Recent zbMATH articles in MSC 62Ehttps://zbmath.org/atom/cc/62E2024-03-13T18:33:02.981707ZWerkzeugThe basic distributional theory for the product of zero mean correlated normal random variableshttps://zbmath.org/1528.600172024-03-13T18:33:02.981707Z"Gaunt, Robert E."https://zbmath.org/authors/?q=ai:gaunt.robert-edwardSummary: The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener-Itô integrals.
{{\copyright} 2022 The Author. \textit{Statistica Neerlandica} published by John Wiley \& Sons Ltd on behalf of Netherlands Society for Statistics and Operations Research.}On recursions for moments of a compound random variable: an approach using an auxiliary counting random variablehttps://zbmath.org/1528.600202024-03-13T18:33:02.981707Z"Kim, Yoora"https://zbmath.org/authors/?q=ai:kim.yooraSummary: We present an identity on moments of a compound random variable by using an auxiliary counting random variable. Based on this identity, we develop a new recurrence formula for obtaining the raw and central moments of any order for a given compound random variable.A new flexible generalized family for constructing many families of distributionshttps://zbmath.org/1528.620112024-03-13T18:33:02.981707Z"Tahir, M. H."https://zbmath.org/authors/?q=ai:tahir.muhammad-hussain"Hussain, M. Adnan"https://zbmath.org/authors/?q=ai:hussain.muhammad-adnan"Cordeiro, Gauss M."https://zbmath.org/authors/?q=ai:cordeiro.gauss-moutinhoSummary: We propose a \textit{new flexible generalized family} (NFGF) for constructing many families of distributions. The importance of the NFGF is that any baseline distribution can be chosen and it does not involve any additional parameters. Some useful statistical properties of the NFGF are determined such as a linear representation for the family density, analytical shapes of the density and hazard rate, random variable generation, moments and generating function. Further, the structural properties of a special model named the \textit{new flexible Kumaraswamy} (NFKw) distribution, are investigated, and the model parameters are estimated by maximum-likelihood method. A simulation study is carried out to assess the performance of the estimates. The usefulness of the NFKw model is proved empirically by means of three real-life data sets. In fact, the two-parameter NFKw model performs better than three-parameter transmuted-Kumaraswamy, three-parameter exponentiated-Kumaraswamy and the well-known two-parameter Kumaraswamy models.Small-sample testing inference in symmetric and log-symmetric linear regression modelshttps://zbmath.org/1528.620132024-03-13T18:33:02.981707Z"Medeiros, Francisco M. C."https://zbmath.org/authors/?q=ai:medeiros.francisco-m-c"Ferrari, Silvia L. P."https://zbmath.org/authors/?q=ai:ferrari.silvia-l-de-paulaSummary: This paper deals with the issue of testing hypotheses in symmetric and log-symmetric linear regression models in small and moderate-sized samples. We focus on four tests, namely, the Wald, likelihood ratio, score, and gradient tests. These tests rely on asymptotic results and are unreliable when the sample size is not large enough to guarantee a good agreement between the exact distribution of the test statistic and the corresponding chi-squared asymptotic distribution. Bartlett and Bartlett-type corrections typically attenuate the size distortion of the tests. These corrections are available in the literature for the likelihood ratio and score tests in symmetric linear regression models. Here, we derive a Bartlett-type correction for the gradient test. We show that the corrections are also valid for the log-symmetric linear regression models. We numerically compare the various tests and bootstrapped tests, through simulations. Our results suggest that the corrected and bootstrapped tests exhibit type I probability error closer to the chosen nominal level with virtually no power loss. The analytically corrected tests as well as the bootstrapped tests, including the Bartlett-corrected gradient test derived in this paper, perform with the advantage of not requiring computationally intensive calculations. We present a real data application to illustrate the usefulness of the modified tests.
{{\copyright} 2017 The Authors. Statistica Neerlandica {\copyright} 2017 VVS.}Bartlett correction to the likelihood ratio test for MCAR with two-step monotone samplehttps://zbmath.org/1528.620292024-03-13T18:33:02.981707Z"Shutoh, Nobumichi"https://zbmath.org/authors/?q=ai:shutoh.nobumichi"Nishiyama, Takahiro"https://zbmath.org/authors/?q=ai:nishiyama.takahiro"Hyodo, Masashi"https://zbmath.org/authors/?q=ai:hyodo.masashiSummary: Assuming that two-step monotone missing data is drawn from a multivariate normal population, this paper derives the Bartlett-type correction to the likelihood ratio test for missing completely at random (MCAR), which plays an important role in the statistical analysis of incomplete datasets. The advantages of our approach are confirmed in Monte Carlo simulations. Our correction drastically improved the accuracy of the type I error in \textit{R. J. A. Little}'s [J. Am. Stat. Assoc. 83, No. 404, 1198--1202 (1988; \url{doi:10.1080/01621459.1988.10478722})] test for MCAR and performed well even on moderate sample sizes.
{{\copyright} 2017 The Authors. Statistica Neerlandica {\copyright} 2017 VVS.}