# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 2 Materials and Methods
### Cyclic Statistical Functions Estimators
#### 2.3.2 Cyclic CDF and pdf (Kernel-Based Estimators)  
Kernel-based estimators of the cyclic CDF and the cyclic pdf are proposed here for the first time. They are obtained by generalizing the definitions for time-invariant CDF and pdf. However, only early results are so far available for these estimators, and their properties are still under investigation.  
The kernel-based estimators of the FOT cyclic pdf (11b) and of the FOT cyclic CDF (9) of a continuous-time signal \(x(t)\) are a generalization of the continuous-time counterparts of the kernel-based estimators of the pdf and CDF proposed for discrete-time processes in [37; 38], [39] (Section 2.1.8, pp. 64-65). They are given by  
\[f_{x}^{(T,b_{T})}(\gamma,\xi;t_{0})=\frac{1}{T}\int_{t_{0}-T/2}^{t_{0}+T/2} \frac{1}{b_{T}}W\Big{(}\frac{\xi-x(t)}{b_{T}}\Big{)}\,e^{-\beta\pi\gamma t}\, \mathrm{d}t \tag{26}\]  
and  
\[F_{x}^{(T,b_{T})}(\gamma,\xi;t_{0}) =\int_{-\infty}^{\xi}f_{x}^{(T,b_{T})}(\gamma,s;t_{0})\,\mathrm{d}s \tag{27a}\] \[=\frac{1}{T}\int_{t_{0}-T/2}^{t_{0}+T/2}W_{c}\Big{(}\frac{\xi-x(t )}{b_{T}}\Big{)}\,e^{-\beta\pi\gamma t}\,\mathrm{d}t \tag{27b}\]  
respectively, where \(W(s)\) is a unit-area smoothing window,  
\[W_{c}(\xi)\triangleq\int_{-\infty}^{\xi}W(s)\,\mathrm{d}s \tag{28}\]  
and \(b_{T}\) is the smoothing parameter. When \(t_{0}=0\) or the dependence on \(t_{0}\) is not of interest, in the left-hand sides of (26) and (27a) such a dependence is omitted for the sake of brevity.  
In the special case \(\gamma=0\), the discrete-time counterparts of the estimators (26) and (27b) are characterized in the stochastic approach in [37; 38], [39] (Section 2.1.8, pp. 64-65) (in which case, \(t_{0}=0\) and the variability (randomness) of the estimate is considered on the ensemble of realizations of a stochastic process). Specifically, under mild assumptions, \(f_{x}^{(T,b_{T})}(0,\xi)\) and \(F_{x}^{(T,b_{T})}(0,\xi)\) are shown to be mean-square consistent estimators of the pdf \(f_{x}^{0}(\xi)\) and the CDF \(F_{x}^{0}(\xi)\), respectively, provided that [38] the following conditions hold:  
\[T\to\infty\qquad b_{T}\to 0\qquad Tb_{T}\to\infty \tag{29}\]  
In addition, for \(\gamma=0\), [PERSON]'s rule of thumb  
\[b_{T}=(4/3)^{1/5}\,\sigma_{x}\,T^{-1/5} \tag{30}\]  
where \(\sigma_{x}\) is the standard deviation of \(x(t)\), assures the minimum mean-squared error of the CDF and pdf estimates [38] (Equation (20)). Furthermore, for \(\gamma=0\), the normalized estimation errors  
\[\sqrt{Tb_{T}}\Big{[}f_{x}^{(T,b_{T})}(0,\xi)-f_{x}^{0}(\xi)\Big{]} \tag{31}\]and  
\[\sqrt{Tb_{T}}\Big{[}F_{x}^{(T,b_{T})}(0,\xi)-F_{x}^{0}(\xi)\Big{]} \tag{32}\]  
are asymptotically normal, provided that \(Tb_{T}^{3}\to\infty\) as \(T\to\infty\)[38] (p. 1822). See [40] for the case of continuous-time processes.  
Numerical experiments have shown that, for \(\gamma\
eq 0\), the estimators (26) and (27b) provide reliable estimates (i.e., exhibiting small bias and variance) of the cyclic pdf and the cyclic CDF, respectively, as the data record length \(T\) increases and the smoothing parameter \(b_{T}\) decreases. This fact suggests that, under appropriate conditions, the mean-square consistence can be proved. In contrast, the asymptotic normality property does not seem to hold. The plausibility of this conjecture is corroborated by the results presented in Section 3.  
An alternative technique for FOT pdf estimation based on linear interpolation of time series is presented in [41].