Statistical Estimation and Hypothesis Testing on Impulse Response Function PDF

Austrian Journal of Statistics 2025, Volume 54, 200–213. http://www.ajs.or.at/ AJS doi:10.17713/ajs.v54i1.1977 Statistical Estimation and Hypothesis Testing on Impulse Response Function Iryna Rozora Department of Applied Statistics, Taras Shevchenko National University of Kyiv Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" Anastasiia Melnyk Department of Applied Statistics, Taras Shevchenko National University of Kyiv Abstract In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function (IRF) which is defined on a bounded domain. A sample input- output cross-correlogram is taken as an estimator of the response function. The input processes are supposed to be zero-mean stationary Gaussian process and can be repre- sented as a finite sum with uncorrelated terms. A rate of convergence of IRF estimator in the space L2 ([0, Λ]) is obtained that gives a possibility to propose a nonparametric goodness-of-fit testing on IRF. Keywords: impulse response function, cross-correlogram, large deviation probability, rate of convergence, statistical criterion, hypothesis testing,nonparametric goodness-of-fit testing. 1. Introduction The problem of estimation of a stochastic linear system driven by impulse response function (IRF) has been a matter of active research in recent years. One of the simplest models considers a black box that receives some input and produces a corresponding output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The range of applications for these models is extensive, ranging from signal processing and automatic control to econometrics (errors-in- variables models) and oceanology. For more details, see Barigozzi, Lippi, and Luciani (2021), Hannan and Deistler (1988), Soderstrom and Stoica (1989), Stern (2018) and Lütkepohl (2010), Takahito and Munehiko (2022). The issue of the estimation of IRF is similar to inverse problem and deconvolution one that are used, for example, for restoring signal or images, signal detection (see Abramovich, Pen- sky, and Rozenholc (2013), Delaigle, Hall, and Meister (2008), Meister (2009), Marteau and Sapatinas (2015)). Austrian Journal of Statistics 201 The estimates in these models have been mostly obtained in usual statistical framework. Optimality results (in the minimax sense) have been given for various loss functions and sequence/function spaces. Many methods have been considered including kernel, local poly- nomial, spline, projection and wavelet methods (see, e.g.Cavalier and Raimondo (2007) ). We are interested in the estimation of the so-called impulse function from observations of responses of a SISO (single-input single-output) system to certain input signals. This problem can be considered both for linear and non-linear systems. To solve this problem, different statistical approaches were used as well as various deterministic methods that are based on a perturbation of the system by stationary stochastic processes and the further analysis of some characteristics of both input and output processes. Let us mention some publications on this problem by Bendat and Piersol (1980) and Schetzen (1980), Nelles (2020). One of the first article on this topic was Akaike (1965) where the author studied a MISO (multiple-input single-output) linear system and obtained estimates of the Fourier transform of the response function in each component. Some methods for estimation of unknown impulse response function of linear system and the study of properties of corresponding estimators were considered in the works of Buldygin and his followers. These methods are based on constructing a sample cross-correlogram between the input stochastic process and the response of the system (see, e.g. Buldygin and Li (1997a), Buldygin and Li (1997b), Blazhievska and Zaiats (2021) ). An inequality for the supremum of the estimation error in the space of continuous functions in the case of integral-type cross-correlogram estimator was obtained by Kozachenko and Rozora (2016). In the paper Rozora and Kozachenko (2016) a time-invariant continuous linear system with a real-valued impulse response function was considered. The input signal process was supposed to be a zero mean Gaussian stochastic process which was represented as a treamed sum with respect to orthonormal basis in L2 (R). The case of Hermite polynomials as orthonormal basis in L2 (R) was studied. A new method for the construction of estimator of the impulse response function was proposed ant the criteria on impulse response function were given. In Rozora (2018) and Rozora (2020) for integral cross–correlogram estimator the algorithm of statistical hypothesis testing for response function was written applying upper estimate of overrunning by square-Gaussian random process the level specified by continuous function. In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function which is defined on bounded domain. An input-output cross-correlogram based on one single observation is taken as an estimator of the response function. The input processes are supposed to be zero-mean stationary Gaussian process and can be represented in trimmed series of Fourier decomposition. In such model the built estimator of impulse function depends on the length of averaging interval for cross-correlogram T and the level of cutting for input signal N. The estimation of large deviation probability for errors in the space L2 (T ) is found. These allow us to develop the hypothesis testing on the shape of the impulse response function. The paper consists of 8 sections and the structure is as follows. The second section covers the main definitions and general properties of the estimator. The input signal process is supposed to be a zero mean Gaussian stochastic process which is represented as a treamed sum with respect to orthonormal basis in L2 ([0, Λ]). In section 3 we suppose that the input process of the system can be represented as a series with respect to the trigonometric basis on [0, Λ]. The upper bounds for mathematical expectation, variance are found that provide asymptotically unbiasedness as N → ∞ and consistency as N, T → ∞. Section 4 deals with square Gaussian random variables and processes. Inequality for Lp (T ) norm of a square Gaussian stochastic process is shown. In the fifth section the convergence rate for the estimator of unknown impulse response 202 On Impulse Response Function function in the space Lp ([0, Λ]) is investigated. In the sixth section the goodness of fit test is developed on the shape of the impulse response function. Section 7 is devoted to software simulation. In one particular case the critical values of the length of averaging interval T are found for different accuracy, reliability and N (the upper limit of the summing in the model) using software environment for statistical computing and graphics R. 2. The estimator of an impulse response function and its properties Consider a time-invariant continuous linear system with a real-valued square integrable im- pulse response function (IRF) H(τ ) which is defined on a finite domain τ ∈ [0, Λ]. This means that the response of the system to an input signal X(t), t ∈ R, has the following form: Z Λ Y (t) = H(τ )X(t − τ )dτ, t∈R (1) 0 and H ∈ L2 ([0, Λ]). One of the problems arising in the theory of linear systems is to estimate the function H from observations of responses of the system to certain input signals. In this paper we use cross-correlogram approach to estimate the unknown IRF H. At first, let’s describe the input process that is performed due to some orthonormal basis. Let the system of functions {φ0 (t), φk (t), ψk (t), k ≥ 1, t ∈ R} be an orthonormal basis in L2 [0, Λ] and the functions φk (t), ψk (t) are continuous and Λ-periodic. Assume that the function φ0 (t) is a constant. This means that it should be equal to φ0 (t) = √1. Λ Consider now as input of the linear system a real-valued Gaussian stationary zero mean stochastic process X = XN = (XN (u), u ∈ R), that can be presented as N N XN (u) = ξk φk (u) + ηk ψk (u), (2) X X u ∈ R, k=1 k=1 where N > 0 is fixed integer number and Gaussian random variables ξk , ηk , k ≥ 0, are uncorrelated with Eξk = Eηk = 0, Eξk2 = Eηk2 = 1. Remark 1. The input signal XN (t) can be considered as a model of some stochastic pro- cess that can be expanded into a series over given orthonormal basis. To read more about model construction with given accuracy and reliability we refer readers to Kozachenko, Pogo- rilyak, Rozora, and Tegza (2016), Kozachenko, Pashko, and Rozora (2007), Vasylyk, Rozora, Ianevych, and Lovytska (2021), Rozora, Ianevych, Pashko, and Zatula (2023), Ianevych, Ro- zora, and Pashko (2022). A covariance function rN (t − s) = EXN (t)XN (s) of the stationary process XN should be equal to N rN (t − s) = (φk (t)φk (s) + ψk (t)ψk (s)). (3) X k=1 If the system (1) is perturbed by the stochastic process XN , then for the output process we obtain Z Λ YN (t) = H(τ )XN (t − τ )dτ, 0 Show that the covariance function of XN will be equal to rN (t − s) N N N N ! ! EXN (t)XN (s) = E ξk φk (t) + ηk ψk (t) ξk φk (s) + ηk ψk (s) X X X X k=1 k=1 k=1 k=1 N = (φk (t)φk (s) + ψk (t)ψk (s)). X k=1 Austrian Journal of Statistics 203 By the condition (3) we obtain that the covariance function of stochastic process XN is equal to EXN (t)XN (s) = rN (t − s). Under a0 we define the output of the system on the constant signal. This means that 1 Z Λ a0 = √ H(t)dt. (4) Λ 0 Set H ∗ (τ ) = H(τ ) − a0. (5) As an estimator of the difference of IRF and a0 H ∗ (τ ) we will consider an integral cross- correlogram 1 T Z Ĥ(τ ) = ĤN,T,Λ (τ ) = ĤN,T,Λ (τ ) = YN (t)XN (t − τ )dt, (6) T 0 where T > 0 is a parameter for averaging. Remark 2. The integral in (1) is considered as the mean-square Riemann integral. The integral in (1) exists if and only if there exists the Riemann integral (see Gikhman and Skorokhod (1996)) Z ΛZ Λ H(τ )rN (s − τ )H(s)dsdτ. (7) 0 0 The covariance function of the process XN is given by (3). Since {φk (t), ψk (t), k ≥ 0} is an orthonormal basis in L2 ([0, Λ]), H ∈ L2 ([0, Λ]), then the integral (7) exists. Therefore, there exists also the integral in (1). Denote now Z Λ Z Λ ak = H(t)φk (t)dt, bk = H(t)ψk (t)dt. (8) 0 0 Since H ∈ L2 ([0, Λ]), then the function H can be expanded into the series by orthonormal basis {φk (t), ψk (t) k ≥ 0} on the domain [0, Λ]. We obtain ∞ ∞ H(t) = ak φk (t) + bk ψk (t), (9) X X k=0 k=1 where ak and bk are from (8). Remark 3. According to the Riesz–Fischer theorem (see Beals (2004)) the series (9) con- verges in the mean squared sense. If the function H(t) and derivative H ′ (t) are continuous then the convergence in (9) can be considered in pointwise and uniform senses. Lemma 1. Kozachenko and Rozora (2023) The following relations hold true: Z Λ N EĤN,T,Λ (τ ) = H(v)rN (v − τ )dv = (φk (τ )ak + ψk (τ )bk ) , τ ∈ [0, Λ], (10) X 0 k=1 and ∞ H ∗ (τ ) − EĤN,T,Λ (τ ) = (φk (τ )ak + ψk (τ )bk ) , τ ∈ [0, Λ]. (11) X k=N +1 Lemma 2. Kozachenko and Rozora (2023) The joint moments of ĤN,T,Λ is equal to Z Λ Z Λ EĤN,T,Λ (τ )ĤN,T,Λ (θ) = H(u)rN (τ − u)du · H(v)rN (θ − v)dv 0 0 1 Z TZ T "Z ΛZ Λ + H(v)H(u)rN (t − s + u − v)dudv · rN (t − s + θ − τ ) T2 0 0 0 0 Z Λ Z Λ # + H(v)rN (t − s + θ − v)dv · H(u)rN (s − t + τ − u)du dtds, (12) 0 0 204 On Impulse Response Function where rN (t − s) = EXN (t)XN (s) is a covariance of XN , the coefficients ak and bk are defined in (8). The variance of the estimator ĤN,T,Λ is equal to 1 Z T Z T "Z Λ Z Λ V arĤN,T,Λ (τ ) = H(v)H(u)rN (t − s + u − v)dudv · rN (t − s) T2 0 0 0 0 Z Λ Z Λ # + H(v)rN (t − s + τ − v)dv · H(u)rN (s − t + τ − u)du dtds.(13) 0 0 3. Fourier series based approach Let now consider the system of functions 1 2 2kπt 2 2kπt ( r r ) √ , cos , sin ,k ≥ 1 (14) Λ Λ Λ Λ Λ that is orthonormal basis in L2 ([0, Λ]). Under notation of previous section 1 2 2kπt 2 2kπt r r φ0 (t) = √ , φk (t) = cos , ψk (t) = sin ,k≥1 Λ Λ Λ Λ Λ and the coefficients ak , bk are equal to 2 2kπτ Z Λ r Z Λ ak = H(τ )φk (τ )dτ = H(τ ) cos dτ k ≥ 1, (15) 0 Λ 0 Λ 2 2kπτ Z Λ r Z Λ bk = H(τ )ψk (τ )dτ = H(τ ) sin dτ k ≥ 1. (16) 0 Λ 0 Λ Suppose now that the input signal processes of the system (1) are zero mean stationary Gaussian stochastic processes that are formed by (14). This means that the process XN (u) is given by: N 2 X 2kπu 2kπu r XN (u) = ξk cos + ηk sin , u ∈ R. (17) Λ k=1 Λ Λ It follows from (3) that the covariance function of stationary Gaussian process XN can be written as N rN (t − s) = (φk (t)φk (s) + ψk (t)ψk (s)) X k=1 N 2 X 2kπt 2kπs 2kπt 2kπs = cos cos + sin sin (18) Λ k=1 Λ Λ Λ Λ N 2 2kπ(t − s) ! = cos X. Λ k=1 Λ Consider the following conditions: Condition A. The function H(τ ) is two times differentiable on [0, Λ]. The functions H(τ ) and H ′ (τ ) are continuous on [0, Λ] and Z Λ I0 = I0 (Λ) = |H(τ )|dτ < ∞, 0 Z Λ !1/2 ′ I1 = I1 (Λ) = |H (τ )| dτ 2 < ∞, 0 Z Λ I2 = I2 (Λ) = |H ′′ (τ )|dτ < ∞. 0 Austrian Journal of Statistics 205 Condition B. The following relation holds true H(0) = H(Λ). Remark 4. Condition B means that the effect of the impulse on the domain [0, Λ] should be completely ended. In case when the condition isn’t fulfilled the rotation of the graph by the angle arctan H(Λ)−H(0) Λ can be applied. Let’s denote d = |H ′ (0)| + |H ′ (Λ)|. (19) Lemma 3. Kozachenko and Rozora (2023) Assume that the conditions A, B are satisfied. Then Λ(d + I2 (Λ)) |H ∗ (τ ) − EĤN,T,Λ (τ )| ≤ , (20) 2π 2 N Λ3 (Λ + 2)I12 1 2 V arĤN,T,Λ (τ ) ≤ 2− , (21) π4T 2 N where I1 (Λ) and I2 (Λ) are defined in condition A. 4. Square Gaussian random variables and processes In this section the definition and some properties of square Gaussian random variables and processes are presented. Let (Ω, L, P ) be a probability space and let (T, ρ) be a compact metric space with metric ρ. Definition 1. Buldygin and Kozachenko (2000) Let Ξ = {ξt , t ∈ T} be a family of joint Gaussian random variables for which Eξt = 0 (e.g., ξt , t ∈ T, is a Gaussian stochastic process). The space SGΞ (Ω) is the space of square Gaussian random variables if any element η ∈ SGΞ (Ω) can be presented as η = ξ¯⊤ Aξ¯ − Eξ¯⊤ Aξ, ¯ (22) where ξ¯⊤ = (ξ1 , ξ2 ,... , ξn ), ξk ∈ Ξ, k = 1,... , n, A is a real-valued matrix or the element η ∈ SGΞ (Ω) is the square mean limit of the sequence (22) η = l.i.m.n→∞ (ξ¯n⊤ Aξ¯n − Eξ¯n⊤ Aξ¯n ). Definition 2. Buldygin and Kozachenko (2000) A stochastic process ξ(t) = {ξ(t), t ∈ T} is square Gaussian if for any t ∈ T a random variable ξ(t) belongs to the space SGΞ (Ω). There are shown in the book by Buldygin and Kozachenko (2000) that SGΞ (Ω) is a Banach space with respect to the norm ∥ζ∥ = Eζ 2 ; p SGΞ (Ω) is a subspace of the Orlicz space LU (Ω) generated by the function U (x) = exp |x| − 1; the norm ∥ζ∥LU (Ω) on SGΞ (Ω) is equivalent to the norm ∥ζ∥. Example 1. Consider a family of Gaussian centered stochastic processes ξ1 (t), ξ2 (t),... , ξn (t), t ∈ T. Let the matrix A(t) be symmetric. Then X(t) = ξ¯⊤ (t)A(t)ξ(t) ¯ − Eξ¯⊤ (t)A(t)ξ(t), ¯ where ξ¯⊤ (t) = (ξ1 (t), ξ2 (t),... , ξn (t)), is a square Gaussian stochastic process. For more information about properties of square Gaussian random processes see Buldygin and Kozachenko (2000), Kozachenko and Moklyachuk (1999), Kozachenko and Moklyachuk (2000), Kozachenko et al. (2007), Kozachenko et al. (2016), Kozachenko and Stus (1998). The following theorem can be found in article Kozachenko and Troshki (2015). 206 On Impulse Response Function Theorem 1. Kozachenko and Troshki (2015) Let {T, A, µ} be a measurable space, where T is a parametric set, and let ξ = {ξ(t), t ∈RT} be a measurable square Gaussian stochastic p process. Assume that the Lebesgue integral T (Eξ 2 (t)) 2 dµ(t) is well defined for p ≥ 1. Then the integral T (Eξ 2 (t))p dµ(t) exists with probability 1, and R v √   Z u x1/p 2  x1/p  |ξ(t)|p dµ(t) > x ≤ 2t1 + exp − √ 1 , (23) u P 1 T p 2Cpp   Cp p q for all x ≥ ( √p2 + ( p2 + 1)p)p Cp , where Cp = T (Eξ 2 (t)) 2 dµ(t). R 5. On the rate of convergence of the estimator of impulse response function This section is devoted to the investigation of the rate of convergence of estimators of unknown IRF in the space of the space L2 ([0, Λ]). The next Lemma is trivial. Lemma 4. Stochastic process ẐN,T,Λ (τ ) = ĤN,T,Λ (τ ) − E ĤN,T,Λ (τ ), τ > 0, is a square Gaussian one. Consider a difference of the estimator ĤN,T,Λ (τ ) and the impulse function H(τ ) nonmetering the response on the constant signal a0 H ∗ (τ ) − ĤN,T,Λ (τ ), τ > 0. Denote Λ(d + I2 (Λ)) h∗N,Λ =. 2π 2 N Then from (20) it follows that |E ĤN,T,Λ (τ ) − H ∗ (τ )| ≤ h∗N,Λ , τ ∈ [0, Λ]. Remark 5. Since h∗N,Λ → 0 as N → ∞ then the estimator ĤN,T,Λ (τ ) is asymptotically unbiased. Put Λ Λ(Λ + 2)I1 1 p γ0 (N, T, Λ) = γ0 = 2 2−. (24) π T N From (21) we have that q sup V arẐN,T,Λ (τ ) ≤ γ0 τ ∈[0,Λ] Since γ0 → 0 as T, N → ∞ then a sufficient condition for consistency of ĤN,T,Λ (τ ) is fulfilled. Consider a parametric set T = [0, Λ] and let {[0, Λ], F, µ} be a metric space with Euclidean measure µ. Theorem 2. Assume that the conditions A, B are satisfied. Then for r p p p 1 ε ≥ (√ + ( + 1)p)Λ p γ0 + Λh∗N,Λ (25) 2 2 the estimate √ v u 1  1  (ε p − Λh∗N,Λ ) 2  ε p − Λh∗  (Z ) Λ u N,Λ |H ∗ (τ ) − ĤN,T,Λ (τ )|p dτ > ε ≤ 2t1 + exp − √ 1 (26) u P 1 0 p Λ γ0  p  2Λ γ0 holds true. Austrian Journal of Statistics 207 Remark 6. Emphasize that the lower bound on ε in (25) can be made arbitrarily small. Really, the values γ0 and h∗N,Λ go to 0 as T, N → ∞. The bigger T and N are, the smaller ε is. Proof. We show first that the result of Theorem 1 can be applied to the process ẐN,T,Λ (τ ), τ > 0. Really, by Lemma 4 ẐN,T,Λ (τ ) is a square Gaussian stochastic process. Prove that the Lebesgue integral Z Λ p (EẐN,T,Λ 2 (τ )) 2 dµ(τ ) 0 is correctly defined. From inequality (21) follows that Z Λ Z Λ p p (EẐN,T,Λ 2 (τ )) 2 dµ(τ ) = (V arẐN,T,Λ (τ )) 2 dτ 0 0 p 4I1 (Λ) · (N + 1) 2 < Λ. Λ q Therefore, inequality (23) for the process ẐN,T,Λ (τ ) as x ≥ ( √p2 + ( p2 + 1)p)p Cp can be rewritten as v √   1/p 2 (Z ) Λ u x  x1/p  |ẐN,T,Λ (τ )|p dµ(τ ) > x ≤ 2t1 + exp − √ 1 , (27) u P 1 0 Cpp 2Cpp   RΛ p where Cp = 0 (EẐN,T,Λ 2 (τ )) 2 dτ. From the Minkowski inequality follows that Z Λ !1 Z Λ !1 p p |H ∗ (τ ) − ĤN,T,Λ (τ )|p dµ(τ ) = |H(τ ) − a0 ± EĤN,T,Λ (τ ) − ĤN,T,Λ (τ )|p dµ(τ ) 0 0 Z Λ !1 p ∗ ≤ |H (τ ) − EĤN,T,Λ (τ )| dµ(τ ) p 0 Z Λ !1 p + |EĤN,T,Λ (τ ) − ĤN,T,Λ (τ )|p dµ(τ ) 0 Z Λ !1 p ≤ h∗N,Λ ·Λ+ |ẐN,T,Λ (τ )| dµ(τ ) p. 0 If ε > (h∗N,Λ Λ)p then from the relationship above we have that (Z )  !1  Λ  Z Λ p 1  |H ∗ (τ ) − ĤN,T,Λ (τ )|p dµ(τ ) > ε = |H ∗ (τ ) − ĤN,T,Λ (τ )|p dµ(τ ) >ε p 0  0   Z Λ !1   p 1 ⊂ Λh∗N,Λ + |ẐN,T,Λ (τ )|p dµ(τ ) > εp  0  (Z ) Λ 1 p = |ẐN,T,Λ (τ )|p dµ(τ ) > ε p − Λh∗N,Λ. 0 1 p Substituting x = ε p − Ah∗N,Λ in (27), we obtain √ v 1  1  (ε p − Λh∗N,Λ ) 2  ε p − Λh∗  (Z ) u Λ u N,Λ P |H ∗ (τ ) − ĤN,T,Λ (τ )|p dµ(τ ) > ε ≤ 2t1 + exp − √ 1. u 1 0 p p 2Cp   Cp (28) 208 On Impulse Response Function Since (21) implies supτ ∈[0,Λ] V arẐN,T,Λ (τ ) ≤ (γ0 )2 , where γ0 is from (24), then Z Λ p Cp = (EẐN,T,Λ 2 (τ )) 2 dτ ≤ Λγ0p. 0 To complete the proof it’s enough to substitute the above value Cp in (28). Hence, √ v u 1  1  (ε p − Λh∗N,Λ ) 2  ε p − Λh∗  (Z ) Λ u N,Λ |H ∗ (τ ) − ĤN,T,Λ (τ )|p dµ(τ ) > ε ≤ 2t1 + exp − √ 1 u P 1. 0 pΛ γ0  p  2Λ γ0 6. Goodness-of-fit testing of IRF Using Theorem 2 it is possible to test hypothesis on the shape of IRF. Let the null hypothesis H0 state that an IRF is H(τ ), τ ∈ [0, Λ], and the alternative Ha implies the opposite statement. Denote √ v u 1  1  u (ε p − Λh∗N,Λ ) 2  ε p − Λh∗  N,Λ g(ε) = g(ε, N ) = 2t1 + exp − √ 1 (29) u 1. p  Λ γ0 p  2Λ γ0 From Theorem 2 follows that if r p p p 1 ε > tN,T,Λ (p) = ( √ + ( + 1)p)Λ p γ0 + Λh∗N,Λ , 2 2 then (Z ) Λ P |H(τ ) − ĤN,T,Λ (τ )| dτ > ε p ≤ g(ε). 0 Let εδ be a solution of the equation g(εδ ) = δ, 0 < δ < 1. Put ε∗δ = max{εδ , tN,T,Λ (p)}. (30) It is clear that g(ε∗δ ) ≤ δ and ( ) P sup |H(τ ) − ĤN,T,Λ (τ )| > ε∗δ ≤ δ. τ ∈[0,Λ] From Theorem 2 it follows that to test the hypothesis H0 , we can use the following testing. Theorem 3. For a given level of confidence 1 − δ, δ ∈ (0, 1), the hypothesis H0 is rejected if Z Λ |H(τ ) − ĤN,T,Λ (τ )|p dτ > ε∗δ , 0 otherwise the hypothesis H0 is accepted, where ε∗δ is from (30). 7. Simulation study We consider a particular case when Λ = 10, α = 1 and p = 2. An IRF is supposed to be H(τ ) = τ (e−τ − e−Λ ). Obviously, that the Conditions A, B are fulfilled for H(τ ) and I0 = 0.9972, I1 = 0.5, I2 = 1.2703, d = 1.0004. The function H ∗ (τ ) = H(τ ) − a0 is equal to H ∗ (τ ) = τ (e−τ − e−Λ ) − 0.3153. Austrian Journal of Statistics 209 Table 1: The minimal values of T for fixed value of N = 1000 with given significant level δ and accuracy ε for the function g(ε, N ) = δ Accuracy, ε T when δ = 0.3 T when δ = 0.2 T when δ = 0.1 0.001 7022 (0.4816) 8185 (0.2900) 10125 (0.1112) 0.005 2386 (0.3312) 2781 (0.2716) 3441 (0.4302) 0.01 1597 (0.4617) 1861 (0.2328) 2302 (0.1395) 0.025 964 (0.3019) 1123 (0.1912) 1389 (0.4428) 0.05 666 (0.1292) 776 (0.3743) 960 (0.4132) 0.075 539 (0.4674) 628 (0.3397) 776 (0.4467) 0.1 464 (0.3408) 540 (0.3954) 668 (0.4956) 0.2 324 (0.2712) 378 (0.0615) 468 (0.4715) 0.3 263 (0.4898) 307 (0.0977) 380 (0.1051) Table 2: The minimal values of T for fixed value of N = 5000 with given significant level δ and accuracy ε for the function g(ε, N ) = δ Accuracy, ε T when δ = 0.3 T when δ = 0.2 T when δ = 0.1 0.001 4820 (0.4189) 5618 (0.2035) 6950 (0.0059) 0.005 2066 (0.1427) 2408 (0.0915) 2979 (0.0723) 0.01 1447 (0.2671) 1686 (0.1688) 2086 (0.1263) 0.025 907 (0.1428) 1057 (0.2759) 1308 (0.0977) 0.05 639 (0.3153) 744 (0.3877) 921 (0.1558) 0.075 520 (0.4909) 607 (0.3672) 750 (0.4346) 0.1 450 (0.2473) 525 (0.2362) 649 (0.1586) 0.2 318 (0.3090) 370 (0.2692) 458 (0.0412) 0.3 259 (0.1477) 302 (0.0369) 374 (0.3654) Table 3: The minimal values of T for fixed value of N = 10000 with given significant level δ and accuracy ε for the function g(ε, N ) = δ Accuracy, ε T when δ = 0.3 T when δ = 0.2 T when δ = 0.1 0.001 4639 (0.3200) 5406 (0.3866) 6688 (0.0339) 0.005 2032 (0.0760) 2368 (0.3868) 2930 (0.1890) 0.01 1430 (0.0315) 1667 (0.3702) 2062 (0.2969) 0.025 901 (0.4603) 1050 (0.4200) 1298 (0.3821) 0.05 635 (0.4138) 741 (0.4245) 916 (0.1283) 0.075 518 (0.3214) 604 (0.1041) 747 (0.3065) 0.1 449 (0.3741) 523 (0.1260) 647 (0.1791) 0.2 317 (0.1124) 369 (0.3328) 457 (0.1171) 0.3 259 (0.3848) 301 (0.4163) 373 (0.1332) 210 On Impulse Response Function Table 4: The minimal values of T for fixed value of N = 50000 with given significant level δ and accuracy ε for the function g(ε, N ) = δ Accuracy, ε T when δ = 0.3 T when δ = 0.2 T when δ = 0.1 0.001 4503 (0.1214) 5248 (0.1099) 6492 (0.1701) 0.005 2006 (0.3767) 2338 (0.4438) 2892 (0.3280) 0.01 1417 (0.1647) 1651 (0.3230) 2043 (0.2322) 0.025 895 (0.3265) 1044 (0.4960) 1291 (0.1343) 0.05 633 (0.1787) 738 (0.4461) 912 (0.3905) 0.075 517 (0.4012) 602 (0.0965) 745 (0.1770) 0.1 447 (0.3373) 521 (0.3721) 645 (0.0370) 0.2 316 (0.2478) 369 (0.4128) 456 (0.0396) 0.3 258 (0.1909) 301 (0.0783) 372 (0.2550) Delta = 0.3 Delta = 0.2 6000 Delta = 0.1 5000 4000 T 3000 2000 1000 0 10 3 10 2 10 1 Epsilon Figure 1: The values of T for fixed value of N = 50000 We use the free software environment for statistical computing and graphics R to find T for different values of accuracy and reliability and the level of cutting N of the input process from the conditions of Theorem 2 for Lp metric. In tables 1-4 the minimal values of T for fixed value of cutting point N with given significant level δ and accuracy ε for the function g(ε, N ) = δ are calculated and the values in parentheses indicate the standard deviation of the estimator ĤN,T,Λ (τ ) with the corresponding values N and T. The values T in tables 1-4 show that the more accurate result we want to obtain the larger term T should be. Figure 1 demonstrates the dependency of accuracy ε and parameter for averaging T in case of tree different significant level and fixed cutting point N = 50000. As we can see the functions are decreasing. 8. Conclusions In this paper we considered a time-invariant continuous linear system in which the IRF is defined on the domain [0, Λ]. The input signal process was supposed to be a zero mean Gaussian stochastic process which was represented as a series with respect to an orthonormal basis in L2 ([0, Λ]. A particular case where the orthonormal basis was given by trigonometric functions was studied in details. Some characteristics of the estimator of impulse function such as mathematical expectation, variance were described that. We also investigated the Austrian Journal of Statistics 211 convergence rate for the estimator of unknown IRF in the space Lp ([0, Λ]). For this reason the theory of square Gaussian random variables and processes was applied, namely we used inequality for Lp (T ) norm of square Gaussian stochastic process. It gave us an opportunity to construct the goodness of fit test on IRF. In one particular case the values T were found for different accuracy and reliability and N (the upper limit of the summing in the model) using software environment for statistical computing and graphics R. In future studies we are planning to consider the cases involving discretely observed processes, which are more realistic scenario in functional data analysis as well. We would like to thank the reviewers for their thorough review of our manuscript and for their valuable feedback, which has helped us to make significant improvements to our work. References Abramovich F, Pensky M, Rozenholc Y (2013). “Laplace Deconvolution with Noisy Observa- tions.” Electronic Journal of Statistics, 7. URL http://dx.doi.org/10.1214/13-ejs796. Akaike H (1965). “On the Statistical Estimation of the Frequency Response Function of a System Having Multiple Input.” Annals of the Institute of Statistical Mathematics, 17(1), 185–210. URL http://dx.doi.org/10.1007/bf02868166. Barigozzi M, Lippi M, Luciani M (2021). “Large-dimensional Dynamic Factor Models: Esti- mation of Impulse–Response Functions with I(1) Cointegrated Factors.” Journal of Econo- metrics, 221(2), 455–482. URL https://www.sciencedirect.com/science/article/ pii/S0304407620302219. Beals R (2004). Analysis: An Introduction. Cambridge University Press. URL http://dx. doi.org/10.1017/cbo9780511755163. Bendat J, Piersol A (1980). “Engineering Applications of Correlation and Spectral Analysis.” The Journal of the Acoustical Society of America, 70, 262–263. URL https://doi.org/ 10.1121/1.386621. Blazhievska I, Zaiats V (2021). “On Cross-correlogram IRF’s Estimators of Two-output Systems in Spaces of Continuous Functions.” Communications in Statistics-Theory and Methods, 50(24), 6024–6048. doi:10.1080/03610926.2020.1738490. Buldygin V, Kozachenko Y (2000). Metric Characterization of Random Variables and Random Processes, volume 188. American Mathematical Society. URL http://dx.doi.org/10. 1090/mmono/188. Buldygin V, Li F (1997a). “On Asymptotic Normality of Estimators of Unit Impulse Re- sponses of Linear Systems. I.” Theory of Probability and Mathematical Statistics, 54, 17–24. Buldygin V, Li F (1997b). “On Asymptotic Normality of Estimators of Unit Impulse Re- sponses of Linear Systems. II.” Theory of Probability and Mathematical Statistics, 55, 29–36. Cavalier L, Raimondo M (2007). “Wavelet Deconvolution with Noisy Eigenvalues.” IEEE Transactions on Signal Processing, 55(6), 2414–2424. URL http://dx.doi.org/10.1109/ tsp.2007.893754. Delaigle A, Hall P, Meister A (2008). “On Deconvolution with Repeated Measurements.” The Annals of Statistics, 36(2). URL http://dx.doi.org/10.1214/009053607000000884. Gikhman I, Skorokhod A (1996). Introduction to the Theory of Random Processes. Dover Publications Inc. doi:10.1007/978-3-642-61921-2_1. 212 On Impulse Response Function Hannan E, Deistler M (1988). “The Statistical Theory of Linear Systems.” 30, 239–242. Ianevych T, Rozora I, Pashko A (2022). “On One Way of Modeling a Stochastic Process with Given Accuracy and Reliability.” Monte Carlo Methods and Applications, 28(2), 135–147. ISSN 1569-3961. URL http://dx.doi.org/10.1515/mcma-2022-2110. Kozachenko Y, Moklyachuk O (1999). “Large Deviation Probabilities for Square-Gaussian Stochastic Processes.” Extremes, 2, 269–293. doi:10.1023/A:1009907019950. Kozachenko Y, Moklyachuk O (2000). “Square-Gaussian Stochastic Processes.” Theory of Stochastic Processes, 6(22), 3–4. Kozachenko Y, Pashko A, Rozora I (2007). Simulation of Stochastic Processes and Fields. Kyiv, Zadruga. ISBN 978-966-432-021-1. Kozachenko Y, Pogorilyak O, Rozora I, Tegza A (2016). Simulation of Stochastic Processes with Given Accuracy and Reliability. Elsevier. ISBN 9781785482175. URL http://dx. doi.org/10.1016/b978-1-78548-217-5.50011-8. Kozachenko Y, Rozora I (2016). “Cross-correlogram Estimators of Impulse Response Func- tions.” Theory of Probability and Mathematical Statistics, 93, 79–91. ISSN 1547-7363. URL http://dx.doi.org/10.1090/tpms/995. Kozachenko Y, Rozora I (2023). On Statistical Properties of the Estimator of Impulse Re- sponse Function, pp. 563–585. Springer International Publishing. URL http://dx.doi. org/10.1007/978-3-031-17820-7_25. Kozachenko Y, Stus O (1998). “Square-Gaussian Random Processes and Estimators of Co- variance Functions.” Mathematical Communications, 3(1), 83–94. Kozachenko Y, Troshki V (2015). “A Criterion for Testing Hypotheses about the Covari- ance Function of a Stationary Gaussian Stochastic Process.” Modern Stochastics: Theory and Applications, 1(2), 139–149. ISSN 2351-6046. URL http://dx.doi.org/10.15559/ 15-vmsta17. Lütkepohl H (2010). Impulse Response Function, pp. 145–150. Palgrave Macmillan UK. ISBN 9780230280830. URL http://dx.doi.org/10.1057/9780230280830_16. Marteau C, Sapatinas T (2015). “A Unified Treatment for Non-asymptotic and Asymptotic Approaches to Minimax Signal Detection.” Statistics Surveys, 9. URL http://dx.doi. org/10.1214/15-ss112. Meister A (2009). Density Deconvolution, pp. 5–105. Springer Berlin Heidelberg. ISBN 9783540875574. URL http://dx.doi.org/10.1007/978-3-540-87557-4_2. Nelles O (2020). Nonlinear Dynamic System Identification, pp. 831–891. Springer In- ternational Publishing. ISBN 978-3-030-47439-3. URL https://doi.org/10.1007/ 978-3-030-47439-3_19. Rozora I (2018). “Statistical Hypothesis Testing for the Shape of Impulse Response Function.” Communications in Statistics - Theory and Methods, 47(6), 1459–1474. URL http://dx. doi.org/10.1080/03610926.2017.1321125. Rozora I (2020). “On the Convergence Rate for the Estimation of Impulse Response Function in the Space Lp(T).” Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, 4, 36–41. doi:10.17721/1812-5409.2018/4.5. Rozora I, Ianevych T, Pashko A, Zatula D (2023). Simulation of Stochastic Processes with Given Reliability and Accuracy, p. 415–452. Nova Science Publishers. ISBN 9781685079826. URL http://dx.doi.org/10.52305/kegg1336. Austrian Journal of Statistics 213 Rozora I, Kozachenko Y (2016). “A Criterion for Testing Hypothesis about Impulse Response Function.” Statistics, Optimization and Information Computing, 4(3). URL http://dx. doi.org/10.19139/soic.v4i3.222. Schetzen M (1980). The Volterra and Wiener Theories of Nonlinear Systems. Wiley, New York. ISBN 9780471044550. Soderstrom T, Stoica P (1989). System Identification. Prentice-Hall, London. Stern D (2018). Energy-GDP Relationship, pp. 3697–3714. Palgrave Macmillan UK. URL http://dx.doi.org/10.1057/978-1-349-95189-5_3015. Takahito I, Munehiko M (2022). “Analytical Solution of Impulse Response Function of Finite- Depth Water Waves.” Ocean Engineering, 249, 110862. URL http://dx.doi.org/10. 1016/j.oceaneng.2022.110862. Vasylyk O, Rozora I, Ianevych T, Lovytska I (2021). “On Some Method on Model Con- struction for Strictly Φ-sub-Gaussian Generalized Fractional Brownian Motion.” Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, 2, 18–25. URL http://dx.doi.org/10.17721/1812-5409.2021/2.3. Affiliation: Iryna Rozora Taras Shevchenko National University of Kyiv Department of Applied Statistics 64/13 Volodymyrska St., Kyiv, 01601, Ukraine E-mail: [email protected] National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" Department of Mathematical Analysis and Probability Theory Peremogy Ave. 37, Kyiv 03056, Ukraine Anastasiia Melnyk Taras Shevchenko National University of Kyiv Department of Applied Statistics 64/13 Volodymyrska St., Kyiv, 01601, Ukraine E-mail: [email protected] Austrian Journal of Statistics http://www.ajs.or.at/ published by the Austrian Society of Statistics http://www.osg.or.at/ Volume 54 Submitted: 2024-06-15 2025 Accepted: 2024-09-10

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