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