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Adv. Appl. Prob. 20, 798-821 (1988) Printed in N. Ireland

? Applied Probability Trust 1988

BIVARIATE EXPONENTIAL AND GEOMETRIC AUTOREGRESSIVE AND AUTOREGRESSIVE MOVING AVERAGE MODELS

H. W. BLOCK, * University of Pittsburgh N. A. LANGBERG, **University of Haifa D. S. STOFFER, *University of Pittsburgh

Abstract

We present autoregressive (AR) and autoregressive moving average (ARMA) processes with bivariate exponential (BE) and bivariate geometric (BG) distributions. The theory of positive dependence is used to show that in various cases, the BEAR, BGAR, BEARMA, and BGARMA models consist of associated random variables. We discuss special cases of the BEAR and BGAR processes in which the bivariate processes are stationary and have well-known bivariate exponential and geometric distributions. Finally, we fit a BEAR model to a real data set.

BIVARIATE EXPONENTIAL AND GEOMETRIC DISTRIBUTIONS; BIVARIATE

AUTOREGRESSIVE AND AUTOREGRESSIVE MOVING AVERAGE MODELS IN EXPONENTIAL

AND GEOMETRIC RANDOM VECTORS; ASSOCIATION; JOINT STATIONARITY

1. Introduction and summary

A primary stationary model in time series analysis is the p x 1 linear process given by

(1.1) X(n)= I A(j)e(n-j), n =0, ?1, ?2, **, j= -00

where A(j), j = 0, ?1, ?2, * *, is a sequence of p x p parameter matrices such that

E=-_ IIA(j)ll <,0 and e(n), n =0, ?1, ?2, *, is a sequence of i.i.d. p x 1 random vectors with mean 0 and common covariance matrix. It is well known that

(1.1) includes the stationary vector autoregressive (AR) process and the stationary and invertible vector autoregressive moving average (ARMA) process. However, in some physical situations where the random vectors X(n) are either positive or discrete, the preceding assumptions on the e(n) sequence are inappropriate (see Lewis (1980), p. 152).

Several researchers, addressing themselves to this problem, have constructed univariate stationary AR type models and stationary ARMA type models where the random variables X(n) have exponential or gamma distributions, and discrete

Received 18 June 1986; revision received 27 August 1987. *Postal address: Department of Mathematics and Statistics, University of Pittsburgh, PA 15260, USA. **Postal address: Department of Statistics, University of Haifa, Mount Carmel, Haifa 31999, Israel. Research partially supported by the Air Force Office of Scientific Research under Contract

AFOSR-84-0113.

798

BEAR, BGAR, BEARMA and BGARMA models

models where X(n) assumes values in a common set. Lawrance and Lewis (1977), (1980), (1981), (1985) and Jacobs and Lewis (1977) present stationary AR and

ARMA-type models where the random variables X(n) have exponential distribu-

tions; Gaver and Lewis (1980) consider stationary ARMA-type models where the random variables X(n) have gamma distributions. Jacobs and Lewis (1978a, b), (1983) construct ARMA-type models where the random variables X(n) are discrete and assume values in a common finite set. The aforementioned models have been used in the various fields of applied probability and time series analysis; for example these models have been used to model and analyze univariate point processes with correlated service and correlated interarrival times (see Jacobs (1978)). Details

concerning bivariate and geometric MA type processes and the corresponding point processes may be found in Langberg and Stoffer (1987).

In this paper we present two classes of AR and of ARMA type sequences of bivariate random vectors. The first class has exponential marginals while the second class has geometric marginals. We denote the first [second] class of models as

BEAR(m) [BGAR(m)] and BEARMA(m1, m2) [BGARMA(ml, m2)] for bivariate

exponential [geometric] autoregressive, order m, and autoregresssive moving average, order (mi, n2), respectively, where m and (ml, m2) parametrize the order

of the dependence on the past. We use the theory of positive dependence to show that in a variety of cases the classes of sequences are associated.

In Section 2, we define the bivariate exponential and geometric distributions which are the underlying distributions of our two classes, and present a variety of

examples of such distributions. Furthermore, in Section 2 we define the concept of association and present a variety of bivariate exponential and geometric distributions that are associated. We conclude Section 2 by describing the bivariate

dependence mechanisms which are used in generating the various models. In Section 3, we construct the general BEAR(m) and BGAR(m) models showing that the sequences have bivariate exponential and bivariate geometric distributions,

respectively. Also, we discuss the autocorrelation structure of the two classes of

sequences. In Section 4, we consider special cases of the BEAR(1) and BGAR(1) sequences. We show that defined appropriately, the bivariate processes are

stationary, and obtain well-known bivariate exponential and geometric distributions. In Section 5, we present the BEARMA(m1, m2) and BGARMA(m1, m2) models. We conclude by describing the association properties of the sequences and

discussing how to utilize association to obtain some probability bounds and moment

inequalities for the bivariate processes and the corresponding point processes. Finally, in Section 6, we apply one of our models to a real data set.

2. Preliminaries

In this section we present definitions and prove some basic results to be used

subsequently. First, we give definitions of bivariate geometric and exponential

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H. W. BLOCK, N. A. LANGBERG AND D. S. STOFFER

distributions and provide some examples. Then we present the concept of association and give some examples. Finally, we discuss some bivariate dependence mechanisms.

First, we present a definition of a bivariate geometric distribution.

Definition 2.1. Let M, N be random variables assuming values in the set {1, 2, * * }. We say that (M, N) has a bivariate geometric distribution if M and N have geometric distributions.

Examples 2.2. (a) Let N be geometric. Then (N, N) is bivariate geometric. (b) Let M and N be independent geometric random variables, then (M, N) is bivariate geometric. (c) Let N1, N2, N3 be independent geometric random variables, and put M= {min (N, N3)}, N = min(N2, N3)}. Then (M, N) has the Esary- Marshall (1974) bivariate geometric distribution. (d) Let poo, pol, Plo, P\i be in [0, 1] such that (i) poo +oi +Plo + Pi = 1, (ii) Pol +Pll < 1 and Pio + Pl < 1, and let M, N be random variables assuming values in the set {1, 2, * } determined by:

(2.3) P(M > m N> n) = Pl [Pol + 1]-m, n> m

!'Pl[Plo +PJl] , n =m, m, n = l,2,, .

Then (M, N) has the Block (1977) fundamental bivariate geometric distribution (see also Block and Paulson (1984)). (e) Let (M1, M2) be bivariate geometric and let (N1(j), N2(j)), j = 1, 2, -, be an i.i.d. sequence of random vectors with bivariate geometric distributions which are independent of (M1, M2). Then (EM_Il N1(j), Ej2i N2(j)) has a bivariate geometric distribution (cf. Lemma 2.14).

In the following remark we show how some of the bivariate geometric distributions are particular cases of Example 2.2(d). Other examples are given in Remark 4.10.

Remarks 2.4. (a) Let pio=Poi=0 in Equation (2.3). Then we obtain the distribution introduced in Example 2.2(a). (b) Let P = (Pll+Plo)(Pll+Pol) in (2.3). Then we obtain the bivariate geometric distribution introduced in Example 2.2(b). (c) Let Pll - (Pl +Plo)(P1i +Pol) in (2.3) and let N1, N2, N3 be independent geometric random variables with parameters Pll(Pl +Pol)-, Pll(Pl +Plo)-1, Pil (Pl +Plo)(PlI +Pol), respectively. Put M = {min (N, N3)} and N= {min (N2, N3)}. Then (M, N) is stochastically equal to the Esary-Marshall bivariate geometric distribution given in Example 2.2(c).

Next, we present a definition of a bivariate exponential distribution.

Definition 2.5. Let E1, E2 be random variables assuming values in (0, oo). We say that (El, E2) has a bivariate exponential distribution if E1 and E2 have exponential distributions.

Examples 2.6. (a) Let E be exponential. Then (E, E) is bivariate exponential. (b) Let E1, E2 be independent exponentials. Then (E1, E2) has a bivariate

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exponential distribution. (c) Let X1, X2, X3 be independent exponentials and put E1 = {min (X1, X3)}, E2 = {min (X2, X3)}. Then (El, E2) has the Marshall-Olkin

(1967) bivariate exponential distribution. (d) Let (M, N) have a bivariate geometric distribution and let (El(j), E2(j)), j = 1, 2, * * *, be an i.i.d. sequence of random vectors with bivariate exponential distributions, independent of M and N. Then (EM1 E1(j), EN E2(j)) has a bivariate exponential distribution (cf. Lemma 2.14). (e) Let 0 ac 1. Then (E1, E2) determined by P{E1 >x, E2>y} = exp {-x -y - axy}, x,y>0, has a Gumbel (1960) bivariate exponential distribution. (f) Let lal - 1. Then (E1, E2) determined by P{E1 <x, E2 y} = (1 -e-X)(1 - e-)(l +

e--Y), x, y >0, has a bivariate Gumbel (1960) exponential distribution. (g) Let a > 1. Then (E1, E2) determined by P{E1 > x, E2 > y} = exp (-(x C + y a)1/), x, y >

0, is bivariate exponential. (h) Let (X, Y) be a random vector with continuous

marginal distributions F and G, respectively. Then the random vector (-ln [1- F(X)], -ln [1 - G(Y)]) is bivariate exponential.

Example 2.6(d) has been used by several researchers to generate bivariate distributions (for example Arnold (1975), Downton (1970), and Hawkes (1972) to mention a few). In the following remarks we illustrate how some of the bivariate exponential distributions are obtained from Example 2.6(d). Other examples are given in Remark 4.5.

Remarks 2.7. (a) M=N and let El(j), E2(j) be independent exponentials, j= 1,2, - . Then we obtain the distribution introduced by Downton (1970). (b) Let (M, N) be as in Example 2.2(d) and let E1(j), E2(j) be independent exponentials, j = 1, 2, .. Then we obtain the bivariate exponential distribution introduced by Hawkes (1972) and Paulson (1973). (c) Let (M, N) be as in Example 2.2(c) and let E(j) = E2(), j = 1, 2, * - -. Then we obtain the Marshall-Olkin

(1967) distribution given in Example 2.6(c) (for details see Marshall-Olkin (1967)). Next, we present a concept of positive dependence.

Definition 2.8. Let T = (Ti, , * Tn), n = 1, 2, * , be a multivariate random vector. We say that the random variables T7, .., Tn are associated if cov (f(T), g(T)) 0 for all f and g monotonically non-decreasing in each argument, such that the expectations exist.

Remark 2.9. (a) Note that independent random variables are associated and that non-decreasing functions of associated random variables are associated (cf. Barlow and Proschan (1981), pp. 30-31). Thus, the components of the vector given in Example 2.2(c) and the components of the vector given in Example 2.6(c) are associated. (b) Let (E1, E2) be as in Example 2.6(e), with a > 0, or as in 2.6(f) with -1 a < 0. Since P{E > x, E2 >y} <P{E > x}P{E2 >y} for x,y >0, E1 and E2 are not associated. (c) Let (X, Y) be as in Example 2.6(h). Then -ln [1- F(X)] and -ln [1 - G(Y)] are associated if and only if X and Y are associated (cf. Barlow and Proschan (1981), Proposition 3, p. 30). (d) The components of the bivariate

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geometric distribution given in Example 2.2(e) are associated provided M1 and M2, and N1(1) and N2(1) are associated (cf. Langberg and Stoffer (1987), Lemma 2.10). (e) The components of the bivariate exponential distribution given in Example 2.6(d) are associated provided that M and N, and El(1) and E2(1) are associated by the same reasoning in Remark 2.9(d).

We are now ready to discuss the various dependence mechanisms used in

obtaining bivariate exponential and geometric distributions. It turns out that many of these mechanisms are related and we describe these relationships. The notation X st will mean that X and Y are random variable (or vectors) with the same distribution.

Lemma 2.10 (random mixing). (a) Let (Xi, X2) and (Z1, Z2) be independent random vectors with exponential marginals where X1 s

Zl, X2 s

Z2, and (X1, X2), (Z1, Z2) have mean vector (A- , A21). Let (I1, 12) be a bivariate Bernoulli random vector independent of (X1, X2), (Z1, Z2). Also, assume

(2.11) P{l= 2i, 12=j}= , i,j=0, 1

such that Ei,jpij =1; 1 - =Po+P11 < 1, and 1 - r2 =P +Pj < 1. Then a random vector given by

(2.12) (Yl, Y2) M (11Z1, 12Z2) + (7r1X1, r2X2)

has a bivariate exponential distribution with the same marginals as (X1, X2) and

(Z1, Z2). (b) Let (M1, M2) and (K1, K2) be independent geometric random vectors such that for k = 1, 2, .* * , P{Mi = k} = (pi/7i)(1 -pi/i)k-l and P{Ki = k} =

pi(l-pi)k-l, i= 1, 2, with 0<pi, p2<1. Let (I1, 2) be defined by (2.11) and be

independent of (M1, M2), (K1, K2). Then a random vector given by

(2.13) (G1, G2) S (I1K1, I2K2) + (M1, M2)

has the same marginals as (K1, K2).

Proof. Parts (a) and (b) follow easily by computing the marginal characteristic functions.

Raftery (1984) gives a special case of Lemma 2.10(a) where Z1 Z2.

Lemma 2.14 (random summation). Let (X1j, X2)[(Mij, M2)], j= 1, 2, , be i.i.d. bivariate exponential [geometric] random vectors with mean vector

(jl/Al, gr2/A2) [(grl/P1, 2), 0<2/2)] , 0 , 2 < 1. Let (N1, N2) have a bivariate geometric distribution with mean vector (jr-1, J2-1) and be independent of the

(X1j, X2j) and the (Mlj, M2j). Then random vectors given by

(2.15) (y, y2) -= 1 j=l1

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and

(2. 16) (Gl, G2) j- M 1

have bivariate exponential and bivariate geometric distributions with mean vectors (A-1, A-') and (p-1, p-1), respectively.

Proof. The proof follows easily by computing the marginal characteristic functions.

Finally, we describe how random mixing and random summation are related in the following lemma. The connection between the two concepts is a key element in the development of the bivariate exponential and geometric AR and ARMA models discussed in Sections 3, 4, and 5.

Lemma 2.17. Let (X11, X2j) and (M1j, M2j), j = 1, 2, be as defined in Lemma 2.14. Let (N1, N2) have the bivariate geometric distribution given by (2.3) with 1 - Jr1 = p10 + p,1 < 1; 1 - .r2 = P01 + P11 < 1, and be independent of the (Xl, X2j) and (M11, M2j). Furthermore, let (X,, X2), (II, I2), (Ml, M2) and (Kl, K2) be as defined in Lemma 2.10. Then (Y1, Y2) = (E'v X1j, E7V21 X2j) has the representation (I,Z1, 12,2) + (.r1Xl, .rlX2), and (G,,y G2) = (X=l M,j, M=1M2) has the representation (I1K1, 12K2) + (Ml, M2), i.e.

(2.18) E[exp {itlYY + it2 Y2} = E [exp {it,(1Y1Z + 7rlX,) + it2(12Z2 + ir2X2)}I and

(2.19) E[exp {it,G1 + it2G2}] = E[exp {it1(I1K, + Ml) + it2(12K2 + M2)}]

where i = VFT and -oc < t1, 2 < Xo

Proof. We prove the lemma for the exponential case, the geometric case being similar. Write

(N1 N2 N1 AT2

x1 > x21) = (Xll, X22)+ ? NI >1) > X11, X(N2 > 1) > X21) 1 = / X j=2 j=2

where x denotes the indicator function. Clearly (Xll, X22) I (~ir1X1, r2X2) and we are left to show that (X(NI > 1) V2 Xlj, X(N2> 1) E X2j) has the same distribution as (I,Z,, 12Z2). Note that

rN1 N2

pX(Nl > 1) Y, xii 2l (2> 1) E 2j<X j=2 j=2 -

1 1 N, N2

(2.20) = P> > X(N > 1) XIjxi,x(N2 > 1) X 2j X2,

il=(02 0 j 2 j=2

X(NI > 1) =i,XN i

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Now, for (i1, i2) = (1, 1) in (2.20) we have

f Nt N2 ]

p{ Xljx, E X2j -X2, N1 > 1, N2 > 1

(2.21) j=2 j=2

E P Xlj <X l, E X2j _x2P{ = nl, N2 = n2}. nl=2 n2=2 j=2 j=2

But P{N1 = n, N2= n2} = P{N = nl-1, N2 = n2- 1}P{N > 1, N2> 1}, so that

(2.21) is equal to

1 N1 N2 4

epX =-<1, X2j x P{I 12 = 1} = P{Z1 xI, Z ; x2}P{i = 1,2 = 1}. j=l j=l

For (i1, 2) = (1, 0) in (2.20) we have

( P Xi '2.) , 0 Xx2, N1 > 1, N2= 1 /=2

(2.22)

-E PE XXlj , xlP{N1 = n, N2= 1}.

n= =2 =2 It is easy to check via (2.3), that

P{N1 = n, N2 = 1} =po(po +Pol)(po +pl)n2 = P{N1 = n - 1}P{N1 > 1, N= 1},

so that (2.22) is equal to

P X x,4P{Il = 1, I2=0 = P{Z <x,}P{I, = 1, I2 = 0). j=l

The cases when (i, i2)= (0, 1) and (i1, i2) = (0, 0) in (2.20) follows similarly.

3. The general BEAR(m) and BGAR(m) models

In this section we construct two classes of AR sequences of bivariate random vectors. In each class the sequences are labeled by the parameter m. We denote the first class of sequences by {X(m, n) = (Xl(m, n), X2(m, n))', n = 0, 1, - } and the second class of sequences by {G(m, n) = (Gl(m, n), G2(m, n))', n = 0, 1, . }, m = 1, 2, - .. We show that the random vectors X(m, n) and G(m, n) have bivariate exponential and geometric distributions, respectively, with mean vectors that do not depend on m or n. Then we discuss the association property for any finite number of random variables belonging to one of the two AR classses. We conclude this section by discussing the autocorrelation structure of the two classes of sequences. Throughout, n ranges over the non-negative integers and I assumes the values 1 or 2. For notational simplicity we suppress the parameter m since it is fixed throughout the section.

First we construct the class of BEAR(m) sequences. Some notation is needed.

Notation 3.1. Let Al, A2 e (0, oo), let ij(n), r2(n) E (0, 1), and let B(n) be a 2 x 2

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diagonal matrix with B(n) = diag {r(1(n), r2(n)}. Further let E'(n) = (El(n), E2(n)) be a sequence of independent bivariate exponential random vectors with mean vector (A'1, i21)', let ej be an m-dimensional vector with component j equal to 1 and the other components equal to 0, j= 1,, m, and let 0 denote the m-dimensional zero vector. Finally let I'(n) = (I(n, 1), * , I,(n, m), I2(n, 1), * * , 12(n, m)) be a sequence of 2m-dimensional independent random vectors with

components assuming values 1 or 0 independent of all E(n), and let A(n, q) be a 2 x 2 random diagonal matrix with A(n, q) = diag {I1(n, q), I2(n, q)}, q = 1, * *, m.

We assume that for 1= 1, 2, m

(3.2) > P{(I(n, 1), * Ii(n, m)) = ej} = 1 - () j=1

and that

(3.3) P{(ll(n, 1), * , I/(n, m)) = 0'} = 7r(n).

We define the BEAR(m) sequences as follows:

E(n), n = 0, * * * m - 1

(3.4) X(n), (3.4) X(n) - A(n, q)X(n -q)+ B(n)E(n), n = m, m + 1, .. 4=1

Raftery (1982) proposed a multivariate exponential autoregressive model which is similar to the BEAR(m) model; however, neither model is a generalization of the other. The major difference between the Raftery (1982) model and the BEAR(m) model is the way that the components of the vector series depend on each other. In the Raftery (1982) model, the components are functionally related to each other with positive probability, so that, for example, the first component can get a contribution from the second component at previous time points. However, in the

BEAR(m) model, the dependence between components comes from the dependence structure inherent in the exponential noise vector E(n).

Next we construct the class of BGAR(m) sequences. Some notation is needed.

Notation 3.5. Let pi, P2, ac(n), a2(n) E (0, 1) such that p, al1(n), let N'(n)= (Ni(n), N2(n)) be a sequence of independent bivariate geometric random vectors with mean vectors (aYl(n)pl1, a2(n)p21), respectively, and let M'(n)= (MI(n), M2(n)) be a sequence of independent bivariate geometric random vectors with a common mean vector (pi , 2 1) independent of all N(n). Further let

J'(n) = (Jl(n, 1), * , Jl(n, m), J2(n, 1), *, J2(n, m)) be a sequence of 2m- dimensional independent random vectors with components assuming the values 1 or 0, independent of all N(n) and M(n). Let C(n, q) be a 2 x 2 random diagonal matrix with C(n, q) = diag {J,(n, q), J2(n, q)}, q = 1, .* , m.

We assume that for l = 1, 2, m

(3.6) E P{(J,(n, 1), * , J(n, m)) = e} = 1 - a,(n) j=l

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and that

(3.7) P{(Jl(n, 1), * * , Jl(n, m)) = O'}= al(n).

We define the BGAR(m) sequences as follows:

M(n), n =0, * m - 1

(3.8) G(n) = m (3.8) G(n)= C(n,q)G(n - q)+N(n), n=m, m + l, .

q=l

Next, we show that X(n) and G(n) have bivariate exponential and geometric distributions, respectively.

Lemma 3.9. For n =0, 1, * * *,X(n)[G(n)] has a bivariate exponential [geometric] distribution with mean vector (A-1', A2')[(pi1 , P21)].

Proof. We prove the results of the lemma by an induction on n. For n= 0, * , m - 1, the results of the lemma follow by the definition of X(n) and of G(n). Let us assume that the results of the lemma hold for all non-negative integers that are less than or equal to r, r m - 1, and prove that the results of the lemma hold for r + 1.

Let E' = (El, E2)[M' = (Ml, M2)] be a bivariate exponential [geometric] random vector with mean vector (A1 l, A31)[(p1 -, P2 )] independent of all E(n)[N(n) and M(n)]. Then, by the induction asssumption, we have for I = 1, 2 that

X(r + 1) st E, + l(r + 1)E(r + 1), w.p. 1 - (r + 1) r( (r + 1)El(r + 1), w.p. Jr1(r + 1)

and that

G (r t + N(r+ 1), w.p. 1 (r + 1), N,(r + 1), w.p. cl(r + 1),

where w.p. stands for 'with probability'. It is easy to check that X,(r + 1)[GC(r + 1)] has an exponential [geometric] distribution with mean A L[p/-1]. Consequently, the results of the lemma follow.

Now we consider the association of any finite collection of the X,(n)'s and of the

Gl(n)'s.

Lemma 3.10. Let us assume that for j = 0,-, m- 1, the random variables

X1(j), X2(j) in (3.4) are associated; let n, < n2 < . * < nr, be non-negative integers and let 11, *, lr {1, 2}, r = 1, 2, - . Then the random variables Xiq(nq), q = 1, -* *, r, are associated.

Proof. Let T2 = X2(j - 1) and let T2j_ = X1(j - 1), j = 1, 2, * . To prove the result of the lemma it suffices, by Barlow and Proschan (1981) Pl, p. 30, to show that

(3.11) the random variables Ti, ... , Tr are associated for all r = 1, 2, - - .

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We prove (3.11) by an induction argument on r. For r '2m (3.11) follows by the lemma assumption and by Barlow and Proschan (1981), P4, p. 30. Let us assume that (3.11) holds for r, r _ 2m and prove that (3.11) holds for r + 1.

By (3.4) the conditional random variable Tr+ T1, * * *, Tr is stochastically non-decreasing in T1, *--, Tr. Thus by Barlow and Proschan (1981), Lemma 4.8, p. 147, there is an r + 1 argument function h, non-decreasing in each argument, and a random variable U independent of Ti, *, Tr, such that

(Ti, ... , Tr+l) t

(T1, ,** Tr, h(U, T1, , Tr)). By Barlow and Proschan (1981), P2, p. 30, U is associated and hence by Barlow and Proschan (1981), P3, p. 30, the random variables U, T1, - , Tr are associated. Consequently, by Barlow and Proschan (1981), P3, p. 30, {T1, ?* * , Tr+j} are associated.

Using a similar proof we obtain the following.

Lemma 3.12. Let us assume that for j =0, -, m - 1, the random variables Gi(j), G2(j) in (3.8) are associated. Let n <n2 <.. <nr, and 11, * *, lr, r=

1, 2, * * *, be as in Lemma 3.10. Then the random variables Glq(nq), q = 1, , r are associated.

Finally, for each class of sequences, we compute the autocorrelation functions in the case when the marginal processes are stationary.

For the exponential models, put jrl(n) = rl, all n; 1 = 1, 2, and let

P{ll(n, q) = 1} = ql(q), I = 1, 2; q = 1, * ? -, m

such that m (i) /l(q) O, and (ii) I 4l(q)= 1- zi, 1=1,2,

q=l

as specified in Equation (3.2). Define Pxl(k)= Corr {Xi(n), Xl(n + k)}, 1 = 1, 2; n =m, m+1, ; k = 1, 2, - . Then

(3.13) px,(k) = 01(1)px,(k - 1) + cl(2)Pxl(k - 2) + * + (Pl(m)px,(k - m)

with Var {Xl(n)} = A-2, 1 = 1, 2. For the geometric models, put acl(n)= a,/, all n; 1 = 1, 2, and suppose that

P{Jl(n, q)= 1} = y!(q), 1= 1, 2; q = 1, * , m such that m

(i) y1(q) -0, and (ii) > yl(q)= l-a,, 1=1,2 q=i

as specified in Equation (3.6). Define

pG,(k) = Corr {Gl(n), Gl(n + k)}, 1 = 1, 2; n = m, m + 1, ? - -,; k = 1, 2, * * .

Then

(3.14) pG,(k) = yl(1)pG,(k - 1) + yl(2)pG,(k - 2) +. + y1(m)pG,(k - m)

with Var {Gl(n)} = (1 - P)p -2, 1 = 1, 2.

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Evidently, the marginal correlation structures of the bivariate exponential and

geometric sequences, as given in (3.13) and (3.14), respectively, are similar to that of the Gaussian AR(m) process. We note that, in general, even when the marginal processes are stationary, the joint process is not stationary. This is easily seen, for

example, by letting m = 1 in (3.4) with jlr(n) = Jrl, all n; I = 1, 2; choosing E(n) to be an i.i.d. sequence of random vectors where E1(n) and E2(n) are i.i.d.

exponential random variables for all n, and letting I(n) be an i.i.d. sequence of random vectors for which P{1I(n) = 1, 12(n) = 1} =p11 (1 - r1)(l - r2). A simple computation shows that Cov {X1(l), X2(1)} : Cov {X1(2), X2(2)} in this example.

In the next section, we develop models in which joint processes are also

stationary.

4. The stationary BEAR(1) and BGAR(1) models

In this section we consider special cases of the BEAR(m) and BGAR(m) models

given in Section 3 in which the joint processes are stationary. Throughout this section we put m = 1, assume that 7rt(n) and aor(n) do not vary with n, and put more structure on the E(n), M(n), and N(n) sequences. We show that for these models, the bivariate distributions of X(n) and of G(n) have a form of the type studied by Arnold (1975). By selecting the E(n), M(n), and N(n) sequences, as defined in Section 3, appropriately, we can obtain well-known bivariate distributions. For a

stationary BEAR(1) model we obtain the (i) Marshall-Olkin (1967), (ii) Downton

(1970), (iii) Hawkes (1972), and (iv) Paulson (1973) bivariate exponential distributions. For a stationary BGAR(1) model we obtain the (i) Esary-Marshall (1973), (ii) Hawkes (1972), and (iii) Paulson-Uppuluri (1972) bivariate geometric distributions. We conclude this section by computing the autocovariance matrices for each model.

First we present the exponential case. Some notation and assumptions are needed. Let m = 1 and let us assume that (Il(n, 1), 12(n, 1)), given in (3.1), is an i.i.d. sequence of bivariate random vectors. For simplicity of notation, denote jr,(n) by .tr, for 1 = 1, 2, and let pij = P{(l1(n, 1), I2(n, 1)) = (i, j)}, i, j = 0, 1. Note that by (3.2) and (3.3)

(4.1) Pio + P1 = 1 - , Po1+P11= 1- 2.

Further let (N1, N2) be a bivariate geometric random vector with parameters Pij, i, j =0, 1 given by (2.3), and let E(r), r = ?1, ?2, * * *, be an i.i.d. sequence of bivariate exponential random vectors with mean vector (A-1, A2 ) independent of

(N1, N2) and all (I1(n, 1), I2(n, 1)). Note that by Lemma 2.14 (E1l 7r1E1(-j), iNV1 2J2E2(-j)) is a bivariate exponential random vector with mean vector

(i1 , 221). We assume that

(N12 () i nN2

(4.2) E(0) = ( rE1(-j), , 2E2(-)) . j=l j=l

808


Define A(n) to be the 2 x 2 diagonal random matrix A(n) = diag {I1(n, 1), 12(n, 1)} and B to be the 2 x 2 diagonal matrix B = diag {fr, Jr2}. The stationary BEAR(1) model is defined as follows:

(4.3) X(n) = E(0), n=0 (4.3) X(n) A(n)X(n - 1) + BE(n), n = 1, 2, ..

We now state and prove a characterization of X(n).

Lemma 4.4. Let X(n) be defined by (4.3). Then for n = 0, 1, * * ,

X(n) t E(O)

where E(0) is given in (4.2).

Proof. We prove the result of the lemma by an induction argument on n. By definition the result of the lemma holds for n = 0. Let us assume that the result of the lemma holds for n, n ? 0. Note that

E'(O) = ( 7lEl(-j), I 2E2(-))

j=1 j=1

N1 N2

= ((N1 > 1)r1, E1(-j), (N2 > 1)r2 2 E2(-j)) j=2 j=2

+ (rlEl(-1), 7r2E2(-1)),

where X(') denotes the indicator function, and that the two summands are independent random vectors. Now by Lemma 2.17, (4.1) and the induction assumption

((N1 > 1)rll > E,(-j), X(N2 > 1)72 E E2(-i))

st Ii(n, 1)71 El(-j), I2(n, 1)m2 E E2(-j)) j=2 j=1

= (I(n, 1)Xi(n), 2(n, )X2(n)).

Further by the definition of E(r)

(YlEl(-l), Jt2E2(-1)) s

(mlEl(n), r2E2(n)).

Since the random vectors (7r1El(n), 7r2E2(n)) and (Il(n, 1)Xl(n), I2(n, 1)X2(n)) are

independent we have by (3.4) that N N2 \

'(i Z El(-j), Y7r2 Z E2(-j) st

(.7lEl(n), '7r2E2(n)) j=1 j=1

+ (II(n, l)Xl(n), 12(n, 1)X2(n)) = X'(n + 1).

Consequently the result of the lemma follows.

809


In the following remark we show that many interesting bivariate distributions are

possible in Lemma 4.4. See Block (1977) for further details.

Remark 4.5. (a) Let E be an exponential random variable with mean 0,

0< 0 < (A1 + A2)-1, let jr1 = AI0, dr2 = A20 and let E(1) = (1 'E, 7 r'E):

(i) If Poo = 0, Poi = Jl, Plo = J2 and pli = 1 - (A, + A2)0, then the resulting X(n)

has independent components. (ii) Let b,, b2, b12 be non-negative real numbers such that Ai = b, + b2 and

A2 = b2 + b12. If poo = Ob12, o10 = Ob2, Poi = Ob, and Pul = 1 - 0(b, + b2 + b12) then

the resulting X(n) has a Marshall-Olkin (1967) bivariate exponential distribution.

(b) Let E(1) have independent components:

(i) If Pll = y(l + y)-l, 0 < y, Plo = Poi = 0, and poo = (1 + y)-' then the resulting

X(n) has the Downton (1970) bivariate exponential distribution.

(ii) With different requirements on Pij, i, j = 0, 1, the resulting X(n) has the

Hawkes (1972) and the Paulson (1973) bivariate exponential distribution. See Block

(1977) for details. Next we present the geometric case. Some notation and assumptions are needed.

Let m = 1 and let us assume that (J1(n, 1), J2(n, 1)), given in (3.5), is an i.i.d.

sequence of bivariate random vectors. For simplicity of notation denote cyl(n) by

at, 1 = 1, 2, and let Pij = P{(Jl(n, 1), J2(n, 1)) = (i, j)}, i, j = 0, 1. Note that by (3.6) and (3.7)

(4.6) Pio + P ll - l, Po +PI = 1- a2.

Further let (Q1, Q2) be a bivariate geometric random vector with parameters Pij, i,j=0, 1, given by (2.3). Let p, P2 E (0, 1) such that p,-- a, and let N(r), r = ?1, ?2, * * *, be an i.i.d. sequence of bivariate geometric random vectors with mean vector (arlP1', a2p21') independent of (Q1, Q2) and all (Jl(n, 1), J2(n, 1)). Note that by Lemma 2.14 (E,-11N,(-j), E i,N2(-j)) is a bivariate geometric random vector with mean vector (p ', p21). We assume that

(4.7) M(0) = EN(-j), ? N 2(-j) -=1 j=1

Define C(n) to be the 2 x 2 diagonal random matrix C(n) = diag {J1(n, 1), J2(n, 1)}. The stationary BGAR(1) model is defined as follows:

(4.8) G(n)= _M(0), n=0 (4.8) ^G(n)=C(n)G(n - 1) +N(n), n = , 2, ..

We now state and prove a characterization of G(n).

Lemma 4.9. Let G(n) be defined by (4.8). Then for n = 0, 1, * * ,

G(n) S M(O)

where M(0) is given in (4.7).

810


Proof. We prove the result of this lemma by using a similar argument to the one used in the proof of Lemma 4.4.

In the following remark we show that many interesting bivariate distributions are

possible in Lemma 4.9. See Block (1977) for further details.

Remark 4.10. (a) Let ac =pl, a2 =P2, thus N(1) = (1, 1):

(i) Let 01, 02, 012 e (0, 1) and let 1 - c1 = 01012, 1 - a2 = 02012. Further let

P11= 0102012, Pio= 01(1 -

02)012, Po = (1 - 01)01012 and poo= 1 -plI -pol -Pio. Then the resulting G(n) has the Esary-Marshall (1973) bivariate geometric distribution in the narrow sense.

(ii) If pl = (1 - ai)(1 - a2), poi = a2(1 - r2), Pio = (1 - C1)at2 and poo = CC2 then the resulting G(n) has independent components.

(iii) With no special requirements on the Pij, i, j = 0, 1, the resulting G(n) has the

Esary-Marshall (1973) bivariate geometric distribution in the wide sense, or

equivalently, the bivariate geometric distribution due to Hawkes (1972). (b) If N(1) has independent components, the resulting G(n) has the Paulson and

Uppuluri (1972) distribution.

Finally, we give the autocovariance matrices for the stationary BEAR(1) and

BGAR(1) models. Let 2x = Var {X(n)} be the variance-covariance matrix of X(n), and 2G = Var {G(n)} be the variance-covariance matrix of G(n). Note that Ex and ZG are independent of n by Lemmas 4.4 and 4.9, respectively. Define Fx(k)= Cov{X(n + k), X(n)}, k =0, 1, 2, * * , and FG(k) = Cov{G(n + k), G(n)}, k =

0, 1, 2, * * *, and note that Fx(0) = Ex and rF(0) = 2c. In view of (4.3) and (4.8), it is easy to see that Fx(k) =AFT(k - 1) and FG(k)= CG(k - 1), k = 1, 2, - , respectively, where A and C are the 2 x 2 diagonal matrices defined by A =

diag {1- il, 1 - r2} and C = diag {1 - ca, 1 - o2}. Hence, for stationary BEAR(1) models we have

(4.11) rx(k) = AkZx; x(-k) = r(k), k=0, 1, 2,

and for the stationary BGAR(1) model we have

(4.12) FG(k) = CkYG; FG(-k) = Fr(k), k = 0, 1, 2, * * .

5. The BEARMA(ml, m2) and BGARMA(ml, m2) models

Using the results of Section 3 and the results of Langberg-Stoffer (1987) for MA

sequences, we construct four classes of ARMA sequences of bivariate random vectors. In each class the sequences are labeled by the parameters ml, m2. We denote the first two classes of sequences by {Z'(j, ml, m2, n) =

(Zl(j, ml, m2, n), Z2(j, ml, m2, n)), n = 0, 1, * - }, j = 1, 2, and the other two classes of sequences by {L'(j, ml, m2, n) = (Li(j, ml, m2, n), L2(j, ml, m2, n)), n =0, 1, * * *}, j= 1, 2. We show that the random vector

811


Z'(j, ml, m2, n)[L'(j, ml, m2, n)] has a bivariate exponential [geometric] distribution with a mean vector that does not depend on j, mi, m2, or n. Then we discuss the association property of any finite number of random variables belonging to one of the four ARMA classes. For notational simplicity we suppress the parameters mi and m2 since they are fixed throughout this section.

First we construct the two classes of BEARMA(ml, m2) sequences. Some notation is needed.

Notation 5.1. Let X(n) be a BEAR(ml) sequence given by (3.4), and let Y(n) be an m2-dependent BEMA sequence as given by Langberg-Stoffer (1987), independent of the X(n) sequence. Further let V'(n)= (Vl(n), V2(n)) be a sequence of

independent bivariate random vectors with components assuming the values 1 or 0,

independent of the X(n) and Y(n) sequences and let P{V,(n)= 1} = r,(n), 0<

,r1(n) <, 1= 1, 2. We define the two BEARMA(m1, m2) sequences as follows:

(5.2) (Zl(1, n), Z2(1, n)) = ((1 - 7rl(n))Yl(n), (1 - 2(n))Y2(n))

+ (Vl(n)Xl(n), V2(n)X2(n))

(Z1(2, n), Z2(2, n)) = ((1 - 2(n))X2(n), (1 - 7r2(n))X2(n))

+ (Vl(n)Yl(n), Vi(n)Y2(n)).

Now we construct the two classes of BGARMA(m1, m2) sequences. Some notation is needed.

Notation 5.4. Let 61, , 62 , , 2 e (0, 1) such that 6 ' fPI, 1 = 1, 2, and let G(1, n) [G(2, n)] be a BGAR(ml) sequence with mean vector (f1, 6T1, ,32621)[(61 , 62')]

given by (3.8). Further let H(1, n)[H(2, n)] be an m2-dependent BGMA sequence with mean vector (61 , 62 )[(,6'l , ,2621)] given by Langberg-Stoffer (1987), independent of all G(1, n)[G(2, n)]. Finally let U(n)= (Ul(n), U2(n)) be an i.i.d.

sequence of bivariate random vectors with components assuming the values 1 or 0

independent of all the previous random vectors such that P{Ui(n)= 1} = 1 - f,, 1= 1, 2. We define the two BGARMA(ml, m2) sequences as follows:

(5.5) (L1(1, n), L2(1, n)) = (G1(1, n), G2(1, n)) + (U1(n)Hi(1, n), U2(n)H2(1, n)),

(5.6) (L1(2, n), L2(2, n)) = (H(2, n), H2(2, n)) + (U1(n)Gl(2, n), U2(n)G2(2, n)).

Next we show that Z(j, n) and L(j, n) have bivariate exponential and geometric distributions, respectively.

Lemma 5.7. For j = 1, 2, and n =0, 1, * , Z(j, n)[L(j, n)] has a bivariate

exponential [geometric] distribution with mean vector (A~11, 621)[(6-1, 621)].

Proof. By Lemma 3.9 X(n)[G(j, n)] has a bivariate exponential [geometric] distribution. By Langberg-Stoffer (1987), Corollary 3.8, Y(n)[H(j, n)] has a

812


bivariate exponential [geometric] distribution. Consequently the result of the lemma follow by the four definitions and by Lemma 2.10.

Now we consider the association property of any finite number of random variables belonging to one of the four ARMA classes. We assume that the assumptions of Langberg-Stoffer (1987), Lemmas 3.12 and 3.13 are satisfied. We need the following lemma.

Lemma 5.8. Let S1, -*-, S* , T1, * * , T be non-negative random variables. Let us assume that S1, * * , Sr and T1, * , Tr are associated, and that the random vectors (Si, * , Sr) and (T1i,, , Tr) are independent. Then the random variables S1 T1, * * *, SrTr are associated.

Proof. Let T= (T1, * , T,), let W =(SiT1, ,S, T,), and let f, g be two non-negative functions each with r arguments, non-decreasing in each argument.

The components of the conditional random vector W IT are non-decreasing functions of the associated random variables S,, * * , Sr. Thus, by Barlow and Proschan (1981), P3, p. 30, the components of W I T are associated. Hence,

E{cov (f(W), g(W)) I T) _0.

E{f(w) I T} and E{g(W) I T are two non-decreasing functions of the associated random variables T1, -? ? , Tr. Thus, by Barlow-Proschan (1981), P3, p. 30, the two random variables E{f(W) I T} and E{g(W) I T} are associated. Hence

cov(E{f(W) | T}, E{g(W) | T}))0.

Note that cov (f(W), g(W)) = E{cov (f(W), g(W)) I T}

+ cov (E{f(W) T}, E{g(W) IT}).

Consequently the result of the lemma follows.

Lemma 5.9. Let us assume that for n = 0, 1, .* , V1(n) and V2(n) are associated. Let n, < n2< . * < nr, and 11, * * , lr, r = 1, 2, * * , be as in Lemma 3.10. Then the

Zlq(1, nq)[ZIl(2, nq)], q = 1, * , r are associated.

Proof. By Lemma 3.10, Xlq(nq), q = 1, * , r are associated. By Langberg- Stoffer (1987), Lemma 3.12, the random variables Ylq(nq), q = 1, , r are associated. By our assumption and by Barlow and Proschan (1981), P4, p. 30, Vlq(nq) = , *, r are associated. Thus, by Lemma 4.8, Vlq(nq)Xlq(nq), q =

1, * *, r and Vlq(nq)Yl(nq), q = 1, * *, r, are associated. Now, by Barlow-Proschan (1981), P4, p. 30, the random variables {Xlq(nq), Vlq(nq)Ylq(nq), q = 1, . * , r} and the random variables {Ylq(nq), Vq(nq)Xlq(nq), q = 1, * * , r} are associated. Conse-

quently the results of lemma follow by (5.2), (5.3) and by Barlow and Proschan (1982), P3, p. 30.

Using a similar argument we obtain the following.

813


Lemma 5.10. Let us assume that U1(0), U2(0) are associated; let nl < n2 < *? < nr,

and 11, * , , r = 1, 2, - *, be as in Lemma 3.10. Then the LlJ(1, nq)[Llq(2, nq)], q = 1, * *, r, are associated.

Langberg and Stoffer (1987), Section 4, present inequalities and probability bounds for the bivariate point processes related to the bivariate exponential or

geometric MA sequences. We note that all the results given by Langberg-Stoffer (1987), Section 4, hold for the bivariate point processes related to the bivariate

exponential or geometric AR sequences, given in Sections 3 and 4, and to the ARMA sequences given in this section, provided that they are associated.

6. An example

As an example of the kind of data that can be handled by the models, we fit a BEAR(1) model (cf. 4.3) to temperature (?F) and wind speed (m.p.h.) measurements in Washington, DC., May-September, 1977; 133 observations made daily at noon. The data set was obtained from Professor R. H. Shumway, Division of Statistics, University of California, Davis, and we are grateful to him for the use of his data.

Figures 1 and 2 show plots of the temperature and wind speed series, respectively.

0

Ln -

i o I I

1 133

Figure 1. Temperature measurements in Washington, D.C. 133 daily observations, May-September, 1977.

814

815 BEAR, BGAR, BEARMA and BGARMA models

o Cu

LO

1 133

Figure 2. Wind speed measurements in Washington, D.C. 133 daily observations, May-September, 1977.

Figures 3 and 4 show a stem-and-leaf diagram of the temperature and wind speed data, respectively, and Table 1 gives a summary of the descriptive statistics for both series. It is clear from Figures 1-4 and Table 1 that the data are highly skewed, and are, of course, positive-valued. Also, the marginal distributions of the temperature and wind speed series appear to be shifted Weibull. This is substantiated by other

investigations of these types of series, for example, Lawrance and Lewis (1985) fit a Weibull model to a wind speed series. It is well known that a Weibull random variable may be transformed to an exponential random variable by a simple power transformation. In particular, we assumed that the wind speed data were shifted (by 5 m.p.h., the minimum of the series) Weibull, and we found the power to transform the data to exponential to be 1.6. A similar technique was used on the temperature

Stem-and-leaf of TEMP Leaf Unit = 1.0

1 2 8

21 49

(35) 49 15

5 5 6 6 7 7 8 8

N = 133

1 5 233444 5566678889999 0011111122222233333333444444 55555555566666666777777888889999999 0000000000111111222333333333444444 555555555666779

Figure 3. Stem-and-leaf diagram of temperature data plotted in Figure 1.


Stem-and-leaf of WIND N = 133 Leaf Unit = 0.10

8 5 05899999 27 6 0002235566668888899 52 7 1111222222335566688889999

(23) 8 11111122333556668899999 58 9 111222444444555889 40 10 11111244455566 26 11 455577799 17 12 122455 11 13 245

8 14 1148 4 15 5 3 16 44 1 17 1 18 8

Figure 4. Stem-and-leaf diagram of wind speed data plotted in Figure 2.

data with a slight modification. Here, we first transformed the data by subtracting the measurements from the maximum temperature (which is 89 ?F) and then found the power to transform these measurements to exponential to be 1.906. Figures 5 and 6 show a stem-and-leaf diagram for the transformed temperature and wind

speed series, respectively, and Table 2 gives a summary of the descriptive statistics for the transformed data. Clearly Figures 5-6 and Table 2 indicate that the transfomed data have exponential marginals. In summary, the temperature and wind speed data were transformed as follows:

T(n) = [89- *(n)]196 and

W(n) = [W*(n) - 5]1.6; n = 1, * *, 133,

where T*(n) and W*(n) are the original temperature and wind speed series, respectively, and T(n) and W(n) denote the corresponding transformed series.

Next, we analyzed the correlation structures of T(n) and W(n) processes. Table 3 shows the estimated autocorrelation function (ACF) and partial autocorrelation function (PACF) for the T(n) series to lag 10. Table 4 shows the estimated ACF and PACF for the W(n) series to lag 10, and Table 5 shows the estimated cross-correlation function (CCF) of the two series to lag 10. Tables 3-5 suggest that the BEAR(l) model given in (4.3) is an appropriate first choice model.

The next step was to obtain estimates of the parameters of the model. Here we assumed that the distribution of the noise vector E'(n)= (El(n), E2(n)) was that of

TABLE 1

Descriptive statistics for the temperature and wind speed data.

N mean median stdev min max

Temperature (?F) 133 76-27 77-00 6-87 51-00 89-00 Wind speed (m.p.h.) 133 9-09 8-60 2-54 5-00 18-80

816

BEAR, BGAR, BEARMA and BGARMA models 817

Stem-and-leaf of T(n) N = 133 Leaf Unit = 10

61 0 0000001111111112222223333333334445555556666666666888888899999 (37) 1 1111113333333355555555577777799999999

35 2 22222244444477 21 3 00003336999 10 4 2266699

3 53 2 6 2 7 2 82 1 9 1 10 2

Figure 5. Stem-and-leaf diagram of transformed temperature data.

the Sarkar (1987) distribution which depends on three parameters Al, A2, and A12.

Henceforth, the temperature series will correspond to subscript 1 and the wind

speed series will correspond to subscript 2. The BEAR(1) model also depends on three other parameters, namely, Pio, pol, and p,1 (cf. 4.1) with jr1 = 1 - (Pio + P1) and r2 = 1 - (poi +P11). Estimation of the six model parameters Al, A2, A12, pio,

Poi, and p, was accomplished using a technique similar to Smith (1986) which we

briefly describe. Let Fx(tn, wn tn-, wn-,) be the conditional survival function of the bivariate

series X(n)= (T(n), W(n))' given T(n-l)=tn-_ and W(n - 1)= w1, n =

2, * , N, that is,

Fx(t, Wn tn- Wn-l)

= Pr {(T(n) > tn, W(n) Wn I T(n - 1) = tn-_, W(n - 1) = w_n-}.

Also, let FE(Xn, Yn) be the survival function of the bivariate exponential noise

process E(n) given by Sarkar (1987), p. 667, and let X(-) denote the indicator

Stem-and-leaf of W(n) N = 133 Leaf Unit = 1.0

44 0 00000000111111112222222222233333333333344444 (37) 0 5555555566666666666777778888888999999

52 1 000000111222333333444 31 1 555559999 22 2 000113334 13 2 558 10 3 0044

6 3 68 4 43 3 4 99 1 5 1 5 1 6 1 6 6

Figure 6. Stem-and-leaf diagram of transformed wind speed data.


TABLE 2

Descriptive statistics for the transformed temperature and wind speed data.

N mean median stdev min max

Transformed

temperature [T(n)I 133 159-3 114-0 159-3 0-0 1025-8 Transformed wind speed [W(n)J 133 11-19 7-64 11-19 0-0 66-65

TABLE 3 Autocorrelation and partial autocorrelation functions of the transformed temperature series.

Lag 1 2 3 4 5 6 7 8 9 10

ACF [T(n)] 0-78 0-57 0-42 0-35 0-42 0-39 0-33 0-23 0-13 0-11 PACF [T(n)I 0-78 -0-08 0-01 0-12 0-28 -0-14 -0.01 -0-07 -0-06 0.04

TABLE 4 Autocorrelation and partial autocorrelation functions of the transformed wind speed series.

Lag 1 2 3 4 5 6 7 8 9 10

ACF [W(n)j 0-40 0-20 0-14 0-17 -0-01 -0-07 0-01 -0-01 0-02 0-02 PACF [W(n)J 0-40 0-05 0-06 0-10 -0-14 -0-07 0-07 -0-03 0.06 0-02

TABLE 5 Cross-correlation function of the transformed temperature and wind speed series.

CCF-Lag 0 1 2 3 4 5 6 7 8 9 10

T(n) leads 0-51 0-47 0-35 0-22 0-13 0-14 0-16 0-14 0-03 -0-01 -0-03 W(n) leads 0-51 0-44 0-33 0-24 0-17 0-06 0-08 0-13 0-07 0-06 0-03

function. Using the model conditions (4.3) we have that

Fx(tn, Wn I tn-11 Wn-1) = POOFE(tn1JT1, Wn/;r2)

(6.1) +POlFE(tfl/Jrl, [w. - w,1I/7r2)X(w , > wn-l)

+PlOFE([tf - tn-1]/lrl, wlIr2)X(tn > t,_,)

+PllFE([tn - t,_1JTI,i, [wn - w,n, IbJ2)X(tn > tn1, w,n> wn-1).

From (6.1) and the density given in Sarkar (1987), p. 670, it is easy to obtain the conditional density of (T(n), W(n)) given T(n - 1)= t, i, W(n - 1) = wn-1, n =2, -- -, N, which we denote by f,(t,, w,, j t,-,, w,-). Given the data

(tl, w1), ..-, (tN, WN), we may compute the likelihood of (t2, wA), ---, (tN, WN)

818


given (t1, w1) as

N

(6.2) l fx(tn, Wn I tn-l, Wn-1)- n=2

The estimates of Al,, A2, A12, Pio, Pol, and pl are then obtained by maximizing (6.2) with respect to the parameters. See Smith (1986) for details.

The values of these estimates are l = 0-0040, 2 = 0-0855, ,12 =0-0023, Pio= 0-065, P oi= 0090, pi1 = 0044 so that i = 1 - (io + P) 0-891 and 2 =-

(pol + n) = 0.866. Note that the T(n) and W(n) series are marginally exponential with parameters Al + A12 and A2 + A12, respectively, which can be estimated as

il + 12 = 0.0063 and X2 + 122 = 0-0878, respectively. As a final check on the appropriateness of the model, we tested for the presence

of the discontinuity in the conditional p.d.f., fx(t,, wn, tn1, wn-1) at tn_, and w,n_. This was accomplished as follows. Let X(n) = (Xl(n), X2(n))', n = 0, * * , N be a bivariate process and define Li(n) and Ui(n), i = 1, 2, such that

Pr {Li(n) X(n) X -n xi(n - 1) Xi(n - 1) = x(n - 1)} = p

Pr {xi(n - 1) < Xi(n) Ui(n) Xi(n - 1) = xi(n - 1)} = p

for a specified value of p. Let

ai(n) 1, if Li(n ) ? Xi(n) _ xi(n- 1)

a =0, otherwise

f 1, if xi(n -1) < Xi(n) _ Ui(n) w t 0, otherwise

and N 2 N 2

A= = E ai(n) and B= > 3 bi(n). n=l i=l n=l i=

Then, if the conditional p.d.f. does have the expected discontinuity at xi(n - 1), i = 1, 2, we would expect A and B to be about the same. If the conditional p.d.f. is smooth at xi(n - 1), i = 1, 2, we would expect B to be smaller than A. We, therefore, base our test on A - B, and reject the null hypothesis that there is a

discontinuity at xi(n- 1), i = 1, 2 for large values of A - B. Under the null

hypothesis, E[A - B] = 0 and asymptotically

(A\ lM2Np (2Np(1l-p) -2Np2 ) (

- MVN [2Np -2Np2 2Np(l-p))

so that A - B - N (0, 4Np). Thus, we reject the model at level a if

A-B A4 Np>

Zt

To implement this procedure for our data, we estimated Li(n) and Ui(n) from the

819

820 H. W. BLOCK, N. A. LANGBERG AND D. S. STOFFER

MLE's of the model parameters. Choosing a value of p = 0-25 we found that for the

temperature series, T(n), 47 observations fell in the lower range and 40 observations fell in the upper range. For the wind speed series, W(n), we found 38 observations in the lower range and 41 observations in the upper range. Thus A = 47 + 38 = 85 and B = 40 + 41 = 81, and the value of the test statistic is

85 -81

V4(132)(0-25)

which is not significant at any reasonable level. Hence we conclude that the BEAR(1) model is adequate for the data.

Acknowledgement

We wish to thank the referee for suggestions which led to a considerable

improvement of the paper, and in particular, for suggesting the test for the presence of the discontinuity in the conditional p.d.f. used in Section 6.

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