A reciprocal relation for Hermite polynomials

For $x\in \mathbb{R}$, the ordinary Hermite polynomial $H_k(x)$ can be written\begin{eqnarray*}\displaystyleH_k(x)= \mathbb{E} \left[ (x + {\rm i} N)^k \right] =\sum_{j=0}^k {k\choose j} x^{k-j} {\rm i}^j \mathbb{E} \left[ N^j \right],\end{eqnarray*}where ${\rm i} = \sqrt{-1}$ and $N$ is a unit normal random variable. We prove the reciprocal relation\begin{eqnarray*}\displaystylex^k=\sum_{j=0}^k {k\choose j} H_{k-j}(x)\ \mathbb{E} \left[ N^j \right].\end{eqnarray*}A similar result is given for the multivariate Hermite polynomial.

USING THE PROGRAMMING LANGUAGE IN STUDYING THE DISTRIBUTION NORMAL RANDOM QUANTITIES FUNCTION

This article uses C++ programming to plot the distribution function of a normal random variable. In addition, it was noted that interdisciplinary integration plays an important role in preparing future programmers for professional activities

Discretisation and Product Distributions in Ring-LWE

A statistical framework applicable to Ring-LWE was outlined by Murphy and Player (IACR eprint 2019/452). Its applicability was demonstrated with an analysis of the decryption failure probability for degree-1 and degree-2 ciphertexts in the homomorphic encryption scheme of Lyubashevsky, Peikert and Regev (IACR eprint 2013/293). In this paper, we clarify and extend results presented by Murphy and Player. Firstly, we make precise the approximation of the discretisation of a Normal random variable as a Normal random variable, as used in the encryption process of Lyubashevsky, Peikert and Regev. Secondly, we show how to extend the analysis given by Murphy and Player to degree-k ciphertexts, by precisely characterising the distribution of the noise in these ciphertexts.

On the use of the Box-Cox transformation in censored and truncated regression models

In this paper we revisit some issues related to the use of the Box-Cox transformation in censored and truncated regression models, which have been overlooked by the econometric and statistical literature. We first analyze the shape of the density function of the random variable which, rescaled by a Box-Cox transformation, leads to a normal random variable. Then, we identify the value ranges of the Box-Cox scale parameter for which a regular expectation of the derived random variable does not exist. This result calls for an extension of the concept of expectation, which can be computed regardless of the value of the scale parameter. For this purpose, we extend the concept of mean of a rescaled series of observations to the case of a random variable. Finally, we run estimates of censored and truncated Box-Cox standard Tobit models to determine the range of the scale parameter most relevant for empirical demand analyzes. These estimates highlight significant deviations from the assumption of normality of the dependent variable towards highly right skewed and leptokurtic distributions with no expectation.

Occupation times of alternating renewal processes with Lévy applications

In this paper we present a set of results relating to the occupation time α(t) of a processX(·). The first set of results concerns exact characterizations of α(t), e.g. in terms of its transform up to an exponentially distributed epoch. In addition, we establish a central limit theorem (entailing that a centered and normalized version of α(t)∕tconverges to a zero-mean normal random variable ast→∞) and the tail asymptotics of ℙ(α(t)∕t≥q). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

New expectation–maximization-type algorithms via stochastic representation for the analysis of truncated normal data with applications in biomedicine

To analyze univariate truncated normal data, in this paper, we stochastically represent the normal random variable as a mixture of a truncated normal random variable and its complementary random variable. This stochastic representation is a new idea and it is the first time to appear in literature. According to this stochastic representation, we derive important distributional properties for the truncated normal distribution and develop two new expectation–maximization algorithms to calculate the maximum likelihood estimates of parameters of interest for Type I data (without and with covariates) and Type II/III data. Bootstrap confidence intervals of parameters for small sample sizes are provided. To evaluate the performance of the proposed methods for the truncated normal distribution, in simulation studies, we first focus on the comparison of estimation results between including the unobserved data counts and excluding the unobserved data counts, and we next investigate the impact of the number of unobserved data on the estimation results. The plasma ferritin concentration data collected by Australian Institute of Sport and the blood fat content data are used to illustrate the proposed methods and to compare the truncated normal distribution with the half normal, the folded normal, and the folded normal slash distributions based on Akaike information criterion and Bayesian information criterion.

Probabilistic representations of matrix variate skew normal models

Azzalini [Scand. J. Stat. 12 (1985), 171–178] first introduced the skew normal distribution family with a shape parameter λ, and then extended this family by adding an additional shape parameters ξ. Basic properties of these two families were studied. Henze [Scand. J. Stat. 13 (1986), 271–275] gave the probabilistic representations for these two families by interpreting it as the linear combination of a normal random variable with another normal random variable truncated at the origin and several properties were illustrated. Chen and Gupta [Statistics 39 (2005), no. 3, 247–253] extended the skew normal distribution family to the matrix variate and proposed the moment generating function and the quadratic form of the matrix variate skew normal models. Motivated by these results, we first study the probabilistic representation for the matrix variate skew normal models and several properties. Then we define the extended skew normal model of the matrix variate, and give the probabilistic representation for this family and its extension.

Normal Limiting Distribution of the Size of Binary Interval Trees

The limiting distribution of the size of binary interval tree is investigated. Our illustration is based on the contraction method, and it is quite different from the case in one-sided binary interval tree. First, we build a distributional recursive equation of the size. Then, we draw the expectation, the variance, and some high order moments. Finally, it is shown that the size (with suitable standardization) approaches the standard normal random variable in the Zolotarev metric space.