Does there exist zero-inflated linear regression?

I have a non-count data with huge number of zeros in the target variable. I need to fit a model being a mixture of Dirac delta function and normal distribution parametrized by mean $X\beta$ and variance $\sigma^2$, with mixing proportion $\pi$, i.e.

$$f(y) = \pi \; \delta(y) + (1 - \pi) \; \mathcal{N}(y \mid X\beta, \sigma^2)$$

to account for the excess zeros. Could you provide me with any references about such models? Or maybe there is some approach that is better, then the above, for continuous, zero-inflated data?

• If you have a genuine expectation that the true distribution is indeed a zero-inflated normal, then just fit that model and be done with it. Whether other approaches are better depends on whether the expectation/evidence that some other choice of distribution is a better approximation to nature. Edited to add: it seems an odd sampling process that both deals with continuous data and has a huge number of integers (zeros) in it. – Jacob Socolar Jun 1 '17 at 14:35
• @user43849 the process that produces such data is very easy to imagine: think of some kind of device that is idle for most of the time, but sometimes fires some continuous signals. – Tim Jun 1 '17 at 14:43
• Interestingly, the Wiki excerpt of the zero-inflation tag says there is zero-inflated normal regression. Not that this would help, but I find it curious. – Richard Hardy Jun 1 '17 at 14:46
• @RichardHardy I wasn't able to find any references dealing with such models, this is how the question emerged... – Tim Jun 1 '17 at 14:51
• When it's non-zero is the response positive? – Glen_b Jun 2 '17 at 4:46