scholarly journals Exactly Computing the Tail of the Poisson-Binomial Distribution

2021 ◽  
Vol 47 (4) ◽  
pp. 1-19
Author(s):  
Noah Peres ◽  
Andrew Ray Lee ◽  
Uri Keich
Keyword(s):  
Binomial Distribution ◽  
R Package ◽  
Tail Probability ◽  
Exact Method ◽  
Tail Probabilities ◽  
Poisson Binomial Distribution ◽  
Relative Errors

We present ShiftConvolvePoibin, a fast exact method to compute the tail of a Poisson-binomial distribution (PBD). Our method employs an exponential shift to retain its accuracy when computing a tail probability, and in practice we find that it is immune to the significant relative errors that other methods, exact or approximate, can suffer from when computing very small tail probabilities of the PBD. The accompanying R package is also competitive with the fastest implementations for computing the entire PBD.

scholarly journals On moderate deviations in Poisson approximation

2020 ◽  
Vol 57 (3) ◽  
pp. 1005-1027
Author(s):  
Qingwei Liu ◽  
Aihua Xia
Keyword(s):  
Random Graphs ◽  
Binomial Distribution ◽  
Poisson Approximation ◽  
Moderate Deviations ◽  
Tail Probabilities ◽  
Matching Problem ◽  
Birthday Problem ◽  
Occupancy Problem ◽  
Poisson Binomial Distribution ◽  
The Right

AbstractIn this paper we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of a Poisson distribution than those of the normal distribution. We then show that the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in [18]. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems via six applications: Poisson-binomial distribution, the matching problem, the occupancy problem, the birthday problem, random graphs, and 2-runs. The paper complements the works [16], [8], and [18].

Six Sigma metrics based on lognormal distribution for life tests

2019 ◽  
Vol 36 (9) ◽  
pp. 1477-1489 ◽  
Author(s):  
Ravichandran Joghee
Keyword(s):  
Six Sigma ◽  
Lognormal Distribution ◽  
Tail Probability ◽  
Tail Probabilities ◽  
Content Type ◽  
Underlying Distribution ◽  
Left Tail ◽  
Long Run ◽  
Life Tests ◽  
The Right

Purpose The purpose of this paper is to propose an approach for studying the Six Sigma metrics when the underlying distribution is lognormal. Design/methodology/approach The Six Sigma metrics are commonly available for normal processes that are run in the long run. However, there are situations in reliability studies where non-normal distributions are more appropriate for life tests. In this paper, Six Sigma metrics are obtained for lognormal distribution. Findings In this paper, unlike the normal process, for lognormal distribution, there are unequal tail probabilities. Hence, the sigma levels are not the same for left-tail and right-tail defects per million opportunities (DPMO). Also, in life tests, while left-tail probability is related to DPMO, the right tail is considered as extremely good PMO. This aspect is introduced and based on which the sigma levels are determined for different parameter settings and left- and right-tail probability combinations. Examples are given to illustrate the proposed approach. Originality/value Though Six Sigma metrics have been developed based on a normality assumption, there have been no studies for determining the Six Sigma metrics for non-normal processes, particularly for life test distributions in reliability studies. The Six Sigma metrics developed here for lognormal distribution is new to the practitioners, and this will motivate the researchers to do more work in this field of research.

Approximating the Student's t-Tail Probability and Its Inverse

1992 ◽  
Vol 6 (1) ◽  
pp. 133-137 ◽  
Author(s):  
Jinn-Tyan Lin
Keyword(s):  
Tail Probability ◽  
Student’S T ◽  
Relative Errors

This paper proposes a logistic approximation to the upper tail of the t distribution and its inverse that is more convenient and more accurate than those of Shah and Allen [3], and Koehler [1] and Lin [2], when measured in terms of the number of keystrokes and absolute relative errors, respectively.

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