Probability Mass Function Research Articles

Discrete Erlang-2 distribution and its application to leukemia and COVID-19

<abstract><p>Via the survival discretization method, this research revealed a novel discrete one-parameter distribution known as the discrete Erlang-2 distribution (DE2). The new distribution has numerous surprising improvements over many conventional discrete distributions, particularly when analyzing excessively dispersed count data. Moments and moments-generating functions, a few descriptive measures (central tendency and dispersion), monotonicity of the probability mass function, and the hazard rate function are just a few of the statistical aspects of the postulated distribution that have been developed. The single parameter of the DE2 distribution was estimated via the maximum likelihood technique. Real-world datasets, leukemia and COVID-19, were applied to analyze the effectiveness of the recommended distribution.</p></abstract>

RECOVERING PROBABILITY FUNCTIONS WITH FOURIER SERIES

The COS method was introduced in Fang & Oosterlee (2008) and then was applied to pricing a variety of stock options for continuous random variables. This paper adapts the Fourier-cosine series (COS) method to recover discrete probability mass functions. We approximate mixture and compound probability distributions with cosine series. Enormous precision and computational speed are the qualities of the function estimates here obtained. We also develop the pricing framework to trade derivatives subject to discrete random variables. We apply the method to calculate, for the first time, the price of an interest rate derivative of recent vintage introduced in the Brazilian financial market. Parameter calibration confirms the quality of the model.

Zero-modified Poisson-Modification of Quasi Lindley distribution and its application

Abstract The Poisson-Modification of Quasi Lindley (PMQL) distribution is a newly introduced mixed Poisson distribution for over-dispersed count data. The aim of this article is to introduce the Zero-modified PMQL (ZMPMQL) distribution as an alternative to the PMQL distribution in order to accommodate zero inflation/deflation. The method of obtaining the ZMPMQL distribution jointly with some of its important properties, namely the probability mass and distribution functions, mean, variance, index of dispersion, and quantile function are presented. Furthermore, some of its special cases are discussed. The maximum likelihood (ML) estimation method is used for the unknown parameter estimation. A simulation study is conducted in order to evaluate the asymptotic theory of the ML estimation method and to show the superiority of the ML method over the method of moments estimation. The applicability of the introduced distribution is illustrated by using a real-world data set.

A combinatorial view of stochastic processes: White noise.

White noise is a fundamental and fairly well understood stochastic process that conforms to the conceptual basis for many other processes, as well as for the modeling of time series. Here, we push a fresh perspective toward white noise that, grounded on combinatorial considerations, contributes to giving new interesting insights both for modeling and theoretical purposes. To this aim, we incorporate the ordinal pattern analysis approach, which allows us to abstract a time series as a sequence of patterns and their associated permutations, and introduce a simple functional over permutations that partitions them into classes encoding their level of asymmetry. We compute the exact probability mass function (p.m.f.) of this functional over the symmetric group of degree n, thus providing the description for the case of an infinite white noise realization. This p.m.f. can be conveniently approximated by a continuous probability density from an exponential family, the Gaussian, hence providing natural sufficient statistics that render a convenient and simple statistical analysis through ordinal patterns. Such analysis is exemplified on experimental data for the spatial increments from tracks of gold nanoparticles in 3D diffusion.

Asymptotic properties of the normalised discrete associated-kernel estimator for probability mass function

ABSTRACT Discrete kernel smoothing is now gaining importance in nonparametric statistics. In this paper, we investigate some asymptotic properties of the normalised discrete associated-kernel estimator of a probability mass function and make comparisons. We show, under some regularity and non-restrictive assumptions on the associated-kernel, that the normalising random variable converges in mean square to 1. We then derive the consistency and the asymptotic normality of the proposed estimator. Various families of discrete kernels already exhibited satisfy the conditions, including the refined CoM-Poisson which is underdispersed and of second-order. Finally, the first-order binomial kernel is discussed and, surprisingly, its normalised estimator has a suitable asymptotic behaviour through simulations.

Exchangeable Bernoulli distributions: High dimensional simulation, estimation, and testing

High dimensional simulation of exchangeable multivariate Bernoulli distributions is a challenging and important issue in applications, for example in credit risk models. The main contributions of this paper are, even for high dimensions, algorithms to sample from exchangeable multivariate Bernoulli distributions and to determine the distributions and the bounds of a wide class of indices and measures of probability mass functions. Unlike the algorithms present in the literature the proposed method gives the possibility to simulate also from negatively correlated distributions. Such a method is based on the geometrical structure of the class of exchangeable Bernoulli probability mass functions, which are points in a convex polytope whose extremal points are analytically known. Estimation and testing are also addressed.

Cpd: An R Package for Complex Pearson Distributions

The complex Pearson (CP) distributions are a family of probability models for count data generated by the Gaussian hypergeometric function with complex arguments. The complex triparametric Pearson (CTP) distribution and its biparametric versions, the complex biparametric Pearson (CBP) and the extended biparametric Waring (EBW) distributions, belong to this family. They all have explicit expressions of the probability mass function (pmf), probability generating function and moments, so they are easy to handle from a computational point of view. Moreover, the CTP and EBW distributions can model over- and underdispersed count data, whereas the CBP can only handle overdispersed data, but unlike other well-known overdispersed distributions, the overdispersion is not due to an excess of zeros but other low values of the variable. Finally, the EBW distribution allows the variance to be split into three uniquely identifiable components: randomness, liability and proneness. These properties make the CP distributions of interest in the modeling of a great variety of data. For this reason, and for trying to spread their use, we have implemented an R package called cpd that contains the pmf, distribution function, quantile function and random generation for these distributions. In addition, the package contains fitting functions according to the maximum likelihood. This package is available from the Comprehensive R Archive Network (CRAN). In this work, we describe all the functions included in the cpd package, and we illustrate their usage with several examples. Moreover, the release of a plugin in order to use the package from the interface R Commander tries to contribute to the spreading of these models among non-advanced users.

SOME RESULTS ON r-TRUNCATED DEGENERATE POISSON RANDOM VARIABLES

The zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers, which are also known as the conditional Poisson distributions or the positive Poisson distributions. Recently, as a natural extension of those distributions, Kim–Kim studied the zero-truncated degenerate Poisson distributions. In this paper, we introduce the [Formula: see text]-truncated degenerate Poisson random variable with parameter [Formula: see text], whose probability mass function is given by [Formula: see text], and investigate various properties of this random variable.

Belief Evolution Network-based Probability Transformation and Fusion

Smets proposes the Pignistic Probability Transformation (PPT) as the decision layer in the Transferable Belief Model (TBM), which argues when there is no more information, we have to make a decision using a Probability Mass Function (PMF). In this paper, the Belief Evolution Network (BEN) and the full causality function are proposed by introducing causality in Hierarchical Hypothesis Space (HHS). Based on BEN, we interpret the PPT from an information fusion view and propose a new Probability Transformation (PT) method called Full Causality Probability Transformation (FCPT), which has better performance under Bi-Criteria evaluation. Besides, we heuristically propose a new probability fusion method based on FCPT. Compared with Dempster Rule of Combination (DRC), the proposed method has more reasonable result when fusing same evidence.

Investigating the Mixture Quality in Multi-Injector Burner Systems—Part I: Experimental Setup

Abstract A new approach to calculate the mixture statistics in multi-injector burner (MIB) systems from a single injector database is presented in the paper. In such systems, the mixture quality is highly sensitive to the flowrate and the skewness of the inflow velocity profile. To determine the mixture fraction statistics from these two parameters for a particular injector in a multi-injector configuration, output-based proper orthogonal decomposition (O-POD) is suggested using results from Reynolds-averaged Navier Stokes (RANS)–computational fluid dynamics (CFD) as the observable. The O-POD mixing database is determined experimentally from two setups: First, the mixture quality in a single injector at ideal inflow conditions is studied. Then the same injector type is investigated in a generic MIB system. To characterize the mixture quality, the mixing probability mass function (PMF) at the injector exit is measured by means of laser-induced fluorescence (LIF) and high speed imaging. The data obtained for both the single injector under ideal inflow and the MIB are presented. These studies of the mixture behavior reveal that an asymmetric inflow velocity profile leads to a significant increase of unmixedness, which is seen as a negative skewness of the mixing PMFs. This effect becomes stronger at higher momentum flux density ratios due to the higher penetration depth of the fuel jets. The application of the O-POD to the database shows that the PMF can be accurately modeled with only three modes.

A Discrete Linear-Exponential Model: Synthesis and Analysis with Inference to Model Extreme Count Data

In this article, a novel probability discrete model is introduced for modeling overdispersed count data. Some relevant statistical and reliability properties including the probability mass function, hazard rate and its reversed function, moments, index of dispersion, mean active life, mean inactive life, and order statistics, are derived in-detail. These statistical properties are expressed in closed forms. The new model can be used to discuss right-skewed data with heavy tails. Moreover, its hazard rate function can be utilized to model the phenomena with a monotonically increasing failure rate shape. Different estimation approaches are listed to get the best estimator for modeling and reading the count data. A comprehensive comparison among techniques is performed in the case of simulated data. Finally, four real data sets are analyzed to prove the ability and notability of the new discrete model.

Risk-Sensitive Portfolio Management by Using C51 Algorithm

Financial trading is one of the most popular problems for reinforcement learning in recent years. One of the important challenges is that investment is a multi-objective problem. That is, professional investors do not act solely on expected profi t but also carefully consider the potential risk of a given investment. To handle such a challenge, previous studies have explored various kinds of risk-sensitive rewards, for example, the Sharpe ratio as computed by a fi xed length of previous returns. This work proposes a new approach to deal with the profi t-to-risk tradeoff by applying distributional reinforcement learning to build a risk awareness policy instead of a simple risk-based reward function. Our new policy, termed C51-Sharpe, is to select the action based on the Sharpe ratio computed from the probability mass function of the return. This produces a signifi cantly higher Sharpe ratio and lower maximum drawdown without sacrifi cing profi t compared to the C51algorithm utilizing a purely profi t-based policy. Moreover, it can outperform other benchmarks, such as a Deep Q-Network (DQN) with a Sharpe ratio reward function. Besides the policy, we also studied the effect of using double networks and the choice of exploration strategies with our approach to identify the optimal training confi guration. We fi nd that the epsilon-greedy policy is the most suitable exploration for C51-Sharpe and that the use of double network has no signifi cant impact on performance. Our study provides statistical evidence of the effi ciency in risk-sensitive policy implemented by using distributional reinforcement algorithms along with an optimized training process.

Time-varying gain controller synthesis of piecewise homogeneous semi-Markov jump linear systems

This paper is devoted to the time-varying gain controller synthesis problem for a class of discrete-time piecewise homogeneous semi-Markov jump linear systems (SMJLSs). Both sojourn-time probability mass functions and embedded Markov chain transition probability are simultaneously regulated by high-level Markov chain, which leads to a more general piecewise homogeneous semi-Markov kernel. A novel class of multivariate-dependent Lyapunov function is constructed, which is mode-dependent, elapsed-time-dependent, and variation-dependent. The objective of this paper is to design two classes of time-varying gain controllers, i.e., elapsed-time-dependent controller and finite memory controller under bound sojourn time. Numerically testable stabilization criteria are established for discrete-time piecewise homogeneous SMJLSs via the proposed novel Lyapunov function and two classes of controllers. Furthermore, two desired stabilizing controllers are designed such that the closed-loop system is mean-square stable. Finally, the proposed control strategies are applied to a numerical example and a practical example of solar thermal receiver to show the effectiveness and applicability of the theoretical results.

Skellam and time-changed variants of the generalized fractional counting process

In this paper, we study a Skellam type variant of the generalized counting process (GCP), namely, the generalized Skellam process. Some of its distributional properties such as the probability mass function, probability generating function, mean, variance and covariance are obtained. Its fractional version, namely, the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator. It is observed that the GFSP is a Skellam type version of the generalized fractional counting process (GFCP) which is a fractional variant of the GCP. It is shown that the one-dimensional distributions of the GFSP are not infinitely divisible. An integral representation for its state probabilities is obtained. We establish its long-range dependence property by using its variance and covariance structure. Also, we consider two time-changed versions of the GFCP. These are obtained by time-changing the GFCP by an independent Lévy subordinator and its inverse. Some particular cases of these time-changed processes are discussed by considering specific Lévy subordinators.

An Overview of Discrete Distributions in Modelling COVID-19 Data Sets.

The mathematical modeling of the coronavirus disease-19 (COVID-19) pandemic has been attempted by a large number of researchers from the very beginning of cases worldwide. The purpose of this research work is to find and classify the modelling of COVID-19 data by determining the optimal statistical modelling to evaluate the regular count of new COVID-19 fatalities, thus requiring discrete distributions. Some discrete models are checked and reviewed, such as Binomial, Poisson, Hypergeometric, discrete negative binomial, beta-binomial, Skellam, beta negative binomial, Burr, discrete Lindley, discrete alpha power inverse Lomax, discrete generalized exponential, discrete Marshall-Olkin Generalized exponential, discrete Gompertz-G-exponential, discrete Weibull, discrete inverse Weibull, exponentiated discrete Weibull, discrete Rayleigh, and new discrete Lindley. The probability mass function and the hazard rate function are addressed. Discrete models are discussed based on the maximum likelihood estimates for the parameters. A numerical analysis uses the regular count of new casualties in the countries of Angola,Ethiopia, French Guiana, El Salvador, Estonia, and Greece. The empirical findings are interpreted in-depth.

NPNT: Non-oscillating Process Negation Transformation of mass functions and a negation-based discounting method in information fusion

Yager proposes a Probability Negation Transformation (PNT) based on the maximum entropy distribution. After continuous PNTs, the Probability Mass Function (PMF) reaches the maximum entropy distribution, i.e., an uniform distribution. In this paper, we propose a new negation transformation method based on Belief Evolution Network (BEN) and Disjunctive Combination Rule (DCR), called Non-oscillating Process Negation Transformation (NPNT). As a cognitive negation transformation, NPNT expresses the negation from known to unknown. The Full Causality Probability Transformation (FCPT) is used to connect the PMF and Basic Belief Assignment (BBA) under NPNT. Based on the proposed method, we extend the Shafer’s discounting method from the perspective of DCR, which realizes less commitment under a same discount coefficient.

The Attraction Indian Buffet Distribution

We propose the attraction Indian buffet distribution (AIBD), a distribution for binary feature matrices influenced by pairwise similarity information. Binary feature matrices are used in Bayesian models to uncover latent variables (i.e., features) that explain observed data. The Indian buffet process (IBP) is a popular exchangeable prior distribution for latent feature matrices. In the presence of additional information, however, the exchangeability assumption is not reasonable or desirable. The AIBD can incorporate pairwise similarity information, yet it preserves many properties of the IBP, including the distribution of the total number of features. Thus, much of the interpretation and intuition that one has for the IBP directly carries over to the AIBD. A temperature parameter controls the degree to which the similarity information affects feature-sharing between observations. Unlike other nonexchangeable distributions for feature allocations, the probability mass function of the AIBD has a tractable normalizing constant, making posterior inference on hyperparameters straight-forward using standard MCMC methods. A novel posterior sampling algorithm is proposed for the IBP and the AIBD. We demonstrate the feasibility of the AIBD as a prior distribution in feature allocation models and compare the performance of competing methods in simulations and an application.

On the Capacity-Achieving Input of the Gaussian Channel With Polar Quantization

The polar receiver architecture is a receiver design that captures the envelope and phase information of the signal rather than its in-phase and quadrature components. Several studies have demonstrated the robustness of polar receivers to phase noise and other nonlinearities. Yet, the information-theoretic limits of polar receivers with finite-precision quantizers have not been investigated in the literature. The main contribution of this work is to identify the optimal signaling strategy for the additive white Gaussian noise (AWGN) channel with polar quantization at the output. More precisely, we show that the capacity-achieving modulation scheme has an amplitude phase shift keying (APSK) structure. Using this result, the capacity of the AWGN channel with polar quantization at the output is established by numerically optimizing the probability mass function of the amplitude. The capacity of the polar-quantized AWGN channel with $b_1$-bit phase quantizer and optimized single-bit magnitude quantizer is also presented. Our numerical findings suggest the existence of signal-to-noise ratio (SNR) thresholds, above which the number of amplitude levels of the optimal APSK scheme and their respective probabilities change abruptly. Moreover, the manner in which the capacity-achieving input evolves with increasing SNR depends on the number of phase quantization bits.

Poisson area-biased Ailamujia Distribution and its applications in environmental and medical sciences

Abstract In this paper, a new Poisson area-biased Ailamujia distribution has been formulated to analyse count data. It was created by combining two distributions: the Poisson and area-biased Ailamujia distributions, using the compounding technique. Several distributional properties of the formulated distribution were studied. Its ageing characteristics were determined and expressed explicitly. A variety of diagrams were used to demonstrate the characteristics of the probability mass function (pmf) and the cumulative distribution function (cdf). The parameter of the developed model was estimated by employing the maximum likelihood estimation approach. Finally, two data sets were used to demonstrate the effectiveness of the investigated distribution.

An Information-Theoretic Perspective on Successive Cancellation List Decoding and Polar Code Design

This work identifies information-theoretic quantities that are closely related to the required list size on average for successive cancellation list (SCL) decoding to implement maximum-likelihood decoding over general binary memoryless symmetric (BMS) channels. It also provides upper and lower bounds for these quantities that can be computed efficiently for very long codes. For the binary erasure channel (BEC), we provide a simple method to estimate the mean accurately via density evolution. The analysis shows how to modify, e.g., Reed-Muller codes, to improve the performance when practical list sizes, e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L\in {[{8, 1024}]}$ </tex-math></inline-formula> , are adopted. Exemplary constructions with block lengths <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N\in \{128,512\}$ </tex-math></inline-formula> outperform polar codes of 5G over the binary-input additive white Gaussian noise channel. It is further shown that there is a concentration around the mean of the logarithm of the required list size for sufficiently large block lengths, over discrete-output BMS channels. We provide the probability mass functions (p.m.f.s) of this logarithm, over the BEC, for a sequence of the modified RM codes with an increasing block length via simulations, which illustrate that the p.m.f.s concentrate around the estimated mean.