# RECURRENCE RELATIONS FOR QUOTIENT MOMENTS OF THE EXPONENTIAL DISTRIBUTION BY RECORD VALUES

• LEE, MIN-YOUNG (Department of Applied Mathematics, Dankook University) ;
• CHANG, SE-KYUNG (Department of Applied Mathematics, Dankook University)
• 투고 : 2004.07.21
• 발행 : 2004.12.25
• 80 4

#### 초록

In this paper we establish some recurrence relations satisfied by quotient moments of upper record values from the exponential distribution. Let $\{X_n,\;n{\geq}1\}$ be a sequence of independent and identically distributed random variables with a common continuous distribution function F(x) and probability density function(pdf) f(x). Let $Y_n=max\{X_1,\;X_2,\;{\cdots},\;X_n\}$ for $n{\geq}1$. We say $X_j$ is an upper record value of $\{X_n,\;n{\geq}1\}$, if $Y_j>Y_{j-1}$, j > 1. The indices at which the upper record values occur are given by the record times {u(n)}, $n{\geq}1$, where u(n)=min\{j{\mid}j>u(n-1),\;X_j>X_{u(n-1)},\;n{\geq}2\} and u(1) = 1. Suppose $X{\in}Exp(1)$. Then $\Large{E\;\left.{\frac{X^r_{u(m)}}{X^{s+1}_{u(n)}}}\right)=\frac{1}{s}E\;\left.{\frac{X^r_{u(m)}}{X^s_{u(n-1)}}}\right)-\frac{1}{s}E\;\left.{\frac{X^r_{u(m)}}{X^s_{u(n)}}}\right)}$ and $\Large{E\;\left.{\frac{X^{r+1}_{u(m)}}{X^s_{u(n)}}}\right)=\frac{1}{(r+2)}E\;\left.{\frac{X^{r+2}_{u(m)}}{X^s_{u(n-1)}}}\right)-\frac{1}{(r+2)}E\;\left.{\frac{X^{r+2}_{u(m-1)}}{X^s_{u(n-1)}}}\right)}$.

#### 참고문헌

1. Commun. Staist. Theor. Meth. v.23 no.1 Recurrence relations for single and product moments of record values from generalized pareto distribution Balakrishnan, N.;Ahsanuallah, M.
2. J. Appl. Statist. Sci Relations for single and product moments of record values from exponential distribution Balakrishnan, N.;Ahsanuallah, M.
3. Commun. Staist.- Theor. Meth. v.22 no.5 Recurrence relations for moments of record values from generalized extreme value distribution Balakrishnan, N.;Ahsanuallah, M.;Chan, P.S.
4. Staist.Probab. Lett. v.15 Relations for single and product moments of record values from Gumbel distribution Balakrishnan, N.;Ahsanuallah, M.;Chan, P.S.
5. J. R. Statist. Soc. B v.14 The distribution and frequency of record values Chandler, K.N.
6. Characterization of probability distributions Galambos, J.;Katz, S.
7. Record Statistics Ahsanuallah, M.