# Record Ranges for Samples From Asymmetrical Laplace Distributions

- DOI
- 10.2991/jsta.2018.17.2.2How to use a DOI?
- Keywords
- Record values; Exponential distribution; Negative exponential distribution; Laplace distribution; Sample ranges
- Abstract
The representations of record ranges via sums of independent identically distributed exponential random variables are obtained for asymmetrical Laplace distributions. This result generalizes the corresponding relations for record values in the cases of exponential and negative exponential distributions

- Copyright
- Copyright © 2018, the Authors. Published by Atlantis Press.
- Open Access
- This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

## Introduction

Let X_{1}, X_{2},… be a sequence of independent identically distributed random variables (r.v.’s) with an absolutely continuous distribution function (d.f.) F(x). For any n=1,2,… let us introduce r.v.’s

Upper record times L(n) and upper record values X(n), n=1,2,…, are defined as follows:

Analogously one can define the so-called lower record times *l* (n) and lower record values *x*(n), n=1,2,… :

Below the notation

In the record theory (see, for example, [1] – [5]) results for the initial sequences of exponentially distributed random variables are very important and popular. We give here two of the corresponding statements.

Let Z_{1}, Z_{2},… be a sequence of independent *E*(1)-distributed r.v.’s with d.f.

_{1}< Z(2) <… be the corresponding upper record values. Let ξ

_{1}, ξ

_{2},… also be independent

*E*(1)-exponential r.v.’s and S

_{k}= ξ

_{1}+ξ

_{2}+…+ ξ

_{k}, k=1,2,…. The following result is valid.

### • **Representation 1.**

*For any n=1,2,…*

Now let us consider r.v.’s V_{k}= − Z_{k}, k=1,2,…, with the negative exponential d.f. H_{2}(x), where

_{1}> v(2)> ….

In this situation the relation, analogous to (5), can be written.

### • **Representation 2.**

*For any n=1,2,… the following equality holds:*

Equalities (5) and (6) together with Smirnov’s transformation allow to obtain some useful results for X’s with any continuous d.f. F(x). For example, one can immediately write that

_{1}, U

_{2},… are independent uniformly

*U*([0,1))-distributed random variables. Hence in this case the following relations are valid for record values:

Analogously,

## Record Ranges and Laplace Distributions

Above the exponential and negative exponential random variables with d.f.’s H_{1}(x)=max{0,1−exp(−x)} and H_{2}(x)= min{e^{x},1} were under our consideration. Let us consider now the distribution which is the mixture with weights q and p = (1−q), 0≤q≤1, of these two distributions. The corresponding d.f. H(x) is taken in this case as H(x) = qH_{1}(x)+pH_{2}(x). Thus,

*E*(1)-exponential distribution. The value p =1 corresponds to the negative exponential distribution. If p = q =1/2 one deals with the classical Laplace pdf h(x) = exp(−|x|)/2. Thus, we can say that relations (10) and (11) present asymmetrical Laplace distributions.

Let now independent r.v.’s X_{1}, X_{2},… have the common distribution function (10) and let us add the degenerate r.v. X_{0} = 0 in the beginning of this sequence. For X_{0}, X_{1}, X_{2}, … we consider maximal and minimal values

Record values W(1) = W_{1} < W(2) <…< W(n) <… in the sequence of ranges W_{1},W_{2},… are the subjects of our interest. It appears that one can express these records via r.v.’s ξ_{1}, ξ_{2},…, which are defined above. The following result is valid.

### • **Representation 3.**

*For any n=1,2,… and 0 ≤ p = 1−q ≤ 1 the relation*

_{k}= ξ

_{1}+ξ

_{2}+…+ ξ

_{k}, k=1,2,….

**1.****Remark 1**. It appears that the RHS of (12) doesn’t depend on*p*and*q*.**2.****Remark 2**. It is easy to see that Representation 1 is the partial (under p = 0) case of (12). Analogously, the result of Representation 2 immediately follows from (12) if to take p =1 and q=0.**3.****Remark 3**. We discuss here record ranges in the sequences of random variables which have different forms of Laplace distributions, which are rather close in some sense to the exponential distributions. Let us note that some analogous results for record ranges were obtained in [6] for the initial sequences of the uniformly distributed r.v.’s.

### Proof of Representation 3

It is evidently that W(1) = W_{1} = max{0,X_{1}} − min{0,X_{1}}= |X_{1}| has the exponential *E*(1)-distribution. Hence one can write that

We examine the behavior of conditional probabilities

Note that the condition {m(n−1) = −y, M(n−1) = v} corresponds to the situation, when W(n−1) = y+v.

Let us denote events

One can see that

Now it is possible to write that

It appears that conditional probabilities R(x,y,v) don’t depend on y and v. Thus, it follows that r.v. T(n) doesn’t depend on W(n−1) and this difference of two neighbouring record values W(n) −W(n−1) has the standard E(1)-exponential distribution. Recalling relation (13) one can write now that for any record value W(n), n=1,2,…, the following presentation is valid:

## The Number of Record Ranges

Let Z_{1},Z_{2},… be a sequence of independent *E*(1)-distributed r.v.’s with d.f. H_{1}(x)=max{0,1−exp(−x)} and N_{1}(n) be the number of the upper records among Z_{1},Z_{2},…,Z_{n}.

It is known that

If independent r.v.’s V_{1},V_{2},… have d.f. H_{2}(x)=min(1,e
^{x}) and N_{2}(n) is the number of the lower record values among V_{1},V_{2},…,V_{n}, then also we come to the relation

Let us consider now the number N(n) of record values in the set of ranges W_{1},W_{2},…,W_{n}. We denote

The number N(n) of records in the set W_{1},W_{2},…,W_{n} is equal to the sum N
^{(1)}(n) + N^{(2)}(n), where

_{1,1},Y

_{2,1},…,Y

_{n,1}and

_{1,2},Y

_{2,2},…,Y

_{n,2}. One can find that

One can see also that

## Acknowledgements

**The authors thank the referees for valuable comments that improved the presentation of the paper**. The work of the third author was partially funded by RFBR grant № 18-01-00393

## References

### Cite this article

TY - JOUR AU - I.V. BELKOV AU - M. AHSANULLAH AU - V. B. NEVZOROV PY - 2018 DA - 2018/06/30 TI - Record Ranges for Samples From Asymmetrical Laplace Distributions JO - Journal of Statistical Theory and Applications SP - 206 EP - 212 VL - 17 IS - 2 SN - 2214-1766 UR - https://doi.org/10.2991/jsta.2018.17.2.2 DO - 10.2991/jsta.2018.17.2.2 ID - BELKOV2018 ER -