# Data distribution for sine time series

Suppose we have a time series $x_t=\sin(0.02\pi t)$. Although this time series is totally deterministic, we can treat it as one realization of a proto/quasi/pseudo-stochastic process and estimate the distribution of $P(x_t)$. For example, if kernel density kernel is used, we will get something like the following image Obviously the estimations at the two ends of the interval [-1,1] are messed up since a sine series can never go outside this range. The image just gives you an idea of what I want.

Question: What is the analytical expression of $P(x_t)$?

Edit 1: The bandwidth I used is the Silverman's rule: B.W. Silverman, “Density Estimation for Statistics and Data Analysis”, Vol. 26, Monographs on Statistics and Applied Probability, Chapman and Hall, London, 1986.

• You might like to vary the kernel width for the KDE plot (I assume this is with the Gaussian kernel, but that is not important), narrower would be interesting – Conrad Turner Dec 21 '14 at 17:05

Suppose we sample uniformly from 1 cycle of the sine wave, then it is quite clear (I would suggest drawing a picture) that the cumulative distribution function is: $$F(x)=\frac{\arcsin(x)}{\pi}+\frac{1}{2},\ \ \textrm{where}\ -1\le x\le1$$ A longer explanation of this can be found here.
Then density can then be found by differentiating this to get: $$f(x)=\frac{1}{\pi\sqrt{1-x^2}},\ \ \textrm{where}\ -1\le x\le1$$ and looks like: