On the Maximum of a Bivariate INMA Model with Integer Innovations.

We study the limiting behaviour of the maximum of a bivariate (finite or infinite) moving average model, based on discrete random variables. We assume that the bivariate distribution of the iid innovations belong to the Anderson's class (Anderson, 1970). The innovations have an impact on the random variables of the INMA model by binomial thinning. We show that the limiting distribution of the bivariate maximum is also of Anderson's class, and that the components of the bivariate maximum are asymptotically independent.

Introduction

Hall (2003) studied the limiting distribution of the maximum term of stationary sequences defined by non-negative integer-valued moving average (INMA) sequences of the formwhere the innovation sequence is an iid sequence of non-negative integer-valued random variables (rvs) with exponential type tails of the formwhere is slowly varying at and denotes binomial thinning with probability . Hall (2003) proved that satisfies Leadbetter’s conditions and , for a suitable real sequence , and thenfor all real x and . Note that plays an important role in this result. This is an extension of Theorem 2 of Anderson (1970), where it is proved that for sequences of iid rvs with an integer-valued distribution function (df) F with infinite right endpoint, the limitis equivalent tofor all real x.

The class of dfs satisfying (1), which is a particular case of (2) (see, e.g., Hall and Temido (2007)) is called Anderson’s class.

In this paper we extend the result of Hall (2003) for the bivariate case of an INMA model. Concretely, we study the limiting distribution of the maximum term of stationary sequences where the two marginals are defined by non-negative integer-valued moving average sequences of the general formwhere and are defined as above with respect to a two-dimensional iid innovation sequence . The binomial thinning operator , due to Steutel and van Harn (1979), is defined by where is an iid sequence of Bernoulli rvs independent of the positive integer rv Z. The possible class of bivariate discrete distributions (see (4)) includes also the bivariate geometric models.

We assume that and are conditionally independent given (V, W), because the binomial thinning with and are independent, X and Y are binomial rv’s. with parameters respectively , i.e.for all events A and B and for all possible values of v and w. We assume that andfor some .

We investigate the limiting behaviour of and want to find out whether the two maxima components are asymptotically dependent, because of the dependence of the innovations . However, we will show that this is not occurring because of the independent thinning, as we believe. We investigate the impact of the dependence of on the limiting distribution and the convergence rate.

Following similar ideas of Hall (2003) for the univariate case, we:Examples: 1.) We may consider the as the number of person newly infected by virus1 (say COVID-19 virus) and virus2 (say the usual seasonal virus) at time i. It is possible that a person is infected by both virus only at the same time point. We count by the total number of infected and still contagious persons at time j adding all infected persons before and at time j. After some time these persons are cured (or died) and are no more counted to the number of infected but still contagious persons. Hence the random numbers  are thinned at each time point, so the contribution to is , and to is for .

Define a bivariate model which contains the bivariate geometric model;

Characterize the tail of and the tail of , in terms of the model ;

Establish the limiting behaviour of the bivariate maximum of the stationary sequence which is defined componentwise; and

Investigate the convergence of the joint distribution of the bivariate maximum to the limiting distribution by simulations.

2.) Another example for bivariate integer valued time series is presented in Pedeli and Karlis (2011) who discuss the bivariate INAR(1) model with negative binomial innovations for the application of road accidents at two different time intervals in Schiphol area. However their bivariate negative binomial innovations are in the case of geometric innovations of a different type herein considered. Similar is the situation in the paper of Silva et al. (2020) who discuss inference of such a bivariate time series with different distribution of the innovations. But their bivariate negative binomial distribution is also not of our type.

3.) A further application of a bivariate time series for count data in finance is given by Quoreshi (2006). He did not specify the bivariate distribution. He derived the mean and variance/covariances of this time series.

Preliminaries Results for Bivariate Innovations

Let (V, W) be a non-negative random vector with bivariate df satisfyingas , for positive real constants , , such that and , some real constants and slowly varying functions , , and where is a positive bounded (say by ) function which converges to a positive constant L as . That converges to L is for simplicity. It has no impact on the results if the limit L would depend on or . By [x] we denote the greatest integer not greater than x.

Remark 2.1

The marginal tails of are of the form:for . Hence, both marginal dfs belong to the Anderson’s class with

From (4), we can derive the probability function (pf) of (V, W). Because the proofs of the following propositions are technical, we move them to Appendix Proofs.

Proposition 2.1

The pf of the random vector (V, W) with df (4) is given byfor v, w large integers, whereand is bounded and converges to positive constants.

Example 2.1

The Bivariate Geometric (BG) distribution is a particular case of the model (4) with margins (5). Consider the bivariate Bernoulli random vector with and success marginal probabilities and . Due to Mitov and Nadarajah (2005), using the construction of a BG, the pf and the df of a random vector (V, W) with BG distribution are given, respectively, byfor , andfor , assuming that . Hence, this df satisfies (4) with the constants , given byand the index associated to the dependence structure of is

The slowly varying functions are constants and , for . The independence case occurs when . For dependence cases, we can have or . Finally, we note that is a constant. For instance, take , , we have with  as in (6).

The marginal df of V and W are obviouslywhich means V and W are geometrically distributed rvs with parameter and , respectively.

In order to characterize the df of we start by establishing the relationship between the probability generating function (pgf) of (V, W) and (X, Y), defined e.g. for (V, W) aswhich exists for in the following region (given in Lemma 2.1).

Taking into account Proposition 2.1, the series converges obviously for any . Even for some the series converges because of the assumption (4). By this assumption, we have if and if . The following lemma gives a condition such that the series exists.

Lemma 2.1

The pgf exists for in

Its more technical proof is given also in the appendix. As consequence of this lemma, the pgf exists for , if in case , and if and in case of . In the following, we use these convenient conditions for the convergence of .

Now the relationship of the two pgf is the following. It holds as long as the pgf’s exist. For our derivations it is convenient to use in the following the given domain . The proof of this relationship is also given in the appendix.

Proposition 2.2

The pgf of is given in terms of the pgf of (V, W):for all such that .

We want to derive an exact relationship of the two distributions and with the help of a suitable transformation, as a modified pgf or a Mellin transform. We define the (bivariate) modified pgf or tail generating function (Sagitov (2017))and analogously for X, Y. The relationship between and is given in the following proposition.

Proposition 2.3

For , we have

Proposition 2.4

The modified pgf of (X, Y) and (V, W) satisfyif the series converge, i.e. .

From Propositions 2.2 and 2.4, we can derive now the tail in terms of

Proposition 2.5

The df is given in terms of the df with :

Hence the tail of can be estimated by the assumption (4).

Proposition 2.6

If the joint df of (V, W) satisfies (4), then for large integers x and ywithwhere are slowly varying functions, beingwithand the bound of .

Note that .

We observe that the stationary bivariate INMA model introduced in our work is an extension of the BINAR model of Pedeli and Karlis (2011) defined bywith an iid innovations sequence . In their paper it is stated that it has also the representation

Hence, considering with for , and for we obtain .

The Bivariate Stationary Sequence

We consider now the stationary bivariate INMA model with iid innovations with df satisfying (4). We establish first the tail behaviour of . The maximal values of and are most important as in the univariate case. Therefore we write and . We assume that they are unique. It may happen in the bivariate case that and occurs at the same index or at different ones. We consider both cases. Furthermore, we use thatwhich holds because of (3).

Suppose first that and are occuring at different indexes and , respectively. We write for any jand

Denote , , , , , and , . Hence, and . Note that and depend on j.

For the proof of the main proposition of this section we need the following lemma.

Lemma 3.1

If the rv V belongs to the Anderson’s class, then

For any set I of integers with , consider the rv Then is finite for any .

The proof of this lemma is given in the appendix. We deal now with the limiting behaviour of the tail of . Besides of the univariate tail distributions we derive only an appropriate positive upper bound for the joint tail which is sufficient for the asymptotic limit distribution of the maxima. We will see that we get asymptotic independence of the components of the bivariate maxima , since this normalized is vanishing, not contributing to the limit.

For the asymptotic behaviour of the tail of the stationary distribution of the sequence , we write simply (X, Y) for any . As mentioned we deal with the two cases that and are occurring at different indexes or at the same one. We start with the first case and the above defined .

For this derivation, we use and such that , with , and given in (9),and

Proposition 3.1

If satisfies (4) and and are unique and taken at different indexes, then as , whereandfor some constant .

for the marginal dfs and

for the joint df with satisfying (12) and (13)

We show also that .

Proof

In fact

We deal with the three terms in (16), separately.

Since , taking the sum in Lemma 3.1, we conclude that is finite. Similarly is finite since with . The tail function of X is given, with , by For the first sum of (17), we get by applying Proposition 2.6 with for the marginal distribution by dominated convergence. For the second sum in (17), we get for x large using the Markov inequality, since is finite for . Since we get by Theorem 4 of Hall (2003) and thus together With the same arguments we characterize the tail . Hence, the statements on the marginal dfs are shown.

Now we deal with the third term in (16). Note that , and in the representation of X and Y are independent. For any and satisfying (12), we use that (18) and (19) imply and The probability in the third term of (16) is split into four summands with satisfying (12), and . We get for x and y large, to simplify the proof. The last sum is bounded by by (20). For the first sum of (22) we use Proposition 2.6 and obtain with such that (13) holds, Note that uniformly for , i.e. . Hence the sum is bounded above by since the last pgf exists due to Lemma 3.1 and (13) Note that The expectations exist by assumption (4) since , and also , for all i, by the choice of in (13), by using the arguments of Lemma 3.1. We consider now the approximation of the second sum in (22). We have with some positive constant C By the arguments used to approximate in (i), we also obtain with some generic constant C. Hence, it implies together with (23) For the third sum in (22), we get analogously to the derivation of the second sum

Combining now the bounds of the four terms , we get the upper bound for which shows our statement.

Suppose now the case that the unique and are taken at the same index , say. Write for any jand

Denote , , , and , as used for Proposition 3.1. Observe that and (S,T) are independent. Then the corresponding statement of Proposition 3.1 holds for this case (letting ) which is given in Proposition 3.2. We omit the proof since it is very similar to the given one with a few obvious changes.

Proposition 3.2

If satisfies (4) and and are unique, occurring at the same index, then the stationary distribution satisfiesas , whereandfor some constant and satisfying (12).

Now we investigate the limiting behaviour for the bivariate maxima, in case of an iid sequence .

Theorem 3.1

Let be such that (4) holds and and are unique, occurring either at the same or not the same index. Let

Define the normalizationsand

Then, for x, y real,

The convergence for the marginal distributions holds by applying Proposition 3.1 or 3.2 with the chosen normalization sequences. Since and are similar in type, we only show the derivation of the first marginal. Because the normalization is not always an integer, we have to consider and . Let us deal with the case. Note thatand

For the normalization we getSo

The derivation of the is similar using .

Now for the joint distribution we use the bounds of of the two propositions. First we consider the case of Proposition 3.1 with and at different indexes. We have to derive the limits of three boundary terms of given in Proposition 3.1 multiplied by n. The last of these terms tends to 0 because (21) holds and due to the fact that from (14), we getwhich is bounded.

The first of the three boundary terms of is smaller thanbecause with B given by (13).

The second boundary term of is smaller thansince and where represents a generic positive constant.

Thus the limiting distribution is proved in case of Proposition 3.1.

Now let us consider the changes of the proof for the case of Proposition 3.2. Again we have to deal with the three boundary terms of where the last two are as in Proposition 3.1. In the first of these terms we have similarlysince . Thus the statements are shown.

Main result

We consider now the stationary sequence . From extreme value theory it is known that the behaviour of their extremes is as in the case of an iid sequence if the following two conditions hold: a mixing condition, called , and a local dependence condition, called . In our bivariate extreme value case we consider the conditions and of Hüsler (1990) (see also Hsing (1989) and Falk et al. (1990)). The condition is a long range mixing one for extremes and means that extreme values occurring in largely separated (by ) intervals of positive integers are asymptotically independent. The condition considers the local dependence of extremes and excludes asymptotically the occurrences of local clusters of extreme or large values in each individual margin of as well as jointly in the two components. We write  for short because  do not play a role in the following proofs.

Definition 4.1

The sequence satisfies the condition if for any integers for which we havefor some with , for some integer sequence .

We use the following condition.

Definition 4.2

Let be a sequence of positive integers such that . The sequence satisfies the condition if

In the following we use the sequences and such that

Such a sequence in (26) exists always. Take e.g. for the given and in condition the sequence . In our proof we use simpler sequences.

Write and . For the stationary sequence satisfying and , the limiting behaviour of the bivariate maxima , under linear normalization, is given in Theorem 3.1, as if the sequence would be a sequence of independent .

In Theorem 3.1 we derived upper and lower bounds of the limiting distribution of the maximum term of non-negative integer-valued moving average sequences which leads to a “quasi max-stable” limiting behavior of the bivariate maximum in the sense of Anderson’s type. So the main result of the maximum of this bivariate discrete random sequence is the following.

Theorem 4.1

Consider the stationary sequences defined by

Suppose that the innovation sequence is an iid sequence of non-negative integer-valued random vectors with df of the form (4), the sequences of and satisfy (3) and and are unique. Then,for all real x and y and where and are defined by (24) and (25).

To prove this theorem, it remains to show that the conditions and hold with and given by (24) and (25).

Proof of :

Let with , with separation , where . We select later. We use the following notation:and

Note thatandare independent.

a) We have as upper boundwhere , , , and .

We split furthermore this upper bound.

The last four terms in (29) tend to 0 as it is proved in Hall (2003) depending on . We show it for one term.for some generic constant C and satisfying (3) with . Selecting , this bound tends to 0. The sum of the bounds of the last four terms in (29) gives the bound , which tends to 0.

b) In the same way we establish the lower bound of (28). In fact, using again the independence mentioned in (27), we getusing (29) and (30). Hence the condition holds.

In the proof of , we need also that . With we select such that , which holds for .

Proof of :

We have to consider first the sums on the terms and on the terms .

We show it for the sum of the first terms, since for the second one the proof follows in the same way. Let with , which implies that . For , we write

Note that for some and for some . For one j we have , i.e. . Hence the maximum terms occur at the same index for and if . If , hence , but this case does not occur in the sum. For all other j’s the maxima is occurring at different indexes. We consider the bound established in Proposition 3.1 and 3.2 for .

For , we showed in the proof of Theorem 3.1 that

For , we have for the terms and deduce from Proposition 3.1 the following upper bound for with defined in (12) and (13). Note that should be such that , for all that (13) is satisfied. It means that the term B in (13) depends on j, i.e. . Note that may be larger or smaller than 1, but is bounded above by . For , we select large such that , which implies that , thus we select . In case , we select also .

It implies that there exists an to select for every such thata) Now the sum of the first term in the bound (31) of multiplied by n, for , is bounded byif also is such that .

The sum of the second term in (31) multiplied by n, for , tends to 0 becauseif also . Hence we choose .

It remains to deal with the sum of the third terms in (31) for . We showed that in (20) with in (12). Let such that . This sum on multiplied with n is bounded byif also and C is a generic positive constant.

Thus combining these three bounds it shows thatif .

b) We consider now the sum on j with and writeand

Note that and are independent. We have, for and some (chosen later, not depending on n),

Similar to Hall (2003), the last two probabilities are sufficiently fast tending to 0. For, we have

We select such that for some constant C. For and and some positive constant , it follows thatby the assumption (3) on the sequence . It implies again thatwhere the expectations exist, and, due to Lemma 3.1,by the choice of . Note that . Now, select k depending on , and such that which holds for . This choice implies that . In the same way we can show that also for such a k, since also for and some constant .

c) In order to deducewe use the same arguments as for . In this case, since and are independent, we get for some positive k

As above we can show that and . In the same way it follows also that

Hence condition holds.

Simulations

We investigate the convergence of the distribution of the bivariate maxima to the limiting distribution as given in Theorem 4.1. We notice that the thinning coefficients and have an impact on the norming values of the bivariate maxima, besides of the distribution of the .

Let us consider the bivariate geometric distribution for mentioned in Example 2.1 and a finite number of positive values and . As mentioned, the bivariate geometric distribution satisfies the conditions of the general assumptions of the joint distribution of . We assumed a strong dependence with and .

We consider quite different models with different and to investigate the convergence rate. Let in the first case and for and in the second case and for

For each of these first two models we simulated 10’000 time series, selected and 500 and derived the bivariate maxima ). Thus we compared the empirical (simulated) distribution functions (cdf) with the asymptotic cdf.

We plotted two cases with where and with given in (24) and (25), respectively, using and 2 (see Figs. 1 and 2).

Fig. 1

Simulated cdf with upper and lower asymptotic cdf, first case, where ,

Fig. 2

Simulated cdf with upper and lower asymptotic cdf, second case, where ,

We notice from these simulations that the convergence rate is quite good, but it depends on the dependence, which is given by the thinning factors and . We find that the convergence rate is slower for the more dependent time series (the first case, Fig. 1) and that the factor has a negligible impact. This is even more clear in the second cases shown in Fig. 2.

In some additional models we considered larger and more thinning factors different from 0. We show the simulations of the cases with , for , and also with , for . These cases are close to a infinite MA series, since are very small for or , respectively. It means that such small values have an impact on the maxima. We figured out that the number of positive values is not so important. However, in these cases the second largest value of or is closer to the maximal value (=1), in particular in the second of these additional models. Considering the results of again 10’000 simulations (Fig. 3), we show that the convergence rates are quite slower than in the first two models (Figs. 1 and 2). We show the results of the two cases with and 500 with only. We also figured out from the simulations of other models and distributions that if the correlation of the two components of the sequence is stronger, then the convergence to the limiting distribution (with asymptotic independence) is slower.

Fig. 3

Simulated cdf with upper and lower asymptotic cdf, third and fourth model where for (third model), and (fourth model), respectively, with and 500 and