Parameter estimation and change-point detection for process AR(p)/ARCH(q) with unknown parameters.pdf The problem of change point detection arises often in different applications connected with time series analysis, financial mathematics, image processing etc. Two types of algorithms are used to detect the change point: a posteriori methods, when the estimation of the change point is conducted in a sample of a fixed size, and sequential methods, when the decision on change point can be taken after obtaining a next observation. Sequential methods include a special stopping rule that determines a stopping time. At this instant a decision on change point can be made. There are two types of errors typical for sequential change point detection procedures: false alarm, when one makes a decision that change is occurred before a change point (type 1 error), and delay, when one makes a decision that change is not occurred after a change point (type 2 error). The properties of the sequential procedures are connected with these errors and include probabilities of the errors, mean delay time and mean time between false alarms. Last decades, autoregressive type processes and autoregressive conditional heteroscedasticity processes are widely used in various applications, such as forecasting of financial indexes, geographic information systems, medical data analysis, etc. For example in paper [1], autoregressive models are used for description of financial data. In the references therein, one can find examples of applications in other fields, including business, economics, finance and quality control. Processes with non-constant parameters also can be used for such purposes. In [2] a piecewise constant model is set off against usual GARCH model for volatility modelling. A two-sample test for a change in variability is proposed, which works well even in case of skewed distributions. Paper [3] describes a usage of mixtures of structured autoregressive models for the analysis of electroencephalogram. On-line posterior estimation of the model parameters and related quantities is achieved using a sequential Monte Carlo algorithm. One of recent papers [4] is devoted to change point detection in casual time series such as AR(w), ARCH(w), etc. The procedure is based on a discrepancy between the historical parameter estimator and the updated parameter estimator, where both these estimators are quasi-likelihood estimators. To construct these estimators historical observations supposed to be available. It is proven that if the change occurs then it is asymptotically detected with the probability one. Asymptotic behavior of the test statistic can be described using the standard Brownian motion. The power of the test is estimated by simulation. In paper [5], change-point detection is applied to analysis of financial data. A fractionally integrated process is considered and changes in the fractional integration parameter supposed to be detected. The authors use AR(p) model, for some large enough p, to approximate the process under consideration. The application of the tests to World inflation rates detected the presence of changes in persistence for most countries. In [6], some historic data set which is stationary and does not contain a change is used to construct an estimator for the initial set of parameters. Then new incoming observations are monitored for a change. It is shown that the algorithm can be applied to mean change model and to non-linear first-order autoregressive time series. Theoretical properties of the described procedures are studied asymptotically when the number of observations before a change point tends to infinity. For small samples, usually simulation study is conducted. In this paper, we develop an alternative approach in the frame of guaranteed sequential methods. Due to a special stopping rule, we construct statistics with variances bounded from above by a known constant. Consequently, we can estimate the probabilities of false alarm and delay non-asymptotically, but we also investigate asymptotic properties of the statistics. 1. Model AR/ARCH We consider scalar autoregressive process AR(p)/ARCH(q) specified by the equation 4 = ^Wi + • • • + ЬрХк-р + Va о + аЛ-1 + ■ ■ ■ + ачх1-ч (1) Here - is a sequence of independent identically distributed random variables with zero mean and unit variance. The density distribution function f (x) of {^is strictly positive for any value of x. Parameters Л = [^1,..., VI and A = [o0,..., aq] are supposed to be unknown. 2. Sequential parameter estimator for AR(p)/ARCH(#) For parameter estimation of the process (1) we use the approach proposed in [7] for classification of autoregressive processes with unknown noise variance bounded from above. At the first stage, we construct a special factor to compensate the influence of the noise variance. Then, we estimate autoregressive parameters by using this factor. Since the noise variance of the observed process is unbounded from above, we transform the model by introducing the following notation m-i = max{1,| xk-i[---\xk-s ll where s = max{p,q}. Dividing equation (1) by mM, we obtain Ук = ZkA + JxkA^k , (2) where 4-1 Xk-p Ук , Zk = m-. 1 xk_q 2 ' 2 '"'7 mk_1 mk_x mk_x к-1 lk-1 "lk-1. It is obvious, that the noise variance of the process (3) is bounded from above by the unknown value a0 + ... + aq. We can construct the compensating factor by first n observations in the following form 2 / N-1 s+n x I n \ r« = Bn Z . j, 2 -ГЛ> Bn = El 2I (3) k=s+1 mm{1, Xk-1,..., xk-q ] У k=1 ) where n observations are taken at the interval where all the values |xk| are sufficiently large. In [8], we use a similar approach to compensate the noise variance of AR(p) process with unknown noise variance; it was proven that the compensating factor satisfies condition analogous to E-F -7-1-i (4) rn \a 0 +а1 + . + а q ) This proof can be generalized for our case with minimum changes so we omit it. We construct the estimator of the parameter vector Л in the form fx \ X Л(я) = С"1(х) £ vkykZTk , C(t) - £ vkZTkZk, (5) Vk=n+s+\ / k=n+s+\ where т is the random stopping time defined as follows x = x(H )= min{t > n + s -1: v mn (t )> H}, (6) vmin (t) is the minimum eigenvalue of the matrix C(t), H is a certain positive parameter. Then we define the weights vk. Let m be the minimum value of t for which the matrix C(n +1) is not degenerate. The weights on the interval [n + s + 1, n + m] are defined as Vk = < (rnZkzT ) ! , if Zk is linearly mdepeMert with {Zn+s+i, ■ ■ ■,Zk-1}; ^ 0, elsewhere. The weights on the interval [n + m,x -1] are defined from the equations Vm»(t) = £ v2ZkZk. (8) n The last weight v is found from condition у £ vkZkZk ■ 1 n k=n+s+1 Vmin(X) > £ v2kZkZTk , Vmin(T) = H. (9) Theorem 1. The stopping time i( //) is finite with probability one and the mean-square accuracy of the estimator A(#) is bounded from above (io) Proof. According to [9], the stopping time x( H) is finite with probability one if да £ vkkzkzk = a.s. k=n+s+1 The equation for vk (14) can be rewritten in the form mjn (X,(C(k -1) + vZT )x) = Vk~1) + vkzZ. n " n It implies that for any vector x || = 1 (x,C(k- 1)x) + Vk(x,ZTTkX) = ft,C(^ 1)x) + Vk(Zkxf 0, then the equation has two roots: non-positive and non-negative. It gives us the equation for the weight (Zkbk )k + iZkbk )4 + 4TnZkZkk ((bk,C (k -1)bk ) - Vmin (k -1)) vk = min-5---. НИН kTnZkZl Consequently, vk tends to zero if and only if Zkbk tends to zero and at the same time bk tends to the eigenvector corresponding to the minimum eigenvalue of the matrix C(k-1) as k tends to infinity. As the first component of the vector Zk depends on Z^ (1) which can take any value then vk does not tend to zero with non-zero probability and the instant x is finite with the probability one. Г n k-n+s-1 For the mean-square accuracy of Л(Я) (5), by using (2), Cauchy-Schwarz-Bunyakovskii inequality, inequality || C(t) ||> vmin(t) and (4), we obtain C"1(t) i vkjxak+z k For the second multiplier, т E I Vk%knZl = E I v2ZkZTk ^ + 2E I I vlVkZlZTk (11) k=n+s+1 k=n+s+1 k=n+s+1l=n+s+2 Consider a truncated stopping instant t(N) = min{T, N} . Consider the sum differing from the first summand only in the upper limit. Let Fk = g( ) be the c-algebra generated by }, then т defined by (6) а, -а. I VkZlS. E = E xH[=p !> yi 2 > xH р{||л(Я)-л| Z ^k+1 k=n+s+1 УH > x < P < Using the result of Theorem 2 and Fubini's theorem to change the order of integration one obtains . ® exp \iX Y1 f i i i j j-i-E expj- X 2X !>dXdY = E j e YYT >xH/B -о 2П 1 2 J J(2п)p YYt >xH/B P jllY||2 > f j exp J-1YTS-1Y idY. The matrix E is symmetric and positive definite; hence, an orthogonal transformation T, resulting in the matrix E to diagonal form E', exists. Using the change of variables S = Y E ~1/2TT one obtains B | //o„ J I 2 j , i=1 £vA2 > xH/B A. We construct a series of sequential estimation plans (тг, Лг), where {т} is the increasing sequence of the stopping instances (то = -1), and Л< is the guaranteed parameter estimator (5) on the interval [т-1 + 1, т]. Then we choose an integer l > 1 and associate the statistic Ji with the i-th interval for all i > l J, = (Л

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