scholarly journals Finite-sample genome-wide regression p-values (GWRPV) with a non-normally distributed phenotype

2017 ◽  
Author(s):  
Gregory Connor ◽  
Michael O’Neill
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Mixture Distribution ◽  
Small Sample ◽  
P Value ◽  
Finite Sample ◽  
P Values ◽  
Genome Wide ◽  
Normally Distributed

AbstractThis paper derives the exact finite-sample p-value for univariate regression of a quantitative phenotype on individual genome markers, relying on a mixture distribution for the dependent variable. The p-value estimator conventionally used in existing genome-wide association study (GWAS) regressions assumes a normally-distributed dependent variable, or relies on a central limit theorem based approximation. The central limit theorem approximation is unreliable for GWAS regression p-values, and measured phenotypes often have markedly non-normal distributions. A normal mixture distribution better fits observed phenotypic variables, and we provide exact small-sample p-values for univariate GWAS regressions under this flexible distributional assumption. We illustrate the adjustment using a years-of-education phenotypic variable.

Multiple comparisons and sums of dissociated random variables

1985 ◽  
Vol 17 (1) ◽  
pp. 147-162 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Multiple Comparisons ◽  
The Central Limit Theorem ◽  
Normally Distributed

Sufficient conditions for a sum of dissociated random variables to be approximately normally distributed are derived. These results generalize the central limit theorem for U-statistics and provide conditions which can be verified in a number of applications. The method of proof is that due to Stein (1970).

scholarly journals Statistical Approximating Distributions Under Differential Privacy

2018 ◽  
Vol 8 (1) ◽  
Author(s):  
Yue Wang ◽  
Daniel Kifer ◽  
Jaewoo Lee ◽  
Vishesh Karwa
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Confidence Intervals ◽  
Central Limit ◽  
Differential Privacy ◽  
Prior Work ◽  
Finite Sample ◽  
The Central Limit Theorem ◽  
Approximating Distributions

Statistics computed from data are viewed as random variables. When they are used for tasks like hypothesis testing and confidence intervals, their true finite sample distributions are often replaced by approximating distributions that are easier to work with (for example, the Gaussian, which results from using approximations justified by the Central Limit Theorem). When data are perturbed by differential privacy, the approximating distributions also need to be modified. Prior work provided various competing methods for creating such approximating distributions with little formal justification beyond the fact that they worked well empirically. In this paper, we study the question of how to generate statistical approximating distributions for differentially private statistics, provide finite sample guarantees for the quality of the approximations.

Multiple comparisons and sums of dissociated random variables

1985 ◽  
Vol 17 (01) ◽  
pp. 147-162 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Multiple Comparisons ◽  
The Central Limit Theorem ◽  
Normally Distributed

Sufficient conditions for a sum of dissociated random variables to be approximately normally distributed are derived. These results generalize the central limit theorem for U-statistics and provide conditions which can be verified in a number of applications. The method of proof is that due to Stein (1970).

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

scholarly journals Combable functions, quasimorphisms, and the central limit theorem

2009 ◽  
Vol 30 (5) ◽  
pp. 1343-1369 ◽  
Author(s):  
DANNY CALEGARI ◽  
KOJI FUJIWARA
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Word Length ◽  
Hyperbolic Groups ◽  
Finite State Automaton ◽  
List Type ◽  
Generating Sets ◽  
Finite State ◽  
Word Hyperbolic Groups

AbstractA function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include:(1)homomorphisms to ℤ;(2)word length with respect to a finite generating set;(3)most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms).We show that bicombable functions on word-hyperbolic groups satisfy acentral limit theorem: if$\overline {\phi }_n$is the value of ϕ on a random element of word lengthn(in a certain sense), there areEandσfor which there is convergence in the sense of distribution$n^{-1/2}(\overline {\phi }_n - nE) \to N(0,\sigma )$, whereN(0,σ) denotes the normal distribution with standard deviationσ. As a corollary, we show that ifS1andS2are any two finite generating sets forG, there is an algebraic numberλ1,2depending onS1andS2such that almost every word of lengthnin theS1metric has word lengthn⋅λ1,2in theS2metric, with error of size$O(\sqrt {n})$.

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