Approximations related to the sums of m-dependent random variables Amit N. Kumar, Neelesh S. Upadhye, P. Vellaisamy Brazilian Journal of Probability and Statistics, 2022 In this paper, we consider the sums of non-negative integer valued $m$-dependent random variables, and its approximation to the power series distribution. We first discuss some relevant results for power series distribution such as Stein operator, uniform and non-uniform bounds on the solution of Stein equation, and etc. Using Stein's method, we obtain the error bounds for the approximation problem considered. As special cases, we discuss two applications, namely, $2$-runs and $(k_1,k_2)$-runs and compare the bound with the existing bounds.
On discrete Gibbs measure approximation to runs A. N. Kumar, N. S. Upadhye Communications in Statistics Theory and Methods, 2022 In this paper, some erroneous results for a dependent setup arising from independent sequence of Bernoulli trials are corrected. Next, a Stein operator for discrete Gibbs measure is derived using PGF approach. Also, an operator for dependent setup is derived and shown as perturbation of the Stein operator for discrete Gibbs measure. Finally, using perturbation technique and explicit form of distributions from discrete Gibbs measure, new error bounds between the dependent setup and Poisson, pseudo-binomial and negative binomial distributions are obtained by matching up to first two moments.
POISSON APPROXIMATION TO THE CONVOLUTION OF POWER SERIES DISTRIBUTIONS Amit Kumar, Palaniappan Vellaisamy, Frederi Viens Probability and Mathematical Statistics, 2022 In this article, we obtain, for the total variance distance, the error bounds between Poisson and convolution of power series distributions via Stein's method. This provides a unified approach to many known discrete distributions. Several Poisson limit theorems follow as corollaries from our bounds. As applications, we compare the Poisson approximation results with the negative binomial approximation results, for the sums of Bernoulli, geometric, and logarithmic series random variables.
Generalizations of distributions related to (k 1 , k 2 )-runs A. N. Kumar, N. S. Upadhye Metrika, 2019 The paper deals with three generalized dependent setups arising from a sequence of Bernoulli trials. Various distributional properties, such as probability generating function, probability mass function and moments are discussed for these setups and their waiting time. Also, explicit forms of probability generating function and probability mass function are obtained. Finally, two applications to demonstrate the relevance of the results are given.
Pseudo-binomial approximation to (k1,k2)-runs N.S. Upadhye, A.N. Kumar Statistics and Probability Letters, 2018 The distribution of ( k 1 , k 2 ) -runs is well-known (Dafnis et al., 2010), under independent and identically distributed (i.i.d.) setup of Bernoulli trials but is intractable under non i.i.d. setup. Hence, it is of interest to find a suitable approximate distribution for ( k 1 , k 2 ) -runs, under non i.i.d. setup, with reasonable accuracy. In this paper, pseudo-binomial approximation to ( k 1 , k 2 ) -runs is considered using total variation distance. The approximation results derived are of optimal order and improve the existing results.
