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Register a new userApproximating the Sum of Independent Non-Identical Binomial Random Variables
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Added by
Boxiang Liu
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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The distribution of sum of independent non-identical binomial random variables is frequently encountered in areas such as genomics, healthcare, and operations research. Analytical solutions to the density and distribution are usually cumbersome to find and difficult to compute. Several methods have been developed to approximate the distribution, and among these is the saddlepoint approximation. However, implementation of the saddlepoint approximation is non-trivial and, to our knowledge, an R package is still lacking. In this paper, we implemented the saddlepoint approximation in the textbf{sinib} package. We provide two examples to illustrate its usage. One example uses simulated data while the other uses real-world healthcare data. The textbf{sinib} package addresses the gap between the theory and the implementation of approximating the sum of independent non-identical binomials.
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We aim to estimate the probability that the sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., $mathbb{P}(sum_{i=1}^{N}{X_i} leq gamma)$, via importance sampling (IS). We are particularly interested in the rare event regime when $N$ is large and/or $gamma$ is small. The exponential twisting is a popular technique that, in most of the cases, compares favorably to existing estimators. However, it has several limitations: i) it assumes the knowledge of the moment generating function of $X_i$ and ii) sampling under the new measure is not straightforward and might be expensive. The aim of this work is to propose an alternative change of measure that yields, in the rare event regime corresponding to large $N$ and/or small $gamma$, at least the same performance as the exponential twisting technique and, at the same time, does not introduce serious limitations. For distributions whose probability density functions (PDFs) are $mathcal{O}(x^{d})$, as $x rightarrow 0$ and $d>-1$, we prove that the Gamma IS PDF with appropriately chosen parameters retrieves asymptotically, in the rare event regime, the same performance of the estimator based on the use of the exponential twisting technique. Moreover, in the Log-normal setting, where the PDF at zero vanishes faster than any polynomial, we numerically show that a Gamma IS PDF with optimized parameters clearly outperforms the exponential twisting change of measure. Numerical experiments validate the efficiency of the proposed estimator in delivering a highly accurate estimate in the regime of large $N$ and/or small $gamma$.
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For a binomial random variable $xi$ with parameters $n$ and $b/n$, it is well known that the median equals $b$ when $b$ is an integer. In 1968, Jogdeo and Samuels studied the behaviour of the relative difference between ${sf P}(xi=b)$ and $1/2-{sf P}(xi<b)$. They proved its monotonicity in $n$ and posed a question about its monotonicity in $b$. This question is motivated by the solved problem proposed by Ramanujan in 1911 on the monotonicity of the same quantity but for a Poisson random variable with an integer parameter $b$. In the paper, we answer this question and introduce a simple way to analyse the monotonicity of similar functions.
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