Filomat 7:7 (013), 1197 103 DOI 10.98/FIL1307197B Publishe by Faculty of Sciences an Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Ranom proucts an prouct auto-regression Hassan S. Bakouch a,b, Miroslav M. Ristić c, E. Sanhya, S. Satheesh e a Statistics Department, Faculty of Sciences, King Abulaziz University, Jeah, Saui Arabia b Mathematics Department, Faculty of Science, Tanta University, Egypt c University of Niš, Faculty of Sciences an Mathematics, Serbia Department of Statistics, Prajyoti Niketan College, Puukka, Inia - 680 301 e Department of Applie Sciences, Viya Acaemy of Science an Technology, Thrissur - 680 501, Inia Abstract. The operation of taking ranom proucts of ranom variables an the notions of infinite ivisibility (ID) an stability of istributions uner this operation are iscusse here. Base on this stationary prouct auto-regressive time series moels are introuce. We investigate some properties of the moels, like autocorrelation function, spectral ensity function, multi-step ahea conitional mean an parameter estimation. 1. Introuction Klebanov et al. [4] consiere the problem of istribution of the prouct of a ranom number of ranom variables (r.vs) an an application of geometric-proucts in mathematical economics. The iscussion therein was base on consiering the log-transform of the r.vs so that a prouct can be treate as a sum (if the range of the r.vs permits this transformation), invoke the result for the sum an then get back to the prouct. McKenzie [6] introuce (perhaps for the first time) a prouct auto-regressive (PAR(1)) moel X n = X α n 1 V n, (1) where α (0, 1) This is the prouct analogue of the AR(1) moel Y n = αy n 1 + ε n, () where Y n = log X n an ε n = log V n. McKenzie [6] notice that the correlation structures of the PAR(1) an AR(1) moels with gamma marginals are the same an given by Corr(X n, X n k ) = α k, k = 0, 1,... Then, he characterize the gamma istribution, among self-ecomposable istributions, as the only one having this correlation structure in the PAR (1) moel. 010 Mathematics Subject Classification. Primary 6M10 Keywors. Auto-regressive process, conitional least squares, log-gamma, ranom proucts, stability Receive: 03 June 01; Accepte: 08 December 01 Communicate by Dejan B. Ilić The secon author acknowleges the grant of MNTR 174013 for carrying out this research. Email aresses: hnbakouch@yahoo.com (Hassan S. Bakouch), miristic7@gmail.com (Miroslav M. Ristić), esanhya@hotmail.com (E. Sanhya), ssatheesh1963@yahoo.co.in (S. Satheesh)
H. Bakouch et al. / Filomat 7:7 (013), 1197 103 1198 Motivations of this paper are represente in the following items. Describing the operation of ranom proucts of ranom variables an the notions of infinite ivisibility an stability of istributions uner this operation. This escription enables us to iscuss first-orer prouct autoregressive moels with ranom coefficients base on this operation. These moels represent nonlinear moels with autoregressive correlation structure. In the next section we iscuss certain aspects of ranom proucts an then in Section 3 two PAR(1) moels relate to them. Section 4 investigates some properties of the moels, like autocorrelation function, spectral ensity function, multi-step ahea conitional mean an parameter estimation.. Ranom proucts an infinite ivisibility Let us note that the ivisibility properties of r.vs uner the operation of sums or ranom-sums reflect the corresponing ivisibility properties of their log-transforms uner the operation of proucts or ranomproucts. In the ensuing iscussion we assume that the range of the r.vs permits this transformation. Hence we may formally have the following. Definition.1. A r.v X is prouct infinitely ivisible (PID) if for each positive integer n, there exists i.i. r.vs {X i,n, i = 1,..., n} such that X = n X i,n. Definition.. A r.v X is prouct stable (PS) if for each positive integer n, there exists a c > 0 such that X = n X c i,n, where X i,n, i = 1,..., n are inepenent copies of X. More generally we have the following. Since Z n = n n X c i log Z n = c log X i if Z n Z as n, then log Zn log Z as n. Similarly if n S n = b Y i e S n = n e by i, then conclusions on the weak limit of S n hol well for log S n as above. We may also invoke the transfer theorem for ranom-sums an obtain results corresponing to ranom-proucts. The following conclusions are now clear. Result.3. Y is ID if an only if X = e Y is PID. Result.4. Y is sum-stable if an only if X = e Y is prouct-stable. Result.5. Y is ranom-sum-stable if an only if X = e Y is ranom-prouct-stable. The notion of ranom infinite ivisible (N-ID) laws is systematically iscusse in Gneenko an Korolev []. To overcome certain limitations in this notion Satheesh [7] introuce φ-id laws an from its easier efinition (Definition.3 in Satheesh et al. [9]) it follows that φ-id laws generalize N-ID laws. Further, notice that the class of geometric-id (GID) laws forms a subclass of the class of Harris-ID (HID) laws which in turn is a subclass of the class of N-ID laws, see e.g. Satheesh et al. [9]. We are thus in a position to formulate the following results. Result.6. Y is φ-id if an only if X = e Y is prouct-φ-id.
H. Bakouch et al. / Filomat 7:7 (013), 1197 103 1199 Corollary.7. Y is N-ID if an only if X = e Y is prouct-n-id. Corollary.8. Y is HID if an only if X = e Y is prouct-hid. Corollary.9. Y is GID if an only if X = e Y is prouct-gid. The following results, which are known in the literature, show the interplay between sums an proucts of r.vs. Theorem.10. ([1]) Let Y 1, Y,... are i.i. positive r.vs with non-zero finite mean an N p a positive geometric(p) N p r.v inepenent of Y 1 for every p (0, 1). Then Y 1 = p Y i if an only if Y 1 is exponential. Theorem.11. ([4]) Let X 1, X,... are i.i. positive r.vs with E(log X 1 ) finite, non-zero an N p a positive geometric(p) r.v inepenent of X 1 for every p (0, 1). Then X 1 = Np Xp i if an only if X 1 is Pareto (logexponential). Theorem.1. ([8]) Let Y 1, Y,... are i.i. positive r.vs with non-zero finite mean an N p a positive Harris(1/p, m) N p r.v inepenent of Y 1 for every p (0, 1). Then Y 1 = p Y i if an only if Y 1 is gamma(1/m). Result.13. Let X 1, X,... are i.i. positive r.vs E(log X 1 ) finite, non-zero an N p a positive Harris(1/p, m) r.v inepenent of X 1 for every p (0, 1). Then X 1 = Np Xp i if an only if X 1 is log-gamma(1/m). 3. PAR(1) moels We will consier two PAR(1) moels here which are generalizations of (1). We have the following AR(1) moel of Lawrance an Lewis [5] Y n = { Vn, with probability p, Y n 1 + V n, with probability 1 p. (3) The prouct analogue of this is X n = { εn, with probability p, X n 1 ε n, with probability 1 p. (4) It is known from [3] that (3) is stationary for each p (0, 1) if an only if Y n is GID. Hence we have Result 3.1. The PAR(1) moel (4) is stationary for each p (0, 1) if an only if X n is prouct-gid. The prouct-stability results, Theorem.11 an Result.13 above, can be use to moel the generalize PAR(1) moels which are the multiplicative analogue of those iscusse in Satheesh et al. [11] an characterize various istributions that are the log-versions of the istributions therein. For a fixe an known m > 0, a generalization of (3) is m Y i,n = m V n, with probability p, m Y n 1 + m V n, with probability 1 p. (5)
H. Bakouch et al. / Filomat 7:7 (013), 1197 103 100 The prouct analogue of this (a generalization of (3)) is given by m m ε n, with probability p, X i,n = m m X i,n 1 ε i,n, with probability 1 p. By Theorem.3 in Satheesh et al. [10] the sequence {Y i,n } efines the moel (5) that is stationary for each p (0, 1) if an only if {Y i,n } is Harris(a, m)-id, a = 1/p. Hence we have Theorem 3.. If for each i fixe, {X i,n } escribes the PAR(1) moel (4) that is stationary for each p (0, 1) an if for each n, the processes {X i,n, i = 1,,..., m} (m > 0 is known) are inepenent, then the istribution of {X i,n } in (6) is prouct-h(a, m)-id, a = 1/p an conversely. We now briefly consier log-gamma istributions in this context. If a r.v Y is gamma, then e Y is log-gamma or if X is log-gamma then log X is gamma. Thus if Y is gamma(θ, β) r.v with p..f f (y) = θβ y β 1 e θy Γ(β) then X = e Y is log-gamma(θ, β) with p..f, y > 0, β > 0, θ > 0, (7) h(x) = θβ (log x) β 1 Γ(β)x θ+1, x > 1, β > 0, θ > 0. (8) Since the gamma(θ, β) istribution is GID for β 1, log-gamma(θ, β) istribution at (8) is prouct-gid an thus gamma(θ, β) istribution can be use to moel the PAR(1) structure (4). Again, since the gamma(θ, β) istribution is H(a, m)-id for β = 1/m, m > 1 integer, log-gamma(θ, 1/m) istribution at (8) is multiplicative- H(a, m)-id an thus gamma(θ, 1/m) istribution can be use to moel the PAR(1) structure (6). (6) 4. Some properties of the PAR(1) moels We now iscuss certain istributional an estimation aspects of the PAR(1) moel (4). Here we assume µ ε = E(ε n ) <. The stationary PAR(1) moel (4) can be rewritten as X n = X A n n 1 ε n, where {A n } is a sequence of i.i. rvs with P(A n = 0) = 1 P(A n = 1) = p inepenent of X n l for l > 0 an {A n } an {ε n } are two mutually inepenent sequences. For the process {X n } given by (9), the autocovariance function γ k = Cov(X n, X n k ) is obtaine as follows. Using (9) an properties of {X n }, we get (9) an E ( X A n n 1 ) ( ( = E E X A n n 1 X n 1)) = p + (1 p)µx E ( X A n n 1 X ) ( ( n k = E E X A n n 1 X )) n k X n 1, X n k = pµx + (1 p)γ k 1 + (1 p)µ X, where µ X = E(X n ) an k > 0. Using the efinition of γ k an results above, we fin that ( ( γ k = µ ε E X A n n 1 X ) n k µx E ( X A n n 1)) = (1 p) k µ k εγ 0. Now, we will show that (1 p)µ ε < 1. From the stationarity of the process {X n }, we obtain that pµ µ X = ε 1 (1 p)µ ε. This implies that µ X exists for 1 (1 p)µ ε 0. Also, from the efinition of the moel an stationarity, we have that µ X + σ X = p(µ ε + σ ε) + (1 p)(µ ε + σ ε)(µ X + σ X ).
H. Bakouch et al. / Filomat 7:7 (013), 1197 103 101 Uner the conition 1 (1 p)(µ ε + σ ε) 0, the variance of {X n } is given by σ X = p(1 p)(1 µ ε) (µ ε + σ ε) + p σ ε ( 1 (1 p)(µ ε + σ ε) ) (1 (1 p)µ ε ). (10) The variance of {X n } is positive for 1 (1 p)(µ ε + σ ε) > 0. We have that 1 (1 p) > 1 1 p > µ ε + σ ε > µ ε, which implies that (1 p)µ ε < 1. Hence we have the following theorem. Theorem 4.1. The autocorrelation function at lag k of the r.vs X n an X n k is given by Corr(X n, X n k ) = (1 p) k µ k ε, k > 0. Further, (1 p)µ ε < 1 an hence the autocorrelation function converges to zero as k. Base on the autocovariance γ k value, the spectral ensity function f XX (λ) = 1 π γ k e iλk, i = 1 k= of the PAR(1) process is given by f XX (λ) = σ X π 1 µ ε(1 p) 1 + µ ε(1 p) µ ε (1 p) cos λ, where σ is given in (10). X Further, for this PAR(1) process the one-step ahea conitional mean is E(X n+1 X n = x) = pµ ε + (1 p)µ ε x, which is linear in x. Also, we can see that E(X n+ X n = x) = pµ ε + p(1 p)µ ε + (1 p) µ εx. Hence using inuction the k-step ahea conitional mean is given by the following theorem. Theorem 4.. The k-step ahea conitional mean is k 1 E(X n+k X n ) = pµ ε (1 p) j µ j ε + (1 p) k µ k εx. j=0 Remark 4.3. It is interesting to note that by virtue of Theorem 4.1 lim E(X pµ ε n+k X n ) = = µ X k 1 (1 p)µ ε which is the unconitional mean of {X n }. We now fin the conitional least squares (CLS) estimators of PAR(1) parameters. Let X 1, X,..., X N be a realization of PAR(1) process an consier the function Q N (p, µ ε ) = N (X n E(X n X n 1 )) = n= N (X n pµ ε (1 p)µ ε X n 1 ). n= Then, the CLS estimators of the parameters p an µ ε are obtaine by solving the system of equations Q N (p,µ ε ) p = 0 an Q N(p,µ ε ) µ ε = 0. The estimators are given in the following theorem.
H. Bakouch et al. / Filomat 7:7 (013), 1197 103 10 Theorem 4.4. The conitional least squares estimators of the PAR(1) parameters are an ˆµ ε = (N 1) n= X n X n 1 n= X n + n= X n n= X n 1 n= X n X n 1 (N 1) n= X n 1 ( ) ˆp = n= X n ˆµ ε ˆµ ε ( N 1 ). Remark 4.5. Let Z n = m X i,n, m > 0 be fixe an known, we can evelop CLS estimators for the moel (6) as one above for (4). Now we will iscuss the asymptotic properties of the obtaine CLS estimators. To erive these properties we will nee the following lemma. Lemma 4.6. The PAR(1) process {X n } given by (4) is a strict stationary an ergoic process. Proof. The strict stationarity of the PAR(1) process {X n } follows from the fact that it is a Markov process of the first orer an that the ranom variables {X n } are ientically istribute ranom variables. The ergoicity of the PAR(1) process follows from the Lemma ([1], pp. 408), the fact that the σ-algebra generate by {X n, X n 1, X n,... } is a subset of the σ-algebra generate by i.i.. ranom variables {ε n, ε n 1, ε n,... } an the fact that n=0 F {ε n, ε n 1, ε n,... } is a tail σ-algebra. Now, the asymptotical properties of the CLS estimators follow from the following theorem. Theorem 4.7. If the PAR(1) process given by (4) has finite moments E(X 4 n), then the CLS estimator ˆθ = ( ˆp, ˆµ ε ) T of the parameter θ = (p, µ ε ) T is a strongly consistent estimator an has asymptotical normal istribution, i.e. we have that N 1( ˆθ θ) converges in istribution to N(0, U 1 RU 1 ), as N, where U = R = [ [ µ εe (1 X n 1 ) µ ε E ( (1 X n 1 )(p + (1 p)x n 1 ) ) µ ε E ( (1 X n 1 )(p + (1 p)x n 1 ) ) E ( p + (1 p)x n 1 ) µ εe ( ν n (1 X n 1 ) ) µ ε E(ν n (1 X n 1 )(p + (1 p)x n 1 )) µ ε E(ν n (1 X n 1 )(p + (1 p)x n 1 )) E(ν n (p + (1 p)x n 1 ) ) an ν n is a conitional preiction error of {X n } given by ν n = p(σ ε + (1 p)µ ε) p(1 p)µ εx n 1 + (1 p)(pµ ε + σ ε)x n 1. Proof. First, we will show that all the conitions of Theorem 3.1 [13] are satisfie. Let n = E(X n X n 1 ). Then n = µ ε (p + (1 p)x n 1 ) an the first erivatives of the function n with respect to p an µ ε are n / p = µ ε (1 X n 1 ) an n / µ ε = p + (1 p)x n 1, respectively. Then all the conitions from Theorem 3.1 [13] except the conition C can be trivially prove. Let us show that the conition C is satisfie. Let us suppose that E a n 1 p + a n µ ε = 0. Then it follows that E a1 µ ε + a p + (a a p a 1 µ ε )X n 1 = 0. From this conition we obtain that a 1 µ ε + a p + (a a p a 1 µ ε )µ X = 0 ] ]
H. Bakouch et al. / Filomat 7:7 (013), 1197 103 103 an a a p a 1 µ ε = 0, which implies that a 1 = a = 0. Thus the conition C is satisfie an from Theorem 3.1 [13] follows that the CLS estimator ˆθ = ( ˆp, ˆµ ε ) T of the parameter θ = (p, µ ε ) T is a strongly consistent estimator. Finally, let us prove that the CLS estimator has asymptotical normal istribution. We have that the conitional preiction error of {X n } is given by ν n E ( (X n n ) X n 1 ) = E(X n X n 1 ) µ ε(p + (1 p)x n 1 ) = p(σ ε + (1 p)µ ε) p(1 p)µ εx n 1 + (1 p)(pµ ε + σ ε)x n 1. Since the PAR(1) process given by (4) has finite moments E(X 4 n), it follows that all the elements of matrix R from the conition D1 ([13], Theorem 3.) are finite. Thus all the conitions of Theorem 3. [13] are satisfie. Then the asymptotical normality of the CLS estimator follows from Theorem 3. [13]. References [1] T.A. Azlarov, Characterization properties of the exponential istribution an their stability, in Limit theorems, Ranom Processes an their Applications, Tashkent, Fan, 3-14 (in Russian), 1979. [] B.V. Gneenko, V.Y. Korolev, Ranom Summation-Limit Theorems an Applications. Boca Raton: CRC Press, 1996. [3] K.K. Jose, R.N. Pillai, Geometric infinite ivisibility an its applications in autoregressive time series moeling, in Stochastic Processes an Its Applications, E. Thankaraj, V, Wiley Eastern, New Delhi, 1995. [4] L.B. Klebanov, J.A. Melame, S.T. Rachev, On the prouct of a ranom number of ranom variables in connection with a problem from Mathematical economics, in Stability problems for Stochastic Moels, Lecture Notes in Mathematics, No.141, Es. Kalashnikov an Zolotarev, Springer-Verlag, Berlin, 1989, 103-109. [5] A.J. Lawrance, P.A.W. Lewis, A new autoregressive time series moeling in exponential variables (NEAR(1)), Avances in Applie Probability 13 (1980) 86-845. [6] E. McKenzie, Prouct autoregression: a time series characterization of the gamma istribution, Journal of Applie Probability 19 (198) 463-468. [7] S. Satheesh, Another look at ranom infinite ivisibility. Statistical Methos 5 (004) 13-144. [8] S. Satheesh, N.U. Nair, E. Sanhya, Stability of ranom sums, Stochastic Moeling an Applications 5 (00) 17-6. [9] S. Satheesh, E. Sanhya, T. Lovely Abraham, Limit istributions of ranom sums of Z + -value ranom variables, Communications in Statistics - Theory an Methos 39 (010) 1979-1984. [10] S. Satheesh, E. Sanhya, K.E. Rajasekharan, A generalization an extension of an autoregressive moel, Statistics an Probability Letters 78 (008) 1369-1374. [11] S. Satheesh, E. Sanhya, S. Sherly, A generalization of stationary AR(1) schemes, Statistical Methos 8 (006) 13-5. [1] A.N. Shiryaev, Probability, n Eition, Springer, 1996. [13] D. Tjøstheim, Estimation in nonlinear time series moels, Stochastic Processes an their Applications 1() (1986) 51 73.