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Open AccessArticle

Reciprocal Data Transformations and Their Back-Transforms

by Daniel A. Griffith

Stats 2022, 5(3), 714-737; https://doi.org/10.3390/stats5030042 - 30 Jul 2022

Cited by 3 | Viewed by 4529

Variable transformations have a long and celebrated history in statistics, one that was rather academically glamorous at least until generalized linear models theory eclipsed their nurturing normal curve theory role. Still, today it continues to be a covered topic in introductory mathematical statistics courses, offering worthwhile pedagogic insights to students about certain aspects of traditional and contemporary statistical theory and methodology. Since its inception in the 1930s, it has been plagued by a paucity of adequate back-transformation formulae for inverse/reciprocal functions. A literature search exposes that, to date, the inequality E(1/X) ≤ 1/(E(X), which often has a sizeable gap captured by the inequality part of its relationship, is the solitary contender for solving this problem. After documenting that inverse data transformations are anything but a rare occurrence, this paper proposes an innovative, elegant back-transformation solution based upon the Kummer confluent hypergeometric function of the first kind. This paper also derives formal back-transformation formulae for the Manly transformation, something apparently never done before. Much related future research remains to be undertaken; this paper furnishes numerous clues about what some of these endeavors need to be. Full article

(This article belongs to the Section Statistical Methods)

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19 pages, 1595 KB

Open AccessArticle

Transformational Approach to Analytical Value-at-Risk for near Normal Distributions

by Puneet Prakash, Vikas Sangwan and Kewal Singh

J. Risk Financial Manag. 2021, 14(2), 51; https://doi.org/10.3390/jrfm14020051 - 26 Jan 2021

Cited by 3 | Viewed by 3384

Abstract

In this paper, we extend the parametric approach of VaR estimation that is based upon the application of two transforms, one for handling skewness and other for kurtosis. These transformations restore normality to data when applied in succession. The transforms are well defined and offer an alternative to VaR models based on the variance–covariance approach. We demonstrate the application of the technique using three pairs of uncorrelated but negatively skewed and fat-tailed stock return distributions, one pair each from recent periods in US and international market, and one from the stressed period of US economic history. Furthermore, we extend the analysis to economic domain by calculating expected shortfalls and risk capital under different estimation methods. For the sake of completion, we compare the estimation results of normal and transformation methods to non-parametric historical simulation. Full article

(This article belongs to the Section Risk)

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20 pages, 1114 KB

Open AccessArticle

An Instrument for In Situ Measuring the Volume Scattering Function of Water: Design, Calibration and Primary Experiments

by Cai Li, Wenxi Cao, Jing Yu, Tiancun Ke, Guixin Lu, Yuezhong Yang and Chaoying Guo

Sensors 2012, 12(4), 4514-4533; https://doi.org/10.3390/s120404514 - 10 Apr 2012

Cited by 12 | Viewed by 8532

Abstract

The optical volume scattering function (VSF) of seawater is a fundamental property used in the calculation of radiative transfer for applications in the study of the upper-ocean heat balance, the photosynthetic productivity of the ocean, and the chemical transformation of photoreactive compounds. A new instrument to simultaneously measure the VSF in seven directions between 20° to 160°, the attenuation coefficient, and the depth of water is presented. The instrument is self-contained and can be automatically controlled by the depth under water. The self-contained data can be easily downloaded by an ultra-short-wave communication system. A calibration test was performed in the laboratory based on precise estimation of the scattering volume and optical radiometric calibration of the detectors. The measurement error of the VSF measurement instrument has been estimated in the laboratory based on the Mie theory, and the average error is less than 12%. The instrument was used to measure and analyze the variation characteristics of the VSF with angle, depth and water quality in Daya Bay for the first time. From these in situ data, we have found that the phase functions proposed by Fournier-Forand, measured by Petzold in San Diego Harbor and Sokolov in Black Sea do not fit with our measurements in Daya. These discrepancies could manly due to high proportion of suspended calcium carbonate mineral-like particles with high refractive index in Daya Bay. Full article

(This article belongs to the Section Physical Sensors)

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