I looked into Shadow Stats, and highly recommend reading these two responses to criticisms of the official BLS CPI measures:

https://www.bls.gov/opub/mlr/2008/08/art1full.pdf This is an incredibly interesting (if a bit dense) report. It clarifies what changes the BLS instituted in their CPI, why they did it, and what impact it had. One example is that the so-called "hedonic regression" only gave rise to ~0.005%/year difference - "To be precise, the use of the models has increased the annual rate of change of the all-items CPI, but by only about 0.005 percent per year."

https://econbrowser.com/archives/2008/10/shadowstats_res This one is really short, and the critical part is that John Williams doesn't recalculate the CPI in a way that he believes is more accurate. Rather, he tacks on to the BLS CPI a set extra increase. If anything, this is a ridiculous (IMO) way of doing what he claims is a "more accurate" measure of inflation.