The STNRW-ECF estimator for stable distributions with application in ECG beat

Abstract

The $\alpha$ -stable distribution can describe a degree of concentration of the observations around the mean as well as their asymmetry, regardless of sample size. As the $\alpha$ stability index is the most interesting parameter in the applications, several estimators have been proposed for $\alpha$ . Here we develop an estimator for $\alpha$ (called STNRW-ECF) based on the empirical characteristic function and the Seismic Trace Noise Reduction by Wavelets and Double Threshold Estimation method (STNRW). We analyze the proposed estimator using Monte Carlo simulations and prove its asymptotic Gaussian distribution. Electrocardiography (ECG) is the process of recording the electrical activity of the heart over a period of time using electrodes placed on the skin. The time intervals between its various peaks, may contain useful information about the nature of disease afflicting the heart. To analyze this kind of data can be tiring and more prone to errors when interpreted by human beings, since there is a huge amount information to be processed. Here we propose the STNRW-ECF estimator to be an additional diagnostic tool that may provide an indication of cardiac arrhythmia, and we also propose a test based on the principal wavelet shrinkage to confirm whether or not the STNRW-ECF estimator should be used for this purpose.

Keywords

Arrhythmia detection α-stable distribution STNRW method ECF wavelet

1. Introduction

Wavelets are becoming increasingly popular in different areas of applied and theoretical sciences (see Vargas & Veiga, 2017; Donoho et al., 1995; Donoho & Johnstone, 1994). This class of function spans the usual Hilbert space $L^{2}(\mathbb{R})$ and possesses local adaptivity, approximation, and computational properties that are not only remarkable, but which give rise to applications of wavelet-based methods in many areas of science and engineering. In statistics, wavelets have been used primarily to deal with problems of a nonparametric character (see Donoho et al., 1995; Donoho & Johnstone, 1994). Some particular problem of interest is de-noising using the Seismic Trace Noise Reduction by Wavelets and Double Threshold Estimation (STNRW) method (see Vargas & Veiga, 2017). The basic aim of the STNRW method is to separate the detailed wavelet coefficients into three groups. The first group contains detailed wavelet coefficients that are considered to be signal information, the second group contains detailed wavelet coefficients that are considered to be noisy information, and the third group contains detailed wavelet coefficients that belong to an indecision region, where the decision-making process is done through a classification function.

The $\alpha$ -stable distribution has been widely used for fitting data in which extreme values are frequent events (see Zhang et al., 2018; Karol et al, 2018; Kateregga et al., 2017; Montillet & Yu, 2015; Salas-Gonzalez et al., 2013; Nolan, 2001; Embrechts et al., 1997). The family of univariate $\alpha$ -stable distributions is well defined as a parametric family of distributions indexed by four real parameters that vary freely in some intervals. The $\alpha$ -stable distribution can describe a degree of concentration of the observations around the mean as well as the asymmetry, regardless of sample size. Another reason to use this distribution class is the generalization of the Central Limit Theorem that ensures that the sum of stable random variables i.i.d. are stable. The $\alpha$ stability index is the most influential parameter in the applications (see Jia et al., 2018; Kateregga et al., 2017; Yang, 2012; Besbeas & Morgan, 2008; Yu, 2004; Nolan, 2001; Koutrouvelis, 1980; McCulloch, 1997; Press, 1972). Recently, Kateregga et al. (2017) shows that empirical characteristic function (ECF) based methods provide the best precision in estimating a wide range of $\alpha$ parameter; also, it is robust and provides better convergence. Here we develop the STNRW-ECF estimator for $\alpha$ , based on the ECF method and using STNRW (see Vargas & Pascoarelli, 2017). We analyze the proposed estimator using Monte Carlo simulations and prove its asymptotic Gaussian distribution.

Electrocardiography (ECG) is the process of recording the electrical activity of the heart over a period of time using electrodes placed on the skin. The time intervals between its various peaks, may contain useful information about the nature of disease afflicting the heart. The RR time series is the series of the heartbeat interval, where R is a peak point with respect to each heartbeat of the electrocardiography (ECG) wave, and RR is the interval between each successive R. To analyze this kind of data can be tiring and more prone to errors when interpreted by human beings, since there is a huge amount information to be processed. In order to cope with such problem, some works have been carried out regarding arrhythmia classification in EEG signals (see Albuquerque et al., 2018; Marwin, 2015). Here we proposed the STNRW-ECF estimator as an additional diagnostic tool that may provide an indication of cardiac arrhythmia. We also propose a test based on the principal wavelet shrinkage (see Donoho & Johnstone, 1995; Donoho et al., 1995) to confirm whether the STNRW-ECF estimator can be appropriately used.

The paper is organized as follows. Section 2 provides background on wavelet analysis and introduces the STNRW method. Section 3 describes the $\alpha$ -stable distributions and presents well-known estimator methods based on the empirical characteristic function to estimate $\alpha$ . The STNRW-ECF estimator and the test based on the principal wavelet shrinkage are proposed in Sections 4 and 5, respectively. The simulation study is presented in Section 6. In Section 7, we present the analysis of 14 healthy and 14 unhealthy (with cardiac arrhythimia) RR time series. Section 8 gives the conclusions.

2. Wavelet

In this section, we present a basic overview of wavelets, including discrete wavelet transforms (DWT), the wavelet shrinkage principal (see Donoho & Johnstone, 1994; Donoho et al., 1995; Vidakovic, 1999) and the STNRW method (see Vargas & Veiga, 2017).

Two functions are very important in wavelet analysis: the mother and father wavelets. These wavelets generate a family of functions that can reconstruct a signal.

(Mother and Father Wavelets).

A mother wavelet $\psi(\cdot)$ and a father wavelet (or scale function) $\phi(\cdot)$ are real functions $\psi,\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that

$\displaystyle\int_{\mathbb{R}}\psi(t)dt=0,\quad\int_{\mathbb{R}}\phi(t)dt=1$

and satisfy the integrability condition, that is, $\psi,\phi\in L^{2}(\mathbb{R})\cap L^{1}(\mathbb{R})$ .

Given the wavelets $\psi(\cdot)$ and $\phi(\cdot)$ , we construct wavelet sequences through translations and dilatations of mother and father wavelets, respectively, given by

$\displaystyle\psi_{j,k}(t)=2^{-j/2}\psi(2^{-j}t-k),$ $\displaystyle\phi_{j,k}(t)=2^{-j/2}\phi(2^{-j}t-k).$

The functions $\{\psi_{j,k}(\cdot),j,k\in\mathbb{Z}\}$ and $\{\phi_{j,k}(\cdot),j,k\in\mathbb{Z}\}$ form bases that are not necessarily orthogonal. The advantage of working with orthogonal bases is that they allow the perfect reconstruction of a signal from the coefficients of the transform. In general, the most used orthogonal wavelets are: Haar, Daublets, Symmlets and Coiflets.

(Discrete Wavelet Transform).

Let $\mathbf{X}=(X_{1},X_{2},\ldots,X_{n})^{\prime}$ be an i.i.d. random sample, with $J=\lfloor\log_{2}(n)\rfloor$ , where $\lfloor\cdot\rfloor$ indicates the integer part function. The discrete wavelet transform (DWT) of $\mathbf{X}$ , with respect to the mother wavelet $\psi(\cdot)$ , is defined as

$\displaystyle d_{j,k}=\sum_{t=1}^{n}X_{t}\psi_{j,k}(t),$ (1)

for all $j=1,2,\ldots,J$ and $k=1,2,\ldots,\lfloor\frac{n}{2^{j}}\rfloor$ . We can write the transform Eq. (1) in matrix form by

$\displaystyle\mathbf{d}_{j}=\mathbf{W}_{j}\mathbf{X},$ (2)

where $\mathbf{W}_{j}=(\psi_{j,k}(t))_{k,t}$ is a $\lfloor\frac{n}{2^{j}}\rfloor\times n$ matrix. Assuming appropriate boundary conditions, the transform is orthogonal and one can obtain the inverse discrete wavelet transform (IDWT) given by

$\displaystyle\mathbf{X}=\mathbf{W}^{\prime}\mathbf{d},$

where $\mathbf{W}^{\prime}$ denotes the transpose of $\mathbf{W}$ .

To compute the DWT, one does not actually perform the matrix multiplication Eq. (2). Instead, one uses a fast “pyramid” algorithm with complexity $O(n)$ (see Meyer, 1993).

2.1 Wavelet shrinkage procedure

Wavelet shrinkage usually refers to reconstructions obtained from the shrunk wavelet coefficients. Let the simplest regression model be

$\displaystyle y_{j}=f(t_{j})+\epsilon_{j},\quad j=1,2,\ldots,n,$ (3)

where the $t_{j}$ ’s are equally spaced points, and the $\epsilon_{j}$ ’s are independent Gaussian random variables with zero mean and variance $\sigma^{2}_{\epsilon}$ . Donoho and Johnstone (1994) and Donoho et al. (1995) have proposed a simple recipe based on thresholding in the wavelet domain. Their wavelet estimation procedure has three steps:

Take the discrete wavelet transform of the observations $y_{j}$ , for all $j\in\{1,2,\ldots,n\}$ , with $J$ levels. The result is the detailed and smooth wavelet coefficients $\mathbf{d}_{1},\mathbf{d}_{2},\ldots,\mathbf{d}_{J}$ , which are contaminated by noise.

At this stage, obtain the coefficients without noise. One shrinks the detailed coefficients at the $j$ finest scales to obtain new detailed coefficients $\tilde{d}_{1}\equiv\delta_{\lambda_{1}\hat{\sigma}_{1}}(d_{1}),\ldots,\tilde{d% }_{j}\equiv\delta_{\lambda_{j}\hat{\sigma}_{j}}(d_{j})$ . The shrinkage function $\delta_{c}(x)$ shrinks $x$ to zero and is parameterized by $c=\lambda\hat{\sigma}$ , where $\lambda$ is the threshold level, and $\hat{\sigma}$ is an estimate of the noise scale.

Apply the inverse discrete wavelet transform using the detailed coefficients $\tilde{d}_{1},\ldots,\tilde{d}_{j},d_{j+1},\ldots,d_{J},s_{J}$ , to recover the estimator of the function, $\hat{f}_{j}(\cdot)$ , for all $j\in\{1,2,\ldots,n\}$ .

The most common shrinkage functions are the soft shrinkage defined by

$\displaystyle\delta_{\lambda}^{S}(x)=\left\{\begin{array}[]{ll}0,&\text{if }|x% |\leqslant\lambda\\ \text{sign}(x)(|x|-\lambda),&\text{if }|x|>\lambda,\end{array}\right.$ (4)

and the hard (H) shrinkage defined by

$\displaystyle\delta_{\lambda}^{H}(x)=\left\{\begin{array}[]{ll}0,&\text{if }|x% |\leqslant\lambda\\ x,&\text{if }|x|>\lambda.\end{array}\right.$ (5)

The universal threshold that is defined by

$\displaystyle\lambda=\lambda_{j}=\hat{\sigma}\sqrt{2\log(n)},$

where $n$ is the signal length and $\hat{\sigma}$ is the estimated level of noise given by

$\displaystyle\hat{\sigma}=\frac{\textit{median}\{|d_{J-1,k}|:0\leqslant k<2^{J% }\}}{0.6745},$

where $J-1$ is the finest scale.

2.2 STNRW method

Here we present the Seismic Trace Noise Reduction by Wavelets and Double Threshold Estimation (STNRW) method, which was proposed by Vargas and Veiga in 2017, where the basic aim of this method is to separate the detailed wavelet coefficients into three groups. The first group contains detailed wavelet coefficients that are considered to be signal information, the second group contains detailed wavelet coefficients that are considered to be noisy information, and the third group contains detailed wavelet coefficients that belong to an indecision region, where the decision-making process is done through a classification function.

To apply the STNRW method to a given $N$ -length signal $\bm{y}$ ( $N=2^{J},J\in\mathbb{N}$ ), it is necessary to implement the following steps.

First, compute the DWT of the signal $\bm{y}$ to obtain the detailed wavelet coefficients $\{d_{j,k}\}$ , where $j\in\{0,\ldots,J-1\}$ and $k\in\{0,\ldots,2^{j}-1\}$ . Next, obtain the threshold

$\displaystyle\lambda_{1}=\hat{\sigma}\sqrt{2\log{N}},$ (6)

where $N$ is the signal length, and $\hat{\sigma}$ is the noise variance estimative given by $\hat{\sigma}=\textit{median}(\{d_{{J-1},k}:k=$ 0, 1, …, $2^{J-1}-1\})$ . Then, obtain the threshold

$\displaystyle\lambda_{2}=\mathop{\arg\min}_{\lambda\in[0,\lambda_{1}]}\{(N-1)% \hat{\sigma}+\sum_{j=0}^{J-1}\sum_{k=0}^{2^{j}-1}\min(d_{j,k}^{2},\lambda^{2})% -2\hat{\sigma}\sum_{j=0}^{J-1}\sum_{k=0}^{2^{j}-1}\mathbb{I}(|d_{j,k}|% \leqslant\lambda)\},$ (7)

where $\mathbb{I}$ is the indicator function. Next, apply the threshoding function given by

$\displaystyle\eta(d_{j,k},\lambda_{1},\lambda_{2})=\left\{\begin{array}[]{ll}d% _{j,k},&\text{if }|d_{j,k}|>\lambda_{1},\\ d_{j,k}\mathbb{I}(||d_{j,k}|-\lambda_{1}|<||d_{j,k}|-\lambda_{2}|),&\text{if }% \lambda_{2}\leqslant|d_{j,k}|\leqslant\lambda_{1},\\ 0,&\text{if }|d_{j,k}|<\lambda_{2}.\end{array}\right.$

Finally, to obtain the $\hat{\bm{y}}$ , apply the Inverse Discrete Wavelet Transform with $\eta(d_{j,k},\lambda_{1},\lambda_{2})$ instead of $\{d_{j,k}\}$ , for all values of $j$ and $k$ .

3. Stable distribution

The $\alpha$ -stable distribution has been widely used for fitting data in which extreme values are prevalent. In general, $\alpha$ -stable distributions do not have closed form expressions for their density and distribution functions but can be described easily by their characteristic function. These distributions are characterized by four parameters: the characteristic exponent $\alpha\in(0,2]$ , the skewness parameter $\beta\in[-1,1]$ , the dispersion parameter $\gamma>$ 0 and the location parameter $\delta\in\mathbb{R}$ . Many different approaches have been proposed for the characteristic exponent estimation, which determines the extreme value occurrence probability of the underlying distribution. Recently, Kateregga et al. (2017) shows that the empirical characteristic function (ECF) based methods provides the best precision in estimating a wide range of $\alpha$ parameter; also, it is robust and provides better convergence. We present, in this section, the $\alpha$ -stable distributions and the well-known ECF-based methods.

(Stable Distribution).

A given random variable $X$ has a 0-parameterization stable distribution, denoted by $X\sim S_{\alpha}(\beta,\gamma,\delta;0)$ , if its characteristic function is given by

$\displaystyle\varphi_{X}(t)=\left\{\begin{array}[]{ll}\exp\left(-\gamma^{% \alpha}|t|^{\alpha}\left[1+i\beta\tan(\frac{\pi\alpha}{2})(\text{sign}(t))(% \gamma(|t|)^{1-\alpha}-1)\right]+i\delta t\right),&\alpha\neq 1,\\ \exp\left(-\gamma|t|\left[1+i\beta\frac{2}{\pi}(\text{sign}(t))(\ln(|t|)+\ln(% \gamma))\right]+i\delta t\right),&\alpha=1.\end{array}\right.$ (8)

(Symmetric $\alpha$ -Stable Distribution).

A given random variable $X$ has a symmetric $\alpha$ -stable distribution with scale $\gamma$ , denoted by $X\sim\mathrm{S}\alpha\mathrm{S}(\gamma)$ , if its characteristic function is given by

$\displaystyle\varphi_{X}(t)=\mathbb{E}(e^{itX})=\exp(-\gamma|t|^{\alpha}),% \text{ for all }t\in\mathbb{R}.$

(Empirical Characteristic Function).

Let $\mathbf{X}=(X_{1},\ldots,X_{n})^{\prime}$ be a random sample. The empirical characteristic function (ecf) of $\mathbf{X}$ is defined by

$\displaystyle\hat{\varphi}_{X}(t)=\frac{1}{n}\sum_{j=1}^{n}e^{itX_{j}}.$ (9)

3.1 Sample characteristic function method

The sample characteristic function method is based on the fact that the characteristic function is an expectation (see Kateregga et al., 2017; Yang 2012 & Press, 1972). It is the expected value of the random variable $e^{itX}$ where $X$ follows an $\alpha$ -stable distribution.

From Definition 4, for all $\alpha$ ,

$\displaystyle\ln(|\varphi_{X}(t)|)=-\gamma|t|^{\alpha},\text{ for all }t\in% \mathbb{R}.$ (10)

Given $t_{1},t_{2}\in\mathbb{R}$ , $t_{1}\neq t_{2}$ ,

$\displaystyle\ln(|\varphi_{X}(t_{1})|)=-\gamma|t_{1}|^{\alpha}\text{ and }\ln(% |\varphi_{X}(t_{2})|)=-\gamma|t_{2}|^{\alpha}.$ (11)

By taking the ratio between the two equations in Eq. (11), we obtain

$\displaystyle\ln\left(\left|\frac{t_{1}}{t_{2}}\right|^{\alpha}\right)=\ln% \left(\frac{\ln(|\varphi_{X}(t_{1})|)}{\ln(|\varphi_{X}(t_{2})|)}\right).$

Hence,

$\displaystyle\alpha=\frac{\ln\left(\left|\frac{\ln(|\varphi_{X}(t_{1})|)}{\ln(% |{\varphi_{X}}(t_{2})|)}\right|\right)}{\ln\left(\left|\frac{t_{1}}{t_{2}}% \right|\right)}.$

Then, the Definition 6 below states an estimator for the stability index parameter.

( $\hat{\alpha}_{\text{ecf}}$ Estimator).

Let $\mathbf{X}=(X_{1},\ldots,X_{n})^{\prime}$ be a random sample. Given $t_{1},t_{2}\in\mathbb{R}$ , $t_{1}\neq t_{2}$ , the estimator for $\alpha$ , based on the empirical characteristic function of $\mathbf{X}$ , is defined by

$\displaystyle\hat{\alpha}_{\text{ecf}}=\frac{\ln\left(\left|\frac{\ln(|\hat{% \varphi}_{X}(t_{1})|)}{\ln(|\hat{\varphi}_{X}(t_{2})|)}\right|\right)}{\ln% \left(\left|\frac{t_{1}}{t_{2}}\right|\right)},$

where $\hat{\varphi}_{X}(\cdot)$ is given in Eq. (9).

In this estimation method, the values $t_{1}=$ 0.2 and $t_{2}=$ 0.8 are proposed in the simulation study (see Fama & Roll, 1971).

Another method is based on regression (see Koutrouvelis, 1980). From Eq. (10), for all $\alpha$ , we obtain

$\displaystyle\ln(-\ln(|\varphi_{X}(t)|))=\ln(\gamma)+\alpha\ln(|t|),\text{ for% all }t\in\mathbb{R}.$ (12)

Note that one can estimate $\alpha$ by regressing $y_{j}=\ln(-\ln(|\varphi_{X}(t_{j})|))$ on $x_{j}=\ln(|t_{j}|)$ in the model

$\displaystyle y_{j}=a+bx_{j}+\epsilon_{j},\text{ for }j\in\{1,\ldots,m\},$

for a set of points $\{t_{j}\}_{j=1}^{m}$ , where $a=\ln(\gamma)$ , $b=\alpha$ and $\epsilon_{j}$ denotes an error term. Therefore, one has the following definition.

( $\hat{\alpha}_{\text{reg}}$ Estimator).

Let $\mathbf{X}=(X_{1},\ldots,X_{n})^{\prime}$ be a random sample. Given a set of points $\{t_{j}\}_{j=1}^{m}$ , $t_{j}\in\mathbb{R}$ , the regression-type estimator based on the ECF of $\mathbf{X}$ for the $\alpha$ stability index parameter, denoted by $\hat{\alpha}_{\text{reg}}$ , is defined by

$\displaystyle\hat{\alpha}_{\text{reg}}=\frac{\sum_{j=1}^{m}(x_{j}-\bar{x})[\ln% (-\ln(|\hat{\varphi}_{X}(t_{j})|))]}{\sum_{j=1}^{m}(x_{j}-\bar{x})^{2}},$ (13)

where $x_{j}=\ln(|t_{j}|)$ , $\bar{x}=\frac{1}{m}\sum_{j=1}^{m}x_{j}$ and $\hat{\varphi}_{X}(\cdot)$ is given in Eq. (9).

One possible set is $t_{j}=\frac{\pi j}{25}$ , $j=1,\ldots,8$ (see Yang, 2012; Koutrouvelis, 1980).

4. The

\hat{\alpha}_{\text{STNRW-ECF}}

estimator

In this section, we develop an estimator for $\alpha$ denoted by $\hat{\alpha}_{\text{STNRW-ECF}}$ , based on the ECF method and on the STNRW method (see Subsection 2.2). We analyze the $\hat{\alpha}_{\text{STNRW-ECF}}$ estimator using Monte Carlo simulations (see Section 6) and prove its asymptotic Gaussian distribution (see Theorem 1).

Given a random sample $\mathbf{X}=(X_{1},\ldots,X_{n})^{\prime}$ , let $\underline{\alpha}=(\alpha_{1},\alpha_{2}\ldots,\alpha_{m})$ with

$\displaystyle\alpha_{i}=\frac{\ln\left(\left|\frac{\ln(|\hat{\varphi}_{X}(t_{(% 1,i)})|)}{\ln(|\hat{\varphi}_{X}(t_{(2,i)})|)}\right|\right)}{\ln\left(\left|% \frac{t_{(1,i)}}{t_{(2,i)}}\right|\right)},$ (14)

where $\hat{\varphi}_{X}(\cdot)$ is given in Eq. (9), $t_{(1,i)}=0.2+\frac{0.01i}{m}$ and $t_{(2,i)}=0.8+\frac{0.01i}{m}$ , for all $i\in\{1,\ldots,m\}$ . To obtain the $\hat{\alpha}_{\text{STNRW-ECF}}$ estimator, it is necessary apply the STNRW method in $\bm{y}=\underline{\alpha}=(\alpha_{1},\alpha_{2}\ldots,\alpha_{m})$ to obtain $\hat{\bm{y}}=\hat{\underline{\alpha}}=(\hat{\alpha}_{1},\hat{\alpha}_{2}\ldots% ,\hat{\alpha}_{m})$ . Then, the estimator for $\alpha$ index, applying the STNRW procedure, is defined as

$\displaystyle\hat{\alpha}_{\text{STNRW-ECF}}=\sum\limits_{i=1}^{m}\frac{\hat{% \alpha}_{i}}{m}.$

Here we consider $m=$ 64.

.

The proposed estimator $\hat{\alpha}_{\text{STNRW-ECF}}$ is consistent, that is, $\hat{\alpha}_{\text{STNRW-ECF}}$ converges in probability to $\alpha$ .

Proof..

By Press (1972), when $m\to\infty$ for all $i\in\mathds{N}$ we have $E(\alpha_{i})=\alpha$ and $\textit{Var}(\alpha_{i})=\frac{\eta_{i}^{2}}{m}$ , where $\eta_{i}^{2}$ does not depend of $m$ . By the STNRW method, we have $E(\hat{\alpha}_{i})=\alpha_{i}$ and $\textit{Var}(\hat{\alpha}_{i})\leqslant\textit{Var}(\alpha_{i})$ , for all $i\in\mathds{N}$ .

Considering Jensen’s inequality and the Cauchy-Schwarz inequality, we obtain that

$\displaystyle\lim_{m\to\infty}\textit{Cov}(\hat{\alpha}_{1},\hat{\alpha}_{2})% \leqslant\lim_{m\to\infty}|\textit{Cov}(\hat{\alpha}_{1},\hat{\alpha}_{2})|% \leqslant\lim_{m\to\infty}\left(\sqrt{\textit{Var}(\hat{\alpha}_{1})\textit{% Var}(\hat{\alpha}_{2})}\right)$ (15) $\displaystyle\leqslant\lim_{m\to\infty}\left(\sqrt{\textit{Var}({\alpha}_{1})% \textit{Var}({\alpha}_{2})}\right)=\lim_{m\to\infty}\sqrt{\left(\frac{\eta_{1}% ^{2}}{m}\right)\left(\frac{\eta_{2}^{2}}{m}\right)}=0,$

then

$\displaystyle\lim_{m\to\infty}\textit{Var}(\hat{\alpha}_{\text{STNRW-ECF}})=% \lim_{m\to\infty}\textit{Var}\left(\frac{\sum_{i=1}^{m}\hat{\alpha}_{i}}{m}% \right)=\lim_{m\to\infty}\left(\frac{\sum\limits_{i=1}^{m}\textit{Var}(\hat{% \alpha}_{i})}{m^{2}}+\frac{2\sum\limits_{j=1}^{m}\sum\limits_{i<j}\textit{Cov}% (\hat{\alpha}_{i},\hat{\alpha}_{j})}{m^{2}}\right)\leqslant\lim_{m\to\infty}% \left(\frac{\sum\limits_{i=1}^{m}\textit{Var}(\alpha_{i})}{m^{2}}+\frac{2\sum% \limits_{j=1}^{m}\sum\limits_{i<j}\textit{Cov}(\hat{\alpha}_{i},\hat{\alpha}_{% j})}{m^{2}}\right)\quad(\textit{Var}(\hat{\alpha}_{i})\leqslant\textit{Var}(% \alpha_{i}))\leqslant\lim_{m\to\infty}\left(\frac{m\left(\mathop{\max}\limits_% {1\leqslant i\leqslant m}\{\eta_{i}^{2}\}\right)}{m^{3}}+\frac{2m-1}{m}\textit% {Cov}(\hat{\alpha}_{1},\hat{\alpha}_{2})\right)\quad\text{(exchangeability)}% \leqslant 0\text{(by \Eq{15}{})}.$

Therefore,

$\displaystyle\lim_{m\to\infty}\textit{Var}(\hat{\alpha}_{\text{STNRW-ECF}})=0.$ (16)

We can note that $\frac{\sum_{i=1}^{m}\alpha_{i}}{m}$ converges in probability to $\alpha$ , then

$\displaystyle\lim_{m\to\infty}E(\hat{\alpha}_{\text{STNRW-ECF}})=\lim_{m\to% \infty}\frac{\sum_{i=1}^{m}E(\hat{\alpha}_{i})}{m}=\lim_{m\to\infty}\frac{\sum% _{i=1}^{m}\alpha_{i}}{m}=\alpha.$ (17)

Therefore, by Eqs (16) and (17), we conclude that $\hat{\alpha}_{\text{STNRW-ECF}}$ is consistent, that is, $\hat{\alpha}_{\text{STNRW-ECF}}$ converges in probability to $\alpha$ . ∎

5. Test L

Let $\{X_{t}\}_{t=1}^{n}$ be an RR time series. To apply the proposed test to $\{X_{t}\}_{t=1}^{n}$ , it is necessary to follow the following steps. In the first step, for each $t\in\{1,2,\ldots,n-1\}$ , we calculate $Z_{t}=|X_{t+1}-X_{t}|$ . In the second step, we compute

$\displaystyle W_{t}=\left\{\begin{array}[]{ll}0,&\text{ if }Z_{t}<2\\ 1,&\text{ elsewhere.}\\ \end{array}\right.$ (18)

In the third step, for each $t\in\{1,2,\ldots,n-1\}$ , we calculate $Y_{t}=\sum_{i=1}^{t}(W_{i}-\overline{W})$ , where $\overline{W}$ is the average value of $\{W_{t}\}_{t=1}^{n-1}$ . The fourth step consists of transforming the observations $Y_{t}$ , $t\in\{1,2,\ldots,n\}$ , into the symmlet wavelet “s8” domain by applying a discrete wavelet transform (see Definition 2), with level $J=\lfloor\log_{2}(n)\rfloor$ , to obtain a sequence of wavelet coefficients $\mathbf{d}_{4},\mathbf{d}_{5},\ldots,\mathbf{d}_{J}$ . Then, the wavelet coefficients are shrunk towards zero, to obtain new detail coefficients $\tilde{d}_{4}\equiv\delta_{-\lambda}^{H}(d_{4}),\ldots,\tilde{d}_{J}\equiv% \delta_{-\lambda}^{H}(d_{J})$ , where $\lambda=\hat{\sigma}\sqrt{2\log(n)}$ , $\hat{\sigma}$ is the estimated level of noise given by $\hat{\sigma}=\frac{\textit{median}\{|d_{J-1,k}|:0\leqslant k<2^{J}\}}{0.6745}$ , and the $\delta_{\lambda}^{H}(x)$ is the hard (H) shrinkage function (see Section 2). Finally, we apply the inverse discrete wavelet transform, to obtain the wavelet shrinkage estimator $\hat{Y}_{t}$ of $Y_{t}$ , for all $t\in\{1,2,\ldots,n-1\}$ . The last step consists of calculating $L=\frac{1}{n-1}\sum_{t=1}^{n-1}\hat{Y}_{t}$ .

Finally,

•

If $L=0$ , then $\{X_{t}\}_{t=1}^{n}$ is a healthy RR time series,

•

if $L=1$ , then $\{X_{t}\}_{t=1}^{n}$ is an unhealthy RR times series.

6. Simulations

We perform Monte Carlo simulations in order to compare the performance of the new estimator $\hat{\alpha}_{\text{STNRW-ECF}}$ with the existing estimators $\hat{\alpha}_{\text{ecf}}$ and $\hat{\alpha}_{\text{reg}}$ . We simulate symmetric $\alpha$ -stable time series, with 200 replications and, sample sizes $n\in\{1000,10000\}$ and $\alpha\in\{0.1,0.2,0.3,0.4,0.5,\ldots,2.0\}$ . We calculate the empirical values of the mean, its bias, its mean square error (mse), and its variance (var) values, for each estimator. We observe that the $\hat{\alpha}_{\text{ecf}}$ , $\hat{\alpha}_{\text{reg}}$ , and $\hat{\alpha}_{\text{STNRW-ECF}}$ estimators provide good and similar results. However, as we can note, when we analyze the bias, the $\hat{\alpha}_{\text{STNRW-ECF}}$ estimator outperforms the $\hat{\alpha}_{\text{ecf}}$ and $\hat{\alpha}_{\text{reg}}$ in, respectively, 65% and 60% of the simulation runs (Table 1). Already in Table 2, the $\hat{\alpha}_{\text{STNRW-ECF}}$ estimator outperforms the $\hat{\alpha}_{\text{ecf}}$ and $\hat{\alpha}_{\text{reg}}$ in, respectively, 60% and 65% of the simulation runs.

.

i)
In Tables 1 and 2 , all values of the mean, bias, mean square error, and variance were rounded to four decimal places.
ii)
We know that $\textit{mse}=\textit{var}+(\textit{bias})^{2}$ . Because of the rounding to four decimal places, for all cases in Tables 1 and 2 , we obtain $(\textit{bias})^{2}=$ 0; therefore, $\textit{mse}=\textit{var}$ .

Table 1
Estimation results for symmetric $\alpha$ -stable time series, when $n\in\{1000\}$

$\alpha$ Estimator mean bias mse var

0.1 $\hat{\alpha}_{\text{ecf}}$ 0.1049 0.0049 0.0025 0.0025

$\hat{\alpha}_{\text{reg}}$ 0.1049 0.0049 0.0025 0.0025

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.1014 0.0014 0.0011 0.0011

0.2 $\hat{\alpha}_{\text{ecf}}$ 0.2007 0.0007 0.0024 0.0024

$\hat{\alpha}_{\text{reg}}$ 0.2006 0.0006 0.0024 0.0024

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.2018 0.0018 0.0019 0.0019

0.3 $\hat{\alpha}_{\text{ecf}}$ 0.2975 $-$ 0.0025 0.0023 0.0023

$\hat{\alpha}_{\text{reg}}$ 0.2976 $-$ 0.0024 0.0023 0.0023

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.2989 $-$ 0.0011 0.0022 0.0022

0.4 $\hat{\alpha}_{\text{ecf}}$ 0.4048 0.0048 0.0024 0.0024

$\hat{\alpha}_{\text{reg}}$ 0.4048 0.0048 0.0024 0.0024

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.4039 0.0039 0.0022 0.0022

0.5 $\hat{\alpha}_{\text{ecf}}$ 0.5024 0.0024 0.0027 0.0027

$\hat{\alpha}_{\text{reg}}$ 0.5023 0.0023 0.0027 0.0027

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.5004 0.0004 0.0027 0.0027

0.6 $\hat{\alpha}_{\text{ecf}}$ 0.6022 0.0022 0.0027 0.0027

$\hat{\alpha}_{\text{reg}}$ 0.6023 0.0023 0.0027 0.0027

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.6014 0.0014 0.0028 0.0028

0.7 $\hat{\alpha}_{\text{ecf}}$ 0.7026 0.0026 0.0026 0.0026

$\hat{\alpha}_{\text{reg}}$ 0.7025 0.0025 0.0026 0.0026

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.7027 0.0027 0.0027 0.0027

0.8 $\hat{\alpha}_{\text{ecf}}$ 0.7999 $-$ 0.0001 0.0026 0.0026

$\hat{\alpha}_{\text{reg}}$ 0.7999 $-$ 0.0001 0.0026 0.0026

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.7990 $-$ 0.0010 0.0028 0.0028

0.9 $\hat{\alpha}_{\text{ecf}}$ 0.9005 0.0005 0.0031 0.0031

$\hat{\alpha}_{\text{reg}}$ 0.9006 0.0006 0.0031 0.0031

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.9018 0.0018 0.0032 0.0032

1 $\hat{\alpha}_{\text{ecf}}$ 0.9968 $-$ 0.0032 0.0032 0.0032

$\hat{\alpha}_{\text{reg}}$ 0.9968 $-$ 0.0032 0.0032 0.0032

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.9966 $-$ 0.0034 0.0034 0.0034

1.1 $\hat{\alpha}_{\text{ecf}}$ 1.0989 $-$ 0.0011 0.0032 0.0032

$\hat{\alpha}_{\text{reg}}$ 1.0989 $-$ 0.0011 0.0032 0.0032

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.0990 $-$ 0.0010 0.0033 0.0033

1.2 $\hat{\alpha}_{\text{ecf}}$ 1.1972 $-$ 0.0028 0.0035 0.0035

$\hat{\alpha}_{\text{reg}}$ 1.1970 $-$ 0.0030 0.0035 0.0035

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.1972 $-$ 0.0028 0.0034 0.0034

1.3 $\hat{\alpha}_{\text{ecf}}$ 1.3022 0.0022 0.0036 0.0036

$\hat{\alpha}_{\text{reg}}$ 1.3023 0.0023 0.0036 0.0036

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.3016 0.0016 0.0036 0.0036

1.4 $\hat{\alpha}_{\text{ecf}}$ 1.4052 0.0052 0.0035 0.0035

$\hat{\alpha}_{\text{reg}}$ 1.4053 0.0053 0.0035 0.0035

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.4049 0.0049 0.0035 0.0035

1.5 $\hat{\alpha}_{\text{ecf}}$ 1.5035 0.0035 0.0040 0.0040

$\hat{\alpha}_{\text{reg}}$ 1.5034 0.0034 0.0040 0.0040

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.5035 0.0035 0.0039 0.0039

1.6 $\hat{\alpha}_{\text{ecf}}$ 1.6015 0.0015 0.0031 0.0031

$\hat{\alpha}_{\text{reg}}$ 1.6015 0.0015 0.0031 0.0031

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.6018 0.0018 0.0031 0.0031

1.7 $\hat{\alpha}_{\text{ecf}}$ 1.6999 $-$ 0.0001 0.0030 0.0030

$\hat{\alpha}_{\text{reg}}$ 1.6999 $-$ 0.0001 0.0030 0.0030

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.6998 $-$ 0.0002 0.0030 0.0030

1.8 $\hat{\alpha}_{\text{ecf}}$ 1.8018 0.0018 0.0023 0.0024

$\hat{\alpha}_{\text{reg}}$ 1.8019 0.0019 0.0024 0.0024

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.8014 0.0014 0.0023 0.0023

1.9 $\hat{\alpha}_{\text{ecf}}$ 1.8996 $-$ 0.0004 0.0016 0.0016

$\hat{\alpha}_{\text{reg}}$ 1.8996 $-$ 0.0004 0.0016 0.0016

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.8997 $-$ 0.0003 0.0015 0.0015

2 $\hat{\alpha}_{\text{ecf}}$ 2.0004 0.0004 0.0001 0.0001

$\hat{\alpha}_{\text{reg}}$ 2.0004 0.0004 0.0001 0.0001

$\hat{\alpha}_{\text{STNRW-ECF}}$ 2.0004 0.0004 0.0001 0.0001

Table 2
Estimation results for symmetric $\alpha$ -stable time series, when $n\in\{10000\}$

$\alpha$ Estimator mean bias mse var

0.1 $\hat{\alpha}_{\text{ecf}}$ 0.0988 $-$ 0.0012 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 0.0990 $-$ 0.0010 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.1005 0.0005 0.0001 0.0001

0.2 $\hat{\alpha}_{\text{ecf}}$ 0.2043 0.0043 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 0.2044 0.0044 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.2029 0.0029 0.0002 0.0002

0.3 $\hat{\alpha}_{\text{ecf}}$ 0.2979 $-$ 0.0021 0.0002 0.0002

$\hat{\alpha}_{\text{reg}}$ 0.2982 $-$ 0.0018 0.0002 0.0002

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.2989 $-$ 0.0011 0.0002 0.0002

0.4 $\hat{\alpha}_{\text{ecf}}$ 0.3990 $-$ 0.0010 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 0.3990 $-$ 0.0010 0.0002 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.4025 0.0025 0.0002 0.0002

0.5 $\hat{\alpha}_{\text{ecf}}$ 0.5019 0.0019 0.0002 0.0002

$\hat{\alpha}_{\text{reg}}$ 0.5019 0.0019 0.0002 0.0002

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.4999 $-$ 0.0001 0.0002 0.0002

0.6 $\hat{\alpha}_{\text{ecf}}$ 0.6044 0.0044 0.0004 0.0003

$\hat{\alpha}_{\text{reg}}$ 0.6044 0.0044 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.6038 0.0038 0.0003 0.0003

0.7 $\hat{\alpha}_{\text{ecf}}$ 0.6982 $-$ 0.0018 0.0002 0.0002

$\hat{\alpha}_{\text{reg}}$ 0.6984 $-$ 0.0016 0.0002 0.0002

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.7000 0.0000 0.0002 0.0002

0.8 $\hat{\alpha}_{\text{ecf}}$ 0.7984 $-$ 0.0016 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 0.7981 $-$ 0.0019 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.7991 $-$ 0.0009 0.0003 0.0003

0.9 $\hat{\alpha}_{\text{ecf}}$ 0.8986 $-$ 0.0014 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 0.8985 $-$ 0.0015 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 0.8979 $-$ 0.0021 0.0003 0.0003

1 $\hat{\alpha}_{\text{ecf}}$ 1.0005 0.0005 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 1.0005 0.0005 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.0008 0.0008 0.0003 0.0003

1.1 $\hat{\alpha}_{\text{ecf}}$ 1.0993 $-$ 0.0007 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 1.0993 $-$ 0.0007 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.1005 0.0005 0.0003 0.0003

1.2 $\hat{\alpha}_{\text{ecf}}$ 1.1998 $-$ 0.0002 0.0004 0.0004

$\hat{\alpha}_{\text{reg}}$ 1.2001 0.0001 0.0004 0.0004

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.2000 0.0000 0.0004 0.0004

1.3 $\hat{\alpha}_{\text{ecf}}$ 1.3008 0.0008 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 1.3011 0.0011 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.3005 0.0005 0.0003 0.0003

1.4 $\hat{\alpha}_{\text{ecf}}$ 1.3988 $-$ 0.0012 0.0004 0.0004

$\hat{\alpha}_{\text{reg}}$ 1.3991 $-$ 0.0009 0.0004 0.0004

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.3985 $-$ 0.0015 0.0004 0.0004

1.5 $\hat{\alpha}_{\text{ecf}}$ 1.5007 0.0007 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 1.5007 0.0007 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.5001 0.0001 0.0003 0.0003

1.6 $\hat{\alpha}_{\text{ecf}}$ 1.6009 0.0009 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 1.6010 0.0010 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.6014 0.0014 0.0003 0.0003

1.7 $\hat{\alpha}_{\text{ecf}}$ 1.7008 0.0008 0.0004 0.0004

$\hat{\alpha}_{\text{reg}}$ 1.7006 0.0006 0.0004 0.0004

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.7012 0.0012 0.0004 0.0004

1.8 $\hat{\alpha}_{\text{ecf}}$ 1.7988 $-$ 0.0012 0.0003 0.0003

$\hat{\alpha}_{\text{reg}}$ 1.7986 $-$ 0.0014 0.0003 0.0003

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.7989 $-$ 0.0011 0.0003 0.0003

1.9 $\hat{\alpha}_{\text{ecf}}$ 1.8998 $-$ 0.0002 0.0002 0.0002

$\hat{\alpha}_{\text{reg}}$ 1.8995 $-$ 0.0005 0.0002 0.0002

$\hat{\alpha}_{\text{STNRW-ECF}}$ 1.8997 $-$ 0.0003 0.0002 0.0002

2 $\hat{\alpha}_{\text{ecf}}$ 2.0008 0.0008 0.0000 0.0000

$\hat{\alpha}_{\text{reg}}$ 2.0007 0.0007 0.0000 0.0000

$\hat{\alpha}_{\text{STNRW-ECF}}$ 2.0009 0.0009 0.0000 0.0000

7. Application

$\alpha$	Estimator	mean	bias	mse	var
0.1	$\hat{\alpha}_{\text{ecf}}$	0.1049	0.0049	0.0025	0.0025
	$\hat{\alpha}_{\text{reg}}$	0.1049	0.0049	0.0025	0.0025
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.1014	0.0014	0.0011	0.0011
0.2	$\hat{\alpha}_{\text{ecf}}$	0.2007	0.0007	0.0024	0.0024
	$\hat{\alpha}_{\text{reg}}$	0.2006	0.0006	0.0024	0.0024
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.2018	0.0018	0.0019	0.0019
0.3	$\hat{\alpha}_{\text{ecf}}$	0.2975	$-$ 0.0025	0.0023	0.0023
	$\hat{\alpha}_{\text{reg}}$	0.2976	$-$ 0.0024	0.0023	0.0023
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.2989	$-$ 0.0011	0.0022	0.0022
0.4	$\hat{\alpha}_{\text{ecf}}$	0.4048	0.0048	0.0024	0.0024
	$\hat{\alpha}_{\text{reg}}$	0.4048	0.0048	0.0024	0.0024
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.4039	0.0039	0.0022	0.0022
0.5	$\hat{\alpha}_{\text{ecf}}$	0.5024	0.0024	0.0027	0.0027
	$\hat{\alpha}_{\text{reg}}$	0.5023	0.0023	0.0027	0.0027
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.5004	0.0004	0.0027	0.0027
0.6	$\hat{\alpha}_{\text{ecf}}$	0.6022	0.0022	0.0027	0.0027
	$\hat{\alpha}_{\text{reg}}$	0.6023	0.0023	0.0027	0.0027
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.6014	0.0014	0.0028	0.0028
0.7	$\hat{\alpha}_{\text{ecf}}$	0.7026	0.0026	0.0026	0.0026
	$\hat{\alpha}_{\text{reg}}$	0.7025	0.0025	0.0026	0.0026
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.7027	0.0027	0.0027	0.0027
0.8	$\hat{\alpha}_{\text{ecf}}$	0.7999	$-$ 0.0001	0.0026	0.0026
	$\hat{\alpha}_{\text{reg}}$	0.7999	$-$ 0.0001	0.0026	0.0026
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.7990	$-$ 0.0010	0.0028	0.0028
0.9	$\hat{\alpha}_{\text{ecf}}$	0.9005	0.0005	0.0031	0.0031
	$\hat{\alpha}_{\text{reg}}$	0.9006	0.0006	0.0031	0.0031
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.9018	0.0018	0.0032	0.0032
1	$\hat{\alpha}_{\text{ecf}}$	0.9968	$-$ 0.0032	0.0032	0.0032
	$\hat{\alpha}_{\text{reg}}$	0.9968	$-$ 0.0032	0.0032	0.0032
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.9966	$-$ 0.0034	0.0034	0.0034
1.1	$\hat{\alpha}_{\text{ecf}}$	1.0989	$-$ 0.0011	0.0032	0.0032
	$\hat{\alpha}_{\text{reg}}$	1.0989	$-$ 0.0011	0.0032	0.0032
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.0990	$-$ 0.0010	0.0033	0.0033
1.2	$\hat{\alpha}_{\text{ecf}}$	1.1972	$-$ 0.0028	0.0035	0.0035
	$\hat{\alpha}_{\text{reg}}$	1.1970	$-$ 0.0030	0.0035	0.0035
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.1972	$-$ 0.0028	0.0034	0.0034
1.3	$\hat{\alpha}_{\text{ecf}}$	1.3022	0.0022	0.0036	0.0036
	$\hat{\alpha}_{\text{reg}}$	1.3023	0.0023	0.0036	0.0036
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.3016	0.0016	0.0036	0.0036
1.4	$\hat{\alpha}_{\text{ecf}}$	1.4052	0.0052	0.0035	0.0035
	$\hat{\alpha}_{\text{reg}}$	1.4053	0.0053	0.0035	0.0035
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.4049	0.0049	0.0035	0.0035
1.5	$\hat{\alpha}_{\text{ecf}}$	1.5035	0.0035	0.0040	0.0040
	$\hat{\alpha}_{\text{reg}}$	1.5034	0.0034	0.0040	0.0040
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.5035	0.0035	0.0039	0.0039
1.6	$\hat{\alpha}_{\text{ecf}}$	1.6015	0.0015	0.0031	0.0031
	$\hat{\alpha}_{\text{reg}}$	1.6015	0.0015	0.0031	0.0031
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.6018	0.0018	0.0031	0.0031
1.7	$\hat{\alpha}_{\text{ecf}}$	1.6999	$-$ 0.0001	0.0030	0.0030
	$\hat{\alpha}_{\text{reg}}$	1.6999	$-$ 0.0001	0.0030	0.0030
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.6998	$-$ 0.0002	0.0030	0.0030
1.8	$\hat{\alpha}_{\text{ecf}}$	1.8018	0.0018	0.0023	0.0024
	$\hat{\alpha}_{\text{reg}}$	1.8019	0.0019	0.0024	0.0024
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.8014	0.0014	0.0023	0.0023
1.9	$\hat{\alpha}_{\text{ecf}}$	1.8996	$-$ 0.0004	0.0016	0.0016
	$\hat{\alpha}_{\text{reg}}$	1.8996	$-$ 0.0004	0.0016	0.0016
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.8997	$-$ 0.0003	0.0015	0.0015
2	$\hat{\alpha}_{\text{ecf}}$	2.0004	0.0004	0.0001	0.0001
	$\hat{\alpha}_{\text{reg}}$	2.0004	0.0004	0.0001	0.0001
	$\hat{\alpha}_{\text{STNRW-ECF}}$	2.0004	0.0004	0.0001	0.0001

$\alpha$	Estimator	mean	bias	mse	var
0.1	$\hat{\alpha}_{\text{ecf}}$	0.0988	$-$ 0.0012	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	0.0990	$-$ 0.0010	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.1005	0.0005	0.0001	0.0001
0.2	$\hat{\alpha}_{\text{ecf}}$	0.2043	0.0043	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	0.2044	0.0044	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.2029	0.0029	0.0002	0.0002
0.3	$\hat{\alpha}_{\text{ecf}}$	0.2979	$-$ 0.0021	0.0002	0.0002
	$\hat{\alpha}_{\text{reg}}$	0.2982	$-$ 0.0018	0.0002	0.0002
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.2989	$-$ 0.0011	0.0002	0.0002
0.4	$\hat{\alpha}_{\text{ecf}}$	0.3990	$-$ 0.0010	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	0.3990	$-$ 0.0010	0.0002	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.4025	0.0025	0.0002	0.0002
0.5	$\hat{\alpha}_{\text{ecf}}$	0.5019	0.0019	0.0002	0.0002
	$\hat{\alpha}_{\text{reg}}$	0.5019	0.0019	0.0002	0.0002
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.4999	$-$ 0.0001	0.0002	0.0002
0.6	$\hat{\alpha}_{\text{ecf}}$	0.6044	0.0044	0.0004	0.0003
	$\hat{\alpha}_{\text{reg}}$	0.6044	0.0044	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.6038	0.0038	0.0003	0.0003
0.7	$\hat{\alpha}_{\text{ecf}}$	0.6982	$-$ 0.0018	0.0002	0.0002
	$\hat{\alpha}_{\text{reg}}$	0.6984	$-$ 0.0016	0.0002	0.0002
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.7000	0.0000	0.0002	0.0002
0.8	$\hat{\alpha}_{\text{ecf}}$	0.7984	$-$ 0.0016	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	0.7981	$-$ 0.0019	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.7991	$-$ 0.0009	0.0003	0.0003
0.9	$\hat{\alpha}_{\text{ecf}}$	0.8986	$-$ 0.0014	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	0.8985	$-$ 0.0015	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	0.8979	$-$ 0.0021	0.0003	0.0003
1	$\hat{\alpha}_{\text{ecf}}$	1.0005	0.0005	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	1.0005	0.0005	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.0008	0.0008	0.0003	0.0003
1.1	$\hat{\alpha}_{\text{ecf}}$	1.0993	$-$ 0.0007	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	1.0993	$-$ 0.0007	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.1005	0.0005	0.0003	0.0003
1.2	$\hat{\alpha}_{\text{ecf}}$	1.1998	$-$ 0.0002	0.0004	0.0004
	$\hat{\alpha}_{\text{reg}}$	1.2001	0.0001	0.0004	0.0004
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.2000	0.0000	0.0004	0.0004
1.3	$\hat{\alpha}_{\text{ecf}}$	1.3008	0.0008	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	1.3011	0.0011	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.3005	0.0005	0.0003	0.0003
1.4	$\hat{\alpha}_{\text{ecf}}$	1.3988	$-$ 0.0012	0.0004	0.0004
	$\hat{\alpha}_{\text{reg}}$	1.3991	$-$ 0.0009	0.0004	0.0004
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.3985	$-$ 0.0015	0.0004	0.0004
1.5	$\hat{\alpha}_{\text{ecf}}$	1.5007	0.0007	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	1.5007	0.0007	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.5001	0.0001	0.0003	0.0003
1.6	$\hat{\alpha}_{\text{ecf}}$	1.6009	0.0009	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	1.6010	0.0010	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.6014	0.0014	0.0003	0.0003
1.7	$\hat{\alpha}_{\text{ecf}}$	1.7008	0.0008	0.0004	0.0004
	$\hat{\alpha}_{\text{reg}}$	1.7006	0.0006	0.0004	0.0004
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.7012	0.0012	0.0004	0.0004
1.8	$\hat{\alpha}_{\text{ecf}}$	1.7988	$-$ 0.0012	0.0003	0.0003
	$\hat{\alpha}_{\text{reg}}$	1.7986	$-$ 0.0014	0.0003	0.0003
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.7989	$-$ 0.0011	0.0003	0.0003
1.9	$\hat{\alpha}_{\text{ecf}}$	1.8998	$-$ 0.0002	0.0002	0.0002
	$\hat{\alpha}_{\text{reg}}$	1.8995	$-$ 0.0005	0.0002	0.0002
	$\hat{\alpha}_{\text{STNRW-ECF}}$	1.8997	$-$ 0.0003	0.0002	0.0002
2	$\hat{\alpha}_{\text{ecf}}$	2.0008	0.0008	0.0000	0.0000
	$\hat{\alpha}_{\text{reg}}$	2.0007	0.0007	0.0000	0.0000
	$\hat{\alpha}_{\text{STNRW-ECF}}$	2.0009	0.0009	0.0000	0.0000

The RR time series is the series of the heartbeat interval, where R is a peak point with respect to each heartbeat of the electrocardiography (ECG) wave, and RR is the interval between each successive R. The discrete series of successive RR intervals (the tachogram) is the simplest signal that can be used to characterize heart rate variability (HRV) and has been applied in various clinical situations (see Marwin, 2015; Taizhi et al., 2014). Cardiac arrhythmia is defined as a change in electrical activity within the heart which manifests as irregular heartbeats. In extreme cases, arrhythmia can be induced by damaged cardiac tissue or abnormal cardiac anatomy, which may lead to a stroke or heart attack.

In this section, in view of the $\hat{\alpha}_{\text{STNRW-ECF}}$ estimator (see Section 4) and the test L based on the principal wavelet shrinkage proposed on Section 5, we analyse the RR time series of 14 healthy and 14 unhealthy (with cardiac arrhythimia) individuals selected from databases available from PhysioBank (https://www.physionet.org/cgi-bin/atm/ATM) in order to identify differences between healthy and unhealthy RR time series. Within the PhysionBank, for the healthy RR time series, we use the MIT-BIH Normal Sinus Rhythm Database (nsrdb) ; for unhealthy RR time series, we use the MIT-BIH Arrhythmia Database (mitdb) . For each RR time series in Table 3, we test the following hypothesis:

$\displaystyle\left\{\begin{array}[]{ll}H_{0}:&\alpha=2,\\ H_{1}:&\alpha<2.\end{array}\right.$ (19)

For Table 3, we consider the following notation:

${}^{*}$ : Rejects $H_{0}$ at 1% significance level.

From Table 3, one observes that for healthy RR time series, the estimates of $\hat{\alpha}_{\text{STNRW-ECF}}$ is always less than 1.8, while for unhealthy RR times series, the estimates of $\hat{\alpha}_{\text{STNRW-ECF}}$ is always approximately equal to 2. For each RR time series, this conclusion is statistically significant to 1%. We can also see in Table 3 that $L=$ 0 for all healthy RR time series, while $L=$ 1 for all unhealthy time series considered here, confirming the decision about the $\hat{\alpha}_{\text{STNRW-ECF}}$ estimates.

Table 3

Estimation results of the $\hat{\alpha}_{\text{STNRW-ECF}}$ and the test L in RR time series of healthy and unhealthy records

Healthy record in the nsrdb			Unhealthy record in the mitdb
Record’s number	$\hat{\alpha}_{\text{STNRW-ECF}}$	$L$	Record’s number	$\hat{\alpha}_{\text{STNRW-ECF}}$	$L$
c16272	$0.6435^{**}$	0	102	1.9962	1
c16420	$1.2938^{**}$	0	103	1.9992	1
c16483	$1.7728^{**}$	0	104	1.9993	1
c16786	$1.7208^{**}$	0	105	1.9916	1
c16795	$1.6363^{**}$	0	106	2.0050	1
c17052	$1.1093^{**}$	0	107	2.0050	1
c17453	$1.7015^{**}$	0	108	1.9848	1
c18177	$0.8518^{**}$	0	109	1.9848	1
c18184	$0.7603^{**}$	0	111	1.9992	1
c19088	$0.6171^{**}$	0	112	1.9980	1
c19090	$1.0001^{**}$	0	113	1.9966	1
c19093	$1.2844^{**}$	0	114	1.9817	1
c19140	$0.9581^{**}$	0	115	1.9990	1
c19830	$0.8248^{**}$	0	116	1.9961	1

Note: ${}^{**}$ denotes rejection of $H_{0}$ at the 1% significance level.

8. Conclusions

Here we develop an estimator $\hat{\alpha}_{\text{STNRW-ECF}}$ for $\alpha$ , based on the empirical characteristic function and the STNRW method. We analyze the proposed estimator using Monte Carlo simulations and prove its asymptotic Gaussian distribution. From the perspective of bias, the $\hat{\alpha}_{\text{STNRW-ECF}}$ estimator outperforms $\hat{\alpha}_{\text{ecf}}$ and $\hat{\alpha}_{\text{reg}}$ . We proposed the $\hat{\alpha}_{\text{STNRW-ECF}}$ estimator as an additional diagnostic tool that may provide an indication of cardiac arrhythmia, and we also proposed a test L based on the principal wavelet shrinkage to confirm whether the $\hat{\alpha}_{\text{STNRW-ECF}}$ estimator could be suitable for that purpose. For healthy RR time series, the estimates of $\hat{\alpha}_{\text{STNRW-ECF}}$ is always less than 1.8, while for unhealthy RR times series the estimates of $\hat{\alpha}_{\text{STNRW-ECF}}$ is always approximately equal to 2. For each RR time series, this conclusion is statistically significant at the 1% significance level.

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The STNRW-ECF estimator for stable distributions with application in ECG beat

Abstract

Keywords

1. Introduction

2. Wavelet

(Mother and Father Wavelets).

(Discrete Wavelet Transform).

(Stable Distribution).

(Symmetric α -Stable Distribution).

(Empirical Characteristic Function).

( α ^ 𝐞𝐜𝐟 Estimator).

( α ^ 𝐫𝐞𝐠 Estimator).

.

Proof..

.

References

(Symmetric $\alpha$ -Stable Distribution).

( $\hat{\alpha}_{\text{ecf}}$ Estimator).

( $\hat{\alpha}_{\text{reg}}$ Estimator).