The accuracy and reliability of gamma-ray bursts (GRBs) as distance indicators are strongly restricted by their systematic errors which are larger than statistical errors. These systematic errors might come from either intrinsic variations of GRBs, o
r systematic errors in observations. In this paper, we consider the possible origins of systematic errors in the following observables, (i) the spectral peak energies (Ep) estimated by Cut-off power law (CPL) function, (ii) the peak luminosities (Lp) estimated by 1 second in observer time. Removing or correcting them, we reveal the true intrinsic variation of the Ep-TL-Lp relation of GRBs. Here TL is the third parameter of GRBs defined as TL ~ Eiso / Lp. Not only the time resolution of Lp is converted from observer time to GRB rest frame time, the time resolution with the largest likelihood is sought for. After removing obvious origin of systematic errors in observation mentioned above, there seems to be still remain some outliers. For this reason, we take account another origin of the systematic error as below, (iii) the contamination of short GRBs or other populations. To estimate the best fit parameters of the Ep-TL-Lp relations from data including outliers, we develop a new method which combine robust regression and an outlier identification technique. Using our new method for 18 GRBs with {sigma}Ep/Ep < 0.1, we detect 6 outliers and find the Ep-TL-Lp relation become the tightest around 3 second.
We reconsider correlations among the spectral peak energy ($E_p$), 1-second peak luminosity ($L_p$) and isotropic energy (Eiso), using the database constructed by citet{yonetoku10} which consists of 109 Gamma-Ray Bursts (GRBs) whose redshifts are kno
wn and $E_p$, $L_p$ and Eiso are well determined. We divide the events into two groups by their data quality. One (gold data set) consists of GRBs with peak energies determined by the Band model with four free parameters. On the other hand, GRBs in the other group (bronze data set) have relatively poor energy spectra so that their peak energies were determined by the Band model with fixed spectral index (i.e. three free parameters) or by the Cut-off power law (CPL) model with three free parameters. Using only the gold data set we found the intrinsic dispersion in $log L_p$ ($=sigma_{rm int}$) is 0.13 and 0.22 for tsutsui correlation ($T_L equiv E_{rm iso}/L_p$) and yonetoku correlation, respectively. We also find that GRBs in the bronze data set have systematically larger $E_p$ than expected by the correlations constructed with the gold data set. This means that the intrinsic dispersion of correlations among $E_p$, $L_p$, and Eiso of GRBs depends on the quality of data set. At present, using tsutsui correlation with gold data set, we would be able to determine the luminosity distance with $sim 16%$ error, which might be useful to determine the nature of the dark energy at high redshift $z > 3$.
