Distributions And Applications Of Two Nested Order Statistics Of The Shape (n~3)\(n-k)|{(2,j)}

Order statistics are statistics frequently used in statistical inference,the distribution of order statistics is an important subject in probability theory.A standard Young tableau(SYT)is a Ferrers diagram which is strictly increasing across each row and each column,and the enumeration of SYTs is an important problem in combinatorial mathematics.By using the enumeration of SYT-type charts,this paper mainly study distributions and applications of two nested order statistics from uniform distribution on interval(0,1)of the shape(n3)\(n-k)| {(2,j)}.Nested order statistics of the shape(n3)\(n-k)\{(2,j)} consists of three groups of independent order statistics(ξ1:n(1),ξ2:n(1),…,ξn:n(1)),(ξ1:k(2),ξ2:k(2),…,ξk:k(2)),(ξ1:n(3),ξ2:n(3),…,ξn:n(3)).In this paper,the probability is called as the distribution of nested order statistic from U(0,1)of the shape(n3)\(n-k)| {(2,j)},which is denoted as P(n3)\(n-k)|{(2,j)}.Firstly,multiple integrals and combinatorial summations are used to evaluate numbers of SYT-type charts of the truncated shapes(n3)\(n-3)| {(2,2)} and(n3)\(n-4)| {(2,3)},which are denoted as H(n3)\(n-3)|{(2,2)},H(n3)\(n-4)|{(2,3)} respectively.And then according to the relationship between nested order statistics and SYT-type charts,distributions of nested order statistics from U(0,1)of the shapes(n3)\(n-3)| {(2,2)} and(n3)\(n-4)| {(2,3)}can be respectively illustrated as:P(n3)\(n-3)|{(2,2)}=H(n3)\(n-3)|{(2,2)}/(n,3,n 2n+3),P(n3)\(n-4)|{(2,3)}=H(n3)\(n-4)|{(2,3)}/(n,4,n 2n+4).At last,these nested order statistics are applied to the problem of voting which has three candidates.