  We study the exponent of the exponential rate of convergence in terms of the
number of degrees of freedom for various non-standard {$p$-version} finite
element spaces employing reduced cardinality basis. More specifically, we show
that serendipity finite element methods and discontinuous Galerkin finite
element methods with total degree $\mathcal{P}_p$ basis have a faster
exponential convergence with respect to the number of degrees of freedom than
their counterparts employing the tensor product $\mathcal{Q}_p$ basis for
quadrilateral/hexahedral elements, for piecewise analytic problems under
$p$-refinement. The above results are proven by using a new $p$-optimal error
bound for the $L^2$-orthogonal projection onto the total degree $\mathcal{P}_p$
basis, and for the $H^1$-projection onto the serendipity finite element space
over tensor product elements with dimension $d\geq2$. These new $p$-optimal
error bounds lead to a larger exponent of the exponential rate of convergence
with respect to the number of degrees of freedom. Moreover, these results show
that part of the basis functions in $\mathcal{Q}_p$ basis {plays} no roles in
achieving the $hp$-optimal error bound in the Sobolev space. The sharpness of
theoretical results is also verified by a series of numerical examples.
