On characterizations of Pareto distribution by
hazard rate of record value

Abstract. Let \(\{ X_n ,n\geq1 \}\) be a sequence of independent and identically distributed non negative random variables which has absolutely continuous distribution function \(F(x)\) with \(F(1)=0\) and probability density function \(f(x)\). Let \(F(x) < 1\) for all \(x>1\) and let \(F\) belong to class \(C_2^+\). Then \(X_k \in PAR(\alpha)\) if and only if for some fixed \(n\), \(n\geq 1\), the hazard rate \(r\) of \(X_k\) is the same as the hazard rate \(r_1\) of \(W_{n,n+1}\) or the hazard rate \(r_2\) of \(Z_{n,n+1}\), where \(W_{n,n+1}=\frac{X_{U(n+1)}}{X_{U(n)}}\), \(Z_{n,n+1}=X_{U(n)}\cdot X_{U(n+1)}\).