  Sketching algorithms use random projections to generate a smaller sketched
data set, often for the purposes of modelling. Complete and partial sketch
regression estimates can be constructed using information from only the
sketched data set or a combination of the full and sketched data sets. Previous
work has obtained the distribution of these estimators under repeated
sketching, along with the first two moments for both estimators. Using a
different approach, we also derive the distribution of the complete sketch
estimator, but additionally consider the error term under both repeated
sketching and sampling. Importantly, we obtain pivotal quantities which are
based solely on the sketched data set which specifically not requiring
information from the full data model fit. These pivotal quantities can be used
for inference on the full data set regression estimates or the model
parameters. For partial sketching, we derive pivotal quantities for a marginal
test and an approximate distribution for the partial sketch under repeated
sketching or repeated sampling, again avoiding reliance on a full data model
fit. We extend these results to include the Hadamard and Clarkson-Woodruff
sketches then compare them in a simulation study.
