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The use of virtual species to test species distribution models is important for understanding how aspects of the model development process influence model performance. Typically, virtual species are simulated by defining their niche as a function of environmental variables and simulating occurrence probabilistically via Bernoulli trials. This approach ignores endogenous processes known to drive species distribution such as dispersal and population dynamics. To understand whether these processes are important for simulating virtual species we compared the probabilistic simulation approach to those incorporating endogenous processes. This comparison was done by evaluating changes in the relationship between species occurrence and habitat suitability over a number of landscapes with varying spatial structure. We found that the combined effects of population dynamics and dispersal meant the probability of occurrence of a single cell was not only dependent on habitat suitability, but also the number of occupied cells nearby. This resulted in a dependence on the size of clusters of high suitability cells (analogous to patch size) to maintain populations, increased residual spatial autocorrelation and non‐stationarity of the species response between landscapes. These data characteristics are attributes of real species distribution data and are not present in probabilistic simulations. Researchers using virtual species should consider the importance of these characteristics to their study objectives to decide whether the inclusion of endogenous processes is necessary.

Availability of a system is a crucial factor for planning and optimization. The concept is more challenging for modern systems such as robots and autonomous systems consisting of a complex configuration of components. In this paper, a reliability evaluation framework is developed for a system of binary state autonomous robots in an automated manufacturing environment. In this framework, the concepts in functional block diagram, table of truth, and sum of state are employed simultaneously to develop a binary state reliability model. Due to inefficacy of the method for larger number of components involved in complex systems, an extension of the Bernoulli trials is proposed. In an implementation study, the effectiveness and computational efficiency of the proposed method are illustrated. In addition, an analysis on the failure rate using the maximum likelihood estimation and confidence interval is reported. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we derive asymptotic formulas for the signless noncentral q-Stirling numbers of the first kind and for the corresponding series. The signless noncentral q-Stirling numbers of the first kind appear as coefficients of a polynomial of q-number [t]q, expressing the noncentral ascending q-factorial of t of order m and noncentrality parameter k. In this paper, we have two main purposes. The first is to give an expression by which we obtain the asymptotic behavior of these coefficients, using the saddle point method. The second main purpose is to derive an asymptotic expression for the signless noncentral q-Stirling of the first kind series by using the singularity analysis method. We then apply our first formula to provide asymptotic expressions for probability functions of the number of successes in m trials and of the number of trials until the occurrence of the nth success in sequences of Bernoulli trials with varying success probability which are both written in terms of the signless noncentral q-Stirling numbers of the first kind. In addition, we present some numerical calculations using the computer program MAPLE indicating that our expressions are close to the actual values of the signless noncentral q-Stirling numbers of the first kind and of the corresponding series even for moderate values of m.

Despite it's popularity and usefulness as a vegetation community descriptor, plant cover is often considered by ecologists to be a qualitative or semiquantitative variable. The relative merit of visual estimation within quadrats versus the use of multiple points to measure cover has been debated. This apparent conflict has been of concern for decades to ecologists in regulatory agencies and in private industry. Coincidently, regulatory agencies often rely on plant cover estimates to form the basis of sound decisions regarding reclamation success. We offer a resolution to these issues. If each sample site within a vegetation community is regarded as a repeated Bernoulli trial of sufficient size, say 100 trials, probability of the number of successes can be closely approximated by the standard normal distribution. There is little to be gained in decreasing or increasing the number of Bernoulli trials away from n = 100. Either quadrats or points may be used to equal effect only if the area formed by the quadrat and placement of points within a point-frame or along a line are the same. Further, even with a moderate sample size of 30 repeated sets of Bernoulli trials (n = 100) randomly located within a vegetation community, there is generally no need to transform the data to approximate a normal distribution prior to conducting hypothesis tests.

This paper provides a test procedure for the problem of testing Bernoulli success probability in the case of costly trials when an inverse sampling is carried out. The proposed test is based on a two population adaptive sampling scheme used in clinical trials. Some exact and asymptotic results related to the test are studied. The proposed procedure is applicable where the alternatives are not too far from the null hypothetical value.

Consider a threshold control policy for an imperfect production system with only a work center handling both regular and rework jobs. An imperfect production system studied here, generates defect jobs by factors other than machine failures. A threshold control or (ω, s) policy sets the guideline for a work center to switch between regular and rework jobs. A production cycle begins with loading and processing of several batches of regular jobs with a lot size equal to s. The outcome of each completed regular job is an independent Bernoulli trial with three possibilities: good, rework, or scrap. Once the work center accumulates more than a threshold ω of rework jobs, it finishes the last batch of regular jobs and switches to rework jobs. The objective of this research is to find a threshold ω and a lot size s that maximize the average long-term profit. The ultimate goal is to construct a simple algorithm to search for ω and s that can be implemented directly in production management systems, as a result of this work. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 273–301, 1999

The paper derives the joint generating function of a collection of pattern statistics associated with binary sequences. The models discussed cover independent and some dependent Bernoulli trials, including Markov dependent ones. The results cover, in particular, the moment generating function of the random search time for certain general binary patterns in the popular Knuth-Morris-Pratt algorithm and hence shed more light into its performance.

We present two time-inhomogeneous search processes for finding the optimal Bernoulli parameters, where the performance measure cannot be evaluated exactly but must be estimated through Monte Carlo simulation. At each iteration, two neighbouring alternatives are compared and the one that appears to be better is passed on to the next iteration. The first search process uses an increasing sample size of each configuration at each iteration. The second search process uses a sequential sampling procedure with increasing boundaries as the number of iterations increases. At each iteration the acceptance of a new configuration depends on the iterate number, therefore, the search process turns out to be inhomogeneous Markov chain. We show that if the increase occurs slower than a certain rate, these search processes will converge to the optimal set with probability one.

SUMMARY Wahba and co-workers introduced the smoothing spline analysis-of-variance (SS ANOVA) method for data from exponential families. In this paper, we estimate the odds ratios based on an SS ANOVA model for binary data and construct Bayesian confidence intervals. We give a calculation using a real data set from the Wisconsin epidemiological study of diabetic retinopathy. We conduct simulations to evaluate the performance of these estimates and their Bayesian confidence intervals. Our simulations suggest that the odds ratio estimates are quite reasonable in general but may be biased towards 1 when comparing estimates at peaks with those in troughs. A bootstrap procedure is proposed to correct possible biases and it works very well in our simulation.

This paper considers a generalized birth process {Xm(t), t > 0} and presents a new stochastic model for the number of eggs laid by a parasite on a host. Also, given an underlying distribution for the number of visits between parasites and a host, this distribution is generalized by the distribution of the number of eggs per visit laid on the host. If a certain number of eggs are already present on the host, a parasite such as a Japanese weevil, may avoid oviposition in subsequent visits (see JANARDAN (1980)) to the same host. A class of generalized distributions are presented to model such situations. The case of a single egg laying parasite and a Poisson distribution for the number of visits of the parasite to the same host yields a distribution of particular interest. In order to develop this model, certain lemmas are derived. Finally a characteristic property of this stochastic model is presented.

When, in a poly-α-olefin, the probability that a given placement be isotactic depends upon the tacticity of only a finite number of immediate predecessors, the resulting diastereosequence distribution obeys the theory of Markoff chains. When this is not the case, one says that the resulting diastereosequence distribution is non-Markoffian. A special case of a Markoffian distribution is given by a simple Markoff chain in which the tacticity of a given placement is assumed to be affected by only the tacticity of the immediately preceding placement. Another special case is, of course, the Bernoulli trial distribution in which the probability that a given placement be isotactic is independent of the tacticity of all other placements. A high resolution NMR spectrum can sometimes yield a quantitative determination of the concentrations of isotactic and syndiotactic placements and the concentrations of the three types of possible adjacent pairs of such placements (i.e., isotactic, syndiotactic, and heterotactic pairs). When this is the case, the spectrum can be used to determine whether or not a given diastereosequence distribution is Bernoullian. However, because the longest diastereosequences whose concentration can be measured by NMR spectroscopy involve only two placements, an NMR spectrum cannot check whether a given non-Bernoullian distribution be simple Markoffian or Markoffian in general. In fact, non-Markoffian distributions are compatible with existing NMR spectra on polymers prepared by anionic polymerizations. In this paper we work within the framework of Kac's theory of stationary statistical processes and point out some general results which are valid for both Markoffian and non-Markoffian processes. The results are applied to NMR spectroscopy and it is pointed out which calculations used to check the self-consistency of NMR data and to obtain the mean length of closed diastereosequences are valid for both Markoffian and non-Markoffian distributions.

Raghavan's paper on derandomized approximation algorithms for 0–1 packing integer programs raised two challenging problems [11]: 1. Are there more examples of NP-hard combinatorial optimization problems for which derandomization yields constant factor approximations in polynomial-time ? 2. The pessimistic estimator technique shows an O(mn)-time implementation of the conditional probability method on the RAM model of computation in case of m large deviation events associated to m unweighted sums of n indepependent Bernoulli trials. Is there a fast algorithm also in case of rational weighted sums of Bernoulli trials ?

Consider a collection of Bernoulli random variables on the two dimensional integer lattice and define the length D of the side of the largest square consisting entirely of successes, which covers the origin of the lattice. The paper gives a method to evaluate the probability distribution function of D. An analogous problem for the Poisson process on the plane is also considered.

This paper discusses five methods for constructing approximate confidence intervals for the binomial parameter Θ, based on Y successes in n Bernoulli trials. In a recent paper, Chen (1990) discusses various approximate methods and suggests a new method based on a Bayes argument, which we call method I here. Methods II and III are based on the normal approximation without and with continuity correction. Method IV uses the Poisson approximation of the binomial distribution and then exploits the fact that the exact confidence limits for the parameter of the Poisson distribution can be found through the x2 distribution. The confidence limits of method IV are then provided by the Wilson-Hilferty approximation of the x2. Similarly, the exact confidence limits for the binomial parameter can be expressed through the F distribution. Method V approximates these limits through a suitable version of the Wilson-Hilferty approximation. We undertake a comparison of the five methods in respect to coverage probability and expected length. The results indicate that method V has an advantage over Chen's Bayes method as well as over the other three methods.

In this paper, a statistical model for clinical trials is presented for the special situation that a varying and unstructered number of binary responses is obtained from each subject. The assumptions of the model are the following: 1.) For each subject there is a (constant) individual Bernoulli parameter determining the distribution of the binary responses of this subject. 2.) The Bernoulli parameters associated with the subjects are realizations of independent random variables with distributions Pg in treatment group g(g = 1, 2, …, G). 3.) Given the value of the Bernoulli parameter, the observations are stochastically independent within each subject. Under these assumptions, a test statistic is derived to test the hypothesis H0:E(P1) = E(P2) = … = E(PG). It is proven and demonstrated by simulations, that the test statistic asymptotically (i.e. for a large number of subjects) follows the X2-distribution.

Given two Bernoulli populations with unknown means, we consider the problem of sampling x 1 , x 2 ,... sequentially from these populations such as to maximize the expected value of the sum Sn=x1,+x 2 +... +x n as n→∞. We obtain an asymptotically efficient rule. Simulation studies show that it performs well for moderate values of n

SUMMARY The atom probe can be used to characterize fine-scale microstructures. The results presented in this and a subsequent paper attempt to develop a methodology for interpreting information concerning 3-D structures from the sequence of ions detected in the atom probe. Probability theory (Bernoulli trials and Markov processes) is used to build appropriate mathematical descriptions of different types of microstructure.

Summary Properties of the random variable representing the number of identical and independent Bernoulli trials necessary to obtain K consecutive successes are investigated. The results are interesting to students in a first course in probability or mathematical statistics.

A series of independent Bernoulli trials is considered in which either an outcome of type A or type B occurs at each trial. The series terminates when n outcomes of one type have occurred. Two observable random variables of interest are the total number of outcomes in the series and the number of outcomes of the “losing kind.” Two methods of approximation of the expectations of these random variables for large n are obtained and compared. The limiting distribution of the number of outcomes of the “losing kind” is considered when a beta distribution is assigned to p.

Almost all statistical estimation problems involving the binomial distribution occur with the number of independent Bernoulli trials, n, being a known parameter and the probability of success, p, being unknown. Applications in which n is unknown and p is known are extremely rare. One such application to a real problem in the physical sciences is the subject of this note.

Summary In these days some sportsmen are given dangerous drugs in order to stimulate them into still higher efforts. The present paper wants to add some poison to the classical approximation of a binomial by a Poisson distribution. The relatively harmless drug of a little extra calculation shall be seen to result in a much better accuracy. This paper reports both on theoretical considerations, following from series expansions of Poisson-type parameters, and on extensive numerical investigations of accuracy by means of an Electrologica X8 computer. The main conclusion is that the probability of at most k successes in n Bernoulli trials with success probability p 5 one should make p < .5 by a reversal, i.e. the interchanging of successes and failures, before the application of a Poisson-type approximation (with one exception mentioned in section 4). For the probability of at least j successes one should approximate the complementary probability of at most j—1 successes. After an introductory section 1 and a discussion of measures of accuracy in section 2, the existing Poisson-type approximations are treated in section 3. The fourth section introduces a new parameter and discusses the advantages of reversal in other situations than p > 5. In section 5 a series expansion for the exact Poisson parameter is derived. Two new parameters, among which is ζ, are introduced in section 6, where the series expansions for parameters given in table 4 lead to conclusions about the various approximations, summarized in table 6. The final section 7 gives some numerical results, which confirm to a large extent the conclusions from the asymptotic expansions.