Negative Hypergeometric Distribution Research Articles

Yet Another Attempt to Classify Positive Univariate Probability Distributions

We propose an original classification of several discrete and continuous probability distributions. We establish links between these distributions, in particular the little known relationship between the negative hypergeometric distribution and the beta distribution. These relations allow us to propose a structure of relations which is summarized in graphic form.Our classification emphasises the analogy between certain discrete and continuous distributions. This analogy makes it possible to establish relations between the theory of point processes and the theory of survey sampling. It also makes it possible to envisage the use of link functions that are little used in generalised regression.

Simple analytical models for estimating the queue lengths from probe vehicles at traffic signals: A combinatorial approach for nonparametric models

This study develops a combinatorial approach for nonparametric short-term queue length estimation in terms of cycle-by-cycle partially observed queues from probe vehicles (PV). The method does not assume random arrivals and does not assume any primary parameters or estimation of any parameters but uses simple algebraic expressions that only depend on signal timing. For an approach lane at a traffic intersection, the conditional queue lengths given probe vehicle location, count, time, and analysis interval (e.g., at the end of the red signal phase) are represented by a Negative Hypergeometric distribution. The simple analytical estimators obtained are compared with parametric methods from literature and highway capacity manual methods using field test data and simulation data involving probe vehicles. The analysis indicates that the nonparametric models presented in this paper match the accuracy of the parametric ones used in the field test and simulated data for estimating queue lengths.

An algebra over the operad of posets and structural binomial identities

We study generating functions of strict and non-strict order polynomials of series–parallel posets, called order series. These order series are closely related to Ehrhart series and \(h^*\)-polynomials of the associated order polytopes. We explain how they can be understood as algebras over a certain operad of posets. Our main results are based on the fact that the order series of chains form a basis in the space of order series. This allows to reduce the search space of an algorithm that finds for a given power series f(x), if possible, a poset P such that f(x) is the generating function of the order polynomial of P. In terms of Ehrhart theory of order polytopes, the coordinates with respect to this basis describe the number of (internal) simplices in the canonical triangulation of the order polytope of P. Furthermore, we derive a new proof of the reciprocity theorem of Stanley. As an application, we find new identities for binomial coefficients and for finite partitions that allow for empty sets, and we describe properties of the negative hypergeometric distribution.

Application of a limit Function Of Negative Hyper-geometric Distribution in Option Pricing.

This work introduces limit function of a negative hyper geometric distribution of the form .where and, which is applied in option pricing using wealth equation and some martingale tools. This paper present a simple discrete time model in compares. This work conclude that limit of negative hyper geometric as can be associated with financial terms that can be used to evaluate option values (non dividend) which gives the same numerical with CRR model.

A number‐between‐events control chart for monitoring finite horizon production processes

AbstractIn this paper, we present a number‐between‐events (NBE) control chart for monitoring the fraction nonconforming in finite horizon production (FHP) processes and related specific performance measures. When monitoring fractions nonconforming in FHP processes, the common binomial ‐chart has two crucial limitations: the underlying distributional assumptions are violated when dealing with low‐volume production and a scarce efficiency in the case of processes characterized by a low fraction nonconforming. Thus, an efficient monitoring of FHP processes requires the selection of the correct underlying statistical model: in this case, a distribution from the hypergeometric family of discrete statistical distributions. An efficient statistical monitoring of processes with low fractions nonconforming can be achieved by means of discrete time‐between‐events (TBE) control charts, which count the number of units up to the appearance of a fixed number of nonconforming units in the sample. Here, we present a discrete TBE‐chart, denoted as NBE‐chart, based on the negative hypergeometric distribution that meets numerous requirements of efficient monitoring of FHP processes. The proposed control chart can be conveniently used for both low‐volume and mass production in processes with frequent changeovers.

Refining One Theorem For The Romanovsky Distribution

The paper considered a refinement of the theorem for a negative-hypergeometric distribution( the Romanovsky distribution), i.e., convergence over variation of the Romanovsky distribution by Erlang distributions. The theorem is proved by the direct asymptotic method.

A non-uniform bound on binomial approximation with w-functions

We use Stein’s method and w-functions to determine a non-uniform bound for approximating the distribution of a non-negative integer-valued random variable X by a binomial distribution with parameters and where Additionally, we also give a new improvement of non-uniform bound for the distance between the cumulative distribution function of X and a binomial cumulative distribution function. For applications, we use the obtained result to approximate the hypergeometric, negative hypergeometric, Pólya and beta binomial distributions.

On the Verge of Life: Distribution of Nucleotide Sequences in Viral RNAs.

The aim of the study is to analyze viruses using parameters obtained from distributions of nucleotide sequences in the viral RNA. Seeking for the input data homogeneity, we analyze single-stranded RNA viruses only. Two approaches are used to obtain the nucleotide sequences; In the first one, chunks of equal length (four nucleotides) are considered. In the second approach, the whole RNA genome is divided into parts by adenine or the most frequent nucleotide as a “space”. Rank–frequency distributions are studied in both cases. The defined nucleotide sequences are signs comparable to a certain extent to syllables or words as seen from the nature of their rank–frequency distributions. Within the first approach, the Pólya and the negative hypergeometric distribution yield the best fit. For the distributions obtained within the second approach, we have calculated a set of parameters, including entropy, mean sequence length, and its dispersion. The calculated parameters became the basis for the classification of viruses. We observed that proximity of viruses on planes spanned on various pairs of parameters corresponds to related species. In certain cases, such a proximity is observed for unrelated species as well calling thus for the expansion of the set of parameters used in the classification. We also observed that the fifth most frequent nucleotide sequences obtained within the second approach are of different nature in case of human coronaviruses (different nucleotides for MERS, SARS-CoV, and SARS-CoV-2 versus identical nucleotides for four other coronaviruses). We expect that our findings will be useful as a supplementary tool in the classification of diseases caused by RNA viruses with respect to severity and contagiousness.

A stratified estimation of a sensitive attribute by using negative binomial and negative hypergeometric distribution as randomization devices

In this paper, when the population is composed of several strata, we deal with the problem of stratified estimation for sensitive attribute by applying the stratified random sampling to the Yennum, Sedory, and Singh (2020) model. In Yennum, Sedory, and Singh (2020) study, they considered the negative binomial and negative hypergeometric distribution as randomization devices. When the size of each stratum was exactly known, the sensitive attribute was estimated by stratification, and the proportional and optimal allocations were examined as a method of allocating samples to each stratum. Also, in case of not knowing the size of each stratum, the sensitive attribute was estimated by using two phase sampling design, and the method of allocating samples to each stratum was also examined. Also, the efficiency between the proposed stratified model of Yennum, Sedory, and Singh (2020) and the existing model of Yennum, Sedory, and Singh (2020) was compared by numerical study for the different choice parameters.

Improved strategy to collect sensitive data by using negative binomial and negative hypergeometric distribution as randomization devices

In this paper, we improve upon the negative binomial randomized response model by implementing a two-stage randomization process. Using the modified estimator of the proportion of members of a population possessing a sensitive attribute, we claim to achieve both better efficiency and better protection. The findings are verified based on extensive simulation study. Then we improvise the technique of use of negative hypergeometric distribution by introducing two-stage randomization response process. Using our new technique, we also assess the efficiency and protection of the respondents as compared to the negative hypergeometric model through extensive simulation studies. We also provide SAS codes used in the evaluations of the strategies considered.

Discrete Distributions in Engineering and the Applied Sciences

This is an introductory book on discrete statistical distributions and its applications. It discusses only those that are widely used in the applications of probability and statistics in everyday life. The purpose is to give a self-contained introduction to classical discrete distributions in statistics. Instead of compiling the important formulas (which are available in many other textbooks), we focus on important applications of each distribution in various applied fields like bioinformatics, genomics, ecology, electronics, epidemiology, management, reliability, etc., making this book an indispensable resource for researchers and practitioners in several scientific fields. Examples are drawn from different fields. An up-to-date reference appears at the end of the book. Chapter 1 introduces the basic concepts on random variables, and gives a simple method to find the mean deviation (MD) of discrete distributions. The Bernoulli and binomial distributions are discussed in detail in Chapter 2. A short chapter on discrete uniform distribution appears next. The next two chapters are on geometric and negative binomial distributions. Chapter 6 discusses the Poisson distribution in-depth, including applications in various fields. Chapter 7 is on hypergeometric distribution. As most textbooks in the market either do not discuss, or contain only brief description of the negative hypergeometric distribution, we have included an entire chapter on it. A short chapter on logarithmic series distribution follows it, in which a theorem to find the kth moment of logarithmic distribution using (k-1)th moment of zero-truncated geometric distribution is presented. The last chapter is on multinomial distribution and its applications. The primary users of this book are professionals and practitioners in various fields of engineering and the applied sciences. It will also be of use to graduate students in statistics, research scholars in science disciplines, and teachers of statistics, biostatistics, biotechnology, education, and psychology.

An approach to negative hypergeometric distribution by generating function for special numbers and polynomials

Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling's approximation for factorials, we investigate some approximation values of the special case of the numbers $Y_{n}\left( \lambda \right) $.

Analytical Expressions and Physics for Single-Cell mRNA Distributions of the lac Operon of E. coli

Mechanistic models of stochastic gene expression are of considerable interest, but their complexity often precludes tractable analytical expressions for messenger RNA (mRNA) and protein distributions. The lac operon of Escherichia coli is a model system with regulatory elements such as multiple operators and DNA looping that are shared by many operons. Although this system is complex, intuition suggests that fast DNA looping may simplify it by causing the repressor-bound states of the operon to equilibrate rapidly, thus ensuring that the subsequent dynamics are governed by slow transitions between the repressor-free and the equilibrated repressor-bound states. Here, we show that this intuition is correct by applying singular perturbation theory to a mechanistic model of lac transcription with the scaled time constant of DNA looping as the perturbation parameter. We find that at steady state, the repressor-bound states satisfy detailed balance and are dominated by the looped states; moreover, the interaction between the repressor-free and the equilibrated repressor-bound states is described by an extension of the Peccoud-Ycart two-state model in which both (repressor-free and repressor-bound) states support transcription. The solution of this extended two-state model reveals that the steady-state mRNA distribution is a mixture of the Poisson and negative hypergeometric distributions, which reflects mRNAs obtained by transcription from the repressor-bound and repressor-free states. Finally, we show that the physics revealed by perturbation theory makes it easy to derive the extended two-state model equations for complex regulatory architectures.

Urns Filled with Bitcoins: New Perspectives on Proof-of-Work Mining

The probability of a miner finding a valid block in the bitcoin blockchain is assumed to follow the Poisson distribution. However, simple, descriptive, statistical analysis reveals that blocks requiring a lot of time to find — long blocks — are won only by miners with a relatively higher hash power per second. This suggests that relatively bigger miners might have an advantage with regard to winning long blocks, which can be understood as a sort of “within block learning”. Modelling the bitcoin mining problem as a race, and by means of a multinomial logit model, we can reject that the time spent mining a particular block does not affect the probability of a miner finding a valid version of this block in a manner that is proportional to her size. Further, we postulate that the probability of a miner finding a valid block is governed by the negative hypergeometric distribution. This would explain the descriptive statistics that emerge from the data and be aligned with the technical aspects of bitcoin mining. We draw an analogy between bitcoin mining and the classical “urn problem” in statistics to sustain our theory. This result can have important consequences for the miners of proof-of-work cryptocurrencies in general, and for the bitcoin mining community in particular.

A Conway Maxwell Poisson type generalization of the negative hypergeometric distribution

Negative hypergeometric distribution arises as a waiting time distribution when we sample without replacement from a finite population. It has applications in many areas such as inspection sampling and estimation of wildlife populations. However, as is well known, the negative hypergeometric distribution is over-dispersed in the sense that its variance is greater than the mean. To make it more flexible and versatile, we propose a modified version of negative hypergeometric distribution called COM-Negative Hypergeometric distribution (COM-NH) by introducing a shape parameter as in the COM-Poisson and COMP-Binomial distributions. It is shown that under some limiting conditions, COM-NH approaches to a distribution that we call the COM-Negative binomial (COMP-NB), which in turn, approaches to the COM Poisson distribution. For the proposed model, we investigate the dispersion characteristics and shape of the probability mass function for different combinations of parameters. We also develop statistical inference for this model including parameter estimation and hypothesis tests. In particular, we investigate some properties such as bias, MSE, and coverage probabilities of the maximum likelihood estimators for its parameters by Monte Carlo simulation and likelihood ratio test to assess shape parameter of the underlying model. We present illustrative data to provide discussion.

A Non-uniform Bound on Negative Binomial Approximation via Stein’s Method and z-functions

In this article, Stein’s method and z-functions are used to determine a non-uniform bound for approximating the cumulative distribution function of a nonnegative integer-valued random variable X by the negative binomial cumulative distribution function with parameters $$r\in {\mathbb {R}}^+$$ and $$p=1-q\in (0,1)$$. This bound is an appropriate criterion for evaluating the accuracy of this approximation. The result obtained in this study is used to approximate cumulative distribution functions including the negative hypergeometric cumulative distribution function, the Polya cumulative distribution function, and the beta negative binomial cumulative distribution function.

Battleship and the negative hypergeometric distribution

SummaryThe negative hypergeometric distribution is often not formally studied in secondary or collegiate statistics in contexts other than drawing cards without replacement. We present a different context with the potential of engaging students in simulating and exploring data.

Sampling Designs on Finite Populations with Spreading Control Parameters

We present new sampling methods in finite population that allow to control the joint inclusion probabilities of units and especially the spreading of sampled units in the population. They are based on the use of renewal chains and multivariate discrete distributions to generate the difference of population ranks between two successive selected units. With a Bernoulli sampling design, these differences follow a geometric distribution, and with a simple random sampling design they follow a negative hypergeometric distribution. We propose to use other distributions and introduce a large class of sampling designs with and without fixed sample size. The choice of the rank-difference distribution allows us to control units joint inclusion probabilities with a relatively simple method and closed form formula. Joint inclusion probabilities of neighboring units can be chosen to be larger, or smaller, compared to those of Bernoulli or simple random sampling, thus allowing to more or less spread the sample on the population. This can be useful when neighboring units have similar characteristics or, on the contrary, are very different. A set of simulations illustrates the qualities of this method.

The Diary of Boima Kiakpomgbo from Mando Town (Liberia): A Quantitative Study of a Vai Text

We analyse a Liberian text from the first quarter of the twentieth century written in Vai, a Mande language of West Africa, using an indigenous syllabic script. Rank–frequency distributions of syllabic symbols and words are studied. The distribution of symbols is well-fitted by the 1-displaced negative hypergeometric distribution, confirming previous studies on alphabetic scripts, with a similar interpretation of parameters. Two regimes are discovered for the saturation of symbols in the text (growth of the number of types approaching the complete set versus the number of tokens). For the distribution of words, the Zipf–Mandelbrot law is found to be a proper model rather than the simple Zipfian dependence.

An improved negative binomial approximation with high accuracy to the negative hypergeometric probability for order-preserving encryption

Encryption is a well established technology to sensitive data. However, once encrypted, data can no longer be easily range queried or indexed efficiently. Order-preserving symmetric encryption (OPE) helps with this problem whose encryption function preserves numerical ordering of the plaintexts. It supports efficient logarithmic-complexity search over encrypted content without sacrificing security. In Eurocrypt 2009, the connection between OPE and the negative hypergeometric (NHG) distribution was given by Boldyreva et al. and the authors pointed out that an efficient sampling algorithm for the NHG distribution was required when designing a practical OPE scheme. However, the existence of such an algorithm is open for it needs a high accuracy approximation to the NHG distribution. By using the Stirling’s formula, we develop a new negative binomial (NB) approximation to the NHG distribution and introduce a series of correction factors to this approximation to improve its accuracy. In addition, a result of the NB approximation to the NHG distribution is given in terms of the variation distance. It is the first time to quantify the quality of the NB approximation for fixed parameters. Some numerical examples are presented to illustrate the results obtained.