the distribution looks like a “triangle”. This must be a result from either some physical constraint or some sample selection bias, or both. For example, a galaxy at *z* $=$ 4.5 with the *i’*-band magnitude *i’* $= 26.0$ has $M_{UV} \lesssim -20.1$, and therefore the number of objects, which are selected from our selection criteria, decrease in the region around $M_{UV} \sim -20$ and $\beta \sim -2.5$. By using only our three data points in $-22.0 \leq M_{\mathrm{UV}} \leq -20.5$, the slope of the relation indeed becomes $-0.09\,\pm\,0.04$ which is quite similar to the value of the previous works. However, the number of the objects in the brightest (most left) bin is much less than the other bins (see column 8 in table \[tab3\]), and the slope is almost estimated from only the two data points. We need to assess the incompleteness of the observed $\beta$ distribution in order to take account of our selection bias, which is discussed in the next section.

Recovery Fraction {#S4s2RF}
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As shown in figure \[fig5\], the observed $\beta$ distribution is restricted in the “triangle” zone. It seems that there are three truncations, namely, (a) at the top left side, (b) at the bottom 