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 
$$ f_{\mathfrak {q},M}(n)=\ell (M/\mathfrak {q}^{n+1}M)-\sum \limits _{i=0}^{d}\text {adeg}_{i}(\mathfrak {q};M) \left (\begin {array}{c} n+i \\ i \end {array}\right ), $$
 where ${\rm adeg}_i(\mathfrak{q};M)$ 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 $f_{\mathfrak{q}, M}(n)\geq 0$ for all $n\geq n_0.$ Moreover, if $M$ is sequentially generalized Cohen-Macaulay then $n_0$ exists independently of the choice of $\frak q$.