Coverage of a Record Value and Related Distribution Problems

Abstract

For any given sequence {t _r } of statistics, $t_r\equiv t_r\left(x_1,x_2,\dots \right),$ i.e. where x _i 's are observations coming from a continuous population with density function f(x), an r-coverage C _r ≡ C(t _r ) has been defined. In particular defining the sequence {Lr} of lower record values, it has been shown that $Z_{r+1}=-\log \left[1-C\left(L_r\right)\right]/-\log \left[1-c\left(L_{r+1}\right)\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}r=1,2,\dots$ are independent of each other. Moreover, $V_{s-r}={\displaystyle \prod _{i=r}^{s-1}{Z}_{i+1}}$ is a $B_1\left(r+1,S-r\right).$ It is interesting to note that $-\log \left\{\left[1-C\left(L_r\right)\right]/\left[1-C\left(L_{r+1}\right)\right]\right\},r=1,2,\dots$ are independent negative exponential variates each with parameter 1.