A Non-Asymptotic Confidence Region with a Fixed Size for a Scalar Function Value : Applications in C-OTDR Monitoring Systems

In this paper we will investigate some non-asymptotic properties of the modified least squares estimates for the non-linear function f(λ*) by observations that nonlinearly depend on the parameter λ*. Non-asymptotic confidence regions with fixed sizes for the modified least squares estimate are used. The obtained confidence region is valid for a finite number of data points when the distributions of the observations are unknown. Asymptotically the suggested estimates represent usual estimates of the least squares. The paper presents the results of practical applications of the proposed method in C-OTDR monitoring systems.


Introduction
In some practical cases there appears a necessity to estimate of the value of the function ( ) f λ * by observa- tions that depend on the parameter λ * .One of such cases is estimation of the value λ * of acceptable thickness of paraffin film on the surface of oil transportation pipes when the permission parameter ( ) f λ * is a nonlinear function λ * .In this case the thickness of the paraffin film λ * is estimated based on the data of the telemetric control.The final solution as to whether the value λ * is acceptable will depend primarily on the value of ( ) f λ * .Another actual example is the problem of estimating the seawater absorption coefficient of the sonar signals in shallow water when sensors of the C-OTDR monitoring system are used for measurements.In this case, the absorption coefficient (target parameter ( ) f λ * ) depends nonlinearly on the water temperature (unob- servable parameter λ * ) and on the frequency of sonar emissions.In this particular case, it is very important to get the guaranteed accuracy estimate of the absorption coefficient using only a limited number of observation steps (non-asymptotic statement of the problem).The importance of non-asymptotic results is dictated by restricted volume of available sample.In addition, the absorption coefficient estimation should be performed as often as possible, to explore its connection with the intensity of very dynamic factors (tidal and bottom currents, turbulence seawater).This condition is realizable only if in each cycle of estimation we spend solely a finite number of steps to provide the guaranteed accuracy estimates.Thus in this particular case the non-asymptotic statement of the problem represents an objective necessity.In this paper we will investigate some non-asymptotic properties of the modified least squares estimates for the non-linear function ( ) f λ * by observations that nonlinearly depend on the parameter λ * .Asymptotically the suggested estimates represent usual estimates of the least squares.Asymptotic properties of nonlinear least squares estimates are well investigated and discussed (Jennrich [1], Ljung [2], Lai [3], Anderson and Taylor [4], Wu [5], Hu [6], Skouras [7]).At the same time, few results addressing the finite sample properties exist, whereas the non-asymptotic solution for the problem of the parameter estimation for stochastic process is practically important because the sample volume is always limited from above.Accurate construction of confidence regions for unknown parameters in a non-asymptotic configuration was obtained for linear models of stochastic dynamic processes (Campi and Weyer [8]- [11], Ooi, Campi and Weyer [12]).Non-asymptotic estimation of scalar parameter of non-linear regression by means of confidence regions was examined by Timofeev [13].Similar estimation of multivariate parameter was researched by Timofeev [14] [15].In this paper a sequential design is suggested that will make it possible to solve the problem of non-linear estimation of the function ( ) f λ * value for a wide class of stochastic processes by means of con- fidence regions in the non-asymptotic setting.The solution was obtained under condition of partial a priori definiteness as regards to the stochastic distribution of the observations

Statement of the Problem
Let us consider estimation of the value of a continuous function ( ) The values of α , β are fixed, the value of the λ * is a priori unknown, but it is definite that the parameter λ * enters into the equation of an observed process Here the non-observed sequence of the noise . The class of models described by ( 1) is wide and includes many linear and nonlinear regression models commonly used.For example, ( ) , a k λ may be a nonlinear function of past ob- servations and any other variables ( ) It is needed to construct the confidence interval of the fixed size for value of the ( ) For the sake of clarity, the following shorthand notation will be used throughout the rest of the paper: ( E instead of

The Main Result
Let us consider the following estimate: where Here ( ) With ( ) 2) is an ordinary LS criterion which is constructed with the sample vo-lume of ( ) 1 t n + .The sequential design for confidence estimation of the parameter ( ) f λ * will be regarded as a pair ( ) ( ) ( ) Let us consider the closed interval ( ) , and we will use the ( ) , S d τ as a confidence interval for the value of ( ) The properties of the sequential design and function 7) There exists a known constant Then for any 0 d > the following assertions hold true: Remarks.The sequences t n are parameters of the estimation procedure.In order to meet the condition 7, the sequences t n may be definite, for example, in the form of ( ) ( ) ( ) 2) If n → ∞ the series ( ) 3) ( ) be a sequence of continuous on the compact K stochastic functions for which the following conditions are met: and limiting function has the unique minimum in the point K λ * ∈ .
Then a constant B exists and for it the following assertion is true: This lemma is a corollary of the Theorem 19 [18].
≥ is a square integrable martingale.From condition 2 of the Theorem 1 we have: Further using the strong law of large numbers for square integrable martingales [17] we have It follows from conditions 2 and 8 of the Theorem 1 for the square integrable martingale ( ) ( ) that the following assertion is true and using the strong law of large numbers for square integrable martingales we have: From ( 3), ( 4) and condition 3 of the Theorem 1 and assertion of the Lemma 1 it follows that λ α β ∈ .Further, using ( 1) and ( 2) we have: 3), ( 5) as well as conditions 3, 4 of the Theorem 1, the following assertions hold true: of the parameter λ * (it follow from the Lemma 3).From here and from the continuity of the function ( ) , Consider the set Taking into account that on the set (7) the following inequalities hold true from (6) it follows that on the same set the following conclusions are true, too: By definition, the estimates ( ) Using (8) we can that on the set (7) the following inequalities are true: Let us set the upper bound of the probability of the event ( 7) not happening: From the conditions 1, 2, 5 of the Theorem 1, the function ( ) ( ) ( ) ( ) Using this property (10) and assertion of the Lemma 3 we have

Practical Example
Let us consider the problem of estimating the seawater absorption coefficient of the sonar signals in shallow water when sensors of a C-OTDR monitoring system are used for measurements.The fiber-optic sensor (FOS) of the C-OTDR monitoring system is laid on the sea bottom; it is an ordinary monomode fiber-optic line stowed inside of a special hygroscopic cable.At the logical level the entire length of this cable is split into equal portions duration of ~5 m.Each of these sections is called a logical C-OTDR channel or DAS (distributed acoustic sensor).Each DAS is able to measure the vibration of the hydrosphere which appeared in the area of its sensitivity.In fact, FOS consists of a huge number of vibration sensitive sensors successively arranged along the cable.A source of the hydroacoustic emissions (SHdE) emits a pulsed narrowband signal at a predetermined frequency.The coordinates of the SHdE are given.Therefore the angle of incidence of the hydroacoustic waves on FOS is = , the suggested method allows us to construct a confidence interval with fixed size for non-known parameter λ * .The results of the practical application showed efficiency of the proposed approach.

Theorem 1 .
Assume that the following statements are true: 1) For some 0 u > stochastic functions

the Theorem 1 .Lemma 1 . [ 1 ] 1 )
The proof is based on the following statements.Let us assume that for a numeric sequence { } k e and a sequence of the continuous on the compact K functions ( ) { } k g t the following conditions are met: If n → ∞ the series 1

3 )
Then there exists a sequence of the random valuesHere ⋅ is the norm of the space in which the compact K is embedded.Lemma 3. [17] Let the ( ) t ξ is a stochastic and continuous on the interval [ ]

Lemma 4 . 4 .
Let us assume that the real function the(1) are continuous on the interval [ ] , α β .If the conditions 2 -4 and 7 of the Theorem 1 hold true, then we have Consider the following representation:

≥
converges uniformly in t K ∈ almost sure.• The limiting function of the series Lemma 4 is proven.■ Let us get back to the proof of the Theorem 1.The ( )

Table 1 .
Estimation results of the absorption coefficient.