  Suppose that $p \in (1,\infty]$, $\nu \in [1/2,\infty)$, $\mathcal{S}_\nu =
\left\{ (x_1,x_2) \in \mathbb{R}^2 \setminus \{(0, 0)\}: |\phi| <
\frac{\pi}{2\nu}\right\}$, where $\phi$ is the polar angle of $(x_1,x_2)$. Let
$R>0$ and $\omega_p(x)$ be the $p$-harmonic measure of $\partial B(0,R) \cap
\mathcal{S}_\nu$ at $x$ with respect to $B(0, R)\cap \mathcal{S}_\nu$. We prove
that there exists a constant $C$ such that \begin{align*}
C^{-1}\left(\frac{|x|}{R}\right)^{k(\nu,p)} \, \leq \omega_p(x) \, \leq C
\left(\frac{|x|}{R}\right)^{k(\nu,p)} \end{align*} whenever $x\in B(0,R) \cap
\mathcal{S}_{2\nu}$ and where the exponent $k(\nu,p)$ is given explicitly as a
function of $\nu$ and $p$. Using this estimate we derive local growth estimates
for $p$-sub- and $p$-superharmonic functions in planar domains which are
locally approximable by sectors, e.g., we conclude bounds of the rate of
convergence near the boundary where the domain has an inwardly or outwardly
pointed cusp. Using the estimates of $p$-harmonic measure we also derive a
sharp Phragmen-Lindel\"of theorem for $p$-subharmonic functions in the
unbounded sector $\mathcal{S}_\nu$. Moreover, if $p = \infty$ then the above
mentioned estimates extend from the setting of two-dimensional sectors to cones
in $\mathbb{R}^n$. Finally, when $\nu \in (1/2, \infty)$ and $p\in (1,\infty)$
we prove uniqueness (modulo normalization) of positive $p$-harmonic functions
in $\mathcal{S}_\nu$ vanishing on $\partial\mathcal{S}_\nu$.
