Dependent Bernoulli Trials Research Articles

The Use of Correlated Binomial Distribution in Estimating Error Rates for Firearm Evidence Identification

In the branch of forensic science known as firearm evidence identification, estimating error rates is a fundamental challenge. Recently, a new quantitative approach known as the congruent matching cells (CMC) method was developed to improve the accuracy of ballistic identifications and provide a basis for estimating error rates. To estimate error rates, the key is to find an appropriate probability distribution for the relative frequency distribution of observed CMCs overlaid on a relevant measured firearm surface such as the breech face of a cartridge case. Several probability models based on the assumption of independence between cell pair comparisons have been proposed, but the assumption of independence among the cell pair comparisons from the CMC method may not be valid. This article proposes statistical models based on dependent Bernoulli trials, along with corresponding methodology for parameter estimation. To demonstrate the potential improvement from the use of the dependent Bernoulli trial model, the methodology is applied to an actual data set of fired cartridge cases.

Quantum-chaotic key distribution in optical networks: from secrecy to implementation with logistic map

In a recent paper, the quantum-chaotic key distribution (QCKD) in optical networks was introduced. In the present work, we extend the QCKD theory in two ways: Firstly, we propose to use the dependent Bernoulli trials to model the key generation in QCKD. Using this model, we show that the key generated by QCKD is far from presenting the observed correlations in chaos-based cryptography, and it is very close to the maximum secrecy offered by ideal quantum cryptography. Secondly, we show a new optical scheme for QCKD in which the optical chaotic scheme using optoelectronic oscillators is substituted by nonlinear discrete equations running in computers and the information carrier used is the phase instead of the light polarization. These changes make much easier its implementation with today technology while keeping the same security level guaranteed by chaotic and quantum rules.

New Non-uniform Bounds on Poisson Approximation for Dependent Bernoulli Trials

The aim of this article is a use of the Stein–Chen method to obtain new non-uniform bounds on the error of the distribution of sums of dependent Bernoulli random variables and the Poisson distribution. The bounds obtained in this study are improved to be more appropriate for measuring the accuracy of Poisson approximation. Examples are provided to illustrate applications of the obtained results.

Corrigenda: Explicit Distributional Results in Pattern Formation II

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.

Theory & Methods: Explicit Distributional Results In Pattern Formation II

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.

A MARKOV CHAIN MODEL FOR MAGAZINE EXPOSURE

SummaryKlotz's (1973) Markov chain model for dependent Bernoulli trials is applied to magazine exposure distributions. Simple parameter estimates are derived and are shown to compare well with the maximum likelihood estimates. The Markov model is fitted to forty magazines from a large print media survey and compares favourably with the most popular non‐proprietary magazine model, the beta‐binomial model. In addition, the Markov model is used to simulate magazine exposure distributions.

Testing Proportions for Markov Dependent Bernoulli Trials

For testing stationary state distribution of positive recurrent Markov dependent Bernoulli trials, Tavaré & Altham (1983) proposed a scale-corrected statistic, XXTA2 which has a X2 limiting distribution. In this paper, we establish the asymptotic optimality of XXTA2 by showing that it is asymptotically equivalent to the optimal test XXTA2 obtained under the modified likelihood approach of Billingsley (1961). Some simulation results are also given to examine finite sample performance of XXTA2 and XXTA2

A Markov Mixture Model for Magazine Exposure

Abstract A magazine-exposure model that mixes Klotz's (1973) dependent Bernoulli-trials model for nonsubscribers with a degenerate distribution for subscribers is proposed. Let Xi = 1 if a person reads an issue of a particular magazine and 0 otherwise. Klotz's parameterization is Pr(Xi = 1) = p and Pr(Xi = 1 |X i–1 = 1) = λ for i = 1, …, k. Using the Markov assumption he obtains the joint distribution of R = Σ k i=2 X i–1 Xi, S = Σ k i=1 Xi , and T = X 1 + Xk , of which we are interested in the marginal distribution of S, the total number of issues a person reads. It is expected that p will be low for nonsubscribers but high for subscribers, so this heterogeneity is modeled by mixing Klotz's Markov model with a point mass of magnitude π at the point S = k. Maximum likelihood estimates of p, λ, and π are used to fit the Markov mixture model to 40 magazines from a large print-media survey. The proposed model is shown to give a much better fit to these data than the beta-binomial model, the most popular nonp...

Testing proportions for Markov dependent Bernoulli trials

For testing stationary state distribution of positive recurrent Markov dependent Bernoulli trials a scale-corrected statistic has been proposed which has a chisquare as limiting distribution. In this article its asymptotic optimality is established by slowly that it is asymptotically equivalent to an optimal test obtained under the modified likelihood approach of Billingsley

Simulating Media Exposure Distributions

We examine three media exposure distribution (e.d.) simulation methods. The first is based on the maximum likelihood estimate of an individual's exposure, the second on ‘personal probability’ (Greene 1970) and the third on a dependent Bernoulli trials model (Klotz 1973). The last method uses population exposure probabilities rather than individual exposure probabilities, thereby markedly reducing computation time. Magazine exposure data are used to compare the accuracy and computation times of the simulation methods with a log–linear e.d. model (Danaher 1988b) and the popular Metheringham (1964) model based on the beta–binomial distribution (BBD). The results show that the simulation methods are not as accurate as the log– linear model but are more accurate than Metheringham's model, However, all the simulation methods take less computation time than the log–linear model for schedules with more than six magazines, making them viable competitors for large schedule sizes

Inference in a waiting-time distribution derived from dependent bernoulli trials

The problems of unbiased estimation of parameters and test of independence are studied for the probability distribution of the number of Markov dependent Bernoulli trials required so as to obtain a preassigned number of successes. It is shown that the maximum likelihood estimate of one parameter is minimum-variance unbiased while that of the other parameter tends to overestimate the true value. For this parameter an alternative estimate is given. A test procedure for the independence of trials is also proposed.