Asymptotic upper bound for multiplier design

1996 ◽  
Vol 32 (5) ◽  
pp. 420 ◽  
Author(s):  
R.G.E. Pinch
Keyword(s):  
Upper Bound ◽  
Asymptotic Upper Bound

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed

Convergence to stationarity in the Moran model

2000 ◽  
Vol 37 (03) ◽  
pp. 705-717 ◽  
Author(s):  
Peter Donnelly ◽  
Eliane R. Rodrigues
Keyword(s):  
Markov Chain ◽  
Total Variation ◽  
Upper Bound ◽  
Separation Distance ◽  
Fixed Size ◽  
Moran Model ◽  
Variation Distance ◽  
Lines Of Descent ◽  
Number Of Individuals ◽  
Asymptotic Upper Bound

Consider a population of fixed size consisting of N haploid individuals. Assume that this population evolves according to the two-allele neutral Moran model in mathematical genetics. Denote the two alleles by A 1 and A 2. Allow mutation from one type to another and let 0 < γ < 1 be the sum of mutation probabilities. All the information about the population is recorded by the Markov chain X = (X(t)) t≥0 which counts the number of individuals of type A 1. In this paper we study the time taken for the population to ‘reach’ stationarity (in the sense of separation and total variation distances) when initially all individuals are of one type. We show that after t ∗ = Nγ-1logN + cN the separation distance between the law of X(t ∗) and its stationary distribution converges to 1 - exp(-γe-γc ) as N → ∞. For the total variation distance an asymptotic upper bound is obtained. The results depend on a particular duality, and couplings, between X and a genealogical process known as the lines of descent process.

Convergence to stationarity in the Moran model

2000 ◽  
Vol 37 (3) ◽  
pp. 705-717 ◽  
Author(s):  
Peter Donnelly ◽  
Eliane R. Rodrigues
Keyword(s):  
Markov Chain ◽  
Total Variation ◽  
Upper Bound ◽  
Separation Distance ◽  
Fixed Size ◽  
Moran Model ◽  
Variation Distance ◽  
Lines Of Descent ◽  
Number Of Individuals ◽  
Asymptotic Upper Bound

Consider a population of fixed size consisting of N haploid individuals. Assume that this population evolves according to the two-allele neutral Moran model in mathematical genetics. Denote the two alleles by A1 and A2. Allow mutation from one type to another and let 0 < γ < 1 be the sum of mutation probabilities. All the information about the population is recorded by the Markov chain X = (X(t))t≥0 which counts the number of individuals of type A1. In this paper we study the time taken for the population to ‘reach’ stationarity (in the sense of separation and total variation distances) when initially all individuals are of one type. We show that after t∗ = Nγ-1logN + cN the separation distance between the law of X(t∗) and its stationary distribution converges to 1 - exp(-γe-γc) as N → ∞. For the total variation distance an asymptotic upper bound is obtained. The results depend on a particular duality, and couplings, between X and a genealogical process known as the lines of descent process.

scholarly journals Cops and robbers on graphs and hypergraphs

2021 ◽  
Author(s):  
William David Baird
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Affine Planes ◽  
Connected Graphs ◽  
Pursuit Game ◽  
Cops And Robbers ◽  
Asymptotic Upper Bound

Cops and Robbers is a vertex-pursuit game played on a graph where a set of cops attempts to capture a robber. Meyniel's Conjecture gives as asymptotic upper bound on the cop number, the number of cops required to win on a connected graph. The incidence graphs of affine planes meet this bound from below, they are called Meyniel extremal. The new parameters mқ and Mқ describe the minimum orders of k-cop-win graphs. The relation of these parameters to Meyniel's Conjecture is discussed. Further, the cop number for all connected graphs of order 10 or less is given. Finally, it is shown that cop win hypergraphs, a generalization of graphs, cannot be characterized in terms of retractions in the same manner as cop win graphs. This thesis presents some small steps towards a solution to Meyniel's Conjecture.

Asymptotic upper bound on the 3-to-3 amplitude

1973 ◽  
Vol 15 (2) ◽  
pp. 249-273 ◽  
Author(s):  
A. Patrascioiu
Keyword(s):  
Upper Bound ◽  
Asymptotic Upper Bound

scholarly journals On the Area Discrepancy of Triangulations of Squares and Trapezoids

10.37236/624 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Bernd Schulze
Keyword(s):  
Upper Bound ◽  
Continued Fractions ◽  
Transcendental Number ◽  
Fast Growing ◽  
Unit Square ◽  
Minimal Difference ◽  
The Difference ◽  
Asymptotic Upper Bound

In 1970 P. Monsky showed that a square cannot be triangulated into an odd number of triangles of equal areas; further, in 1990 E. A. Kasimatis and S. K. Stein proved that the trapezoid $T(\alpha)$ whose vertices have the coordinates $(0,0)$, $(0,1)$, $(1,0)$, and $(\alpha,1)$ cannot be triangulated into any number of triangles of equal areas if $\alpha>0$ is transcendental. In this paper we first establish a new asymptotic upper bound for the minimal difference between the smallest and the largest area in triangulations of a square into an odd number of triangles. More precisely, using some techniques from the theory of continued fractions, we construct a sequence of triangulations $T_{n_i}$ of the unit square into $n_i$ triangles, $n_i$ odd, so that the difference between the smallest and the largest area in $T_{n_i}$ is $O\big(\frac{1}{n_i^3}\big)$. We then prove that for an arbitrarily fast-growing function $f:\mathbb{N}\to \mathbb{N}$, there exists a transcendental number $\alpha>0$ and a sequence of triangulations $T_{n_i}$ of the trapezoid $T(\alpha)$ into $n_i$ triangles, so that the difference between the smallest and the largest area in $T_{n_i}$ is $O\big(\frac{1}{f(n_i)}\big)$.

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