  In this paper, we investigate ${\mathcal F}$-hypercyclicity of linear, not
necessarily continuous, operators on Fr\' echet spaces. The notion of lower
$(m_{n})$-hypercyclicity seems to be new and not considered elsewhere even for
linear continuous operators acting on Fr\' echet spaces. We pay special
attention to the study of $q$-frequent hypercyclicity, where $q\geq 1$ is an
arbitrary real number. We present several new concepts and results for lower
and upper densities in a separate section, providing also a great number of
illustrative examples and open problems.
