathematical and statistical properties of the HTQL distribution are derived, including explicit expressions for the quantile function, ordinary and incomplete moments, order statistics, and reliability measures. Important tail-based risk measures such as Value-at-Risk, Tail Value-at-Risk, Tail Variance, Tail Variance Premium, and Expected Shortfall are also obtained. Parameter estimation is investigated using maximum likelihood, ordinary least squares, Cramér–von Mises, and Bayesian approaches, together with bootstrap confidence intervals. A Monte Carlo simulation study is conducted to evaluate the finite-sample performance of the proposed estimators. The practical usefulness of the HTQL distribution is illustrated using three real-world datasets from pharmacokinetics, engineering reliability, and environmental studies. The empirical results show that the HTQL distribution provides highly competitive fits compared with several classical, Lindley-type, and heavy-tailed distributions. Overall, the proposed model constitutes a flexible and parsimonious alternative for modelling positive heavy-tailed data.

The rapid growth of high-dimensional predictive models in science and industry has intensified the need for statistically rigorous interpretability tools. Although model-agnostic feature importance methods are widely used to explain black-box models, they lack formal uncertainty quantification, leading to unreliable conclusions in high-dimensional settings where spurious correlations are common. We propose a Bootstrap-of-Bootstrap (BoB) inference framework that enables valid uncertainty quantification and hypothesis testing for any model-agnostic feature importance measure. To overcome the high computational cost of nested resampling, we develop an efficient analytical approximation based on influence function theory. The proposed approach provides calibrated confidence intervals and a stability score for each feature, strengthening the statistical foundations of explainable AI. Simulation studies and real-world applications in cancer genomics and credit risk modeling demonstrate its effectiveness, providing reliable, auditable explanations for high-stakes decision-making.