A large number of functional forms have been estimated for income distribution in Japan with an emphasis on selecting appropriate functional forms. Five estimation procedures, namely (1) Pearsonian β₁ and β₂ method, (2) matching moments, (3) least squares, (4) minimum chi-square, and (5) maximum likelihood. Some adjustments have been made for correcting the influence of the number of parameters in the functions, namely (1) PC and (2) AIC criteria have been considered. Some statistical test has been attempted to examine whether it was possible to eliminate one parameter. Attention has been paid to investigating the difference in parameter values and in inequality by several occupations, and between primary income figures and redistributed income figures.
Followings are the summary of the results. The Pearsonian β₁ and β₂ method gives us a preliminary idea on estimation. It is, however, risky to rely on this method. Method of matching moments is preferable to least squares method or minimum chi-square method for the functions which are supposed to be symmetric a priori. We relied mainly on minimum chi-square method as an estimation procedure to select appropriate functional forms. It is feasible to conclude that the three parameters' functions provide better fits than the two parameters' functions except for a few cases. Statistical test, however, suggests that there are several cases such that one parameter could be eliminated without losing the substance. Since a fewer parameters are definitely desirable for the interpretation purpose, reduction in parameters is recommended in such cases. As for the fitting, following results have been obtained. The normal and lognormal provide in general the poorest fit except for a rare case, i.e., Self-employed. The Gamma and Generalized Gamma give better results than the previous two functions. The Beta provides considerably better fit which is consistent with the preliminary result by the Pearsonian β₁ and β₂ method. The J-shaped Beta is especially useful for the particular case (i.e., farmers' primary income). The Singh-Maddala function gives in general excellent performances. In some case, however, the functions can be replaced by either the log-logistic or the Weibull statistically. A trade-off between them was argued. The Johnson (S_B) shows a unique result in the sense that when the distribution is highly skewed, it is considerably useful. The sample experimentations due to various occupations and primary/redistributed incomes suggest, finally, that those are considerably crucial for the determination of the most appropriate density functions.

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