On distribution of upper marginal records in bivariate random sequences

Gülder Kemalbay,İsmihan Bayramoğlu

doi:10.3906/mat-1901-54

Gülder Kemalbay, İsmihan Bayramoğlu

Open Access

https://doi.org/10.3906/mat-1901-54

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Abstract

The theory of record values has been extensively studied in the statistical literature. However, there are not many papers devoted to the theory of records for bivariate and multivariate random sequences. This paper presents the marginal record values and record times in extended sequence of bivariate random vectors. The joint distributions of some upper marginal records are derived. Some results on joint probability mass function of upper record time vectors and distribution function of upper record value vectors are given via copula functions. Moreover, the numerical and graphical applications of considered upper records using flood data and prediction of rainfall variables such as intensity, depth, and duration are provided.

Highlights

Interest in the theory of records has increased rapidly since the appearance of the first pioneering paper by Chandler [11]
We are interested in marginal records of each of the sequences {Xk}k≥1 and {Yk}k≥1 and we investigate the joint distributions of marginal record times and record values of these sequences
The obtained pmfs of record times and cdfs of record values come into prominence while predicting future records based on past observations

Summary

Introduction

Interest in the theory of records has increased rapidly since the appearance of the first pioneering paper by Chandler [11]. Some papers study the characterization of distributions through the properties of record values. The theory of records has been well developed, only a few papers in the statistical literature deal with bivariate and multivariate records. Some of these works introduce the definition of records according to different ordering principles of multivariate random vectors. In recent years, [7] considered records of bivariate sequences by using conditional ordering and has gave the definition of multivariate records according to component-wise ordering. For example, knowledge of exact distribution functions of the record value sequence is sufficient to characterize the common distribution function of the underlying observations. While predicting the value or time of the record, we need the joint pmf of record times and joint cdf of record values, which are the main subjects of this paper

Records of extended bivariate random sequence

Joint probability mass function of marginal records

Conclusion