Quantile Process with Applications : Chirp Signal Model

The parameters in the well-known chirp signal model controls the frequency ﬂuctuations of the signals, and consequently, the estimation of the parameters has received considerable attention in the literature of statistical signal processing. In the same spirit with a broader view, this article investigates the quantile estimator of parameters involved in the chirp signal model, which enables us to provide basic features of the entire distribution of the signals. In the course of this study, we establish the limiting behaviour of the associated stochastic process, which we call quantile process. As the applications of this result, we obtain the limiting distributions of various quantile based measures of descriptive statistics, which give us summarized features of the ﬂuctuations of the signals in various senses. Finally, along with ex-tensive simulation study, the practicability of the proposed methodology is shown on a few benchmark real datasets closely related with various chirp signal models.


Introduction
Very often the scientists in various interdisciplinary and engineering sciences experience a situation, where the frequency of a signal is time dependent, i.e., the frequency of the signal varies over time, and such signal is known as chirp signal in the literature of signal processing. Exemplary we mention that the chirp signals are visible in radar, laser, active and passive sonar systems etc (see, e.g, Bender et al. (2006), Nakahira et al. (2005), Caloz et al. (2013) and a few relevant references therein). A real valued one-dimensional chirp signal model in additive noise can be written as follows : Y n = A cos(αn + βn 2 ) + B sin(αn + βn 2 ) + X n ; n = 1, . . . , N, where Y n is is a real valued signal observed at n = 1, . . . , N , n denotes the time point, A and B are amplitudes, α and β are frequency and frequency rate, respectively. Here X n is a sequence of independent and identically distributed (i.i.d.) random variables with certain properties. To know about this model, the readers may refer to Dhar et al. (2019) and a few references in that article.
In many applications in signal processing, the frequency of signals can be modelled by (1.1). For instance, sonar (i.e., Sound Navigation And Ranging) signals very often are treated as chirp signals, and as an illustration, one may consider the following example. The frequency generated by different materials are different on or under the surface of the water. For example, very often a specific material absorbs the energy of some frequencies faster, and consequently, emit less information of return. In order to model such phenomena, a wide range of survey signals covering a band of frequencies can be fitted by the chirp signal model described in (1.1). Other than sonar signals, there are many examples in various sciences, where the signals can be fitted by model (1.1). For example, the receiving antenna of an aircraft receives Radar (i.e., Radio Detection And Ranging) signals from a target, and at the same time, the transmitting antenna transmits Radar signals to the target. As the frequencies of the Radar received by the receiving antenna and the transmitted by transmitting antenna vary over time, the respective Radar signals can be well fitted by the model (1.1). For such type of signals, the parameters A, B, α and β involved in the model (1.1) can be estimated by the least squares or the least absolute deviation methodology, and one may work on various statistical methodologies using those point estimators (see, e.g., Wu et al. (2008), Yetik and Nehorai (2003), Nandi and Kundu (2004) and a few references therein). However, those estimators only give us an insight about the mean or the median surface of the response variable given the covariates; not the whole feature of the surface of the response variable conditioning on the covariates. The concept of the quantiles (see, e.g., Serfling (1980), p. 74 andKoenker (2005)) was introduced in the literature to capture the whole distributional feature of a random variable.
As an alternative approach to the least squares estimation, Koenker and Bassett (1978) proposed the idea of quantile regression in the multiple linear regression model, which provides us the quantile surface of the response variable conditioning on the covariates. The concept of the quantile regression since then has been amply available in the literature of Statistics, Economics and other interdisciplinary sciences. The idea of quantile regression is easy to interpret and because of its good robustness property, it has become a powerful toolkit in interpreting data. Moreover, like the mean and the median, the quantile can also be obtained from a certain minimization problem, where the check function is a convex function, and this fact enables researchers to extend the concepts of quantiles in many topics in Statistics. Specifically, using the similar idea of the regression quantiles, the next paragraph will describe the concept of quantiles for the chirp signal model described in (1.1). For details about quantile regression, the readers may refer to Koenker (2005).
As mentioned earlier, Statistical signal processing literature has paid considerable attention on the point estimation of A, B, α and β in model (1.1) but to the best of our knowledge, the quantiles of A, B, α and β or in other words, the quantile surface of the response variable Y n in model (1.1) has not yet been studied in the statistical signal processing literature. This article explores the quantiles of the unknown parameters of model (1.1), which are obtained from a certain minimization problem and studies a number of key results related to quantiles with its application. Apart from the theoretical advantage of studying quantile estimates, it has a numerous direct advantage in applications also. For instance, as described earlier, the receiving and the transmitting antenna of an aircraft receive the Radar, and the frequencies of the Radar is likely to follow the model (1.1); see also Gini et al. (2000). The least squares or the least absolute deviation estimates of A, B, α and β only provides us the idea of the central tendency of the signals, and the central tendency of the signals may be highly influenced by the extreme signals occurring due to turbulence. However, even under such extreme situations, various quantile estimates of A, B, α and β enable us to see the effect of extreme signals on the whole distributional feature of the signals. Besides, one can also propose various robust measure like the trimmed mean (see, e.g., Dhar and Chaudhuri (2012) and a few references therein) or inter quartile range (see, e.g., Warr and Erich (2013) and a few reference therein) based on quantiles to interpret the various properties of the observed signals.
The main contribution of this article is three fold. The first fold is introducing the concept of quantiles in the well-known chirp signal model described in (1.1). In the literature of statistical signal processing, there were a few attempts on L 1 or least absolute deviation estimates of A, B, α and β; however, to the best of our knowledge, the statistical signal processing literature has not studied the concept quantile estimators of the parameters in chirp signal model such as (1.1). This article introduces the idea of quantile estimates of A, B, α and β in model (1.1). The second fold is the development of results on the convergence of the associated sample quantile process to a certain Gaussian process. The convergence of the quantile process associated with the estimators of the unknown parameters involved in the model have been studied in the literature (see, e.g., Jureckova et al. (2020), Volgushev (2020, Zwingmann and Holzmann (2020), Volgushev et al. (2019), Parker (2019), Hsieh and Wang (2018), Yuan et al. (2017), Tse (2005), Tse (2009), Qu and Yoon (2015), Wagener et al. (2012) and many more articles); however, any kind of process convergence has not been received any attention in the statistical signal processing literature; particularly, for the parameters involved in the chirp signal model like (1.1). Along with proposing the concepts of quantiles in the chirp signal model (1.1), this article thoroughly investigates the various properties of the sample quantile process. Exemplary, the result on the weak convergence of the quantile process to a certain Gaussian process implies the uniform convergence of the quantile estimates to their population counterpart, i.e., corresponding unknown parameters, where uniformity is respect to the quantile index. It is needless to mention that this fact has numerous statistical implications, which will be elaborated in the subsequent sections. The third fold of the contribution is related to the applications of the quantile estimators of A, B, α and β in model (1.1). We here propose the quantile based measures of the location, the scatter and the skewness of the observed signals, and using the result on convergence of the sample quantile process, the asymptotic properties of the sample version of the proposed quantile based measure are derived. The performance of those measures are also investigated on various simulated and real data as well along with a thorough study to see how good the proposed measures compared to well-known classical measures associated with classical descriptive Statistics.
In the course of exploring the aforesaid points, a number of mathematical challenges arise in deriving the results. The first challenge is associated with the non-differentiability of the criterion function of the quantile, and to overcome this problem, in the neigh-bourhood of the point of non-differentiability of the original criterion function, a three degree polynomial is considered, which approximates the original criterion function, i.e., a smoothed approximation of the original criterion function by an approximated criterion function. Afterwards, as the smoothed approximated criterion function is infinitely times differentiable, the asymptotic analysis is carried out on that approximated criterion function. Finally, the result is obtained since the approximated criterion function converges to the original criterion function uniformly. To be summarized, the proof of Theorem 2.1 deals with substantial Mathematical complexity related to Analysis and Probability theory. The next level mathematical difficulty is involved in deriving the limiting stochastic process of the sample quantile process of A, B, α and β indexed by quantile index, which has not been studied in the statistical signal processing literature to the best of our knowledge. In order to establish the process convergence of the sample quantile process, first we prove the arbitrary finite dimensional weak convergence, which is itself a new result in the literature. Afterwards, we establish the complicated tightness property of the sample quantile process, and the process convergence follows from the tightness property of the sample quantile process along with the weak convergence of the arbitrary finite dimensional distributions of the sample quantiles.
The rest of the article is organized as follows. Section 2 describes the model (1.1) and proposes the quantile estimator. The properties of the estimator are discussed in Section 2.1, and the results of the quantile estimator and the associated processes are also stated in that section. Section 2.2 investigates various applications of the quantile based estimator. Two well-known data sets are analysed in Section 3, and Section 4 studies the performance of the quantile based estimators when the number of signals is small or moderately small. A few concluding remarks are discussed in Section 5. Finally, Section 6 contains all technical details.

Preliminaries and model
Recall model (1.1), i.e., Y n = A cos(αn + βn 2 ) + B sin(αn + βn 2 ) + X n ; n = 1, . . . , N, where Y n is a real valued signal at n = 1, . . . , N , A and B are real valued amplitudes, α and β are frequency and frequency rate, respectively, and X n is a sequence of i.i.d. error random variables. A few other technical conditions will be stated before the statements of the results in the subsequent sections. For notational convenience, let us denote θ = [A, B, α, β], and for any γ ∈ (0, 1), θ γ = [A γ , B γ , α γ , β γ ], where A γ , B γ , α γ and β γ are γ-th quantile of A, B, α and β, respectively, which is defined as follows. Here for the sake of technicalities, we are assuming that the parameter space of where L is an arbitrary large constant.
The existences of θ γ andθ N,γ follow from their forms in (2.2) and (2.3), respectively as Q(θ) (see (2.2)) and Q N (θ) (see (2.3)) are convex functions of θ. Moreover, the uniqueness of θ γ can be ensured once the probability density function of the random variable X N is non-degenerate, and for the uniqueness ofθ N,γ , one needs to assume that all observed signals y 1 , . . . , y n are not the same. Regarding the computation of the estimatorθ N,γ , it is an appropriate place to mention that the closed form ofθ N,γ is not tractable in this case as |.| in R is not a differentiable function, and hence, to computeθ N,γ for a given observations of the signals, one needs to implement numerical methods such as Newton-Raphson method to find the solution of the minimization problem in (2.3).
Next, note that at γ = 1 2 , the estimatorθ N,γ will be the well-known least absolute deviation (LAD) estimator of θ = [A, B, α, β], and a version of this estimator is studied in Lahiri et al. (2014). One can expect that the LAD estimator is expected to be more robust than the least squares estimator (LSE), and consequently, more efficient than the LSE estimator when the signals are generated from any heavy tailed distribution. In the same spirit, one can expect that the quantile based estimator will be more effective than the moment based estimators in the presence of outliers in the data. In order to study the large sample properties of the quantile based estimator, one needs to investigate the large sample properties of the the sequence of stochastic processθ N,γ indexed by γ ∈ [0, 1], which is mathematically more challenging than deriving the limiting properties ofθ N,γ at a fixed γ ∈ [0, 1]. All these challenging issues along with the results are studied in Section 2.1..

Properties ofθ N,γ and Results
This section discusses various Statistical properties of the γ-th quantileθ N,γ . Firstly, note thatθ N,γ is not equivariant under location transformation unlike the quantiles for univariate data because of the presence of trigonometric functions in (2.3) through x n . Secondly, due to the same reason,θ N,γ is not equivariant under scale transformation. Hence, one can conclude thatθ N,γ is not equivariant under arbitrary affine transformation, and this fact indicates thatθ N,γ has complex geometric structure, which motives us to study the distributional feature ofθ N,γ .
In order to study the distributional properties ofθ N,γ , one may be first interested to know the exact distribution ofθ N,γ . However, because of the complex structure ofθ N,γ , the exact distribution ofθ N,γ is intractable, and for this reason, our aim here is to study the large sample properties ofθ N,γ , i.e., strictly speaking, the distributional feature ofθ N,γ as N → ∞. Now, note that as the components of theθ N,γ , i.e.,Â N,γ ,B N,γ ,α N,γ andβ N,γ are random variables, the random vectorθ N,γ has its own sampling distribution for a fixed γ ∈ [0, 1], and hence, one may be interested to study distribution of the random vectorθ N,γ for a fixed γ as N → ∞. This study is also supposed to add a significant contribution by its own worth. However, such pointwise (with respect to γ) distribution ofθ N,γ does not lead us to derive the distribution of any functional depending onθ N,γ . In order to resolve this issue, one may consider the sequence of random (i.e., stochastic) processθ N,γ indexed by γ ∈ [0, 1], and the large sample properties of the sequence of the random processθ N,γ (γ ∈ [0, 1]) in l ∞ ([0, 1]) enables us to study the large sample properties of any functional depending onθ N,γ as the metric space l ∞ ([0, 1]) is equipped with supremum norm. The following technical assumptions will be required to establish the limiting distribution of the sequence of the stochastic processθ N,γ in the metric space l ∞ ([0, 1]).
(A2) The vector of unknown parameters, i.e., θ = [A, B, α, β] is an interior point in where L is an arbitrary constant.
Remark 2.1 The assumptions (A1) and (A2) are realistic in nature. Note that many well-known probability density functions such as normal, Cauchy, Exponential satisfy the conditions (A1). The assumption (A2) is also common in practice. It is expected that the unknown parameters A and B will lie in a bounded set, i.e., they will not be infinite in practice. Moreover, for the parameters α and β, the parameter space as [0, π] is justifiable as π and 0 are the largest and the smallest possible values of α (and β), respectively.
The assumption (A3) indicates that both A and B cannot be equal with zero, which is reasonable as the deterministic part involving A, B, α and β in model (1) disappears when A = B = 0, and consequently, model (1) contains only the random part X n that makes the model irrelevant.
Let us now consider the following 4 × 4 dimensional matrix for a fixed γ, which is denoted as Σ γ .
The next theorem describes the asymptotic normality ofθ N,γ for a fixed γ.
Remark 2.2 The assertion in Theorem 2.1 indicates thatθ N,γ has asymptotically (i.e., as N → ∞) normal distribution after appropriate normalization. Moreover, it is also implied by this this result is thatθ N,γ is a consistent estimator of θ γ as N → ∞. In this context, the rate of convergence ofθ N,γ should be discussed as well. Note that the rates of convergence ofÂ N,γ andB N,γ are as usual N −1/2 but the rates of convergence ofα N,γ and β N,γ are N −3/2 and N −5/2 , respectively. The reason of unusual rates of convergences of α N,γ andβ N,γ is lies in the fact that the unknown parameters α and β are associated with the multipliers n and n 2 , respectively in model 1.1. Here we want to discuss the outline of the proof of Theorem 2.1 also. Note the criterion function in (2.3) is not a differentiable function of θ as f (x) = |x| is not a differentiable function at x = 0, and hence, one cannot adopt the approach based on derivative to tackle the problem related to arg min of a function. In order to avoid this problem, the non-differentiable criterion function is approximated by a suitable differentiable function, and the rest of the proof follows from the long and complex derivation of the Bahadur type expansion ofθ N,γ . The technical details of the proof of Theorem 2.1 is provided in Section 6.
Next, we would like to discuss the process convergence of the sequence of stochastic processesθ N,γ indexed by γ ∈ [0, 1]. Let us now consider a random element W γ associated with a Gaussian process with zero mean and the covariance kernel Σ γ 1 ,γ 2 = Covariance (W γ 1 , W γ 2 ), where γ 1 , γ 2 ∈ K ⊂ [0, 1] are two arbitrary constants, and K is the same compact set as defined in the statement of Theorem 2.1. The explicit form of Σ γ 1 ,γ 2 is as follows.
. We now state the result on the convergence of the sequence of the stochastic processθ N,γ .
K is the same as defined in the statement of Theorem 2.1, and W γ is a four dimensional random element associated with a Gaussian process with zero mean and the covariance kernel Σ γ 1 ,γ 2 = Covariance (W γ 1 , W γ 2 ). The explicit expression of Σ γ 1 ,γ 2 is provided above.
is the same as defined in Theorem 2.1.

Remark 2.3
The assertion of Theorem 2.2 indicates that the sequence of stochastic processθ N,γ converges weakly to a certain Gaussian process in l ∞ (.) space after appropriate normalization. This fact (see Corollary 2.1) implies thatθ N,γ converges weakly to θ γ as N → ∞ over uniform choice of γ ∈ K ⊂ [0, 1]. Such Donsker type of result associated witĥ θ N,γ have many applications like the usual Donsker type result of the empirical distribution function (see, e.g., van der Vaart (1998)), and a few of those applications will be studied in the subsequent section.
Remark 2.4 Here we want to discuss a few issues briefly related to the proof of Theorem 2.2. In order to prove the weak convergence of the sequence of the stochastic process in l ∞ (.), there are two steps. In the first step, one needs to establish the weak convergence of the sequence of the stochastic process at arbitrary finitely many time points (i.e., here γ is the time parameter), and in the second step, it is required to establish that the sequence of the stochastic process is uniformly bounded almost surely. The first step can be established by a tactful application of Cramer-Wold device (see, e.g., Serfling (1980), p. 18) on the Bahadur expansion ofθ N,γ (see the proof of Theorem 2.1), and the second step requires application of inequalities like Bernstein inequality (see, e.g., van der Vaart (1998), p. 285) to find the uniform bound almost surely. The details of the proof is provided in Section 6.

Applications
In the literature of Statistical Signal Processing, as said before, the estimation of the parameters in model (1.1) had received considerable attention but they only studied the point estimation of the parameters based on well-known methodologies (see, e.g, Gini et al. (2000), Yetik and Nehorai (2003) etc). However, none of them investigated the quantiles of the parameter estimation in the Chirp signal model, and to the best of our knowledge, this topic is an entirely untouched part in the Statistical Signal Processing literature. Recently, Dhar et al. (2019) extensively studied one-sample testing of hypothesis problem using the points estimators of parameters involved in (1.1) but their work also do not have any study related to quantile estimators of the parameters in model (1.1). We here study a few applications based onθ N,γ .

LAD Estimation
As said earlier, there were a few research articles on the point estimation of the parameters A, B, α and β in model (1.1), and the least absolute deviation (LAD) estimator of θ is one of them (see, e.g., Lahiri et al. (2014)). Note that at γ = 1 2 ,θ N,γ coincides with the LAD estimator of θ, which follows from (2.3). As a consequence of this fact, when γ = 1 2 , the asymptotic normality ofθ N,γ after appropriate normalization stated in Theorem 2.1 coincides with the result of the asymptotic normality of the LAD estimator of θ; see Theorem 2 in Lahiri et al. (2014). Strictly speaking, the result in Theorem 2.1 generalizes the result in Theorem 2 in Lahiri et al. (2014) through the varying choices of the index parameter γ. The advantage of using the LAD estimator as a point estimator of θ is that the LAD estimator is likely to be more efficient than the least squares estimator (LSE) when the signals are obtained from any heavy tailed distribution. In this context, the readers may look at any standard text book of robust Statistics (see, e.g., Huber (2004)).
We now state the asymptotic distribution of S 1,N (γ) as N → ∞.
) converges weakly to a Gaussian distribution with zero mean and Variance = 1, where Remark 2.5 The assertion in Theorem 2.3 indicates that the proposed scatter measure S 1,N (γ) has asymptotically normal distribution after appropriate normalization. Using this asymptotic normality result, one can construct a confidence interval of S 1 (γ) for a fixed γ or carry out a certain testing of hypothesis problem based on S 1,N (γ). In this context, we would also like to mention that one may be tried to derive the weak convergence of the sequence of stochastic process S 1,N (γ) indexed by γ ∈ [0, 1] for theoretical interest; however, Statistically speaking, it is of more interest to see the performance of S 1,N (γ) for a fixed γ, and for that reason, we here skip the investigation on the process convergence result of S 1,N (γ).
We now want to propose an alternative estimator of the scatter based on the spread of distribution ofθ N,γ . In order to measure the spread of the distribution ofθ N,γ or any suitable function ofθ N,γ , one may consider the volume of the support associated with the distribution ofθ N,γ or any suitable function ofθ N,γ . Let S 2,N denotes the proposed measure, i.e., Mathematically speaking, one can write Here we should point out that instead of the Euclidean norm ofθ N,γ in the expression of S 2,N , one may consider any other norm to accumulate the information, and the multiplier D −1 N appears only because of sake of technicalities. We now state the asymptotic distribution of S 2,N as N → ∞.
where W γ is the same as defined in Theorem 2.2, and ||.|| denotes the usual Euclidean norm.
Remark 2.6 The construction of S 2,N indicates that S 2,N is essentially the normalized sum of squares of the components ofθ N,γ , i.e., in other words, it is equivalent to the variance ofθ N,γ . Theorem 2.4 studies the distributional behaviour ofθ N,γ when the number of signals is large. Precisely speaking, it follows from the statement of Theorem 2.4 that the proposed measure S 2,N converges weakly to the distribution of a certain random variable involved integration over a bounded set [0, 1]. It is to be noted that in practice, the exact computation of this integral is intractable but some Riemann type approximation can be done. This issue is discussed in Section 3.1.2, where the performance of the proposed measure S 2,N is carried out.

Interquartile Range
As we proposed measures of scatter in Section 2.2.2, one can propose alternative measures of scatter similar to the concept of interquartile range. In this context, one may consider S 1,N (0.25) as the interquartile range. The following Corollary states the asymptotic distribution of S 1,N (0.25).
) converges weakly to a Gaussian distribution with zero mean and Variance = 1, where Using the result stated in Corollary 2.2, one can carry out the test based on S N,1 (0.25) to capture the spread of the data. If the data has some outliers/influential observations, S N,1 (0.25) may work well.
An alternative measure of previously discussed interquartile measure, one may also consider as an interquartile based spread measure. The following corollary states the asymptotic distribution of S 2,N,0.25 .
where W γ is the same as defined in Theorem 2.2, and ||.|| denotes the usual Euclidean norm.
Corollary 2.3 asserts that S 2,N,0.25 converges weakly to the distribution of a certain random variable, which involves integration. As we discussed in Section 2.2.2, the exact computation of the integration may not be tractable in practice, and to resolve this issue, one may adopt the idea of Riemann approximation of the integral. This measure is also expected to perform well in the presence of outliers/influential observations.

Skewness
In order to measure the skewness of the data, i.e., to check how much the data is deviated from the symmetry, one may propose a measure of skewness as The formulations of SK N,1 and SK N,2 are motivated by the fact that ||θ 1−γ +θ γ −2θ 0 || = 0 for all γ ∈ [0, 1] if the distribution of the data is symmetric. The next theorem describes the distributional feature of SK N,1 and SK N,2 as N → ∞.
Theorem 2.5 Suppose that SK 1 = sup Remark 2.7 Theorem 2.5 asserts that the proposed measures of the skewness, namely, SK N,1 and SK N,2 converges weakly to the distribution of certain random variables, which involves sup and integration, respectively. In practice, to compute SK N,1 , consider a partition T = [γ 0 , γ 1 , . . . , γ k ], where 0 ≤ γ 0 < γ κ ≤ 1, where κ is a sufficiently large number, and afterwards, compute the maximum value of SK N,1 over the partition T . Next, in order to compute SK N,2 , one can approximate the integration by the Riemann approximation over the partition T . The performance of the proposed measures SK N,1 and SK N,2 is studied in Sections 3.1.3 and 3.2.3.

Data Analysis
As we discussed in Section 1, there are many practical situations, where the frequency of signals changes over time, and in those cases, the chirp signal model described in (1.1) can be well fitted to the given signals. In this section, we analyse a few real data, which are well-known in Statistical signal processing literature and investigate the performance of various quantile based measures described in Section 2.2.

Connectionist Bench (Sonar, Mines vs. Rocks) Data Set
This data set has two hundred eight many signals observed at various 208 angles; among them, one hundred eleven signals observed of a metal cylinder at various angles, and ninety seven signals are obtained from rocks at various angles, and this well-known data set is available at https://archive.ics.uci.edu/ml/496datasets/Connectionist+Bench+ (Sonar,+Mines+vs.+Rocks). This data set was earlier analysed in Gorman and Sejnowski (1988), and Figure 1 illustrates the observed signals. In this figure, the red coloured observations are the signals from the metal, and the blue coloured observations are the signals from the rock. For both type of signals, we first try to check whether the model described in (1.1) fits the data or not. In order to investigate it, we plot the residuals of both type of signals in Figure 2, where the i-th residuals is defined aŝ x n = y n −Â N,0.5 cos(α N,0.5 n +β N,0.5 n 2 ) −B N,0.5 sin(α N,0.5 n +β N,0.5 n 2 ).
The diagrams in Figures 2 indicate that the residuals are randomly distributed, and hence, one can conclude that the data is well fitted by the model (1.1). Here we would like to briefly discuss about the formula ofx n provided above. Note thatÂ N,0.5 ,B N,0.5 ,α N,0.5 and β N,0.5 are the LAD estimators of A, B, α and β, respectively (see Section 2.2.1). However, in principle, one may use any other appropriate estimators of A, B, α and β such as the least squares estimators of A, B, α and β as well. Here, we consider the LAD estimators as they are consistent estimators, which follows from the assertion in Theorem 2.1.

Analysis associated with application in Section 2.2.1
Now, suppose that we want to check whether the parameters involved in the model for signals from the metal and the parameters involved in the model for signals from the rocks are the same or not. With notations, one can describe the problem in the following way. Let θ 1 = [A 1 , B 1 , α 1 , β 1 ] be the parameters involved in the model for signals from the metal, and θ 2 = [A 2 , B 2 , α 2 , β 2 ] is the parameters involved in the model for signals from where " = " indicates the componentwise equality, and " = " indicates that at least one of the components are unequal. To test H 0 against H 1 , one may consider the test statistic as a suitable difference betweenθ 1,N 1 , 1 2 andθ 2,N 2 , 1 2 , whereθ 1,N 1 , 1 2 andθ 2,N 2 , 1 2 are the LAD estimators (see Section 2.2.1 for detail on the LAD estimator) of θ 1 and θ 2 , respectively, and here N 1 = 111 and N 2 = 97. Let the test statistic be where ||.|| denotes the usual Euclidean norm. For this data, we obtain Therefore, for this data, T N 1 ,N 2 = θ 1,N 1 , 1 2 −θ 2,N 2 , 1 2 = 1.308. Next, in order to validate the null hypothesis H 0 , we compute the p-value using the technique of residual Bootstrap for this data set in the following way, which can also be used as a generic algorithm.

Analysis associated with application in Section 2.2.2
In Section 3.1.1, we have seen that the parameters involved in the model for the signals obtained from the metal and the rock are different. Now, one may be interested to know whether the spread of the signals obtained from the metal and the rock are the same or not. Notationally speaking, we want to check where θ 1,γ and θ 2,γ are γ-th quantiles of θ 1 and θ 2 , respectively. The definitions of θ 1 and θ 2 are provided at the beginning of Section 3.1.1.
In order to test H 0 against H 1 , we consider the test statistic S N 1 ,N 2 as For detailed interpretation of the construction of S 2,N 1 and S 2,N 2 , we refer the readers to look at the construction of S 2,N in Section 2.2.2. Here we want to discuss the computation of S 2,N 1 and S 2,N 2 . Note that where the random variable γ follows uniform distribution over [0, 1], and by law of large number, one can claim that as R → ∞, where γ 1 , . . . , γ L are i.i.d. sequence of random variables having the same distribution of the random variable γ, i.e., the uniform distribution over [0, 1]. Therefore, in order to compute S 2,N 1 , we generate γ 1 , . . . , γ 1000 from uniform distribution over [0, 1], and 1 1000 1000 i=1 ||D −1 N 1θ N 1 ,γ i || 2 is considered as an estimate of S 2,N 1 for this data. The similar approach is adopted to compute S 2,N 2 as well for this data.
For this data, we obtain S N 1 ,N 2 = 0.091, and as described in Section 3.1.1, we also compute the p-value, which equals with 0.537. This large value p-value indicates that this data favours the null hypothesis H 0 , i.e., the signals obtained from the metal and the rock do not have any significant difference in terms of spread for this data. In this context, we should mention that one may formulate the test statistic based on the measure S 1,N (γ) (studied in Section 2.2.2) for various choices of γ but for the sake of concise presentation, we here skip this study.

Analysis associated with application in Section 2.2.4
In Section 3.1.2, we have observed that the spread of the signals generated by the metal and the rock do not have any significant difference. This fact motivates us to study whether the signals from the metal and the rock are generated from the symmetric distribution or from skewed distribution. In order to check it, for both types of signals, we consider the following two testing of hypothesis problems: The expressions of SK 1 and SK 2 are provided in the statement of Theorem 2.5. To test H 0 against H 1 , SK N,1 (see Section 2.2.4) is considered as the test statistic, and for testing H * 0 against H * 1 , SK N,2 (see Section 2.2.4) is considered as the test statistic. Note that N = N 1 = 111 for the signals obtained from the metal, and N = N 2 = 97 for the signals obtained from the rock. In order to compute SK N,2 , the same procedure described for computing S 2,N 1 (see Section 3.1.2) is adopted here.
To test H 0 against H 1 , we obtain SK N 1 ,1 = 0.784 for the signals obtained from the metal, and the p-value equals with 0.033, where the p-value is computed following the procedure described in Section 3.1.1. At the same time, the same test is carried out for the signals obtained from the rock as well. In this case, we obtain SK N 2 ,1 = 0.725, and the corresponding p-value = 0.084. These small p-values indicate that the signals obtained from the metal and the rock do not favour the null hypothesis H 0 , i.e., the distribution of the signals from both the metal and the rock have the feature of skewness. The same phenomenon we observe for the testing of hypothesis problem H * 0 against H * 1 . For the signals obtained from the metal, we obtain SK N 1 ,2 = 0.923 and the p-value = 0.065, and for the signals obtained from the rock, we obtain SK N 2 ,2 = 0.901 and the associated pvalue = 0.083. Overall, all these investigations indicate that the distributional feature of the signals obtained from both the metal and the rock is skewed; not symmetric.

Sonar Data
This data set contains 477 sonar signals obtained from an experiment conducted in the Department of Electrical Engineering at the IIT Kanpur, India. The data set is submitted to Journal's repository and illustrated in Figure 3. As we analysed various aspects in Section 3.1, here also we investigate all these issues for this data set. This data set is well fitted by the model (1.1) as Dhar et al. (2019) empirically argued that the estimated residuals follow some light tailed distribution using the technique of QQ plot (see, e.g., Wilk and Gnanadesikan (1968) and Doksum and Sievers (1976)), i.e., the residuals do not have any fixed pattern, and hence, one can carry out the analysis on this data assuming that the data is associated with the model (1.1).

Analysis associated with application in Section 2.2.1
We here want to see the central tendency of the signals using the approach of LAD techniques discussed in Section 2.2.1. Suppose that we want to test where " = " denotes the componentwise equality, and " = " denotes at least one inequality among the components. Now, to test H 0 against H 1 , we consider the test statistic as

Analysis associated with application in Section 2.2.2
In Section 3.1.2, we studied the testing of hypothesis problem regarding the equality of scatters of two different types of signals; however, this data has only one kind of sonar signal, and we want to know whether the spread equals some specified value or not. Formally speaking, suppose that we want to test To test H 0 against H 1 , we consider the test statistic as motivated by the fact thatθ N,γ is a consistent estimator of θ γ , which follows from Theorem 2.2. The integral involved in S N is calculated using the same approach described in Section 3.1.2. For this data, we obtain S N = 1.233 and p-value = 0.038, where p-value is computed using the algorithm described in Section 3.1.1. This small p-value indicates that the scatter of the signals is significantly different than one, i.e., the spread of the signals is not standardized to one in some sense.

Analysis associated with application in Section 2.2.4
In Section 3.2.2, we have seen that the spread of the signals is different from unity. Afterwards, we here would like to know whether the distributional feature of the signals is symmetric in nature or not. In order to study it, as we did in Section 3.1.3, we want to test The expressions of SK 1 and SK 2 are provided in the statement of Theorem 2.5. In order to test H 0 against H 1 , we consider the test statistic as SK N,1 , and for testing H * 0 against H * 1 , SK N,2 is considered. For the expressions of SK N,1 and SK N,2 , the readers are advised to see Section 2.2.4. For this data, we obtain SK N,1 = 0.862 and the corresponding p-value = 0.037, and we have SK N,2 = 0.936 and the corresponding p-value = 0.057. The p-values are computed following the algorithm provided in Section 3.1.1. These small pvalues indicate that the signals do not follow the null hypotheses, and hence, the signals are likely to be generated from a skewed distribution.

Finite Sample Study
In Section 3, we analysed two real data, which are associated with the chirp signal model characterized by the regression equation (1.1), and as usual, the sizes of both data sets are large enough. In this section, we now want to see the performance of various quantile based estimators when the sample size is small or moderately small.

Comparison : LAD and LSE estimators
As we have already seen in Section 2.2.1 that the LAD estimator of θ = [A, B, α, β] is a special case ofθ N,γ when γ = 1 2 while the least squares estimator (LSE)θ N,LSE is defined asθ where x n = y n − A cos(αn + βn 2 ) − B sin(αn + βn 2 ). In order to study the performance of θ N, 1 2 andθ N,LSE , we compute the finite sample efficiency ofθ N, 1 2 relative toθ N,LSE and carry out the study in the following way.
Suppose that in the model (1.1), A = B = 1 and α = β = π 4 , and to generate the signals y n , the observations of the error random variable X n are generated from various distributions like normal distribution, t-distribution with 4 degrees of freedom and Cauchy distribution. Here N = 10, 50 and 100. In order to compute the efficiency ofθ N, 1 2 relative toθ N,LSE , we replicate this experiment M times, and letθ N, 1 2 ,i andθ N,LSE,i be the estimate ofθ N, 1 2 andθ N,LSE for i-th replicate, respectively, where i = 1, . . . , M . Then the empirical mean squared error (EMSE) ofθ N, 1 2 is defined as , and the empirical mean squared error (EMSE) ofθ N,LSE is defined as The finite sample efficiency ofθ N, 1 2 relative toθ N,LSE is defined as . In our study, we choose M = 1000, and the summarized result is as follows.
(i) Let X n follows standard normal distribution. All these studies in (i), (ii) and (iii) clearly indicate that the LAD estimator is more efficient compared to the LSE estimator when the signals are obtained from the heavy tailed distribution in view of the fact that Cauchy distribution has heavier tail than t-distribution with 4 degrees of freedom, and t-distribution with 4 degrees of freedom has heavier tail than normal distribution. Strictly speaking, as the tail of the distribution becomes heavier, the efficiency of the LAD estimator increases. This also gives us an impression of having advantage in using a quantile based estimator when the data is generated from any heavy tailed distribution.

Comparison: S 2,N and Variance-Covariance matrix
In Section 4.1, we have studied the performance of the quantile based location estimator, namely, the LAD estimator and the moment based location estimator, namely, the LSE estimator, and we have seen that for the signals generated from the heavy tailed distributions, the quantile based LAD estimator performs better, i.e., in other words, the quantile based estimator is more robust against the outliers/influential observations than the moment based estimator. One may be now interested to see whether the same phenomena is true for the scatter estimator or not. In order to investigate it, we here consider the same setting of simulation as we did in Section 4.1, which is briefly described in the following for the sake of completeness.
Consider the model (1.1), and choose A = B = 1 and α = β = π 4 . Here also, N = 10, 50 and 100, and three cases are investigated when the error random variable X n follows normal distribution, t-distribution with 4 degrees of freedom and Cauchy distribution. As a quantile based estimator of scatter, we consider as we proposed in Section 2.2.2, and as a moment based estimator of scatter, we here consider where θ 0 = [1, 1, π 4 , π 4 ] is specified,θ N,LSE is defined in Section 4.1, and det(.) denotes the determinant of the matrix (.). As earlier, the experiment is replicated M = 1000 times, and SK 2,N,i denotes the value of SK 2,N for i-th replication, and S N,i denotes the value of S N for i-th replication. The finite sample efficiency of SK 2,N relative to S N is defined as Now the results are summarized in the following. S N,i = 34.736. Hence, the finite sample efficiency of S 2,N relative to S N is 9.048. Overall, these finite sample efficiency values indicate that S 2,N is substantially more efficient than S N .
All inclusive, as we have observed in Section 4.1, one can conclude that the quantile based estimator S 2,N performs better than the moment based estimator S N when the signals are obtained from the heavy tailed distribution, i.e, to be summarized, the quantile based scatter estimator is also more robust against the presence of outliers/influential observations.

Concluding Remarks
This article studies the quantile estimators of the unknown amplitude parameters A and B, frequency parameter α and frequency rate parameter β. At first, we here establish the limiting distribution of those quantile estimators at a fixed value of quantile index. Next, the major contribution of this article is deriving the Donsker-type result of the quantile process indexed by the quantile index parameter, which is formally stated in Theorem 2.2. Strictly speaking, this theorem enables us to derive the limiting distribution of any continuous functional of the quantiles based estimator, which has wide applications, and a few of them are studied in Section 2.2. Moreover, we implement those applications on real and simulated data as well.
We assume the condition A 2 + B 2 > 0 to establish the results in Theorems 2.1 and 2.2. This means that both A and B cannot be equal to zero in order to satisfy the aforementioned condition on A and B. However, note that model 1.1 reduces to a complete random model when A = B = 0, and hence, the results in Theorems 2.1 and 2.2 are incapable to test whether the signals are generated from any chirp signal model or a random model. The derivation of the similar results asserted in Theorems 2.1 and 2.2 may be of interest for future research when A = B = 0. The choice of model (1.1) may also be of interest to many practitioners. For instance, one may be interested to consider the model as where (A i , B i , α i , β i ), i = 1, . . . , p is a 4p-dimensional unknown parameter, and X n is a sequence of i.i.d. random errors. For such models, one of the major issues is the appropriate choice of p and to resolve it, one may adopt or propose variable selection methodology to choose appropriate p. Afterwards, for such models also, one may consider the quantile estimators of (A i , B i , α i , β i ), i = 1, . . . , p as we have done here for p = 1. However, for a large p, the curse of dimensionality may arise, and it is also of interest to see whether the limiting distribution of the quantile based estimators is possible or not when p equals with some order of N .
As we consider the LAD estimator as a location estimator in Section 2.2.1, one can use the trimmed mean also a location estimator (see, e.g., Dhar and Chaudhuri (2012) and a few references therein), and the concept of the trimmed mean is extended for various regression models (see, e.g., Welsh (1987) and Dhar (2016)). To the best of our knowledge, the approach of trimmed mean to estimate the unknown parameters in the chirp signal model has not been paid any attention in the literature. To study this research problem, one may use the results presented in Theorems 2.1 and 2.2 in view of the fact that the trimmed is a certain linear combination of the order statistic, and hence, there is a oneto-one correspondence with the quantiles as well. In the same spirit, one may derive the limiting distribution of the L-estimator (see, e.g., Serfling (1980), Chapter 8 and Jureckova and Welsh (1990) in the context of the linear model) also using the results in Theorems 2.1 and 2.2. Moreover, one can conceive the idea of rank as well using the concept of quantiles studied here. g(x n ) (see (2.3)), where x n = y n − A cos(αn + βn 2 ) − B sin(αn + βn 2 ), g : R → R, and g(x) = |x| + (2γ − 1)x for any x ∈ R, and γ ∈ (0, 1). Let us now consider a new sequence of functions g N (.), which is the following.
whereas recall thatθ Let us now try to show that where a.s. denotes the almost sure convergence. Note that for an arbitrary > 0, we have Hence, we firstly have (6.5) where o p denotes the convergence in probability.
Next, note that As it is true for any > 0, by first Borel-Cantelli Lemma (see, e.g., Serfling (1980), p. 351) and in view of compactness of Θ, the claim in (6.4) is established.
Finally, in view of (6.4), we have The analogous convergence in probability result follows from (6.5). Moreover, arguing in a similar way as Kim et al. (2000) studied and in view of (6.6), we further have Now, we want to establish that , θ := [θ 1 , θ 2 , θ 3 , θ 4 ] = [A, B, α, β] is a generic variable, and D N and Σ γ are the same as defined in Section 2.1.
In order to establish it, recall that Hence, we have g N (X n ), and note that . Therefore, the key term involves in In the above expression, observe that F is a continuous distribution function, and due to the same reason along with the fact of (A1), 1 − F 1 → 0 as N → ∞. (6.10) Afterwards, we consider Hence, using (6.9), (6.10) and (6.11), we have Observe now the followings.
Arguing exactly in a similar way as in (6.10) and (6.11), we have Next, consider N 12 dx → 0 as N → ∞ using the fact that f (x) is bounded by some constant G. The exact arguments also leads to Hence, all these above facts imply that E d dx {g N (x)}| x=Xn 2 → (1 − γ) + (γ − 1)γ = γ 2 , and hence, . Therefore, (6.12) and (6.16) imply that (6.17) where " p → " denotes the convergence in probability.
We now want to work on the distribution of {D N ∇Q * N (θ γ )}, where
We next want to establish that sup N sup γ∈K ||D −1 N (θ N,γ 1 − θ γ 1 )|| is bounded almost surely, where K is a compact subset of [0, 1]. Note that as K is a compact set, it is enough to establish that sup N ||D −1 N (θ N,γ − θ γ )|| is bounded almost surely for a fixed γ since K can be covered by finitely many sub-covers. Next, observe that β γ )| are bounded almost surely for a fixed γ, which follows from (6.34).