North-east bivariate records
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North-east bivariate records N. Balakrishnan1 · A. Stepanov2 · V. B. Nevzorov3 Received: 1 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Bivariate records have many application in different fields such as hydrology, economy, finance and weather forecasting; see, for example, the works of Bayramoglu (Metrika 79:725–747, 2016) and Kemalbay and Bayramoglu (Turk J Math 43:1474– 1491, 2019). It should be noted that there are various definitions of bivariate records. In this paper, we discuss the concept of north-east bivariate records, originally proposed by Arnold et al. (Records, Wiley, New York, 1998). Even though the concept was introduced more than two decades ago, no serious research has been made on it due to its complexity. In the present paper, we obtain distributional and limit results for the north-east bivariate records. We also discuss generation techniques for these bivariate records. Keywords Univariate records · North-east bivariate records · Limit results · Generation techniques
1 Introduction Let Z = (X , Y ), Z 1 = (X 1 , Y1 ), Z 2 = (X 2 , Y2 ), . . . be independent and identically distributed random vectors with a continuous bivariate distribution F(x, y) = P(X ≤ ¯ x, Y ≤ y), survival function F(x, y) = P(X > x, Y > y), corresponding marginal distributions H (x) = P(X ≤ x), G(y) = P(Y ≤ y) and marginal survival functions ¯ H¯ (x) = P(X > x), G(y) = P(Y > y), respectively. When the corresponding densities exist, they will be denoted by f (x, y), h(x) and g(y), respectively.
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A. Stepanov [email protected]
1
Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4K1, Canada
2
Institute of Physics, Mathematics and Information Technology, Immanuel Kant Baltic Federal University, A. Nevskogo 14, Kaliningrad, Russia 236041
3
Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya nab., 7/9, St. Petersburg, Russia 199034
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The sequences of univariate X record times L(n, X ) (n ≥ 1) and record values X (n) (n ≥ 1) are defined as follows: L(1, X ) = 1, L(n + 1, X ) = min j : j > L(n, X ), X j > X L(n,X ) (n ≥ 1), X (n) = X L(n,X ) (n ≥ 2). Let X m = X (k) (1 ≤ k ≤ m), with m = 1 when k = 1. Then, Y [k] = Ym is referred to as the concomitant of the univariate record value X (k). The concept of concomitants of records was first proposed by Houchens (1984). In the same way, using the sequence of Y1 , Y2 , . . ., one can define the sequences of univariate Y record times and record values and their concomitants. The mathematical theory of univariate records is well developed; see the books of Arnold et al. (1998), Nevzorov (2001), Ahsanullah (2015) and the references therein. Univariate records have been used extensively in different fields such as sport, finance, reliability and hydrology. In the bivariate case, the situation becomes more complex. There are many definitions of bivariate records; see the works of Goldie and Resnick (1989, 1995), Gnedi
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