In this present analysis, I use two data sources--first, from the Guardian's, The Counted, and second, from the Ross, Plos One article above. Both have publicly available raw data, whereas the NBER paper does not. The Guardian data is available at GitHub, and is from all of 2015 through July 15, 2016. The Ross' data is available from Google Docs, and is from 2011-2014.
I limited my analysis to just those incidents where the victim was shot dead by police, and where the victim was either unarmed, or armed with a gun (or what could be misinterpreted as a gun, such as a realistic-looking toy gun). I use the phrase "shot dead" to specifically refer to the fact that the victim was killed by a firearm. The Guardian data lists all persons "killed" by police or in police custody by any means. The Ross data only lists police "shootings", but includes victims who were shot but did not die, and victims who were shot and died. The results are in Table 1 below.
In top half of Table 1, from the column labelled "X/White:Firearm," the Guardian data shows that Blacks are 2.3 times more likely than Whites to be shot dead by police if the victim is carrying a firearm, and 4.1 times more likely if they are not carrying a firearm (Guardian data). In the bottom half of the table, the Ross data (PLOS One), shows that Blacks are 3.3 times more likely than Whites to be shot dead by police if the victim is carrying a firearm, and 4.8 times more likely if they are not carrying a firearm. Hispanics are also at some greater levels of risk in both sets of data, while Asians are far less likely to be shot dead by police in any circumstance, while native Americans are at far greater risk if they are carrying a firearm (the Ross data only looks at Black, White & Hispanic).
Both sets of data shows that Whites are shot dead by police more frequently than Blacks, and Blacks are shot dead by police more frequently than Hispanics. This holds true whether or not the victim had a firearm, although the Ross data shows that from 2011-2014, the same number of unarmed Blacks and Whites were shot dead by police. Columns 5 & 6 show the rates at which Whites, Blacks & Hispanics are shot dead by police per million of their race/ethnic group. So Blacks with firearms are shot dead by police at a rate of 5.04 per million Blacks, and Blacks without firearms are shot dead by police at a rate of 1.4 per million Blacks. The final two columns show rates of Black and Hispanic deaths by police shootings in reference to White shooting deaths by police.
Both of these data sets fail to support findings published by Fryer in NBER. His study focused only on 10 specific communities, and his core analysis focuses only on Houston. He also asks very specific questions other than "rates at which Whites vs Blacks vs Hispanics are shot dead by police." The New York Times discussion of his results is here. Criticisms of the study can be found at Vox, by Feldman, and by Simonsohn.
Table two shows the population values I used to calculate the rates per million. This data was retrieved July 15, 2016, using Census FactFinder. One of the difficulties of these types of race-ethnicity analyses, is that while the Guardian and Ross create three categories of Black, White & Hispanic, the Census has two categories for race, Black & White, and a category for Hispanic ethnicity. This means that there are actually four categories for what the Guardian and Ross list--Black Hispanic, Black not-Hispanic, White Hispanic and White non-Hispanic. The Ross data does actually provide a way to separate these out, however, it is left unclarified how race & ethnicity are determined. In this case, I calculated White using non-Hispanic White, Black as non-Hispanic Black, and Hispanic as all categories noting Hispanic ethnicity. In other words, summing White Hispanic, Black Hispanic, Asian Hispanic, etc.