Different Algorithms for Obtaining Upper Bounds to Multivariate Normal Areas Outside of Origin Centered Rectangles Using Joint Marginal Probabilities

1988 ◽  
Author(s):  
Donald R. Hoover
Keyword(s):  
Upper Bounds ◽  
Multivariate Normal

Bonferroni bounds revisited

1989 ◽  
Vol 26 (2) ◽  
pp. 233-241 ◽  
Author(s):  
Stratis Kounias ◽  
Kiki Sotirakoglou
Keyword(s):  
Normal Distribution ◽  
Multivariate Normal Distribution ◽  
Upper Bounds ◽  
Multivariate Normal ◽  
Lower And Upper Bounds

Lower and upper bounds of degree m for the probability of the union of n not necessarily exchangeable events are established. These bounds may be constructed to improve the Bonferroni and the Sobel–Uppuluri bounds.An application to equi-correlated multivariate normal distribution is given.

An upper bound for the probability of a union

1976 ◽  
Vol 13 (03) ◽  
pp. 597-603 ◽  
Author(s):  
David Hunter
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Spanning Tree ◽  
Upper Bounds ◽  
Minimal Spanning Tree ◽  
Multivariate Normal ◽  
Tail Probabilities ◽  
Tree Algorithm ◽  
New Family ◽  
The Family

The problem of bounding P(∪ Ai ) given P(A i) and P(A i A j) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.

An upper bound for the probability of a union

1976 ◽  
Vol 13 (3) ◽  
pp. 597-603 ◽  
Author(s):  
David Hunter
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Spanning Tree ◽  
Upper Bounds ◽  
Minimal Spanning Tree ◽  
Multivariate Normal ◽  
Tail Probabilities ◽  
Tree Algorithm ◽  
New Family ◽  
The Family

The problem of bounding P(∪ Ai) given P(Ai) and P(AiAj) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.

Bonferroni bounds revisited

1989 ◽  
Vol 26 (02) ◽  
pp. 233-241 ◽  
Author(s):  
Stratis Kounias ◽  
Kiki Sotirakoglou
Keyword(s):  
Normal Distribution ◽  
Multivariate Normal Distribution ◽  
Upper Bounds ◽  
Multivariate Normal ◽  
Lower And Upper Bounds

Lower and upper bounds of degree m for the probability of the union of n not necessarily exchangeable events are established. These bounds may be constructed to improve the Bonferroni and the Sobel–Uppuluri bounds. An application to equi-correlated multivariate normal distribution is given.

Upper bounds for the probability of a union by multitrees

2001 ◽  
Vol 33 (2) ◽  
pp. 437-452 ◽  
Author(s):  
József Bukszár
Keyword(s):  
Graph Theory ◽  
Normal Distribution ◽  
Multivariate Normal Distribution ◽  
Upper Bounds ◽  
Distribution Functions ◽  
Multivariate Normal ◽  
Common Generalization

The problem of finding bounds for P(A1 ∪ ⋯ ∪ An) based on P(Ak1 ∩ ⋯ ∩ Aki) (1 ≤ k1 < ⋯ < ki ≤ n, i = 1,…,d) goes back to Boole (1854), (1868) and Bonferroni (1937). In this paper upper bounds are presented using methods in graph theory. The main theorem is a common generalization of the earlier results of Hunter, Worsley and recent results of Prékopa and the author. Algorithms are given to compute bounds. Examples for bounding values of multivariate normal distribution functions are presented.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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