• Communications for Statistical Applications and Methods
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• v.31 no.4
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• pp.459-466
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• 2024

Multivariate regular variation is a popular framework of multivariate extreme value analysis. However, a suitable parametric model needs to be introduced for efficient estimation of its spectral measure. In such a view, elliptical distributions have been employed for deriving such models. On the other hand, the second order behavior of multivariate regular variation has to be specified for investigating the property of the estimator. This paper derives such a behavior by imposing a widely adopted second order regular variation condition on the representation of elliptical distributions. As result, the second order variation for the convergence to spectral measure is characterized by a signed measure with a regular varying index. Moreover, it leads to the asymptotic bias of the estimator. For demonstration, multivariate t-distribution is considered.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1463-1481
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• 2018

Let $X=\{X_t\}_{t{\geq}0}$ be a $L{\acute{e}}vy$ process in ${\mathbb{R}}^d$ and ${\Omega}$ be an open subset of ${\mathbb{R}}^d$ with finite Lebesgue measure. The quantity $H_{\Omega}(t)={\int_{\Omega}}{\mathbb{P}}^x(X_t{\in}{\Omega})$ dx is called the heat content. In this article we consider its generalized version $H^{\mu}_g(t)={\int_{\mathbb{R}^d}}{\mathbb{E}^xg(X_t){\mu}(dx)$, where g is a bounded function and ${\mu}$ a finite Borel measure. We study its asymptotic behaviour at zero for various classes of $L{\acute{e}}vy$ processes.

• Korean Journal of Agricultural Science
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• v.46 no.4
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• pp.941-953
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• 2019

Biodiesels are being explored as a clean energy alternative to regular diesel, which causes pollution. In this study, the optimum conditions for producing biodiesel (BD) by combining beef tallow, an animal waste resource with a high saturated fatty acid content, and corn oil, a vegetable oil with a high unsaturated fatty acid content, were investigated, and the fuel properties were analyzed. Furthermore, Multivariate Analysis of Variance (MANOVA) was used to verify the optimum conditions for producing biodiesel. The influences of control factors, such as the oil blend ratio and methanol to oil molar ratio, on the fatty acid methyl ester and biodiesel production yield were investigated. As a result, the optimum condition for producing blended biodiesel was verified to be tallow to corn oil blend ratio of 7 : 3 (TACO7) and a methanol to oil molar ratio of 14 : 1. Moreover, the interaction between the oil blend ratio and the methanol to oil molar ratio has the most crucial effects on the production of oil blended biodiesel. In conclusion, the analysis results of the fuel properties of TACO7 BD satisfied the BD quality standard, and thus, the viability of BD blended with waste tallow as fuel was verified.