Generalized Bernoulli process with long-range dependence and fractional binomial distribution

Bernoulli process is a finite or infinite sequence of independent binary variables, X i , i = 1, 2, · · ·, whose outcome is either 1 or 0 with probability P(X i = 1) = p, P(X i = 0) = 1 – p, for a fixed constant p ∈ (0, 1). We will relax the independence condition of Bernoulli variables, and develop a generalized Bernoulli process that is stationary and has auto-covariance function that obeys power law with exponent 2H – 2, H ∈ (0, 1). Generalized Bernoulli process encompasses various forms of binary sequence from an independent binary sequence to a binary sequence that has long-range dependence. Fractional binomial random variable is defined as the sum of n consecutive variables in a generalized Bernoulli process, of particular interest is when its variance is proportional to n 2 H , if H ∈ (1/2, 1).

A note on negative λ-binomial distribution

In this paper, we introduce one discrete random variable, namely the negative λ-binomial random variable. We deduce the expectation of the negative λ-binomial random variable. We also get the variance and explicit expression for the moments of the negative λ-binomial random variable.

Finiteness of Record values and Alternative Asymptotic Theory of Records with Atom Endpoints

Asymptotic theories on record values and times, including central limit theorems, make sense only if the sequence of records values (and of record times) is infinite. If not, such theories could not even be an option. In this paper, we give necessary and/or sufficient conditions for the finiteness of the number of records. We prove, for example for iid real valued random variable, that strong upper record values are finite if and only if the upper endpoint is finite and is an atom of the common cumulative distribution function. The only asymptotic study left to us concerns the infinite sequence of hitting times of that upper endpoints, which by the way, is the sequence of weak record times. The asymptotic characterizations are made using negative binomial random variables and the dimensional multinomial random variables. Asymptotic comparison in terms of consistency bounds and confidence intervals on the different sequences of hitting times are provided. The example of a binomial random variable is given.

The Stochastic Delta Rule: Faster and More Accurate Deep Learning Through Adaptive Weight Noise

Multilayer neural networks have led to remarkable performance on many kinds of benchmark tasks in text, speech, and image processing. Nonlinear parameter estimation in hierarchical models is known to be subject to overfitting and misspecification. One approach to these estimation and related problems (e.g., saddle points, colinearity, feature discovery) is called Dropout. The Dropout algorithm removes hidden units according to a binomial random variable with probability [Formula: see text] prior to each update, creating random “shocks” to the network that are averaged over updates (thus creating weight sharing). In this letter, we reestablish an older parameter search method and show that Dropout is a special case of this more general model, stochastic delta rule (SDR), published originally in 1990. Unlike Dropout, SDR redefines each weight in the network as a random variable with mean [Formula: see text] and standard deviation [Formula: see text]. Each weight random variable is sampled on each forward activation, consequently creating an exponential number of potential networks with shared weights (accumulated in the mean values). Both parameters are updated according to prediction error, thus resulting in weight noise injections that reflect a local history of prediction error and local model averaging. SDR therefore implements a more sensitive local gradient-dependent simulated annealing per weight converging in the limit to a Bayes optimal network. We run tests on standard benchmarks (CIFAR and ImageNet) using a modified version of DenseNet and show that SDR outperforms standard Dropout in top-5 validation error by approximately 13% with DenseNet-BC 121 on ImageNet and find various validation error improvements in smaller networks. We also show that SDR reaches the same accuracy that Dropout attains in 100 epochs in as few as 40 epochs, as well as improvements in training error by as much as 80%.

Compound Poisson point processes, concentration and oracle inequalities

This note aims at presenting several new theoretical results for the compound Poisson point process, which follows the work of Zhang et al. (Insur. Math. Econ. 59:325–336, 2014). The first part provides a new characterization for a discrete compound Poisson point process (proposed by Aczél (Acta Math. Hung. 3(3):219–224, 1952)), it extends the characterization of the Poisson point process given by Copeland and Regan (Ann. Math. 37:357–362, 1936). Next, we derive some concentration inequalities for discrete compound Poisson point process (negative binomial random variable with unknown dispersion is a significant example). These concentration inequalities are potentially useful in count data regression. We give an application in the weighted Lasso penalized negative binomial regressions whose KKT conditions of penalized likelihood hold with high probability and then we derive non-asymptotic oracle inequalities for a weighted Lasso estimator.

Force of infection ofHelicobacter pyloriin Mexico: evidence from a national survey using a hierarchical Bayesian model

Helicobacter pylori(H. pylori)is present in the stomach of half of the world's population. The force of infection describes the rate at which susceptibles acquire infection. In this article, we estimated the age-specific force of infection ofH. pyloriin Mexico. Data came from a nationalH. pyloriseroepidemiology survey collected in Mexico in 1987–88. We modelled the number of individuals withH. pyloriat a given age as a binomial random variable. We assumed that the cumulative risk of infection by a given age follows a modified exponential catalytic model, allowing some fraction of the population to remain uninfected. The cumulative risk of infection was modelled for each state in Mexico and were shrunk towards the overall national cumulative risk curve using Bayesian hierarchical models. The proportion of the population that can be infected (i.e. susceptible population) is 85.9% (95% credible interval (CR) 84.3%–87.5%). The constant rate of infection per year of age among the susceptible population is 0.092 (95% CR 0.084–0.100). The estimated force of infection was highest at birth 0.079 (95% CR 0.071–0.087) decreasing to zero as age increases. This Bayesian hierarchical model allows stable estimation of state-specific force of infection by pooling information between the states, resulting in more realistic estimates.

Poisson and negative binomial item count techniques for surveys with sensitive question

Although the item count technique is useful in surveys with sensitive questions, privacy of those respondents who possess the sensitive characteristic of interest may not be well protected due to a defect in its original design. In this article, we propose two new survey designs (namely the Poisson item count technique and negative binomial item count technique) which replace several independent Bernoulli random variables required by the original item count technique with a single Poisson or negative binomial random variable, respectively. The proposed models not only provide closed form variance estimate and confidence interval within [0, 1] for the sensitive proportion, but also simplify the survey design of the original item count technique. Most importantly, the new designs do not leak respondents’ privacy. Empirical results show that the proposed techniques perform satisfactorily in the sense that it yields accurate parameter estimate and confidence interval.