Limit Distributions of Ordered Random Variables in Mixture of Two Gaussian Sequences

Barakat, H. M.; Dwes, M. H.

doi:https://doi.org/10.1155/2022/7956195

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Research Article | Open Access

Volume 2022 | Article ID 7956195 | https://doi.org/10.1155/2022/7956195

Limit Distributions of Ordered Random Variables in Mixture of Two Gaussian Sequences

H. M. Barakat¹and M. H. Dwes²

Academic Editor: Tuncer Acar

Received13 Feb 2022

Accepted23 Mar 2022

Published31 Aug 2022

Abstract

In this paper, we study the limit distributions of extreme, intermediate, and central order statistics, as well as record values, of the mixture of two stationary Gaussian sequences under equicorrelated setup.

1. Introduction

Because the mixture model, which is a convex combination of two distribution functions (DFs) and incorporates the qualities of the separate DFs, it is a strong and flexible tool for modelling complex data (e.g., see Barakat [1] and Barakat et al. [2]). Apart from this benefit, mixture models are widely employed in a variety of applications, as evidenced by Titterington et al. [3] and Doğru and Arslan [4]. For example, a two-dimensional mixture model arises in the theory of reliability when individuals belong to one of two distinct types, with proportions and , of two populations which have lifetime DFs and . An individual is randomly selected from the population and then has the lifetime DF:

It is known that model (1) is not derivable (cf. Nelson [5]), i.e., there does not exist any two-place function such that, for any pair of random variables (RVs) with respective DFs and , the mixture is the DF of the RV . However, the mixture model (1) can be represented in terms of RVs rather than DFs (cf. Bernard and Vanduffel [6] and Barakat et al. [2]) aswhere is a Bernoulli distributed RV with parameter , , , and the RVs and are independent of (and ). Representation (2) was recently used in literature to derive the quantile function of the mixture model (1) (e.g., Barakat et al. [2]). On the other hand, this representation will be an essential pillar of our study. Our objective is to study the limit distributions of important ordered RVs, which are order statistics (OSs) (extreme, intermediate, and central OSs) and record values of a mixture of two stationary Gaussian sequences (SGSs) under an equicorrelated setup.

Despite the fact that we consider the equicorrelated setup, the paper’s results are of considerable theoretical and practical interest due to the remarkable paucity of works on the limit of OSs and record values arising from the mixture model and the absence of any work concerning the mixture model of two SGSs. Moreover, there are some scientific evidences that a deviation from this “neat”constant sequence of the correlation will not dramatically affect the limiting form of the statistics that are considered in this paper. Actually, for the maximum OSs, based on SGS, a deviation from constant to varying correlation does not affect the limiting form of the maximum OSs when properly normalized (e.g., see Theorems 3.8.1 and 3.8.2 in Galambos [7]). For the limit DFs of order RVs under the equicorrelated setup, the literature is abounded with some variant works [8–14].

Let and be two SGSs. Furthermore, for , let have zero mean, unit variance, and constant correlation coefficient , written . The sequence , can be exemplified by , where are i.i.d standard normal variables (cf. [7]).

It is worth mentioning that both representations (1) and (2) show that the dependence structure between the two RVs and does not influence these representations. Therefore, without any loss of generality, we assume that the two SGSs and are independent. Therefore, relying on representation (2), the mixture of these sequences iswhere and are independent and each of them is independent of (and ). On the other hand, sequence (3) can also be exemplified bywhere, is the standard normal DF, and are i.i.d RVs with common DF as follows:

Now, for any , the th OS based on the sequence , represented by (3), can be expressed aswhere is the th OS based on the sequence , represented by (5).

For fixed rank , with respect to , and are called the th lower and upper extremes, respectively. Any result for the lower OSs may be inferred from the upper OSs using the well-known relation and vice versa. Besides extreme OSs, there are two types of OSs according to their rank nature. A sequence is called a sequence of OSs with variable rank if , as . Consequently, two particular variable ranks are of special interest:(1) (or ), as , which will be referred to as the lower intermediate rank case (or the upper intermediate rank case).(2), as , which we shall call the case of central ranks. A familiar example of the central OSs is the th sample quantile, where , and denotes the largest integer not exceeding .

Record values in a sequence of RVs are successive maxima or values that strictly exceed all preceding values. Let be i.i.d RVs with common DF . Then, is an (upper) record value if , and therefore is a record value. The record time sequence is times at which records appear. Therefore, and . Thus, the record value sequence is defined by . For more details about the record value model, see Arnold et al. [15]. Thus, the records based on the sequence , represented by (3), can be expressed aswhere is the record value based on the sequence , represented by (5).

In the second section of this paper, we study the asymptotic distribution of upper extreme OSs concerning the mixture of two SGSs represented by (3), under mild conditions. In the third section, we obtain the parallel results for the central OSs. In the fourth section, the asymptotic behavior of the intermediate OSs of the mixture of two SGSs is studied. In the last section, the asymptotic behavior of the record in the mixture of two SGSs, represented in (8), is studied.

Everywhere in what follows, the symbols , , and stand for convergence, converge weakly, and converge in probability, as , respectively. Also, stands for the convolution operation.

2. Limit Distributions of Extreme OSs in a Mixture of Two SGSs

We start with a slight generalization of a result of [16], which will be needed for our study in this section.

Lemma 1. Let be a given sequence of nondegenerate DFs and suppose are i.i.d RVs with mixture DF,

Furthermore, let be the th OS of the sequence . Then, for some suitable linear transformation , we getif belongs to the max-domain of attraction of the nondegenerate max-type , written .

Remark 1. According to the Extremal Type theorem (cf. [7, 17]), there are only three possible max-types, namely, max-Weibull, Fréchet, and Gumbel types. The Gumbel type is an exclusively important type in our study in this section, , because of the well-known fact , where and .

Proof of Lemma 1. The proof follows by using Theorems 2.2.1 and 2.2.2 in [17] and the obvious relation:

In [16], Lemma 1 was proved for and . Moreover, AL-Hussaini and El-Adll [16] claimed that the converse of Lemma 1 is true, but Sreehari and Ravi [18] proved that this allegation is false. Moreover, Barakat et al. [19] refined the results of [16, 18] by revealing that any scale normalizing constant in determines uniquely (up to scale changes) the max-stable type. The following theorem gives the asymptotic distribution of upper extreme OSs concerning sequence (3) (and consequently sequence (7)).

Theorem 1. Concerning sequence (7), let and be defined as in Remark 1. Furthermore, let , . If , thenwhere whereas if , thenwherewhere is the indicator function of set . On the other hand, let . Then,(1) if , , and , or if and .(2) if , , and , or if and .(3) and if , , and .

Proof. Under the condition , , the distribution of can be written aswhere and are independent. From equation (5), , but , thenTherefore, by using the law of total probability and some simple algebra, we getThus,On the other hand, we havewhere is the th upper extreme OS based on the sequence , represented by (5), and are i.i.d RVs with common DF (6). Therefore,Now, the asymptotic distribution of can be found from Lemma 1 after determining the domain of attraction of the DF and . To do that, we use Khinchin’s type theorem and Extreme Value Theorem. First, we note that, from the assumption (thus, ), we get . On the other hand, using the relations and and bearing in mind that (by using L’Hopital’s rule), we get . Thus, Khinchin’s type theorem and Extreme Value theorem yieldFrom (19)–(21) and Lemma 1, we getTherefore, from (15) and Lemma 2.2.1 in [7], a combination of (18) and (22) yields the first part of the theorem. Now, turning to the case , , and , or and . From representation (7), we getFrom (5), we have since . Therefore,While, for every , we getsince . Finally, from (23) and Lemma 2.2.1 in [7], a combination of (24) and (25) yieldsIn the same manner, under the condition , , and , or and , we getsince and . Again, in the same manner, under the conditions , , and , we getsince and . Finally, we getsince and . This completes the proof of the theorem.

3. Limit Distributions of Central OSs in a Mixture of Two SGSs

We will need two essential results (Lemmas 2 and 3) due to [19, 20]. These two results will be formulated in a slightly different way that fits our study in this section, where the modifications made in these results are very easy to prove.

Lemma 2 (see [20]). If are the OSs corresponding to the i.i.d RVs , with common DF and probability density function (PDF) , such that , where and , then for any central rank satisfying the regular condition , we havewhere .

Lemma 2 was originally given in [20] for with the same limit and the normalizing constants and , where and .

Lemma 3 (see [19]). Let be a given sequence of nondegenerate DFs and suppose are i.i.d RVs with common mixture DF:

Furthermore, let (where ) be the th OS of the sequence . Then, for some suitable linear transformation , we get

if belongs to the central-domain of attraction of the nondegenerate type , written .

It should be noted that if , for , then .

Remark 2. In [19], Lemma 3 was proved for . In addition, according to the result of [20], there are only four possible limit types for . Among these nondegenerate types, the normal type is an exclusively important type in our study due to Lemma 2.

The following theorem gives the asymptotic distribution of central OSs concerning sequence (3) (and consequently sequence (7)).

Theorem 2. Concerning sequence (7), let and , where is the standard normal PDF. Then, for any central rank satisfying the regular condition , we have(1), if , for . Note that , if .(2), if , , and , or if and .(3), if , , and , or if and .(4) and , if , , and .

Proof. Under the condition , the distribution of can be written aswhere and are independent. From equation (5), , but , thenOn the other hand, we havewhere is the th central OS based on the sequence , represented by (5), and are i.i.d RVs with common DF (6). Therefore,By relying on Khinchin’s type theorem and Lemma 2, we are going to proveImplication (37) holds since and , where and . The first limit is evident from the assumption (thus,). Hence, only the latter limit needs explanation. Using and , the result follows. Therefore, from (35)–(37) and Lemma 3, we getThus, from (33), a combination of (34) and (38) yields the first part of the theorem under the condition . Now, turn to the conditions , , and or and . From relation (7), we haveFrom (5), we get since . Therefore,On the other hand, for every , we getsince . Finally, from (39) and Lemma 2.2.1 in [7], a combination of (40) and (41) yields . In the same manner under the conditions , and or and , we get , since and , . Also, under the condition , , and , we get , since and , . Finally, since and , .

4. Limit Distributions of Intermediate OSs in a Mixture of Two SGSs

Again in this section, we will need two essential results (Lemmas 4 and 5) due to [19, 21]. These two results will be formulated in a slightly different way that fits our study, where the modifications made in these results are very easy to prove.

Lemma 4 (see [21]). If are the OSs corresponding to the i.i.d RVs , with common standard normal distribution , then for any upper intermediate rank satisfying the condition , we havewhere , , and , as .

Lemma 5 (see [19]). Let be a given sequence of nondegenerate DFs and suppose are i.i.d RVs with common mixture DF,

Furthermore, let (where ) be the th OS of the sequence . Then, for some suitable , we getif belongs to the intermediate-domain of attraction of the nondegenerate type , written . Note that if , for , then .

Remark 3. In [21], Lemma 4 was proved for (lower intermediate OSs). Moreover, in [19], Lemma 5 was proved for . In addition, according to the result of [22], there are only three possible limit types for . Among these nondegenerate types, the normal type is an exclusively important type in our study due to Lemma 4.

The next theorem gives the limit DF of the th OS (upper intermediate OSs) of the sequence (7), under the condition , . It is worth mentioning that this condition is quite general, e.g., it is easily to check that it holds with when , (the Chibisov rank sequence [19, 22]). Also, this condition will hold with when . Finally, it will hold with when .

Theorem 3. Concerning sequence (7), let , , , , and , as , where be the standard normal PDF. Then, for any upper intermediate rank , we have(1), if , for .(2), if , , and , or if and .(3), if , and , or if and .(4) and , if , , and .

Proof. Under the condition , for , the distribution of concerning sequence (7) can be written aswhere and are independent. In view of (5), we get ,but since . Therefore,Also,where is the th intermediate OS based on the sequence , represented by (5), and are i.i.d RVs with common DF (6). Thus,By using Khinchin’s type theorem and Lemma 4, we are going to proveImplication (49) holds, since and , where and . The first limit is evident from the assumption (thus, ). Hence, only the latter limit needs explanation. Applying that , thus , but (from the assumption ) and , which proves the second limit relation. Therefore, from (47)–(49) and Lemma 5, we getThus, from (45), a combination of (46) and (50) yields the first part of the theorem under the condition . Now, turning to the conditions , , and , or and . From relation (45), we haveFrom equation (5), we get since . Therefore,On the other hand, for every , we havesince . Thus, from (51) and Lemma 2.2.1 in [7], a combination of (52) and (53) yieldsIn the same manner, under the conditions , and , or and , we get since and , . Also, under the conditions , , and , we get , since and , . Finally, , since and , . This completes the proof of the theorem.

5. Limit Distributions of Record Values in a Mixture of Two SGSs

Chandler [23] published the key study on statistical treatment of the record values. He looked at the stochastic behavior of the random record values coming from i.i.d observations from a continuous DF . The basic properties of the record values depend on the cumulative hazard function and its inverse , e.g., the DF of the upper record value based on the DF may be expressed in terms of as (cf. [15]), where is the incomplete gamma ratio function. Resnick [24] derived the class of all possible limit laws of the upper record . He connected between these limit laws and the max-limit laws through what is known as the Duality theorem via the associated DF . It is noteworthy that these limit laws are the same as the three possible limit laws for the intermediate OSs. For further interesting work related to the link between the OSs and record values, see [25]. Again, the normal type is an exclusively important type in our study due to the fact that the upper record based on the standard normal distribution belongs to the domain of attraction of the normal type (cf. [15]). Namely,where is the usual inverse function of . Moreover, an equivalent simpler form of this limiting result is obtained by an application of the mean value theorem (cf. example 2.3.4 of [15]) asi.e., , where and .

The following lemmas will be needed in our study. The first lemma (Lemma 6) individually expresses an interesting fact concerning the mixture of normal DFs.

Lemma 6. Let . Then, , as , where and

Proof. By using the obvious relations and and L’Hospital’s rule, we get (after some obvious algebra)where is the PDF of the standard normal distribution.

Lemma 7. Let be the upper record value based on the DF , given by equation (6). Then,where , , as , and .

Proof. Let . Then, Lemma 6 yieldsTake . Note that , since (cf. [10]). Now, we can easily get . Thus, . From relation (56), Corollary 6.4.1 in [7] (Tata [26] is credited with this corollary), and the last asymptotic relation, we getAgain, by Corollary 6.4.1 in [7], we get , as required to prove.

The following theorem, which is the main result in this section, gives the asymptotic distribution of record values concerning sequence (8).

Theorem 4. Let be the upper record value based on , given by equation (3). Let , , as , and .(1)If , for , then , where (2)If , , , and , or if , and , then (3)If , , , and , or if , , and , then (4)If , , , and , then

Proof. Under the condition , for , by using the representation (8), the distribution of the record value can be written aswhere and are independent. Clearly, , since from the assumption . On the other hand, by using Lemma 7 and Khinchin’s type theorem, we are going to prove the limit relationEquation (64) holds, since and , where , , , and . The first limit is evident from the assumption (thus, ). Hence, only the second limit relation needs explanation. Since and , ,as , which verifies the second limit relation. Now, from equations (63) and (64) and since , we get . This completes the proof of the first part of the theorem. Turning now to prove the second part, i.e., we adopt the conditions , , , and , or , , and . First, we note that under the conditions of this case, we have , for all large . Thus, by using representation (8), the distribution of can be written aswhere and . Clearly, andThe remaining part of the proof is obvious. On the other hand, the proofs of the third and fourth parts of the theorem are identical to the proof of the second part, with the exception of the obvious changes, and we omit their proofs for brevity. The proof of the theorem is complete.

Corollary 1. Under the conditions of the first part of Theorem 4 and by using Khinchin’s type theorem, we can easily prove the following two relations:

Moreover, under the conditions of the second or third parts of Theorem 4, if , by using Khinchin’s type theorem, we can easily prove the following relation:

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2022 H. M. Barakat and M. H. Dwes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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