Class Of Probability Distributions Research Articles

ON THE MOMENTS OF THE 3-PARAMETER GOMPERTZ DISTRIBUTION

Gompertz distribution is a classical probability distribution extensively used in actuarial science, reliability, and survival analysis. Gompertz distribution also plays a role in various fields, such as biology, economics, and marketing analysis. Some extensions of this distribution have been studied and applied to various problems. In this article, we are concerned with some statistical properties of a 3-parameter Gompertz distribution. This extension of the Gompertz distribution introduced has been used in studying competing risk survival analysis. Our main results are the derivation of moments of this distribution and other statistical properties related to moments, such as moment generating function, mean residual life function, mean inactivity time and Lorenz curve. These results will serve as a complement to the theoretical aspect of the analysis of the distribution.

Quantitative bounds to propagation of quantum correlations in many-body systems

We investigate how much information about a quantum system can be simultaneously communicated to independent observers, by establishing quantitative limits to bipartite quantum correlations in many-body systems. As recently reported in Girolami et al. (2022) [16], bounds on quantum discord and entanglement of formation between a single quantum system and its environment, e.g., a large number of photons, dictate that independent observers which monitor environment fragments inevitably acquire only classical information about the system. Here, we corroborate and generalize those findings. First, we calculate continuity bounds of quantum discord, which establish how much states with a small amount of quantum correlations deviate from being embeddings of classical probability distributions. Also, we demonstrate a universally valid upper bound to the bipartite entanglement of formation between an arbitrary pair of components of a many-body quantum system. The results confirm that proliferation of classical information in the Universe suppresses quantum correlations.

A New Truncated Lindley-Generated Family of Distributions: Properties, Regression Analysis, and Applications.

We present the truncated Lindley-G (TLG) model, a novel class of probability distributions with an additional shape parameter, by composing a unit distribution called the truncated Lindley distribution with a parent distribution function G(x). The proposed model's characteristics including critical points, moments, generating function, quantile function, mean deviations, and entropy are discussed. Also, we introduce a regression model based on the truncated Lindley-Weibull distribution considering two systematic components. The model parameters are estimated using the maximum likelihood method. In order to investigate the behavior of the estimators, some simulations are run for various parameter settings, censoring percentages, and sample sizes. Four real datasets are used to demonstrate the new model's potential.

Modeling and simulation of financial returns under non-Gaussian distributions

It is well known that the probability distribution of high-frequency financial returns is characterized by a leptokurtic, heavy-tailed shape. This behavior undermines the typical assumption of Gaussian log-returns behind the standard approach to risk management and option pricing. Yet, there is no consensus on what class of probability distributions should be adopted to describe financial returns and different models used in the literature have demonstrated, to varying extent, an ability to reproduce empirically observed stylized facts. In order to provide some clarity, in this paper we perform a thorough study of the most popular models of return distributions as obtained in the empirical analyses of high-frequency financial data. We compare the statistical properties and simulate the dynamics of non-Gaussian financial fluctuations by means of Monte Carlo sampling from the different models in terms of realistic tail exponents. Our findings show a noticeable consistency between the considered return distributions in the modeling of the scaling properties of large price changes. We also discuss the convergence rate to the asymptotic distributions of the non-Gaussian stochastic processes and we study, as a first example of possible applications, the impact of our results on option pricing in comparison with the standard Black and Scholes approach.

An extended exponential hyper-Poisson distribution: Properties and applications

Here we propose a new class of probability distributions as an extended version of the exponential hyper-Poisson distribution and Weibull Poisson distribution. We investigate several important aspects of the distribution through deriving expressions for its probability density function (pdf), cumulative distribution function, survival function, failure rate function, pdf of the order statistics, r-th raw moments, etc. The method of maximum likelihood estimation procedures along with EM algorithm is discussed for estimating the parameters of the distribution and a test procedure is suggested for testing the significance of the additional parameters of the proposed model. The use of the proposed distribution is illustrated through real-life data sets. Further, a brief simulation study is carried out for evaluating the performance of the estimators obtained for the parameters of the distribution.

Confinement in non-Abelian lattice gauge theory via persistent homology

We investigate the structure of confining and deconfining phases in SU(2) lattice gauge theory via persistent homology, which gives us access to the topology of a hierarchy of combinatorial objects constructed from given data. Specifically, we use filtrations by traced Polyakov loops, topological densities, holonomy Lie algebra fields, as well as electric and magnetic fields. This allows for a comprehensive picture of confinement. In particular, topological densities form spatial lumps which show signatures of the classical probability distribution of instanton-dyons. Signatures of well-separated dyons located at random positions are encoded in holonomy Lie algebra fields, following the semiclassical temperature dependence of the instanton appearance probability. Debye screening discriminating between electric and magnetic fields is visible in persistent homology and pronounced at large gauge coupling. All employed constructions are gauge-invariant without a priori assumptions on the configurations under study. This work showcases the versatility of persistent homology for statistical and quantum physics studies, barely explored to date.

A new entropic quantum correlation measure for adversarial systems

Quantum correlation often refers to correlations exhibited by two or more local subsystems under a suitable measurement. These correlations are beyond the framework of classical statistics and the associated classical probability distribution. Quantum entanglement is the most well-known of such correlations and plays an important role in quantum information theory. However, there exist non-entangled states that still possess quantum correlations which cannot be described by classical statistics. One such measure that captures these non-classical correlations is discord. Here we introduce a new measure of quantum correlations which we call entropic accord that fits between entanglement and discord. It is defined as the optimised minimax mutual information of the outcome of the projective measurements between two parties. We show a strict hierarchy exists between entanglement, entropic accord and discord for two-qubit states. We study two-qubit states which shows the relationship between the three entropic quantities. In addition to revealing a class of correlations that are distinct from discord and entanglement, the entropic accord measure can be inherently more intuitive in certain contexts.

Bridging the gap between classical and quantum many-body information dynamics

The fundamental question of how information spreads in closed quantum many-body systems is often addressed through the lens of the bipartite entanglement entropy, a quantity that describes correlations in a comprehensive (nonlocal) way. Among the most striking features of the entanglement entropy are its unbounded linear growth in the thermodynamic limit, its asymptotic extensivity in finite-size systems, and the possibility of measurement-induced phase transitions, all of which have no obvious classical counterpart. Here, we show how these key qualitative features emerge naturally also in classical information spreading, as long as one treats the classical many-body problem on par with the quantum one, that is, by explicitly accounting for the exponentially large classical probability distribution. Our analysis is supported by extensive numerics on prototypical cellular automata and Hamiltonian systems, for which we focus on the classical mutual information and also introduce a ``classical entanglement entropy.'' Our study sheds light on the nature of information spreading in classical and quantum systems, and opens avenues for quantum-inspired classical approaches across physics, information theory, and statistics.

Neuromorphic quantum computing.

Quantum computation builds on the use of correlations. Correlations could also play a central role for artificial intelligence, neuromorphic computing or "biological computing." As a step toward a systematic exploration of "correlated computing" we demonstrate that neuromorphic computing can perform quantum operations. Spiking neurons in the active or silent states are connected to the two states of Ising spins. A quantum density matrix is constructed from the expectation values and correlations of the Ising spins. We show for a two qubit system that quantum gates can be learned as a change of parameters for neural network dynamics. These changes respect restrictions which ensure the quantum correlations. Our proposal for probabilistic computing goes beyond Markov chains and is not based on transition probabilities. Constraints on classical probability distributions relate changes made in one part of the system to other parts, similar to entangled quantum systems.

A New Family of Distributions in the Class of the Alpha Power Transformation with Applications to Income

In this paper, a new technique of the alpha-power transformation (APT) is used to propose a new class of probability distribution which is particularly useful for analysis of lifetime data. For the illustrative purposes of the new proposal, a special sub-model of the proposed family is considered in details. Various mathematical properties of the proposed model including quantile function, moments and moment generating function, residual life, mean residual life and order statistics are derived. The maximum likelihood method is used to estimate the unknown parameters. Finally A real data set is analyzed using the new distribution, and a simulation study is carried out to asses the performance of the new family.

The Energy probability distribution of quantum levels of a particle imprisoned in a three dimensional box

This work was trigerred by the earlier achivements of Yarman et al, aiming to bridge themordynamics and quantum mechanics, whence, Planck constant came to replace Boltzmann constant, and “average quantum level number” came to replace “temperature”. This evoked that the classical Maxwell energy probability distribution p(E) with respect to energy E of gas molecules might be taken care of, by the “energy probability distribution of the quantum levels” of a particle imprisoned in a given volume, assuming that in the case we have many particles, following Pauli exclusion principle, no pair of particles can sit at the same level. Thereby, the energy probability distribution of the quantum levels of a particle imprisoned in three dimensions, will be the subject of this essay. Such an outlook becomes interesting from several angles: i) It looks indeed very much like a classical Maxwellian distribution. ii) In the case we have as many free particles in the box as the number of levels depicted by the number of quantum levels in between the predetermined lower bound energy level and the upperbound energy level, all the while assuming that the Pauli principle holds, the distribution we disclose becomes the energy probability distribution of the ensemble of particles imprisoned in the given box. iii) It can even be guessed that, if elastic collisions between the free particles were allowed, and still assuming quantization and the Pauli principle, the outcome we disclose should be about the same as that of the energy probability distribution, molecules in a room would display in equilibrium. iv) The quantized energy being proportional to the sum of three squared integers associated with respectively, each of the spatial dimensions; the property we reveal certainly becomes remarkable from the point of view of mathematics of integer numbers. All the more, we further disclose that, to the probability distribution outlook remains the same, be this qualitatively for higher dimensions than 3.

A practical approach for sampling the Longuet–Higgins distribution for engineering applications

The Longuet–Higgins distribution is a classical joint probability distribution of wave periods and amplitudes still being used for comparison against current approaches in coastal engineering and other applications. However, comparisons are carried out against the theoretical distribution rather than against jointly simulated values. Some formulae to implement the generation of jointly distributed wave amplitudes and periods is unavailable. A practical approach is proposed in this study to cope with this issue. Expressions for the marginal and conditional cumulative distribution functions of the Longuet–Higgins​ distribution based on the error function, not reported in the literature, are provided. Resulting simulations are in good agreement with the theoretical distributions. It is concluded that the developed equations are formal distributions based on probability theory, that the proposed method leads to adequate simulated values of jointly distributed wave heights and periods and that it is easy to implement in current software and suitable for engineering applications.

Point forecasting and forecast evaluation with generalized Huber loss

Huber loss, its asymmetric variants and their associated functionals (here named Huber functionals) are studied in the context of point forecasting and forecast evaluation. The Huber functional of a distribution is the set of minimizers of the expected (asymmetric) Huber loss, is an intermediary between a quantile and corresponding expectile, and also arises in M-estimation. Each Huber functional is elicitable, generating the precise set of minimizers of an expected score, subject to weak regularity conditions on the class of probability distributions, and has a complete characterization of its consistent scoring functions. Such scoring functions admit a mixture representation as a weighted average of elementary scoring functions. Each elementary score can be interpreted as the relative economic loss of using a particular forecast for a class of investment decisions where profits and losses are capped. The relevance of this theory for comparative assessment of weather forecasts is also discussed.

Efficient Process Monitoring Under General Weibull Distribution

Product testing is a key ingredient in maintaining the quality of a production process. The production process is considered an efficient process if it is capable of quick identification of faulty products. The items produced by any production process are usually packed and acceptance or rejection of the pack depends upon its conformity to some specified quality level. Generally, the specified quality level is based upon the number of defective items found in the inspected number of items. Such decisions are based upon some rules and usually acceptance of the pack is based upon a fewer number of defective items in the pack. Such questions can be answered by using acceptance sampling plans. The acceptance sampling plans assume the fact that the quality level of the item follows some probability distribution. The sampling plans based upon some classical probability distributions are available but often it happens that the quality behavior of the product does not follow a simple probability model and hence the available sampling plans fail. In this paper, we have developed acceptance sampling plans when the product life follows a general Weibull distribution. The sampling plans have been constructed by considering the crisp and fuzzy behavior of the acceptance probability. These sampling plans have been constructed by assuming an infinite lot size. It has been found that the number of items required for inspection decreases with an increase in some parameters.

Quantum logical entropy: fundamentals and general properties

Logical entropy gives a measure, in the sense of measure theory, of the distinctions of a given partition of a set, an idea that can be naturally generalized to classical probability distributions. Here, we analyze how this fundamental concept and other related definitions can be applied to the study of quantum systems with the use of quantum logical entropy. Moreover, we prove several properties of this entropy for generic density matrices that may be relevant to various areas of quantum mechanics and quantum information. Furthermore, we extend the notion of quantum logical entropy to post-selected systems.

Probability distribution for the quantum universe

We determine the inner product on the Hilbert space of wavefunctions of the universe by imposing the Hermiticity of the quantum Hamiltonian in the context of the minisuperspace model. The corresponding quantum probability density reproduces successfully the classical probability distribution in the ħ → 0 limit, for closed universes filled with a perfect fluid of index w. When −1/3 < w ≤ 1, the wavefunction is normalizable and the quantum probability density becomes vanishingly small at the big bang/big crunch singularities, at least at the semiclassical level. Quantum expectation values of physical geometrical quantities, which diverge classically at the singularities, are shown to be finite.

A New Neutrosophic Negative Binomial Distribution: Properties and Applications

Many problems in real life exist that are full of confusion, vagueness, and ambiguity. The quantification of such issues in a scientific way is the need of time. The negative binomial distribution is an important discrete probability distribution from the account of classical probability distribution theory. The distribution was used to study the chance of kth success in n trials before n − 1 failures for crisp data. The literature lacks in dealing with the situations for interval-valued data under negative binomial distribution. In this research, the neutrosophic negative binomial distribution is proposed to generalize the classical negative binomial distribution. The generalized proposed distribution considers the indeterminacy and crisp form from interval-valued. Several properties of the proposed distribution, such as moment generating function, characteristic function, and probability generating function, are also derived. Furthermore, the derivation of reliability analysis properties such as survival, hazard rate, reversed hazard rate, cumulative hazard rate, mills ratio, and odds ratio are also presented. In addition, order statistics for the proposed distribution, including w th , joint, median, minimum, and maximum order statistics are part of the paper. The proposed distribution is discussed from the real data applications perspective by considering the different case studies. This research opens the way to deal with the problems that follow conventional conveyances and include nonprecisely determined details simultaneously.

On quantization dimensions of idempotent probability measures

The concept of quantization dimensions (upper and lower) of idempotent probability measures on a metric compactum is introduced. These dimensions, which are defined using a general functorial scheme, are analogous to the quantization dimensions of classical probability distributions. It is proved that the quantization dimensions of an idempotent measure do not exceed the corresponding box dimensions of its support. At the same time, for any non-negative number b not exceeding the upper box dimension of a metric compactum X, there exists an idempotent probability measure μb with support equal to X and the upper quantization dimension equal to b. For the lower quantization dimension, a similar intermediate value theorem is valid under some additional restrictions on the compactum X.

Matrix-variate Lindley distributions and its applications

Restring on the fact that the definition of multivariate analogs of the real gamma distribution is replaced by the Wishart distribution on symmetric matrices, and based on the notion of mixture models which is a flexible and powerful tool for treating data taken from multiple subpopulations, we set forward a multivariate analog of the real Lindley distributions of the first and second kinds on the modern framework of symmetric cones which can be used to model waiting and survival times matrix data. Within this framework, we first construct a new probability distributions, named the matrix-variate Lindley distributions. Some fundamental properties of these new distributions are established. Their statistical properties including moments, the coefficient of variation, skewness and the kurtosis are discussed. We then create an iterative hybrid Expectation-Maximization Fisher-Scoring (EM-FS) algorithm to estimate the parameters of the new class of probability distributions. Through simulation as well as comparative studies with respect to the Wishart distribution, the effectiveness and reliability of the proposed distributions are proved. Finally, the usefulness and the applicability of the new models are elaborated and illustrated by means of two real data sets from biological sciences and medical image segmentation which is one of the most important and popular tasks in medical image analysis.

Schmidt Decomposition and Coherence of Interfering Alternatives

The Schmidt decomposition and the correlational analysis based on it make it possible to identify statistical dependences between various subsystems of a single physical system. The systems under consideration can be both quantum states and classical probability distributions. In this study, two different physical systems are considered: quantum Schrödinger cat states and double-slit interference of microparticles. It is shown that the considered systems have a single internal structure and can be described in general terms of interfering alternatives. An effective approach is developed that allows us to calculate optical characteristics of interference such as visibility and coherence. It is shown that the scalar product of the states of the environment of interfering alternatives acts as a natural generalization of the classical complex parameter of the coherence of light oscillations, which determines the visibility of the interference pattern. A simple quantitative relationship is obtained between the visibility of the interference pattern and the Schmidt number, which determines the level of connection between a quantum system and its environment. The developed approaches are generalized to the case of multidimensional Schrödinger cat states.