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A Geometric Approach to Confidence Sets for Ratios: Fieller's Theorem, Generalizations, and Bootstrap

TL;DR: In this article, the authors presented a geometric method to determine confidence sets for the ratio E(Y)/E(X) of the means of random variables X and Y. This method is valid in a large variety of circumstances.
Abstract: We present a geometric method to determine confidence sets for the ratio E(Y)/E(X) of the means of random variables X and Y. This method reduces the problem of constructing confidence sets for the ratio of two random variables to the problem of constructing confidence sets for the means of one-dimensional random variables. It is valid in a large variety of circumstances. In the case of normally distributed random variables, the so constructed confidence sets coincide with the standard Fieller confidence sets. Generalizations of our construction lead to definitions of exact and conservative confidence sets for very general classes of distributions, provided the joint expectation of (X,Y) exists and the linear combinations of the form aX + bY are well-behaved. Finally, our geometric method allows to derive a very simple bootstrap approach for constructing conservative confidence sets for ratios which perform favorably in certain situations, in particular in the asymmetric heavy-tailed regime.
Citations
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Journal ArticleDOI
06 Aug 2008-PLOS ONE
TL;DR: An algorithm is presented which is mathematically sound and computationally efficient to accurately analyze CNV in a DNA sample utilizing a nanofluidic device, known as the digital array, based on results from probability theory and statistics.
Abstract: Copy Number Variations (CNVs) of regions of the human genome have been associated with multiple diseases. We present an algorithm which is mathematically sound and computationally efficient to accurately analyze CNV in a DNA sample utilizing a nanofluidic device, known as the digital array. This numerical algorithm is utilized to compute copy number variation and the associated statistical confidence interval and is based on results from probability theory and statistics. We also provide formulas which can be used as close approximations.

333 citations


Cites background or methods from "A Geometric Approach to Confidence ..."

  • ...If WR = WL = zcsx and HT = HB = zcsy, it is symmetric case [20,21]....

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  • ...Theoretical Predictions Simulation Results Mean 311.5 311.48 Standard Deviation 13.59 13.58 Percent of times M = 400 lies in the computed 95% confidence interval 95% 94.44% doi:10.1371/journal.pone.0002876.t001 PLoS ONE | www.plosone.org 5 August 2008 | Volume 3 | Issue 8 | e2876 Fieller’s Theorem and its geometric interpretation Let the sampling distributions of the test gene and the reference gene be g1 l̂1 and g2 l̂2 , respectively....

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  • ...show in [20,21] how a confidence ellipse in the two-dimensional plane can be constructed....

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  • ...If these distributions were normal, then one can make use of Fieller’s Theorem [18,19]....

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  • ...For given estimations l̂1 and l̂2, assuming that distributions are normal, it follows from the analysis in [20,21] that the boundary of the confidence ellipse for a given confidence level zc would be defined by...

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Journal ArticleDOI
TL;DR: Comparison of measured data with calculated isotopic ratios as well as analysis using atmospheric transport modeling indicate that it is likely that the xenon measured was created in the underground nuclear test conducted by North Korea on February 12, 2013, and released 7-8 weeks later.

110 citations


Cites methods from "A Geometric Approach to Confidence ..."

  • ...The uncertainties in the activity ratios are calculated using Fiellers’s theorem (von Luxburg and Franz, 2009) which provides an exact solution to the problem of calculating the confidence intervals of the ratio of two random variables, if they are jointly normally distributed....

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Proceedings ArticleDOI
Alex Deng1, Ulf Knoblich1, Jiannan Lu1
19 Jul 2018
TL;DR: In this article, the authors provide a practical guide to applying the Delta method, one of the most important tools from the classic statistics literature, to solve big data problems via closed-form formulas using distributed algorithms at a fraction of the cost of simulation-based procedures.
Abstract: During the last decade, the information technology industry has adopted a data-driven culture, relying on online metrics to measure and monitor business performance. Under the setting of big data, the majority of such metrics approximately follow normal distributions, opening up potential opportunities to model them directly without extra model assumptions and solve big data problems via closed-form formulas using distributed algorithms at a fraction of the cost of simulation-based procedures like bootstrap. However, certain attributes of the metrics, such as their corresponding data generating processes and aggregation levels, pose numerous challenges for constructing trustworthy estimation and inference procedures. Motivated by four real-life examples in metric development and analytics for large-scale A/B testing, we provide a practical guide to applying the Delta method, one of the most important tools from the classic statistics literature, to address the aforementioned challenges. We emphasize the central role of the Delta method in metric analytics by highlighting both its classic and novel applications.

54 citations

Journal ArticleDOI
TL;DR: The mechanisms behind such synergistic effects of binary mixtures in bees are known to involve direct cytochrome P450 (CYP) inhibition, resulting in an increase in internal dose and toxicity of the binary mixture, which is known to have the lowest number of CYP copies and other detoxification enzymes in the insect kingdom.

50 citations

Journal ArticleDOI
TL;DR: The authors compared proxy-based reconstructions and climate model simulations of past millennium temperature variability to provide insights into climate sensitivity and feedback mechanism, and found that the latter is more robust than the former.
Abstract: Systematic comparisons of proxy-based reconstructions and climate model simulations of past millennium temperature variability offer insights into climate sensitivity and feedback mechanism...

34 citations

References
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Book ChapterDOI
01 Jan 2008

6,615 citations


"A Geometric Approach to Confidence ..." refers background or methods in this paper

  • ..., Efron and Tibshirani (1993)). We also tried several other standard methods such as the percentile or the bias corrected and accelerated (BCA) method (cf., Efron and Tibshirani (1993)), but did not observe qualitatively different behavior. To deal with heavy-tailed distributions, we applied methods based on subsampling self-normalizing sums, as introduced by Hall and LePage (1996), see also Romano and Wolf (1999)....

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  • ..., Efron (1979), Efron and Tibshirani (1993), Shao and Tu...

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  • ..., Efron and Tibshirani (1993)). We also tried several other standard methods such as the percentile or the bias corrected and accelerated (BCA) method (cf., Efron and Tibshirani (1993)), but did not observe qualitatively different behavior. To deal with heavy-tailed distributions, we applied methods based on subsampling self-normalizing sums, as introduced by Hall and LePage (1996), see also Romano and Wolf (1999). Here one has to choose a single parameter, the...

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  • ..., Efron and Tibshirani (1993)). We also tried several other standard methods such as the percentile or the bias corrected and accelerated (BCA) method (cf., Efron and Tibshirani (1993)), but did not observe qualitatively different behavior....

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  • ...A natural candidate to construct approximate confidence sets for ratios are bootstrap procedures (e.g., Efron, 1979; Efron and Tibshirani, 1993; Shao and Tu, 1995; Davison and Hinkley, 1997)....

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6,420 citations

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Book
01 Nov 1989
TL;DR: In this article, the authors define marginal distributions, moments and density marginal distributions moments density the relationship between (phi and f) conditional distributions properties of elliptically symmetric distributions mixtures of normal distributions robust statistics and regression model robust statistics regression model log-elliptical and additive logistic elliptical distributions multivariate log elliptical distribution additive logistics elliptical distribution complex elliptical symmetric distribution.
Abstract: Part 1 Preliminaries: construction of symmetric multivariate distributions notation of algebraic entities and characteristics of random quantities the "d" operator groups and invariance dirichlet distribution problems 1. Part 2 Spherically and elliptically symmetric distributions: introduction and definition marginal distributions, moments and density marginal distributions moments density the relationship between (phi) and f conditional distributions properties of elliptically symmetric distributions mixtures of normal distributions robust statistics and regression model robust statistics regression model log-elliptical and additive logistic elliptical distributions multivariate log-elliptical distribution additive logistic elliptical distributions complex elliptically symmetric distributions. Part 3 Some subclasses of elliptical distributions: multiuniform distribution the characteristic function moments marginal distribution conditional distributions uniform distribution in the unit sphere discussion symmetric Kotz type distributions definition distribution of R(2) moments multivariate normal distributions the c.f. of Kotz type distributions symmetric multivariate Pearson type VII distributions definition marginal densities conditional distributions moments conditional distributions moments some examples extended Tn family relationships between Ln and Tn families of distributions order statistics mixtures of exponential distributions independence, robustness and characterizations problems V. Part 6 Multivariate Liouville distributions: definitions and properties examples marginal distributions conditional distribution characterizations scale-invariant statistics survival functions inequalities and applications.

2,106 citations


"A Geometric Approach to Confidence ..." refers background in this paper

  • ...For a brief overview of spherical and elliptical distributions see Eaton (1981), for an extensive treatment see Fang, Kotz, and Ng (1990)....

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Journal ArticleDOI
TL;DR: In this article, the fiducial distributions of a simple equation (i.e., a ratio), and the roots of a quadratic equation with variable coefficients with respect to the region of the (aC, t2) plane lying above the curve are investigated.
Abstract: THE object of this paper is to propose for discussion the following topic: b1, b2, ... are unbiased estimates of P1, ,2, ... , distributed normally with variances and covariances jointly estimated, with f degrees of freedom and independently of bl, b2, . .. , as vll, v12, V22, . .. , and the functions F,(cc) do not involve the parameters P3, What can we say about the roots of the equation in F(r, M) = rlFl (a) + P2 F2 (a) + . . . = 0? Numerical examples are discussed in detail to illustrate the problems of determining the fiducial distributions of (i) the root of a simple equation (i.e., a ratio), (ii) the roots of a quadratic equation with variable coefficients. The solutions proposed are based on a consideration of the region of the (aC, t2) plane lying above the curve

949 citations


"A Geometric Approach to Confidence ..." refers background or result in this paper

  • ...To this end let us first state Fieller’s result according to Subsection 4, p. 176-177 of (Fieller, 1954)....

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  • ...Now we want to compare them to the classic confidence sets constructed by Fieller (1932, 1940, 1944, 1954)....

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  • ...In the case where X and Y are jointly normally distributed, an exact solution to this problem has been derived by Fieller (1932, 1940, 1944, 1954); for more detailed discussions see Kendall and Stuart (1961), Finney (1978), Miller (1986), and Buonaccorsi (2001)....

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