Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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A New Integral Representation of the Coverage Probability of a Random Convex Hull

  • Son, Won (Department of Statistics, Seoul National University) ;
  • Ng, Chi Tim (Department of Statistics, Chonnam National University) ;
  • Lim, Johan (Department of Statistics, Seoul National University)
  • Received : 2014.11.25
  • Accepted : 2014.12.30
  • Published : 2015.01.31
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Abstract

In this paper, the probability that a given point is covered by a random convex hull generated by independent and identically-distributed random points in a plane is studied. It is shown that such probability can be expressed in terms of an integral that can be approximated numerically by function-evaluations over the grid-points in a 2-dimensional space. The new integral representation allows such probability be computed efficiently. The computational burdens under the proposed integral representation and those in the existing literature are compared. The proposed method is illustrated through numerical examples where the random points are drawn from (i) uniform distribution over a square and (ii) bivariate normal distribution over the two-dimensional Euclidean space. The applications of the proposed method in statistics are are discussed.

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References

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