  Local differential privacy has become the gold-standard of privacy literature
for gathering or releasing sensitive individual data points in a
privacy-preserving manner. However, locally differential data can twist the
probability density of the data because of the additive noise used to ensure
privacy. In fact, the density of privacy-preserving data (no matter how many
samples we gather) is always flatter in comparison with the density function of
the original data points due to convolution with privacy-preserving noise
density function. The effect is especially more pronounced when using
slow-decaying privacy-preserving noises, such as the Laplace noise. This can
result in under/over-estimation of the heavy-hitters. This is an important
challenge facing social scientists due to the use of differential privacy in
the 2020 Census in the United States. In this paper, we develop density
estimation methods using smoothing kernels. We use the framework of
deconvoluting kernel density estimators to remove the effect of
privacy-preserving noise. This approach also allows us to adapt the results
from non-parameteric regression with errors-in-variables to develop regression
models based on locally differentially private data. We demonstrate the
performance of the developed methods on financial and demographic datasets.
