A new prediction interval for binomial random variable based on inferential models

2020 ◽  
Vol 205 ◽  
pp. 156-174
Author(s):  
Hezhi Lu ◽  
Hua Jin
Keyword(s):  
Prediction Interval ◽  
Random Variable ◽  
Binomial Random Variable

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

2020 ◽  
Vol 32 (5) ◽  
pp. 1018-1032 ◽  
Author(s):  
Noah Frazier-Logue ◽  
Stephen José Hanson
Keyword(s):  
Prediction Error ◽  
Saddle Points ◽  
Model Averaging ◽  
Random Variable ◽  
Mean Values ◽  
Delta Rule ◽  
Binomial Random Variable ◽  
Optimal Network ◽  
Training Error ◽  
Feature Discovery

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%.

scholarly journals A refinement of normal approximation to Poisson binomial

2005 ◽  
Vol 2005 (5) ◽  
pp. 717-728 ◽  
Author(s):  
K. Neammanee
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Standard Normal Distribution ◽  
Bernoulli Random Variables ◽  
Binomial Random Variable ◽  
Standard Normal ◽  
Correction Terms ◽  
Better Than

LetX1,X2,…,Xnbe independent Bernoulli random variables withP(Xj=1)=1−P(Xj=0)=pjand letSn:=X1+X2+⋯+Xn.Snis called a Poisson binomial random variable and it is well known that the distribution of a Poisson binomial random variable can be approximated by the standard normal distribution. In this paper, we use Taylor's formula to improve the approximation by adding some correction terms. Our result is better than before and is of order1/nin the casep1=p2=⋯=pn.

Negative binomial sums of random variables and discounted reward processes

1998 ◽  
Vol 35 (3) ◽  
pp. 589-599
Author(s):  
William L. Cooper
Keyword(s):  
Differential Equation ◽  
Negative Binomial ◽  
Infinite Horizon ◽  
Random Variables ◽  
Random Variable ◽  
Random Time ◽  
Sums Of Random Variables ◽  
Total Reward ◽  
Binomial Random Variable ◽  
Binomial Sums

Given a sequence of random variables (rewards), the Haviv–Puterman differential equation relates the expected infinite-horizon λ-discounted reward and the expected total reward up to a random time that is determined by an independent negative binomial random variable with parameters 2 and λ. This paper provides an interpretation of this proven, but previously unexplained, result. Furthermore, the interpretation is formalized into a new proof, which then yields new results for the general case where the rewards are accumulated up to a time determined by an independent negative binomial random variable with parameters k and λ.

A note on stochastic ordering of order statistics

1997 ◽  
Vol 34 (03) ◽  
pp. 785-789 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Order Statistics ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stochastic Comparisons ◽  
Homogeneous Population ◽  
Independent Components ◽  
Binomial Random Variable ◽  
Necessary And Sufficient

A necessary and sufficient condition is obtained for a Poisson binomial random variable to be stochastically larger (or smaller) than a binomial random variable. It is then used to deal with the stochastic comparisons of order statistics from heterogeneous populations with those from a homogeneous population. The result has obvious applications in the stochastic comparisons of lifetimes of k-out-of-n systems having independent components.

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

2021 ◽  
Vol 9 (1) ◽  
pp. 1-12
Author(s):  
Jeonghwa Lee
Keyword(s):  
Long Range ◽  
Infinite Sequence ◽  
Binary Sequence ◽  
Random Variable ◽  
Long Range Dependence ◽  
Bernoulli Process ◽  
Binomial Random Variable ◽  
Fixed Constant ◽  
Bernoulli Variables ◽  
Auto Covariance

Abstract 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).

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

2014 ◽  
Vol 26 (2) ◽  
pp. 931-947 ◽  
Author(s):  
Guo-Liang Tian ◽  
Man-Lai Tang ◽  
Qin Wu ◽  
Yin Liu
Keyword(s):  
Confidence Interval ◽  
Negative Binomial ◽  
Survey Design ◽  
Random Variable ◽  
Parameter Estimate ◽  
Original Design ◽  
Bernoulli Random Variables ◽  
Binomial Random Variable ◽  
Original Item ◽  
Item Count Technique

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.

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

2020 ◽  
Vol 15 (3) ◽  
pp. 2371-2385
Author(s):  
Gane Samb Lo ◽  
Harouna Sangaré ◽  
Cherif Mamadou Moctar Traoré ◽  
Mohammad Ahsanullah
Keyword(s):  
Negative Binomial ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Record Values ◽  
Cumulative Distribution ◽  
Hitting Times ◽  
Record Times ◽  
Binomial Random Variable ◽  
Upper Record Values

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.

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