The limit distribution for maxima of ‘weighted' rth-nearest-neighbour distances

1982 ◽  
Vol 19 (02) ◽  
pp. 344-354
Author(s):  
Norbert Henze
Keyword(s):  
Limit Distribution ◽  
Goodness Of Fit ◽  
Continuity Condition ◽  
Weak Continuity ◽  
Nearest Neighbour ◽  
Multivariate Test ◽  
Maximum Value ◽  
Special Case ◽  
Distance To The Boundary ◽  
Independent Identically Distributed

Let X 1, X 2, · ··, Xn be independent identically distributed random points with common density f(x), taking values in a bounded region (p ≧ 1). We obtain the limit distribution, as n → ∞, for the maximum value of the suitably ‘weighted' (according to f(x)) rth-nearest-neighbour distances of Χ 1, · ··, Χ n (r ≧ 1 fixed) provided that f(x) is bounded from below by a positive constant and a weak continuity condition holds. This is achieved by refining an argument used by the author (Henze (1981)) to derive the limit distribution in the special case r = 1. Edge-effects are eliminated by defining, for each Xi , the distance to the boundary of G to be the ‘rth-nearest-neighbour distance' if it is smaller than the distance to the rth nearest neighbour among the remaining points. Applications to a multivariate test of goodness of fit are given.

The limit distribution for maxima of ‘weighted' rth-nearest-neighbour distances

1982 ◽  
Vol 19 (2) ◽  
pp. 344-354 ◽  
Author(s):  
Norbert Henze
Keyword(s):  
Limit Distribution ◽  
Goodness Of Fit ◽  
Continuity Condition ◽  
Weak Continuity ◽  
Nearest Neighbour ◽  
Multivariate Test ◽  
Maximum Value ◽  
Special Case ◽  
Distance To The Boundary ◽  
Independent Identically Distributed

Let X1, X2, · ··, Xn be independent identically distributed random points with common density f(x), taking values in a bounded region (p ≧ 1). We obtain the limit distribution, as n → ∞, for the maximum value of the suitably ‘weighted' (according to f(x)) rth-nearest-neighbour distances of Χ1, · ··, Χ n (r ≧ 1 fixed) provided that f(x) is bounded from below by a positive constant and a weak continuity condition holds. This is achieved by refining an argument used by the author (Henze (1981)) to derive the limit distribution in the special case r = 1. Edge-effects are eliminated by defining, for each Xi, the distance to the boundary of G to be the ‘rth-nearest-neighbour distance' if it is smaller than the distance to the rth nearest neighbour among the remaining points. Applications to a multivariate test of goodness of fit are given.

scholarly journals Testing in Nonparametric Accelerated Life Time Models

2016 ◽  
Vol 37 (1) ◽  
Author(s):  
Hannelore Liero
Keyword(s):  
Limit Distribution ◽  
Goodness Of Fit ◽  
Bootstrap Method ◽  
Life Time ◽  
Test Statistic ◽  
Goodness Of Fit Test ◽  
Time Model ◽  
Power Of The Test ◽  
Accelerated Life ◽  
Type Test

A goodness-of-fit test for testing the acceleration function in a nonparametric life time model is proposed. For this aim the limit distribution of an L2-type test statistic is derived. Furthermore, a bootstrap method is considered and the power of the test is studied.

The limit distribution of the largest nearest-neighbour link in the unit d-cube

1989 ◽  
Vol 26 (01) ◽  
pp. 67-80 ◽  
Author(s):  
Holger Dette ◽  
Norbert Henze
Keyword(s):  
Limit Distribution ◽  
Nearest Neighbour

Let· p denote the p-norm on . For n points drawn independently and uniformly from the unit d-cube, we obtain the limit distribution of the largest nearest-neighbour link (with respect to | · | p ) in the cases ; (2) d = 3, p = 1, 2, ∞and (3) 4, p =∞.

Convergence of Markov chains in the relative supremum norm

2000 ◽  
Vol 37 (4) ◽  
pp. 1074-1083 ◽  
Author(s):  
Lars Holden
Keyword(s):  
Markov Chain ◽  
Detailed Balance ◽  
Continuity Condition ◽  
Supremum Norm ◽  
Weak Continuity ◽  
Norm Convergence ◽  
Detailed Balance Condition ◽  
Balance Condition ◽  
Qualitative Understanding ◽  
Doeblin Condition

It is proved that the strong Doeblin condition (i.e., ps(x,y) ≥ asπ(y) for all x,y in the state space) implies convergence in the relative supremum norm for a general Markov chain. The convergence is geometric with rate (1 - as)1/s. If the detailed balance condition and a weak continuity condition are satisfied, then the strong Doeblin condition is equivalent to convergence in the relative supremum norm. Convergence in other norms under weaker assumptions is proved. The results give qualitative understanding of the convergence.

Dielectric conductivity of a bounded plasma and its rate of convergence towards its infinite-geometry value

2003 ◽  
Vol 69 (5) ◽  
pp. 449-463
Author(s):  
PASCAL OMNES
Keyword(s):  
Distribution Function ◽  
Electric Field ◽  
Rate Of Convergence ◽  
Dielectric Function ◽  
Bounded Geometry ◽  
First Order ◽  
First Order Correction ◽  
Time Periodic ◽  
Special Case ◽  
Distance To The Boundary

This paper deals with the linear response of a plasma in a one-dimensional bounded geometry under the action of a time-periodic electric field. The nonlinear Vlasov equation is solved by following the characteristic curves until they reach the boundary of the domain, where the distribution function of the incoming particles is supposed to be known and independent of time. Then, a first-order Taylor expansion in the velocity variable is performed, thanks to an approximation of the exact characteristics by the unperturbed ones. The resulting first-order correction to the distribution function is finally integrated over velocities to yield the dielectric function. The special case of a plane wave for the electric field is examined and the results are compared with the more usual unbounded case: the integral does not present any singularity in the vicinity of resonant particles and the dielectric function depends on the distance to the boundary and tends to the usual infinite-geometry value when this distance tends to infinity, with a rate of convergence proportional to its inverse square root. Numerical examples are provided for illustration.

scholarly journals Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution

2014 ◽  
Vol 51 (2) ◽  
pp. 466-482 ◽  
Author(s):  
Marcus C. Christiansen ◽  
Nicola Loperfido
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Distribution Functions ◽  
Skew Normal Distribution ◽  
Random Vectors ◽  
Multivariate Distributions ◽  
Uniform Distance ◽  
Special Case ◽  
Independent Identically Distributed ◽  
Skew Normal

We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n-2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.

scholarly journals Homogeneous Premium Calculation Principles

1984 ◽  
Vol 14 (2) ◽  
pp. 123-133 ◽  
Author(s):  
Axel Reich
Keyword(s):  
Utility Function ◽  
General Theorem ◽  
Random Variables ◽  
Continuity Condition ◽  
Weak Continuity ◽  
Local Homogeneity ◽  
Diophantine Approximations ◽  
Premium Calculation

AbstractA premium calculation principle π is called positively homogeneous if π(cX) = cπ(X) for all c > 0 and all random variables X. For all known principles it is shown that this condition is fulfilled if it is satisfied for two specific values of c only, say c = 2 and c = 3, and for only all two point random variables X. In the case of the Esscher principle one value of c suffices. In short this means that local homogeneity implies global homogeneity. From this it follows that in the case of the zero utility principle or Swiss premium calculation principle, the underlying utility function is of a very specific type.A very general theorem on premium calculation principles which satisfy a weak continuity condition, is added. Among others the proof uses Kroneckers Theorem on Diophantine Approximations.

A limit theorem for the maximum term in a particular EARMA(1, 1) sequence

1980 ◽  
Vol 17 (3) ◽  
pp. 869-873 ◽  
Author(s):  
Michael R. Chernick
Keyword(s):  
Limit Theorem ◽  
Limit Distribution ◽  
Random Variables ◽  
Extreme Value ◽  
Maximum Term ◽  
Exponential Random Variables ◽  
Independent Identically Distributed

The EARMA(1, 1) process was described by Jacobs and Lewis (1977). Chernick (1978) showed that the limit for the maximum term is the same as for a sequence of independent, identically distributed exponential random variables when the parameter ρ is less than 1. When ρ = 1, a different limit theorem is obtained. The resulting limit distribution is not an extreme-value type. It is, however, of the general form given by Galambos (1978). The sequence is exchangeable.

Research on Generalized Compatibility Equations of Quasi-Conforming Element Methods

2011 ◽  
Vol 94-96 ◽  
pp. 1651-1654
Author(s):  
Ke Wei Ding
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Continuity Condition ◽  
Weak Continuity ◽  
The Finite Element Method ◽  
Weak Formulation ◽  
Equilibrium Conditions ◽  
Stress Equilibrium ◽  
Generalized Compatibility ◽  
Element Method

Brief development process of the finite element method, foundation of quasi-conforming element has been analyzed from weak formulation generalized compatibility equations and its weak continuity condition in this paper. The quasi-conforming element methods are the exact solution of generalized compatibility equations and satisfy the weak continuity requirement naturally. The quasi-conforming element methods do not satisfy stress equilibrium conditions and concision calculating process of matrix’s athwart. The discrete precision can be predicted in prior. It also extends space of original finite element method and is landmark in computational mechanics.

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