Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$.  Let $\mathfrak{q}$ be a parameter ideal of $M$. Consider an adjusted Hilbert-Samuel function in $n$ defined by 
$$f_{\mathfrak{q},M}(n)=\ell (M/{\mathfrak{q}}^{n+1}M)-\sum _{i=0}^d{\text{adeg}}_i(\mathfrak{q};M)\left(\begin{array}{c}n+i\\ i\end{array}\right),$$
 where ${\mathrm{a}\mathrm{d}\mathrm{e}\mathrm{g}}_i(\mathfrak{q};M)$ is the $i$-th arithmetic degree of $M$ with respect to $\mathfrak{q}$.  In this paper, we prove that if $\mathfrak{q}$ is a distinguished parameter ideal  then there exists an integer $n_0$ such that $f_{\mathfrak{q},M}(n)\ge 0$ for all $n\ge n_0.$ Moreover, if $M$ is sequentially generalized Cohen-Macaulay then $n_0$ exists independently of the choice of $\mathfrak{q}$.