On sequentional estimation of the location parameter for families of distributions with discontinuous densities

We consider sequential estimation of the location parameter $\theta$ from independent observations $X_1,X_2,\dots$ with a common probability density function $f(x-\theta)$; $x,\theta\in R^1$.
Under the conditions:
(i) the only discontinuities of $f(x)$ are jumps at points $x_1,\dots,x_r$,
(ii) $\displaystyle{\int_{-\infty}^\infty|f'(x)|\,dx<\infty}$,
(iii) $\displaystyle{\biggl(\sum_if^2(x_i+0)\biggr)\biggl(\sum_if^2(x_i-0)\biggr)>0}$,
we construct two invariant sequential procedures $[d,\tau]$, $\mathbf E_\theta\tau\le n$, such that
$$ \varlimsup_n\mathbf E_\theta|d_\tau-\theta|^a/\mathbf E_\theta|\widetilde t_n-\theta|^a<1,\quad a>1, $$
and $\widetilde t_n$ is the best invariant estimator of $\theta$ corresponding to the loss function $|u-\theta|^a$.