In this paper we solve the estimation problem of the drifting parameter of ARARCH(1,1) model with guaranteed accuracy in the mean square sense by sample of fixed size. Cases of known and unknown noise variances are considered. For the first case we apply the truncated sequential estimation method and for the second one we use the truncated estimation method. It is shown that the obtained results can be applied to more general model

Dogadova, Tatiana V.

Vasilev, Vyacheslav A.

English

Tomsk State University Repository

ИЗВЕСТИЯ ВЫСШИХ УЧЕБНЫХ ЗАВЕДЕНИЙТ. 58, № 11/2 ФИЗИКА 2015УДК 519.233.22T.V. DOGADOVA*, V.A. VASILIEV**GUARANTEED PARAMETER ESTIMATION OF ARARCH(1,1) WITH DRIFTINGPARAMETERIn this paper we solve the estimation problem of the drifting parameter of ARARCH(1,1) model with guaranteedaccuracy in the mean square sense by sample of fixed size. Cases of known and unknown noise variances are considered.For the first case we apply the truncated sequential estimation method and for the second one we use the truncated esti-mation method. It is shown that the obtained results can be applied to more general models.Keywords: parameter estimation, ARARCH model, drifting parameter, truncated sequential estimators, guaranteed accu-racy.Nowadays mathematical statistics is turned to development of data processing methods by depend-ent sample of finite size. One of such possibilities gives a well-known sequential estimation method,which was successfully applied to parametric and non-parametric problems, see, e.g. [1, 2].To obtain sequential estimators with an arbitrary accuracy one needs to have a sample of unboundedsize. However, in practice the observation time of a system is usually not only finite but fixed. One of thepossibilities for finding estimators with the guaranteed accuracy of inference using a sample of fixed sizeis provided by the approach of truncated sequential estimation. The truncated sequential estimationmethod was developed in [3–5] and others for parameter estimation problems in discrete-time dynamicmodels. In these papers, estimators of dynamic system parameters with known variance by sample offixed size were constructed. Another but very similar approach was proposed in [6], where truncated es-timation method was developed for estimation of ratio type functionals.It is known that nonlinear stochastic systems are being widely used for describing real processes ineconomics, technics, medicine etc. We consider as a model the combination of two well-known models –AR with drifting parameters and ARARCH model. Such model can describe more accurately the realprocesses with complicated behavior.The estimation problem of drifting parameter in ARARCH(1,1) model is considered. In the case ofknown noise variances truncated sequential estimation method developed in [3, 4] is applied. It makespossible to solve the estimation problem even for unstable observable process, while the truncated esti-mation method proposed in [6] works in the case of unknown variances and can be applied for morecomplicated multivariate models.1. Problem statementConsider one-dimensional process ARARCH(1,1) with drifting parameter12 2 21 1 0 1( ) nn n n nx s x x −− −= λ + + σ + σ ⋅ ⋅ ξ ,                            (1)with the initial zero mean variable 0x  having the forth moment; ( )ns  and ( )nξ  are sequences of i.i.d.zero mean random variables such that 0x , ( )ns  and ( )nξ  are mutually independent and2 2 4 2 4, , 1, .s n n n nEs Es E Eσ = < ∞ ξ = ξ < ∞Our aim is to estimate with guaranteed accuracy the unknown parameterλ . Cases of known and un-known noise variances will be considered.2. Case of known noise variancesWe suppose that the parameters 2 2 20 1, , sσ σ σ  are known and the noise nξ  satisfies the following addi-tional condition 21 1( ) ,D E mα = α − < ∞? ?  where211 12 21, .1sm Eξα == = ασ ξ +? ??  Moreover, noises ( )nξ  havecentral symmetric density ( )f xξ  such that the set { : ( ) }x f x uξ ≥  is convex for every u.Guaranteed parameter estimation of ARARCH(1,1) with drifting parameter 277Similar to general estimation procedure in [5] the truncated sequential estimator of parameter λ  hasthe form21 12 2 2 2 2 2 2 21 10 1 1 0 1 11 ,( ) ( )N Nn n n nN Nn nN s n s nx x x HH x xτ − −= =− −⎡ ⎤β ⋅λ = χ ≥⎢ ⎥σ + σ + σ σ + σ + σ⎢ ⎥⎣ ⎦∑ ∑?          (2)where , (0, ),NH hN h m= ∈  the stopping time Nτ  and weights nβ  have respectively the following forms:[ ] 2 21 12 2 2 2 2 2 2 21 10 1 1 0 1 1212 2 2 21 0 1 1inf 1, : , ,( ) ( ), ,( )k Nn nN Nn ns n s nNNnNn s nx xk N H Hx xxN Hx− −= =− −−= −⎧ ⎧ ⎫⎪ ⎪∈ ≥ ≥⎪ ⎨ ⎬σ + σ + σ σ + σ + σ⎪ ⎪⎪ ⎩ ⎭τ = ⎨⎪ <⎪ σ + σ + σ⎩∑ ∑∑(3)212 2 2 21 0 1 1212 2 2 21 0 1 11, 1 , 1,  ,  ,( ),  ,  ,( )NNnn N Nn s nNnN N Nn s nnxn Hxxn Hx−= −−= −⎧⎪ ≤ < τ⎪⎪⎪β = = τ <⎨ σ + σ + σ⎪⎪⎪α = τ ≥⎪ σ + σ + σ⎩∑∑221 112 2 2 2 2 2 2 21 0 1 1 0 1 1.( ) ( )NNnN Nn s n s nxxHx xτ − τ −−= − −⎛ ⎞α = −⎜ ⎟⎜ ⎟σ + σ + σ σ + σ + σ⎝ ⎠∑According to Theorem 1 from [5] we have22 2 12 2 2 21 0 1 1( ) .( )NnN Nn s nxCE P HN x−λ λ= −⎛ ⎞λ − λ ≤ + λ <⎜ ⎟⎜ ⎟σ + σ + σ⎝ ⎠⋅ ∑?Now we estimate the probability in the right-hand side of the inequality. Denote 2 2 21 ,s sσ = σ + σ?211 2 2 21 0,nns nxx−−−α = σ + σ?   0 0 1( , ,..., , ,..., ),N N NF x s s= σ ξ ξ  12 11NN N nnH H−− −== − α∑?  and similar to [2, 4], usingthe conditions on the density (.),fξ  estimate the conditional probability:212 1 2 1 2 22 2 2 21 10 1 1| | ( | )( )N NnN N n N N N N Nn ns nxP H F P H F P H Fx−− − − − − −= =−⎛ ⎞ ⎛ ⎞< = α < = α < =⎜ ⎟ ⎜ ⎟⎜ ⎟σ + σ + σ ⎝ ⎠⎝ ⎠∑ ∑ ?2 2 2 2 21 0 1 2 12 2 2 22 2 2 2 2 2 2 2 21 0 0 1 2 1 0( )| |( )N N NN N N Ns N s N Nx xP H F P H Fx x− − −− − − −− − −⎛ ⎞ ⎛ ⎞σ + σ ⋅ ξ= < ≤ < ≤⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟σ + σ σ σ + σ ⋅ ξ + σ⎝ ⎠ ⎝ ⎠? ?? ?2 112 2 1 1 22 221| | ,1NNN N N n N Nns NP H F P H F−−− − − − −=−⎛ ⎞ξ ⎛ ⎞≤ < ≤ α + α <⎜ ⎟ ⎜ ⎟⎜ ⎟σ ξ + ⎝ ⎠⎝ ⎠∑? ??where                                                    22 2 .1NNs Nξα = σ ξ +? ?Analogously, we obtain1 21 1 3 1 2 1 32 2| |N NN n N N N N n N Nn nP H F P H F− −− − − − − − −= =⎛ ⎞ ⎛ ⎞α + α < ≤ α + α + α <⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑? ? ?etc. So that finally we get1 1 11 2 2( ) ( 1)N N Nn N n N n Nn n nP H P H P m H m N− − −= = =⎛ ⎞ ⎛ ⎞ ⎛ ⎞α < ≤ α < = α − < − − ≤⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠∑ ∑ ∑? ?T.V. Dogadova, V.A. Vasiliev278211 12 22 2( 1)1( ) ( ) ( ) .[( ) ] [( ) ]N Nn nn nN D CP m m h N m E mNm h N m m h N m− −= =⎛ ⎞ − α⎡ ⎤≤ α − > − − ≤ α − = ≤⎜ ⎟ ⎢ ⎥− − − −⎣ ⎦⎝ ⎠∑ ∑?? ?Thus, it was proved that for every L  2( ) .sup NLCENλλ ≤λ − λ ≤?Similar result can be obtained if the noise variances are unknown but their upper bounds are known2 2 2 2 2 20 0 1 1, ,s sσ ≤ σ σ ≤ σ σ ≤ σ  and 2 2.sσ ≤ σ?  In this case the condition 1Dα < ∞?  must be changed tothe fol-lowing one:2 20 12 2 2 20 1 0.D⎛ ⎞σ ⋅ ξ < ∞⎜ ⎟⎜ ⎟σ ⋅σ ⋅ξ + σ⎝ ⎠3. Case of unknown noise variancesWhen the noise variances as well as their upper bounds are unknown the truncated estimationmethod in the represented form cannot be applied. That is why we use the truncated estimation methodpresented in [6].In addition to general conditions on the noises and parameters of the observable process we supposethat there exists the known lower bound for the parameter 2 20 * 0σ ≥ σ >  and the following stability condi-tion4 2 2 2 2 41 1 1 1( ) 6 ( ) 1sE s Eλ + + σ σ + σ ξ <                        (4)holds. Define the set Θ of vectors 2 2 20 1( , , , )sθ = λ σ σ σ  satisfying the condition (4).It is easy to verify that under this condition the process ( )nx has the following property41sup nnEx≥< ∞and positive limits exist2 42 4, .lim limn nn nm Ex m x→∞ →∞= =According to the general estimation procedure in [6] we define the truncated estimator as follows:21 12 1 1111ˆ ,N NN n n n NNn nnnx x x Hx− −= =−=⎡ ⎤λ = ⋅ ⋅ χ ≥⎢ ⎥⎣ ⎦∑ ∑∑where 2* .NH N= σ ⋅From (1) it follows that the deviation of the estimator ˆ Nλ  has the form2 2 2 2 21 1 0 1 1 1 1 12 1 1 1111ˆ .N N NN n n n n n n N n NNn n nnnx s x x x H x Hx− − − − − −= = =−=⎡ ⎤ ⎡ ⎤⎡ ⎤λ − λ = + σ + σ ⋅ξ ⋅χ ≥ − λ ⋅χ <⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦∑ ∑ ∑∑Using this representation similar to [6] we can prove for every θ∈Θ the following property2ˆ( ) ,NCENθλ − λ ≤where the number  C depends on 2m  and 4.mRemark.  Similar results can be obtained using the truncated estimation method for ARARCH(p,q)model with drifting parameters of the form2 2 2 2 21 1 1 0 1 1( ( 1)) ... ( ( 1)) ... n qn n p p n p n q nx s n x s n x x x −− − −= λ + − ⋅ + + λ + − ⋅ + σ + σ ⋅ + + σ ⋅ ⋅ ξand  using the truncated sequential estimation method for ARARCH (1, q).Guaranteed parameter estimation of ARARCH(1,1) with drifting parameter 2794. SimulationsTo confirm the convergence of the truncated sequential estimators we made the simulations of theesti-mator Nλ? defined in (2), (3) (Tables 1, 2) with NH h N= ⋅ in case of known 2 20 11,σ = σ =20.1, 0.05.s= σ =For this purpose we used the software package MATLAB. In Tables 1, 2 the average( ) ( )10011100 NkN k=λ = λ∑? ?of estimators (2) for the k-th realization ( ) ( )( ) , 1...100k knx x k= =  of the process (1) and their quality char-acteri-stics( )( )100 2211( )100 NkS N kλ== λ − λ∑ ?for different N are given.T a b l e  1Estimation of the parameter λ with h = 0.2N = 100 N = 200 N = 500λ ( )Nλ? ( )S Nλ ( )Nλ? ( )S Nλ ( )Nλ? ( )S Nλ0.2 0.2136 0.1630 0.2118 0.0871 0.2057 0.0393–0.2 –0.2004 0.1722 –0.2373 0.0999 –0.2010 0.05890.9 0.8876 0.2494 0.9199 0.1164 0.9156 0.0413–0.9 –0.8886 0.2781 –0.8726 0.1307 –0.8908 0.05891 1.0128 0.2392 0.9820 0.1312 1.0341 0.0577–1 –0.9811 0.2065 –1.0105 0.1246 –1.0004 0.05424 3.9711 0.2661 4.0862 0.2083 4.0575 0.0868–4 –4.0118 0.2901 –3.9984 0.1512 –4.0366 0.0533T a b l e  2Estimation of the parameter λ with h = 0.6N = 100 N = 200 N = 500λ ( )Nλ? ( )S Nλ ( )Nλ? ( )S Nλ ( )Nλ? ( )S Nλ0.2 0.2234 0.0721 0.1966 0.0249 0.2103 0.0149–0.2 –0.1918 0.0613 –0.1837 0.0389 –0.2058 0.01230.9 0.8958 0.0665 0.9182 0.0341 0.8920 0.0191–0.9 –0.8884 0.0520 –0.8829 0.0350 –0.8914 0.02291 1.0234 0.0714 1.0076 0.0539 1.0050 0.0212–1 –1.0159 0.0755 –1.0318 0.0453 –1.0064 0.03384 4.0079 0.1023 4.0451 0.0544 4.0165 0.0202–4 –4.0581 0.1098 –3.9939 0.0724 –3.9829 0.0184Looking at the simulation results we can say that the deviation becomes less with growth of thesample size. It means that the estimator’s value becomes closer to the true meaning of the parameter. Thisfact proves that these estimation procedures are quite effective. Moreover, the considered estimator worksin the unstable case as well.REFERENCES1. K o n e v  V . V .  Sequential parameter estimation of stochastic dynamical systems. – Tomsk: Tomsk Univ. Press, 1985.2. K o n e v  V . V .  The boundaries for the mean number of observations in problems of sequential parameter estimation forrecurrent processes // Avtomatica i Telemehanica. – 1983. – No. 8. – P. 64–73.3. F o u r d r i n i e r  D . ,  K o n e v  V . ,  P e r g a m e n s h c h i k o v  S .  Truncated sequential estimation of the parame-ter of a first order autoregressive process with dependent noises // Mathematical Methods of Statistics. – 2009. – V. 18. –No. 1. – P. 43–58.T.V. Dogadova, V.A. Vasiliev2804. K o n e v  V . V . ,  P e r g a m e n s h c h i k o v  S . M .  Truncated sequential estimation of the parameters in random re-gression // Sequential Analysis. – 1990. – V. 9. – No. 1. – P. 19–41.5. D o g a d o v a  T . V . ,  V a s i l i e v  V . A .  Guaranteed parameter estimation of stochastic linear regression linear re-gression by sample of fixed size // Tomsk State University: Journal of Control and Computer Science. – 2014. – V. 26. –No. 1. – P. 39–52.6. V a s i l i e v  V . A .  A truncated estimation method with guaranteed accuracy // Annals of the Institute of StatisticalMathematics. – 2014. – V. 66. – No. 1.– P. 141–163.*National Research Tomsk State University, Tomsk, Russia Article submitted October 14, 2015**National Research Tomsk Polytechnic University, Tomsk, RussiaE-mail: aurora1900@mail.ru, vas@mail.tsu.ru_______________Dogadova Tatiana Valerievna, Student;Vasil'ev Vyacheslav Arturovich, Dr., Professor, Professor Department of Higher Mathematics and Mathematical Modelling.

Guaranteed parameter estimation of ARARCH(1,1) with drifting parameter

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Guaranteed parameter estimation of ARARCH(1,1) with drifting parameter

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