A generalization of Student’s t-distribution from the viewpoint of special functions

Student’s t-distribution has found various applications in mathematical statistics. One of the main properties of the t-distribution is to converge to the normal distribution as the number of samples tends to infinity. In this paper, by using a Cauchy integral we introduce a generalization of the t-distribution function with four free parameters and show that it converges to the normal distribution again. We provide a comprehensive treatment of mathematical properties of this new distribution. Moreover, since the Fisher F-distribution has a close relationship with the t-distribution, we also introduce a generalization of the F-distribution and prove that it converges to the chi-square distribution as the number of samples tends to infinity. Finally some particular sub-cases of these distributions are considered.

Study of individual lactation patterns of Iranian dairy buffaloes

The aims of the current study were 1) to investigate the effects of some environmental factors on lactation curve traits (LCTs) including initial milk yield (A), peak yield (PY), days to attain peak yield (PD), inclining- and declining slope of lactation (B and C, respectively), persistency (Per), and 240-d milk yield, and 2) to estimate pairwise phenotypic correlations between these traits in two Iranian buffalo ecotypes (Khuzestani and Azeri buffaloes). The dataset consisted of 15396 and 9283 lactations from 6632 Khuzestani and 3558 Azeri buffaloes, respectively (collected during 1992–2009). The results revealed that almost all of the factors had significant effects on the majority of the LCTs, whereby age group, parity and season of calving had greater influence on 240-d milk yield and PY than the other LCTs in both of the ecotypes. These effects were more apparent in Khuzestani buffaloes than in Azeri buffaloes. In the Khuzestani ecotype, the LCTs were significantly correlated with each other. However, in the Azeri ecotype the 240-d milk yield showed no significant relationship with parameters B, PD and Per. In conclusion, the studied factors play an important role in determining both the shape of the lactation curve and the overal performance of Iranian dairy buffaloes.