  The hazard function is a ratio of a density and survival function, and it is
a basic tool of the survival analysis. In this paper we propose a kernel
estimator of the hazard ratio function, which are based on a modification of
\'{C}wik and Mielniczuk's method. We study nonparametric estimators of the
hazard function and compare those estimators by means of asymptotic mean
squared error ($AMSE$). We obtain asymptotic bias and variance of the new
estimator, and compare them with a naive estimator. The asymptotic variance of
the new estimator is always smaller than the naive estimator's, so we also
discuss an improvement of $AMSE$ using Terrell and Scott's bias reduction
method. The new modified estimator ensures the non-negativity, and we
demonstrate the numerical improvement.
