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The problem of characterizing a probability distribution is an important problem which has attracted the attention of many researchers in the recent years. To understand the behavior of the data obtained through a given process, we need to be able to describe this behavior via its approximate probability law. This, however, requires to establish conditions which govern the required probability law. In other words we need to have certain conditions under which we may be able to recover the probability law of the data. So, characterization of a distribution plays an important role in applied sciences, where an investigator is vitally interested to find out if their model follows the selected distribution. In this short note, certain characterizations of three recently introduced discrete distributions are presented to complete, in some way, the works ofHussain(2020), Eliwa et al.(2020) and Hassan et al.(2020).

We consider nonparametric estimation of a mixed discrete-continuous distribution under anisotropic smoothness conditions and a possibly increasing number of support points for the discrete part of the distribution. For these settings, we derive lower bounds on the estimation rates. Next, we consider a nonparametric mixture of normals model that uses continuous latent variables for the discrete part of the observations. We show that the posterior in this model contracts at rates that are equal to the derived lower bounds up to a log factor. Thus, Bayesian mixture of normals models can be used for (up to a log factor) optimal adaptive estimation of mixed discrete-continuous distributions. The proposed model demonstrates excellent performance in simulations mimicking the first stage in the estimation of structural discrete choice models.

Seven versions of the discrete Weibull distribution are characterized via conditional expectation of function of the random variable as well as based on the hazard or reverse hazard function.

Fractured rocks, as typical engineering materials, are commonly influenced by the distinct arrangement of joints and the angle of their inclination. In this study, uniaxial compression tests were conducted on three series of marble samples (a-series, b-series, and c-series), each containing a dip angle of 45°. The mechanical properties and crack propagation characteristics were subsequently analyzed. Furthermore, three series of numerical simulations of the fractured rocks were performed using the discrete element method. A comparison was made between the failure patterns and crack distribution observed in the laboratory tests and those obtained from the PFC simulations, including crack distribution, particle displacement magnitude, and displacement distribution maps for each specimen. Additionally, the strength, deformation characteristics, and failure processes of the rock mass were examined. The results demonstrate that the distribution of joints significantly influences the mechanical behavior of marble. The numerical simulations reveal that both the joint dip angle and joint distribution affect the strength and deformation characteristics of the rock mass. The trend of peak strength variation with dip angle is consistent across all three series, with peak strength increasing as the dip angle increases. Both the joint dip angle and joint distribution impact the failure process of the rock samples; however, the influence of joint distribution is found to be more significant than that of joint dip angle.

In this article we present a stochastic ordering verification algorithm between multivariate discrete distributions implemented in the C++ programming language. This algorithm is essential in problems of finding the optimal portfolio when dealing with discrete distributions.

We introduce a probabilistic language and an efficient inference algorithm based on distributional clauses for static and dynamic inference in hybrid relational domains. Static inference is based on sampling, where the samples represent (partial) worlds (with discrete and continuous variables). Furthermore, we use backward reasoning to determine which facts should be included in the partial worlds. For filtering in dynamic models we combine the static inference algorithm with particle filters and guarantee that the previous partial samples can be safely forgotten, a condition that does not hold in most logical filtering frameworks. Experiments show that the proposed framework can outperform classic sampling methods for static and dynamic inference and that it is promising for robotics and vision applications. In addition, it provides the correct results in domains in which most probabilistic programming languages fail.

A new one-parameter discrete distribution, namely the Poisson Haq (PH) distribution , is proposed by a mixture of the Poisson variable and an independently distributed Haq random variable. This model effectively analyzes over-dispersed count datasets by extending Poisson distribution. Various useful statistical properties of the PH distribution are derived and discussed. The failure rate of the proposed distribution is “increasing” and “upside bathtub” shaped. The model parameter estimation is performed using renowned estimation approaches, method of moments, and method of maximum likelihood estimation. A parametric regression model tailored for count datasets is also developed using the proposed distribution. A simulation study is conducted to demonstrate the performance and behavior of the proposed estimators. The present study validates that the new count model adequately explains the medical datasets, which are the number of infected patients with the Nipah virus, the number of mammalian cytogenetic dosimetry lesions, and the Length of Hospital Stay. Additionally, we also estimate the model parameter using the Bayesian approach with gamma prior. Compared to widely used alternatives such as the Poisson (AIC = 145.16, BIC = 147.19 ) , Poisson moment exponential (AIC = 137.53, BIC = 139.56 ) , Poisson-XLindley (AIC = 135.86, BIC = 137.88 ) distributions and others, our model demonstrates improved fitting accuracy, as evidenced by lower AIC (135.78 ) and BIC (137.81 ) values for first data and similarly for second data applications. Finally, to validate the fit of the PH regression model, it is applied to the Length of Hospital Stay dataset.

In this article, new properties of the Poisson distribution of order k with parameter λ are found. Based on them, the modes of the Poisson distributions of order k=3 and 4 are derived for λ in (0,1). They are 0, 3, 5, and 0, 4, 7, 8, respectively, for λ in specified subintervals of (0, 1). In addition, using Mathematica, computational results for the modes of the Poisson distributions of order k=2,3, and 4 are presented for λ in specified subintervals of (0,2).

The poblem of characterizing of discrete probability distributions is an important problem. Recently many new results are obtained in characterization of distributions using k th records. Based on the distributional properties of kth weak and ordinary records some characterizations of geometric and discrete pareto distributions are given.

Spatial domain identification is crucial in spatial transcriptomics analysis. Existing methods excel with continuous and clustered distributions but struggle with discrete ones. We present spaMGCN, an innovative approach specifically designed for identifying spatial domains, especially in discrete tissue distributions. By integrating spatial transcriptomics and spatial epigenomic data through an autoencoder and a multi-scale adaptive graph convolutional network, spaMGCN outperforms baseline methods. Our evaluations demonstrate its effectiveness in recognizing discrete T cell zones in mouse spleens and follicular cells in human lymph nodes, as well as effectively distinguishing capsule structures from surrounding tissues.