We present a detail analysis on a general class of holographic type dark energy models characterized by the length scale L = \frac { 1 } { a ^ { n } ( t ) } \int _ { 0 } ^ { t } dt ^ { \prime } ~ { } a ^ { m } ( t ^ { \prime } ) .
We show that n \geq 0 is required by the recent cosmic accelerated expansion of universe .
In the early universe dominated by the constituent with constant equation of state w _ { m } , we have w _ { de } \simeq - 1 - \frac { 2 n } { 3 } for n \geq 0 and m < 0 , and w _ { de } \simeq - \frac { 2 } { 3 } ( n - m ) + w _ { m } for n > m \geq 0 .
The models with n > m \geq 0 become single-parameter models like the \Lambda CDM model due to the analytic feature \Omega _ { de } \simeq \frac { d ^ { 2 } } { 4 } ( 2 m + 3 w _ { m } +3 ) ^ { 2 } a ^ { 2 ( n - m ) } at radiation- and matter-dominated epoch .
Whereas the cases n = m \geq 0 should be abandoned as the dark energy can not dominate the universe forever and there might be too large fraction of dark energy in early universe , and the cases m > n \geq 0 are forbidden by the self-consistent requirement \Omega _ { de } \ll 1 in the early universe .
Thus a detailed study on the single-parameter models corresponding to cases n > m \geq 0 is carried out by using recent observations .
The best-fit analysis indicates that the conformal-age-like models with n = m + 1 , i.e . L \propto \frac { 1 } { Ha } in early universe , are more favored and also the models with smaller n for the given n - m are found to fit the observations better .
The equation of state of the dark energy in models with n = m + 1 > 0 transits from w _ { de } < -1 during inflation to w _ { de } > -1 in radiation- and matter-dominated epoch , and then back to w _ { de } < -1 eventually .
The best-fit result of the case ( n = 0 ,m = -1 ) which is so-called \eta HDE model proposed in [ [ 1 ] ] is the most favorable model and compatible with the \Lambda CDM model .