Nonparametric estimation of a periodic function

TLDR: In this article, nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times were studied. But the shape of the periodic function is unknown and the first-order properties of the amplitude functions are identical to those that would obtain if the period were known.

Abstract: SUMMARY Motivated by applications to brightness data on periodic variable stars, we study nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times It is shown that nonparametric estimators of period converge at parametric rates and attain a semiparametric lower bound which is the same if the shape of the periodic function is unknown as if it were known Also, first-order properties of nonparametric estimators of the amplitude function are identical to those that would obtain if the period were known Numerical simulations and applications to real data show the method to work well in practice

Figures

Table 1: Standardized bias, variance, and MSE of the frequency estimates

Figure 2: Phase plot of star 77043:4317 at period 2.47133 days.

Figure 1: Brightness plotted against time for the red and blue bands of star 77043:4317.

Figure 3: Phase plots of simulated light curves from model A at period 1.4432 days.

Frequently asked questions

Q1. What have the authors contributed in "Nonparametric estimation of a periodic function" ?

Motivated by applications to brightness data on periodic variable stars, the authors study nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times. 

Q2. What is the r’th order regression estimator?

If h ∼ Cn−1/(2r) for a constant C > 0, and ĝ is an r’th order regression estimator, then n3/2 (θ̂ − θ0) remains asymptotically Normally distributed but its asymptotic bias is no longer zero. 

Q3. What is the limiting Fisher information for the estimate of the period?

Then if the regression function is known except12for phase, or if it is unknown, the information bound for the estimate of the period is given by (2.6), where σ2 is replaced by the reciprocal of Fisher information, I, on a location parameter in the error density; the scaling factor is n3/2 as at (2.5). 

Q4. What is the order of approximation at (2.7)?

despite the Nadaraya-Watson estimator g̃(·|θ) being susceptible to edge effects, the order of approximation at (2.7) is available uniformly in x ∈ (0, θ0], since the effects of errors at boundaries are of order n |θ̂ − θ0| = op{(nh0)−1/2}.7 

Q5. What is the infinite sequence of numbers Xj modulo?

In particular, if the distribution is defined on an integer lattice, and if θ0 is a rational number, then the infinite sequence of numbers 

Q6. What is the problem with the Xj ’s?

These difficulties vanish if the authors assume that the Xj ’s are generated by (2.3) where the distribution of V > 0 is absolutely continuous with an integrable characteristic function and that all moments of V are finite. 

Q7. What is the simplest way to calculate a moment?

In the case of model (MX,2), where the Xi’s represent ordered values of independent random variables, moment calculations are relatively straight forward. 

Q8. how do the authors calculate the k’th moment of fxi?

writing ∑′i to denote summation over nη < i ≤ n, the authors calculate the k’th moment of ∑′i f{Xi(θ)}, where f(u) might for example denote The author[ u ∈ (x, x + h1)] , by writing the moments as∑i1′ . . . ∑ ik ′ E [ f { Xi1(θ) } . . . f { Xik (θ) }] .15This series may be approximated up to a remainder of order n−C , for any given C > 0, by using an Edgeworth expansion of the joint density of (Xi1 , . . . ,Xik ). 

Q9. Why is it not necessary to use a function estimationmethod?

In view of the periodicity of g it is not necessary to use a function estimationmethod, such as a local linear smoother, which accommodates boundary effects.