Nikolai Kolev - Academia.edu (original) (raw)
Papers by Nikolai Kolev
In this note we obtain reliability function of two-component system under the Marshall-Olkin fail... more In this note we obtain reliability function of two-component system under the Marshall-Olkin failure model in terms of Laplace transform. The problem of its sensitivity to the shape of the system components repair times is investigated as well.
is the total claim amount. In the classical theory it is assumed that (i) N and (Y1, Y2, . . .) a... more is the total claim amount. In the classical theory it is assumed that (i) N and (Y1, Y2, . . .) are independent random variables (r.v.’s); (ii) Y1, Y2, . . . are independent and (iii) Y1, Y2, . . . have the same distribution. The assumptions (ii) and (iii) of mutual independence between the components of the sum is very convenient, mainly because the mathematics is easier. In many situations, the individual claims (risks) are dependent since they are influenced by the same economic environment. In this study we relax the condition (ii) supposing that the claim amounts are dependent random variables. The observed claims are usually correlated because they contain a common random factor. This kind of correlation can be presented by the following model: For i = 1, 2, . . ., define Yi = J(Ui, V ), where U1, U2, . . . and V are independent r.v.’s, and Ui ∼ F (u), V ∼ G(v), and J(u, v) is some correlated link function. Therefore, the distribution of Yi’s is determined by the link function...
Hence, the information in the joint distribution H(x, y) is decomposed into those of marginal dis... more Hence, the information in the joint distribution H(x, y) is decomposed into those of marginal distributions and that of copula function C(u, v) which captures the dependence structure between X and Y. On the other hand, for any copula function C(u, v) and any univariate continuous distribution functions F (x) and G(y), the function C(F (x), G(y)) is a bivariate distribution function H(x, y) as given by H(x, y) = C(F (x), G(y)), x, y ∈ (−∞, ∞). Consequently, copulas allow one to model the marginal distributions and the dependence structure of multivariate random variable separately. The copula function is therefore a class of multivariate distributions being functionally independent of its marginals.
ASTIN Bulletin, 2019
In this paper we suggest a modeling of joint life insurance pricing via Extended Marshall–Olkin (... more In this paper we suggest a modeling of joint life insurance pricing via Extended Marshall–Olkin (EMO) models and related copulas. These models are based on the combination of two approaches: the absolutely continuous copula approach, where the copula is used to capture dependencies due to environmental factors shared by the two lives, and the classical Marshall–Olkin model, where the association is given by accounting for a fatal event causing the simultaneous death of the two lives. New properties of the EMO model are established and applied to a sample of censored residual lifetimes of couples of insureds extracted from a data set of annuities contracts of a large Canadian life insurance company. Finally, some joint life insurance products are analyzed.
Mathematical Communications, 1999
In this paper we study the blocking time of an unreliable single-server queueing system Geo/GD/1.... more In this paper we study the blocking time of an unreliable single-server queueing system Geo/GD/1. The service can be interrupted upon explicit or implicit breakdowns. For the successful finish of the service we use a special service discipline dividing the pure service time X (assumed to be a random variable with known distribution) in subintervals with deterministically selected time-points 0 = t0 < t1 < . . . < tk < tk+1; tk < X ≤ tk+1, and making a copy at the end of each subinterval (if no breakdowns occur during it) we derive the probability generating function of the blocking time of the server by a customer. As an application, we consider an unreliable system Geo/D/1 and the results is that the expected blocking time is minimized when the timepoints t0, t1, . . . are equidistant. We determine the optimal number of copies and the length of the corresponding interval between two consecutive copies.
Sankhya A, 2020
A new measure of the bivariate asymmetry of a dependence structure between two random variables i... more A new measure of the bivariate asymmetry of a dependence structure between two random variables is introduced based on copula characteristic function. The proposed measure is represented as the discrepancy between the rank-based distance correlation computed over two complementary orderpreserved sets. Generalproperties of the measure are established, as well as an explicit expression for the empirical version. It is shown that the proposed measure is asymptotically equivalent to a fourth-order degenerate V-statistics and that the limit distributions have representations in terms of weighted sums of an independent chi-square random variables. Under dependent random variables, the asymptotic behavior of bivariate distance covariance and variance process is demonstrated. Numerical examples illustrate the properties of the measures.
Brazilian Journal of Probability and Statistics
Following the methodology developed by Hyman and Shashkov (1997), we define a discrete version of... more Following the methodology developed by Hyman and Shashkov (1997), we define a discrete version of gradient vector and associated line integral along arbitrary path connecting two nodes of uniform grid. An exponential representation of joint survival function of bivariate discrete non-negative integer-valued random variables in terms of discrete line integral is established. We apply it to generate a discrete analogue of the Sibuya-type aging property, incorporating many classical and new bivariate discrete models. Several characterizations and closure properties of this class of bivariate discrete distributions are presented.
We discuss our practice related to classical hypothesis testing about unknown parameters of a nor... more We discuss our practice related to classical hypothesis testing about unknown parameters of a normal population offered to undergraduates in the University of São Paulo. We consider the tests for the population mean and variance when the sample size is “large” and “small” as well as the well-known tests comparing the means and variances for independent samples. We suggest an algorithmic approach, which our students appreciate.
Brazilian Journal of Probability and Statistics
We suggest a modification of the classical Marshall-Olkin's bivariate exponential distribution co... more We suggest a modification of the classical Marshall-Olkin's bivariate exponential distribution considering a possibility of a singularity contribution along arbitrary line through the origin. It serves as a base of a new weaker version of the bivariate lack of memory property, which might be both "aging" and "non-aging" depending on the additional inclination parameter. The corresponding copula is obtained and we establish its disagreement with Lancaster's phenomena. Characterizations and properties of the novel bivariate memory-less notion are obtained and its applications are discussed. We characterize associated weak multivariate version. The weak bivariate lack of memory property implies restrictions on the marginal distributions. Starting from pre-specified marginals we propose a procedure to build bivariate distributions possessing a weak bivariate lack of memory property and illustrate it by examples. We complement the methodology with closure properties of the new class. We finish with a discussion and suggest several related problems for future research.
Analytical and Computational Methods in Probability Theory
Journal of Statistical Distributions and Applications
International Journal of Statistics and Probability
We construct and characterize bivariate extreme value distributions with exponential marginals ge... more We construct and characterize bivariate extreme value distributions with exponential marginals generated by the stochastic representation (X1,X2) = (min(T1,T3), min(T2,T3)) where the random variable T3 is independent of random variables T1 and T2 which are assumed to be dependent. A building procedure is suggested when the joint distribution of (T1,T2) is absolutely continuous and Ti's are not necessarily exponentially distributed, i=1,2,3. The Pickands representation of the vector (X1,X2) is computed. We illustrate the general relations by examples.
Http Dx Doi Org 10 1080 03610929908832417, Jun 27, 2007
Run and frequency quotas studied by Balasubramanian et al. (1993) for Markov correlated Bernoulli... more Run and frequency quotas studied by Balasubramanian et al. (1993) for Markov correlated Bernoulli trials are generalized to the time-homogeneous multi-state Markov chain {Xnn≧0} with state labeled as &amp;quot;0&amp;quot; (Success) and &amp;quot;f&amp;quot; (failure), f=1,2,…
Hence, the information in the joint distribution H(x, y) is decomposed into those of marginal dis... more Hence, the information in the joint distribution H(x, y) is decomposed into those of marginal distributions and that of copula function C(u, v) which captures the dependence structure between X and Y. On the other hand, for any copula function C(u, v) and any univariate continuous distribution functions F (x) and G(y), the function C(F (x), G(y)) is a bivariate distribution function H(x, y) as given by H(x, y) = C(F (x), G(y)), x, y ∈ (−∞, ∞). Consequently, copulas allow one to model the marginal distributions and the dependence structure of multivariate random variable separately. The copula function is therefore a class of multivariate distributions being functionally independent of its marginals.
AIP Conference Proceedings, 2008
latent multinomial variable, probability generating function. Abstract. Let us consider a collect... more latent multinomial variable, probability generating function. Abstract. Let us consider a collective insurance contract in some fixed time period (0,T). Let N denote the number of claims in (0,T) and Y1,Y2,...YN the correspond- ing claims. Then SN = PN i=1 Yi is the total claim amount. In the classical theory it is assumed that (i) N and (Y1,Y2,...) are independent
Springer Proceedings in Mathematics & Statistics, 2015
In this note we obtain reliability function of two-component system under the Marshall-Olkin fail... more In this note we obtain reliability function of two-component system under the Marshall-Olkin failure model in terms of Laplace transform. The problem of its sensitivity to the shape of the system components repair times is investigated as well.
is the total claim amount. In the classical theory it is assumed that (i) N and (Y1, Y2, . . .) a... more is the total claim amount. In the classical theory it is assumed that (i) N and (Y1, Y2, . . .) are independent random variables (r.v.’s); (ii) Y1, Y2, . . . are independent and (iii) Y1, Y2, . . . have the same distribution. The assumptions (ii) and (iii) of mutual independence between the components of the sum is very convenient, mainly because the mathematics is easier. In many situations, the individual claims (risks) are dependent since they are influenced by the same economic environment. In this study we relax the condition (ii) supposing that the claim amounts are dependent random variables. The observed claims are usually correlated because they contain a common random factor. This kind of correlation can be presented by the following model: For i = 1, 2, . . ., define Yi = J(Ui, V ), where U1, U2, . . . and V are independent r.v.’s, and Ui ∼ F (u), V ∼ G(v), and J(u, v) is some correlated link function. Therefore, the distribution of Yi’s is determined by the link function...
Hence, the information in the joint distribution H(x, y) is decomposed into those of marginal dis... more Hence, the information in the joint distribution H(x, y) is decomposed into those of marginal distributions and that of copula function C(u, v) which captures the dependence structure between X and Y. On the other hand, for any copula function C(u, v) and any univariate continuous distribution functions F (x) and G(y), the function C(F (x), G(y)) is a bivariate distribution function H(x, y) as given by H(x, y) = C(F (x), G(y)), x, y ∈ (−∞, ∞). Consequently, copulas allow one to model the marginal distributions and the dependence structure of multivariate random variable separately. The copula function is therefore a class of multivariate distributions being functionally independent of its marginals.
ASTIN Bulletin, 2019
In this paper we suggest a modeling of joint life insurance pricing via Extended Marshall–Olkin (... more In this paper we suggest a modeling of joint life insurance pricing via Extended Marshall–Olkin (EMO) models and related copulas. These models are based on the combination of two approaches: the absolutely continuous copula approach, where the copula is used to capture dependencies due to environmental factors shared by the two lives, and the classical Marshall–Olkin model, where the association is given by accounting for a fatal event causing the simultaneous death of the two lives. New properties of the EMO model are established and applied to a sample of censored residual lifetimes of couples of insureds extracted from a data set of annuities contracts of a large Canadian life insurance company. Finally, some joint life insurance products are analyzed.
Mathematical Communications, 1999
In this paper we study the blocking time of an unreliable single-server queueing system Geo/GD/1.... more In this paper we study the blocking time of an unreliable single-server queueing system Geo/GD/1. The service can be interrupted upon explicit or implicit breakdowns. For the successful finish of the service we use a special service discipline dividing the pure service time X (assumed to be a random variable with known distribution) in subintervals with deterministically selected time-points 0 = t0 < t1 < . . . < tk < tk+1; tk < X ≤ tk+1, and making a copy at the end of each subinterval (if no breakdowns occur during it) we derive the probability generating function of the blocking time of the server by a customer. As an application, we consider an unreliable system Geo/D/1 and the results is that the expected blocking time is minimized when the timepoints t0, t1, . . . are equidistant. We determine the optimal number of copies and the length of the corresponding interval between two consecutive copies.
Sankhya A, 2020
A new measure of the bivariate asymmetry of a dependence structure between two random variables i... more A new measure of the bivariate asymmetry of a dependence structure between two random variables is introduced based on copula characteristic function. The proposed measure is represented as the discrepancy between the rank-based distance correlation computed over two complementary orderpreserved sets. Generalproperties of the measure are established, as well as an explicit expression for the empirical version. It is shown that the proposed measure is asymptotically equivalent to a fourth-order degenerate V-statistics and that the limit distributions have representations in terms of weighted sums of an independent chi-square random variables. Under dependent random variables, the asymptotic behavior of bivariate distance covariance and variance process is demonstrated. Numerical examples illustrate the properties of the measures.
Brazilian Journal of Probability and Statistics
Following the methodology developed by Hyman and Shashkov (1997), we define a discrete version of... more Following the methodology developed by Hyman and Shashkov (1997), we define a discrete version of gradient vector and associated line integral along arbitrary path connecting two nodes of uniform grid. An exponential representation of joint survival function of bivariate discrete non-negative integer-valued random variables in terms of discrete line integral is established. We apply it to generate a discrete analogue of the Sibuya-type aging property, incorporating many classical and new bivariate discrete models. Several characterizations and closure properties of this class of bivariate discrete distributions are presented.
We discuss our practice related to classical hypothesis testing about unknown parameters of a nor... more We discuss our practice related to classical hypothesis testing about unknown parameters of a normal population offered to undergraduates in the University of São Paulo. We consider the tests for the population mean and variance when the sample size is “large” and “small” as well as the well-known tests comparing the means and variances for independent samples. We suggest an algorithmic approach, which our students appreciate.
Brazilian Journal of Probability and Statistics
We suggest a modification of the classical Marshall-Olkin's bivariate exponential distribution co... more We suggest a modification of the classical Marshall-Olkin's bivariate exponential distribution considering a possibility of a singularity contribution along arbitrary line through the origin. It serves as a base of a new weaker version of the bivariate lack of memory property, which might be both "aging" and "non-aging" depending on the additional inclination parameter. The corresponding copula is obtained and we establish its disagreement with Lancaster's phenomena. Characterizations and properties of the novel bivariate memory-less notion are obtained and its applications are discussed. We characterize associated weak multivariate version. The weak bivariate lack of memory property implies restrictions on the marginal distributions. Starting from pre-specified marginals we propose a procedure to build bivariate distributions possessing a weak bivariate lack of memory property and illustrate it by examples. We complement the methodology with closure properties of the new class. We finish with a discussion and suggest several related problems for future research.
Analytical and Computational Methods in Probability Theory
Journal of Statistical Distributions and Applications
International Journal of Statistics and Probability
We construct and characterize bivariate extreme value distributions with exponential marginals ge... more We construct and characterize bivariate extreme value distributions with exponential marginals generated by the stochastic representation (X1,X2) = (min(T1,T3), min(T2,T3)) where the random variable T3 is independent of random variables T1 and T2 which are assumed to be dependent. A building procedure is suggested when the joint distribution of (T1,T2) is absolutely continuous and Ti's are not necessarily exponentially distributed, i=1,2,3. The Pickands representation of the vector (X1,X2) is computed. We illustrate the general relations by examples.
Http Dx Doi Org 10 1080 03610929908832417, Jun 27, 2007
Run and frequency quotas studied by Balasubramanian et al. (1993) for Markov correlated Bernoulli... more Run and frequency quotas studied by Balasubramanian et al. (1993) for Markov correlated Bernoulli trials are generalized to the time-homogeneous multi-state Markov chain {Xnn≧0} with state labeled as &amp;quot;0&amp;quot; (Success) and &amp;quot;f&amp;quot; (failure), f=1,2,…
Hence, the information in the joint distribution H(x, y) is decomposed into those of marginal dis... more Hence, the information in the joint distribution H(x, y) is decomposed into those of marginal distributions and that of copula function C(u, v) which captures the dependence structure between X and Y. On the other hand, for any copula function C(u, v) and any univariate continuous distribution functions F (x) and G(y), the function C(F (x), G(y)) is a bivariate distribution function H(x, y) as given by H(x, y) = C(F (x), G(y)), x, y ∈ (−∞, ∞). Consequently, copulas allow one to model the marginal distributions and the dependence structure of multivariate random variable separately. The copula function is therefore a class of multivariate distributions being functionally independent of its marginals.
AIP Conference Proceedings, 2008
latent multinomial variable, probability generating function. Abstract. Let us consider a collect... more latent multinomial variable, probability generating function. Abstract. Let us consider a collective insurance contract in some fixed time period (0,T). Let N denote the number of claims in (0,T) and Y1,Y2,...YN the correspond- ing claims. Then SN = PN i=1 Yi is the total claim amount. In the classical theory it is assumed that (i) N and (Y1,Y2,...) are independent
Springer Proceedings in Mathematics & Statistics, 2015