  The goal of subsampling is to select an informative subset of all
observations, when using the full data for statistical analysis is not viable.
We construct locally $ D $-optimal subsampling designs under a Poisson
regression model with a log link in one covariate. A Representation of the
support of locally $ D $-optimal subsampling designs is established. We make
statements on scale-location transformations of the covariate that require a
simultaneous transformation of the regression parameter. The performance of the
methods is demonstrated by illustrating examples. To show the advantage of the
optimal subsampling designs, we examine the efficiency of uniform random
subsampling as well as of two heuristic designs. Further, the efficiency of
locally $ D $-optimal subsampling designs is studied when the parameter is
misspecified.
