Negative Binomial Approximation Research Articles

A negative binomial approximation to the distribution of the sum of maxima of indicator random variables

In a sequence of independent indicator random variables, we consider the sum of maxima of k consecutive indicator random variables that represent a specific type of k-runs in the theory of runs and patterns. Intriguingly, it can be expressed as the sum of locally dependent random variables. Our focus is on employing Stein’s method to provide a negative binomial approximation to the distribution of this specific random variable under certain conditions on moments. The derived bounds exhibit an optimal order, reflecting the precision and efficiency of the proposed approximation.

Efficient Posterior Sampling for Bayesian Poisson Regression

ABSTRACT Poisson log-linear models are ubiquitous in many applications, and one of the most popular approaches for parametric count regression. In the Bayesian context, however, there are no sufficient specific computational tools for efficient sampling from the posterior distribution of parameters, and standard algorithms, such as random walk Metropolis-Hastings or Hamiltonian Monte Carlo algorithms, are typically used. Herein, we developed an efficient Metropolis-Hastings algorithm and importance sampler to simulate from the posterior distribution of the parameters of Poisson log-linear models under conditional Gaussian priors with superior performance with respect to the state-of-the-art alternatives. The key for both algorithms is the introduction of a proposal density based on a Gaussian approximation of the posterior distribution of parameters. Specifically, our result leverages the negative binomial approximation of the Poisson likelihood and the successful Pólya-gamma data augmentation scheme. Via simulation, we obtained that the time per independent sample of the proposed samplers is competitive with that obtained using the successful Hamiltonian Monte Carlo sampling, with the Metropolis-Hastings showing superior performance in all scenarios considered. Supplementary materials for this article are available online.

Big data ordination towards intensive care event count cases using fast computing GLLVMS

BackgroundIn heart data mining and machine learning, dimension reduction is needed to remove multicollinearity. Meanwhile, it has been proven to improve the interpretation of the parameter model. In addition, dimension reduction can also increase the time of computing in high dimensional data.MethodsIn this paper, we perform high dimensional ordination towards event counts in intensive care hospital for Emergency Department (ED 1), First Intensive Care Unit (ICU1), Second Intensive Care Unit (ICU2), Respiratory Care Intensive Care Unit (RICU), Surgical Intensive Care Unit (SICU), Subacute Respiratory Care Unit (RCC), Trauma and Neurosurgery Intensive Care Unit (TNCU), Neonatal Intensive Care Unit (NICU) which use the Generalized Linear Latent Variable Models (GLLVM’s).ResultsDuring the analysis, we measure the performance and calculate the time computing of GLLVM by employing variational approximation and Laplace approximation, and compare the different distributions, including Negative Binomial, Poisson, Gaussian, ZIP, and Tweedie, respectively. GLLVMs (Generalized Linear Latent Variable Models), an extended version of GLMs (Generalized Linear Models) with latent variables, have fast computing time. The major challenge in latent variable modelling is that the function fleft(varTheta right)=int fleft(uvarTheta right)hleft(uright)du is not trivial to solve since the marginal likelihood involves integration over the latent variable u.ConclusionsIn a nutshell, GLLVMs lead as the best performance reaching the variance of 98% comparing other methods. We get the best model negative binomial and Variational approximation, which provides the best accuracy by accuracy value of AIC, AICc, and BIC. In a nutshell, our best model is GLLVM-VA Negative Binomial with AIC 7144.07 and GLLVM-LA Negative Binomial with AIC 6955.922.

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English

Effects of cell cycle variability on lineage and population measurements of messenger RNA abundance.

Many models of gene expression do not explicitly incorporate a cell cycle description. Here, we derive a theory describing how messenger RNA (mRNA) fluctuations for constitutive and bursty gene expression are influenced by stochasticity in the duration of the cell cycle and the timing of DNA replication. Analytical expressions for the moments show that omitting cell cycle duration introduces an error in the predicted mean number of mRNAs that is a monotonically decreasing function of η, which is proportional to the ratio of the mean cell cycle duration and the mRNA lifetime. By contrast, the error in the variance of the mRNA distribution is highest for intermediate values of η consistent with genome-wide measurements in many organisms. Using eukaryotic cell data, we estimate the errors in the mean and variance to be at most 3% and 25%, respectively. Furthermore, we derive an accurate negative binomial mixture approximation to the mRNA distribution. This indicates that stochasticity in the cell cycle can introduce fluctuations in mRNA numbers that are similar to the effect of bursty transcription. Finally, we show that for real experimental data, disregarding cell cycle stochasticity can introduce errors in the inference of transcription rates larger than 10%.

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.

An improved bound for negative binomial approximation with [formula omitted]-functions

In this article, we use Stein’s method together with z-functions to give an improved bound for the total variation distance between the distribution of a non-negative integer-valued random variable X and the negative binomial distribution with parameters r∈R+ and p=1−q∈(0,1), where rqp is equal to the mean of X, E(X). The improved bound is sharper than that mentioned in Teerapabolarn and Boondirek (2010). We give three examples of the negative binomial approximation to the distribution of X concerning the negative hypergeometric, Pólya and negative Pólya distributions.

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.

Stein factors for negative binomial approximation in Wasserstein distance

The paper gives the bounds on the solutions to a Stein equation for the negative binomial distribution that are needed for approximation in terms of the Wasserstein metric. The proofs are probabilistic, and follow the approach introduced in Barbour and Xia (Bernoulli 12 (2006) 943-954). The bounds are used to quantify the accuracy of negative binomial approximation to parasite counts in hosts. Since the infectivity of a population can be expected to be proportional to its total parasite burden, the Wasserstein metric is the appropriate choice.

AN IMPROVEMENT OF POINTWISE NEGATIVE BINOMIAL APPROXIMATION BY w-FUNCTIONS

The result in pointwise negative binomial approximation for a non- negative integer-valued random variable proposed by (5) could not be applied to the case of r ∈ (0,1). In this study, we determine a non-uniform bound on this approximation for r ∈ (0,1).

A uniform bound on negative binomial approximation with w-functions

This paper uses Stein’s method and w-functions to determine a uniform bound for the Kolmogorov distance between the cumulative distribution function of a non-negative integer-valued random variable X and the negative binomial cumulative distribution function with parameters 0 r > and 1( 0,1) pq =− ∈ , where () rq p EX = and () EX is the mean of X. Two examples are provided to

Performance of Interval Estimators for the Inverse Hypergeometric Distribution

The inverse hypergeometric distribution is of interest in applications of inverse sampling without replacement from a finite population where a binary observation is made on each sampling unit. Thus, sampling is performed by randomly choosing units sequentially one at a time until a specified number of one of the two types is selected for the sample. Assuming the total number of units in the population is known but the number of each type is not, we consider the problem of estimating this parameter. We use the Delta method to develop approximations for the variance of three parameter estimators. We then propose three large sample confidence intervals for the parameter. Based on these results, we selected a sampling of parameter values for the inverse hypergeometric distribution to empirically investigate performance of these estimators. We evaluate their performance in terms of expected probability of parameter coverage and confidence interval length calculated as means of possible outcomes weighted by the appropriate outcome probabilities for each parameter value considered. The unbiased estimator of the parameter is the preferred estimator relative to the maximum likelihood estimator and an estimator based on a negative binomial approximation, as evidenced by empirical estimates of closeness to the true parameter value. Confidence intervals based on the unbiased estimator tend to be shorter than the two competitors because of its relatively small variance but at a slight cost in terms of coverage probability.

NEGATIVE BINOMIAL APPROXIMATION TO THE GENERALIZED HYPERGEOMETRIC DISTRIBUTION

This paper uses Stein's method and the w-function associated with the generalized hypergeometric random variable to determine a bound for the total variation distance between the generalized hypergeometric distribu- tion with parameters �, � and N and the negative binomial distribution with parameters r = � + 1 and p = 1 − q = �+�+2 �+�+N+1 . In view of this bound, it is observed that the desired result gives a good negative binomial approximation whenis large.

A modelling and simulation approach to assessment of a negative binomial approximation in a multi-echelon inventory system

Some studies in the multi-echelon inventory systems literature have used a negative binomial distribution to approximate a critical random variable arising in the model. Graves (1996) developed a multi-echelon inventory model with fixed replenishment intervals, where each site follows a base stock policy. He proposed in the one-warehouse, N-retailer case a negative binomial distribution to approximate a random variable he referred to as ‘uncovered demand’. Computational evidence was provided to demonstrate the effectiveness of the approximation. Graves then suggested search procedures for approximately optimal base stock levels at the warehouse and N identical retailers under two customer service criteria: 1) probability of no stockout; 2) fill rate. A separate analytical evaluation of the negative binomial approximation has been preliminarily reported elsewhere. In the current study, we apply a modelling and simulation approach to assess whether the approximation-based search procedures, in fact, lead to optimal or near-optimal stock levels.

Approximating the negative hypergeometric distribution

An improvement to a suggested negative binomial approximation to the Negative Hypergeometric Distribution (NHGD) is advanced and the accuracy of this approximation is also quantified in terms of variation distance. The method to obtain the improved approximation is to expand the probability function of NHGD. Some numerical examples are presented to compare with the established negative binomial approximation. The result shows our approximation is more accurate almost everywhere than the negative binomial. A gamma approximation to the cumulative distribution of the negative hypergeometric is initiated. Our gamma approximation is attractive in practice because of its arithmetic simplicity and accuracy over a wide range of parameter values. Some numerical examples are presented to illustrate the result obtained.

A new improvement of negative binomial approximation for the negative hypergeometric distribution

In this paper, we give a new improved negative binomial distribution with parameters r and M N p = to approximate the negative hypergeometric distribution with parameters ,

An Improved Negative Binomial Approximation for Negative Hypergeometric Distribution

This paper gives an improved negative binomial approximation for negative hypergeometric probability. Some numerical examples are presented to illustrate that in most practical cases the effect of our approximation is almost uniformly better than the negative binomial approximation.

On Negative Binomial Approximation

This paper deals with negative binomial approximation to sums of independent ${\bf Z}_+$-valued random variables. Stein's method is employed to obtain the error bounds. Convolution of negative binomial and Poisson distribution is used as a three-parametric approximation.

On exceedances of high levels

The distribution of the excess process describing heights of extreme values can be approximated by the distribution of a Poisson cluster process. An estimate of the accuracy of such an approximation has been derived in [4] in terms of the Wasserstein distance. The paper presents a sharper estimate established in terms of the stronger total variation distance. We derive also a new bound to the accuracy of negative Binomial approximation to the distribution of the number of exceedances.

Improved Base-Stock Approximations for Independent Stochastic Lead Times with Order Crossover

Order crossovers occur when replenishment orders arrive in a sequence that is different than the one in which they were placed. Order crossovers require that optimal reorder levels be set with regard to the inventory shortfall distribution rather than the lead-time demand distribution. Assuming periodic review and independent lead times, this paper suggests simple approximations of the shortfall distribution by showing that the variance of the number of orders outstanding is bounded above by the standard deviation of lead time divided by √3. Using this bound in a normal approximation improves significantly upon the common practice of basing policies on the lead-time demand distribution. A negative binomial approximation of the shortfall, based on its exact variance, offers even greater improvement, at the cost of some additional informational and computational requirements.