The limiting behavior of a modified maximal symmetric \(2s\)-spacing with applications
The unknown periodic density \(f\) of the folded (modulo 1) arrival times can be represented as \[ f(x)= 1-p+p f_s(x),\;0<x<1, \] where \(p\), \(0<p<1\), is the unknown strength of the pulsed signal and \(f_s\) is the unknown source function that characterizes the radiation pattern of the source (pulsar). The estimation of \(p\) is reduced to the estimation of \(f(\theta)= \min_{0<x<1}f(x)\). There is given a nonparametric estimator \(\widehat f(\theta)\) for \(f(\theta)\), and then an estimator for \(p\) has the form \(\widehat p_n=\max \{0,1-\widehat f(\theta)\}\). The presented construction of \(\widehat f( \theta)\) uses order statistics \(Y_1\leq Y_2\leq\cdots\leq Y_n\) of a sample \(X_1, X_2, \dots, X_n\). Let \(\{s_n\}\) be a nonrandom sequence of positive integers. The maximal \(s_n\)-spacing is defined for \(1\leq s_n\leq n-1\) by \(M_n= \max_{1\leq j\leq n-s_n} (Y_{j+s_n} -Y_j)\). The following modification of \(M_n\) leads to a statistic \(V_n\) allowing to estimate \(f(\theta)\). Suppose that \(2s_n \leq n-1\) for \(n=1,2, \dots\), and \(s_n\uparrow\infty\) as \(n\to\infty\). For each \(n\) let \(K_n\) be positive integer-valued random variables defined by \[ Y_{K_n+ s_n}-Y_{K_n-s_n}= \max_{s_n+1\leq j\leq n-s_n} (Y_{j+s_n}-Y_{j-s_n}). \] Let \(\{r_n \}\) be another nonrandom sequence of positive integers such that \(r_n \leq s_n\) for \(n=1,2, \dots\), and \(r_n\uparrow\infty\) as \(n\to\infty\). Now, define \(V_n= Y_{K_n+r_n} -Y_{K_n-r_n}\). The author suggests estimating \(f(\theta)\) by \(\widehat f(\theta) =2r_n (nV_n)^{-1}\). It is shown to be strongly consistent and asymptotically normally distributed. S simulation study compares this estimator with the minimum value of the kernel density estimator.
scientific article; zbMATH DE number 1359627