  Linear Fractional Stable Motion (LFSM) of Hurst parameter $H$ and of
stability parameter $\al$, is one of the most classical extensions of the
well-known Gaussian Fractional Brownian Motion (FBM), to the setting of
heavy-tailed stable distributions \cite{SamTaq,EmMa}. In order to overcome some
limitations of its areas of application, coming from stationarity of its
increments as well as constancy over time of its self-similarity exponent,
Stoev and Taqqu introduced in \cite{stoev2004stochastic} an extension of LFSM,
called Linear Multifractional Stable Motion (LMSM), in which the Hurst
parameter becomes a function $H(\cdot)$ depending on the time variable $t$.
Similarly to LFSM, the tail heaviness of the marginal distributions of LMSM is
determined by $\al$; also, under some conditions, its self-similarity is
governed by $H(\cdot)$ and its path roughness is closely related to
$H(\cdot)-1/\al$. Namely, it was shown in \cite{stoev2004stochastic} that
$H(t_0)$ is the self-similarity exponent of LMSM at a time $t_0\neq 0$;
moreover, very recently, it was established in \cite{hamonier2012lmsm}, that
the quantities $\min_{t\in I} H(t)-1/\al$, and $H(t_0)-1/\al$, are respectively
the uniform H\"older exponent of LMSM on a compact interval $I$, and its local
H\"older exponent at $t_0$.
  The main goal of our article, is to construct, using wavelet coefficients of
LMSM, strongly consistent (i.e. almost surely convergent) statistical
estimators of $\min_{t\in I} H(t)$, $H(t_0)$, and $\al$; our estimation
results, are obtained when $\al\in (1,2)$, and, $H(\cdot)$ is a H\"older
function smooth enough, with values in a compact subinterval
$[\underline{H},\bar{H}]$ of $(1/\al,1)$.
