We consider two-parametric compound binomial approximation of the generalized Poisson binomial distribution. We show that the accuracy of approximation essentially depends on the symmetry or shifting of distributions and construct asymptotic expansions. For the proofs, we combine the properties of norms with the results for convolutions of symmetric and shifted distributions. In the lattice case, we use the characteristic function method. In the case of almost binomial approximation, we apply Stein’s method.

Two-parametric compound binomial approximations

Using Stein's method, we derive explicit upper bounds on the total variation
distance between a Poisson-binomial law (the distribution of a sum of
independent but not necessarily identically distributed Bernoulli random
variables) and a Polya distribution with the same support, mean, and
variance; a nonuniform bound on the pointwise distance between the probability
mass functions is also given. A numerical comparison of alternative
distributional approximations on a somewhat representative collection of case
studies is also exhibited. The evidence proves that no single one is uniformly
most accurate, though it suggests that the Polya approximation might be
preferred in several parameter domains encountered in practice.

A Pólya approximation to the Poisson-binomial law

In this paper, we introduce the median-of-means approach for estimating unknown parameters in multiple linear regression model and the regression model with autocorrelated error. The proposed estim...

Median-of-means approach for repeated measures data

This chapter describes an algorithm for computing the PDF of the product of two independent continuous random variables. This algorithm has been implemented in the Product procedure in APPL. The algorithms behind the Transform and BiTransform procedures from the two previous chapters differ fundamentally from the algorithm behind the Product procedure in that the transformation algorithms are more general whereas determining the distribution of the product of two random variables is more specific. Some examples given in the chapter demonstrate the algorithm’s application.

Products of Random Variables

In mathematics and statistics, the desire to eliminate mathematical tedium and facilitate exploration has lead to computer algebra systems. These computer algebra systems allow students and researchers to perform more of their work at a conceptual level. The design of generic algorithms for tedious computations allows modelers to push current modeling boundaries outward more quickly. 
Probability theory, with its many theorems and symbolic manipulations of random variables is a discipline in which automation of certain processes is highly practical, functional, and efficient. There are many existing statistical software packages, such as SPSS, SAS, and S-Plus, that have numeric tools for statistical applications. There is a potential for a probability package analogous to these statistical packages for manipulation of random variables. The software package being developed as part of this dissertation, referred to as “A Probability Programming Language” (APPL) is a random variable manipulator and is proposed to fill a technology gap that exists in probability theory. 
My research involves developing algorithms for the manipulation of discrete random variables. By defining data structures for random variables and writing algorithms for implementing common operations, more interesting and mathematically intractable probability problems can be solved, including those not attempted in undergraduate statistics courses because they were deemed too mechanically arduous. Algorithms for calculating the probability density function of order statistics, transformations, convolutions, products, and minimums/maximums of independent discrete random variables are included in this dissertation.

Algorithms for operations on probability distributions in a computer algebra system

(2000). Getting Normal Probability Approximations without Using Normal Tables. The College Mathematics Journal: Vol. 31, No. 1, pp. 51-54.

Getting Normal Probability Approximations without Using Normal Tables

https://ir.nctu.edu.tw:443/bitstream/11536/26875/1/000220782400001.pdf

Generalized and pseudo-generalized trimmed means for the linear regression with AR(1) error model

The parameter n in the binomial distribution is a positive integer. What happens if we allow it to be any positive number? The new distribution we get is not only easy to work with but is useful in approximation situations that beginning statistics students could encounter. In addition, applications involving the distribution provide a good source for undergraduate research problems. We begin by setting g(x) = ()pX(l p)n-x, where (n) = n(n-1)(n-x+l) When p < 0.5, we observe that many of the features of the binomial distribution still hold. In particular, -Ex= o g(x) = 1, which follows from the binomial series,

Almost-Binomial Random Variables

A two-stage symmetric regression quantile is considered as an alternative for estimating the population quantile for the simultaneous equations model. We in- troduce a two-stage symmetric trimmed least squares estimator (LSE) based on this quantile. It is shown that, under mixed multivariate normal errors, this trimmed LSE has asymptotic variance much closer to the Cramer-Rao lower bound than some usual robust estimators. It can even achieve the Cramer-Rao lower bound when the contaminated variance goes to infinity. This suggests that the symmetric- type quantile function is as efficient in other statistical applications, such as outlier detection. A Monte Carlo study under asymmetric error distribution and a real data analysis are also presented. The two-stage LSE, developed by Anderson and Rubin (1949) in the simul- taneous equations model, is highly efficient under certain conditions. However, in the presence of heavy-tail errors, the two-stage LSE's efficiency can be markedly reduced. To overcome this, some robust techniques have been applied. Amemiya (1982) and Powell (1983) investigated the large sample property of a two-stage � 1-norm estimator. Krasker (1985) considered a two-stage bounded influence estimator. Chen and Portnoy (1996) applied the regression quantile technique of Koenker and Bassett (1978) to develop a two-stage trimmed LSE. However, from the comparison of the estimator's asymptotic variance with the Cramer- Rao (C-R) lower bound under mixed normal distributions (shown in Table 1 of this paper), the existing robust estimators are not really efficient under such heavy-tail errors. Is there an efficient technique for identifying outliers so that the corresponding trimmed LSE is efficient in such a way that the asymptotic variances are much closer to the C-R lower bound than the usual robust estima- tors?

/pdf/the-symmetric-type-two-stage-trimmed-least-squares-estimator-92k8iz6pcv.pdf

The symmetric type two-stage trimmed least squares estimator for the simultaneous equations model

SUMMARY We present asymptotic distributions of the Mallow0s type bounded-influence regression quantile for the linear regression model and also the simultaneous equations model Monte Carlo simulation comparing mean squared errors shows that the bounded-influence one is more ecient than the unbounded-influence one (Koenker and Bassett (1978)) when gross errors occur in the independent-variables-space Analysis of examples involving real data have also been provided

Mallow's Type Bounded Influence Regression Quantile for Linear Regression Model and Simultaneous Equations Model

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