Why income inequality is very difficult to analyze?

There are several major agencies reporting various measures of personal income. The Census Bureau, CB, measures personal incomes in household surveys (CPS ASEC) at an annual rate. This measure is called Money Income, MI, and includes various types of personal income. The CB provides these estimates to the Bureau of Labor Statistics in form of distributions over age/race/sex. 
The Bureau of Economic Analysis also carries out annual estimates of gross personal income, GPI, as based on administrative records but does not provide any dependence on age/race/sex. In that sense the BEA reports only the cumulative number and does not allow inferring any evolution of personal income distribution in time. The most important similarities and differences of the CB and BEA measures are discussed in depth in this CB document.   
The IRS also measures and reports personal incomes filed for tax purposes. Since 1996, the IRS has been publishing detailed tables of personal incomes distribution is various income bins. This is similar to the CB reports but includes capital gains as income source.  
Unfortunately, multiple purposes and multiple agencies reporting personal incomes make it difficult to follow up actual evolution of income distribution and income inequality in the US. There is no unique way to merge all data in one consistent table and to estimate the distribution of income over age/race/sex and to calculate any quantitative measure of inequality like Gini coefficient or Thile index.  We illustrate the difficulties with two plots. Figure 1 shows the portion of personal income reported by three agencies in nominal GDP. The BEA reports around 85% of GDP as personal incomes but does not include capital gains. The CB and IRS both report only 55% to 62% of GDP as personal incomes with a very large difference in sources of income.
Figure 2 shows the portion of population with income as defined by the CB and IRS. There is a dramatic difference of 30% between these agencies. In other word, the IRS does not count as personal income what approximately 30% of the total population define as money income. This might be not a big difference in total income when all personal incomes are summarized over this 30 per cent of population.
One may conclude that the way these three major agencies consider and resolve the problem of personal income and income inequality is counterproductive and confusing for any quantitative analysis.  This also means that the speculations about income inequality are mostly qualitative and thus emotional.

Figure 1. Portion of personal income in GDP.
Figure 2. Portion of people with personal income in total population.  

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