  In wall-bounded turbulence, the moment generating functions (MGFs) of the
streamwise velocity fluctuations $\left<\exp(qu_z^+)\right>$ develop power-law
scaling as a function of the wall normal distance $z/\delta$. Here $u$ is the
streamwise velocity fluctuation, $+$ indicates normalization in wall units
(averaged friction velocity), $z$ is the distance from the wall, $q$ is an
independent variable and $\delta$ is the boundary layer thickness. Previous
work has shown that this power-law scaling exists in the log-region {\small
$3Re_\tau^{0.5}\lesssim z^+$, $z\lesssim 0.15\delta$}, where $Re_\tau$ is the
friction velocity-based Reynolds numbers. Here we present empirical evidence
that this self-similar scaling can be extended, including bulk and
viscosity-affected regions $30<z^+$, $z<\delta$, provided the data are
interpreted with the Extended-Self-Similarity (ESS), i.e. self-scaling of the
MGFs as a function of one reference value, $q_o$. ESS also improves the scaling
properties, leading to more precise measurements of the scaling exponents. The
analysis is based on hot-wire measurements from boundary layers at $Re_\tau$
ranging from $2700$ to $13000$ from the Melbourne
High-Reynolds-Number-Turbulent-Boundary-Layer-Wind-Tunnel. Furthermore, we
investigate the scalings of the filtered, large-scale velocity fluctuations
$u^L_z$ and of the remaining small-scale component, $u^S_z=u_z-u^L_z$. The
scaling of $u^L_z$ falls within the conventionally defined log region and
depends on a scale that is proportional to {\small $l^+\sim Re_\tau^{1/2}$};
the scaling of $u^{S}_z$ extends over a much wider range from $z^+\approx 30$
to $z\approx 0.5\delta$. Last, we present a theoretical construction of two
multiplicative processes for $u^L_z$ and $u^S_z$ that reproduce the empirical
findings concerning the scalings properties as functions of $z^+$ and in the
ESS sense.
