Let $ X_1, X_2, cdots, X_n $ be i.i.d. random variables having a probability density function $ f(x) $ and $ f_n(x) $ be a nonparametric density estimator of $ f(x) $. We investigate the property of a location shift random variable $ a_n $ which minimizes integrated squared error $ mathrm{ISE}_n(a) $: $ mathrm{ISE}_n(a) = int_{-x}^{x}{mid f_n(x) - f(x - a) mid^2}dx $. The asymptotic normality and the order of strong convergence of the $ mathrm{r.v.}a_n $ and those of $ mathrm{ISE}_n(a_n) $ are studied. We also give some numerical examples and some simulations which show the effectiveness of using the $ a_n $ when one estimates $ f(x) $ by $ f_n(x) $.