This is where Jason and I do disagree and think that inflation has actually been overestimated. And the two areas, housing and college, along with health are the areas that have risen faster than the other inflation which has skewed inflation up would not have as big of impact on most min wage earners and that in min wage earners do purchase more than there predecessors.
Inflation is the rate of change in the general level of prices. The different price indices all measure inflation. And there is no dispute that resulting measurements are inaccurate: they all measure the rate of change in the general level of prices in their indices.
There are reasons indicating that the resulting measures may be inaccurate in estimating the actual effects on well-being. The generally accepted estimates are that fixed weight indices over-estimate the effect on well-being by 1 to 2 percentage points (i.e. that a reported 3% increase in the CPI is really a 1 to 2 percent reduction in well-being). To my knowledge, there is no generally accepted estimates for the inaccuracy of chain-weighted indices.
In 1967, the federal minimum wage was 2.00 and the CPI as currently measured was 33.40. In 2016, the federal minimum wage was $7.25 and the CPI was 240.008. So, the annual rate of increase of the CPI (as measured) over those 39 years is
(240/33.4)\(1/39\)-1 =5.18%. Using the 1 to 2 percentage point over-estimate interval, that means that if the 2016 federal minimum wage were to give the same level of estimated well being as $2.00 per hour in 1967, it would be between
$2.00(1+0.0318)\(39\) = $6.78 and $2.00(1+0.0418)\(39\)=$9.87.
That suggests to me there is little reason to believe that the federal minimum wage today provides more well-being than it did in 1967.
Furthermore, your implicit argument that health care and housing "inflation" should be excluded from the estimates because those items are not large in the consumed market basket of most minimum wage earners is not terribly convincing, since we do not know what the market basket of minimum wage earners is. And, if one is being consistent, one would have to construct a market basket for minimum wage earners.