Denote the well-known product limit estimator by \(\hat F_ n\), and the corresponding quantile function by \(\hat Q_ n(p)=\inf \{t\); \(\hat F{}_ n(t)\geq p\}\). The kernel type empirical quantile function studied by the second author [J. Am. Stat. Assoc. 81, 215-222 (1986; Zbl 0596.62043)] is defined as \[ Q_ n(p)=h^{-1}\int^{1}_{0}\hat Q_ n(t)K((t-p)/h)dt \] for a kernel function K and bandwidth h. The authors study and compare the mean square errors of these two quantile estimators.