Robert Mnatsakanov - Academia.edu (original) (raw)

Papers by Robert Mnatsakanov

Journal of mathematical sciences, Jul 2, 2024

We consider estimation of the structural distribution function of the cell probabilities of a mul... more We consider estimation of the structural distribution function of the cell probabilities of a multinomial sample in situations where the number of cells is large. We review the performance of the natural estimator, an estimator based on grouping the cells and a kernel type estimator. Inconsistency of the natural estimator and weak consistency of the other two estimators is derived by Poissonization and other, new, technical devices.

Theory of Probability and Its Applications, 1988

arXiv (Cornell University), Sep 25, 2018

Moment methods to reconstruct images from their Radon transforms are both natural and useful. The... more Moment methods to reconstruct images from their Radon transforms are both natural and useful. They can be used to suppress noise or other spurious effects and can lead to highly efficient reconstructions from relatively few projections. We establish a modified Radon transform (MRT) via convolution with a mollifier and obtain its inversion formula. The relationship of the moments of the Radon transform and the moments of its modified Radon transform is derived and MRT data is used to provide a uniform approximation to the original density function. The reconstruction algorithm is implemented, and a simple density function is reconstructed from moments of its modified Radon transform. Numerical convergence of this reconstruction is shown to agree with the derived theoretical results.

Theory of Probability and Its Applications, Sep 1, 1986

Journal of Computational and Applied Mathematics, Nov 1, 2018

We propose three modified approximations of the the ruin probability and its inverse function usi... more We propose three modified approximations of the the ruin probability and its inverse function using the inversion of the scaled values of Laplace transform suggested by Mnatsakanov et al. (2015). We consider the classical risk model for the evaluation of the probability of ultimate ruin and its inverse function, The problem of evaluating numerically the tail Value at Risk of an insurance portfolio will be discussed briefly, Performances of the proposed constructions are demonstrated via graphs and tables using several examples. Adetokunbo Ibukun, FADAHUNSI and Robert MNATSAKANOV (2018) Recovery of ruin probability and Value at Risk from the scaled Laplace transform inversion May 19, 2018 2 / 34 where, c = lnb, for some b > 1.

arXiv (Cornell University), Nov 8, 2015

A modified Radon transform for noisy data is introduced and its inversion formula is established.... more A modified Radon transform for noisy data is introduced and its inversion formula is established. The problem of recovering the multivariate probability density function f from the moments of its modified Radon transform Rf is considered.

CRC Press eBooks, Jul 10, 2023

Theory of Probability and Its Applications, 1988

Let * (v*, *) be any function satisfying the conditions of Theorem 1. Then by Jensen's inequality... more Let * (v*, *) be any function satisfying the conditions of Theorem 1. Then by Jensen's inequality ETp(17(u*)) ETET{p(17(*)) *} => ETp (ET{ 17(r*) *}) w(ai(dP), a,_,(*))= (.O(Oi((I)e), Ol_i((I)e)), i=0, 1, and therefore to prove 3 it suffices to establish that to(Ceo, eel) decreases in each argument. Since-1 ci> > c_i, i=0, 1, and p is convex, it follows from (12) that r(c, cx_) increases in each argument. Since by (4) and (8) c=p(1-ai)/p_iai, i=0, 1, we find that to decreases in each argument. As a corollary to Theorem 1, we give an assertion on the "asymptotic admissibility" of the test .W e note that a similar result in a nonasymptotic setting in [4] only concerns likelihood ratio tests "stopped on the boundary". COROLLARY. Under the conditions of Theorem there exist no tests dp* (u*, *), such that P(u* u)= for at least one if or which li-e_,oOti(dP*)<-ai with strict inequality for at least one i-0, 1. Example. We consider the problem of testing two hypotheses/z _-< 0 and/x > 0 about the mean value /z of the normal distribution from the sample X,..., X, (the variance of the observations is assumed known and equal to 1). Suppose that the a priori distribution of the parameter/z is also normal with mean 0 and variance e 2. Then conditions (C1)-(C6) are evidently satisfied and by Theorem the asymptotically optimal (as e->0) test =(u,) is based on the statistic T(X,...,X,)=(ff-O)/x/n+e-2 and has the form u= min{n" T(-A(al), A(a0))}, =x{T>-A(ao)}, where A(p) is the (1-p)-quantile of the standard normal distribution.

Methodology and Computing in Applied Probability

Transactions of A. Razmadze Mathematical Institute, 2020

Consistent estimators of the mixing distribution in Poisson mixture models are constructed for bo... more Consistent estimators of the mixing distribution in Poisson mixture models are constructed for both the right censored and the uncensored case. The estimators are based on a kind of Laplace inversion via factorial moments. The rate of convergence of the mean integrated squared error of these estimators is (log n/ log log n) 2. It is also shown that there do not exist estimators for which this rate is better.

Statistics, 2012

An unknown moment-determinate cumulative distribution function or its density function can be rec... more An unknown moment-determinate cumulative distribution function or its density function can be recovered from corresponding moments and estimated from the empirical moments. This method of estimating an unknown density is natural in certain inverse estimation models like multiplicative censoring or biased sampling when the moments of unobserved distribution can be estimated via the transformed moments of the observed distribution. In this paper, we introduce a new nonparametric estimator of a probability density function defined on the positive real line, motivated by the above. Some fundamental properties of proposed estimator are studied. The comparison with traditional kernel density estimator is discussed.

Georgian Mathematical Journal, 1996

The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for ... more The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.

We present the first results from a new population study of the radio pulsar content in globular ... more We present the first results from a new population study of the radio pulsar content in globular clusters. Our goal is to develop a set of publicly available tools to constrain the underlying population distribution functions based on the sample of 140 pulsars in 26 clusters. In this work, we will present our main statistical techniques and apply them to

Applied Mathematics and Computation, 2015

The problem of recovering the ruin probability in the classical risk model based on the scaled La... more The problem of recovering the ruin probability in the classical risk model based on the scaled Laplace transform inversion is studied. It is shown how to overcome the problem of evaluating the ruin probability at large values of an initial surplus process. Comparisons of proposed approximations with the ones based on the Laplace transform inversions using a fixed Talbot algorithm as well as on the ones using the Trefethen-Weideman-Schmelzer and maximum entropy methods are presented via a simulation study.

Journal of Computational and Applied Mathematics

Abstract The problems of recovering a multivariate function f from the scaled values of its Lapla... more Abstract The problems of recovering a multivariate function f from the scaled values of its Laplace and Radon transforms are studied, and two novel methods for approximating and estimating the unknown function are proposed. Moreover, using the empirical counterparts of the Laplace transform of the underlying function, a new estimate of the Radon transform itself is obtained. Under smoothed conditions on the underlying function the uniform convergence of the proposed constructions are established, and their accuracy is illustrated graphically with several simple examples.

Statistics & Probability Letters, 2018

The problem of recovering quantiles and quantile density functions of a positive random variable ... more The problem of recovering quantiles and quantile density functions of a positive random variable via the values of frequency moments is studied. The uniform upper bounds of the proposed approximations are derived. Several simple examples and corresponding plots illustrate the behavior of the recovered approximations. Some applications of the constructions are discussed as well. Namely, using the empirical counterparts of the constructions yield the estimates of the quantiles, and the quantile density functions. By means of simulations, the average errors in terms of L 2-norm are evaluated to justify the consistency of the estimate of the quantile density function. As an application of the constructions, the question of estimating the so-called expected shortfall measure in risk models is also studied.

Journal of Mathematical Analysis and Applications, 2020

Abstract This article deals with characterizations of a function in terms of its circular mean Ra... more Abstract This article deals with characterizations of a function in terms of its circular mean Radon transform. We present a new approach (the consistency method) showing how to describe the class of real-valued, planar functions f which have the given circular mean Radon transform M f over circles centered on the unit circle. Also, expressions are derived for the geometric moments of an unknown function in terms of its circular mean Radon transform.

Transactions of A. Razmadze Mathematical Institute, 2018

New nonparametric procedure for estimating the probability density function of a positive random ... more New nonparametric procedure for estimating the probability density function of a positive random variable is suggested. Asymptotic expressions of the bias term and Mean Squared Error are derived. By means of graphical illustrations and evaluating the Average of L 2-errors we conducted comparisons of the finite sample performance of proposed estimate with the one based on kernel density method. c

Journal of mathematical sciences, Jul 2, 2024

We consider estimation of the structural distribution function of the cell probabilities of a mul... more We consider estimation of the structural distribution function of the cell probabilities of a multinomial sample in situations where the number of cells is large. We review the performance of the natural estimator, an estimator based on grouping the cells and a kernel type estimator. Inconsistency of the natural estimator and weak consistency of the other two estimators is derived by Poissonization and other, new, technical devices.

Theory of Probability and Its Applications, 1988

arXiv (Cornell University), Sep 25, 2018

Moment methods to reconstruct images from their Radon transforms are both natural and useful. The... more Moment methods to reconstruct images from their Radon transforms are both natural and useful. They can be used to suppress noise or other spurious effects and can lead to highly efficient reconstructions from relatively few projections. We establish a modified Radon transform (MRT) via convolution with a mollifier and obtain its inversion formula. The relationship of the moments of the Radon transform and the moments of its modified Radon transform is derived and MRT data is used to provide a uniform approximation to the original density function. The reconstruction algorithm is implemented, and a simple density function is reconstructed from moments of its modified Radon transform. Numerical convergence of this reconstruction is shown to agree with the derived theoretical results.

Theory of Probability and Its Applications, Sep 1, 1986

Journal of Computational and Applied Mathematics, Nov 1, 2018

We propose three modified approximations of the the ruin probability and its inverse function usi... more We propose three modified approximations of the the ruin probability and its inverse function using the inversion of the scaled values of Laplace transform suggested by Mnatsakanov et al. (2015). We consider the classical risk model for the evaluation of the probability of ultimate ruin and its inverse function, The problem of evaluating numerically the tail Value at Risk of an insurance portfolio will be discussed briefly, Performances of the proposed constructions are demonstrated via graphs and tables using several examples. Adetokunbo Ibukun, FADAHUNSI and Robert MNATSAKANOV (2018) Recovery of ruin probability and Value at Risk from the scaled Laplace transform inversion May 19, 2018 2 / 34 where, c = lnb, for some b > 1.

arXiv (Cornell University), Nov 8, 2015

A modified Radon transform for noisy data is introduced and its inversion formula is established.... more A modified Radon transform for noisy data is introduced and its inversion formula is established. The problem of recovering the multivariate probability density function f from the moments of its modified Radon transform Rf is considered.

CRC Press eBooks, Jul 10, 2023

Theory of Probability and Its Applications, 1988

Let * (v*, *) be any function satisfying the conditions of Theorem 1. Then by Jensen's inequality... more Let * (v*, *) be any function satisfying the conditions of Theorem 1. Then by Jensen's inequality ETp(17(u*)) ETET{p(17(*)) *} => ETp (ET{ 17(r*) *}) w(ai(dP), a,_,(*))= (.O(Oi((I)e), Ol_i((I)e)), i=0, 1, and therefore to prove 3 it suffices to establish that to(Ceo, eel) decreases in each argument. Since-1 ci> > c_i, i=0, 1, and p is convex, it follows from (12) that r(c, cx_) increases in each argument. Since by (4) and (8) c=p(1-ai)/p_iai, i=0, 1, we find that to decreases in each argument. As a corollary to Theorem 1, we give an assertion on the "asymptotic admissibility" of the test .W e note that a similar result in a nonasymptotic setting in [4] only concerns likelihood ratio tests "stopped on the boundary". COROLLARY. Under the conditions of Theorem there exist no tests dp* (u*, *), such that P(u* u)= for at least one if or which li-e_,oOti(dP*)<-ai with strict inequality for at least one i-0, 1. Example. We consider the problem of testing two hypotheses/z _-< 0 and/x > 0 about the mean value /z of the normal distribution from the sample X,..., X, (the variance of the observations is assumed known and equal to 1). Suppose that the a priori distribution of the parameter/z is also normal with mean 0 and variance e 2. Then conditions (C1)-(C6) are evidently satisfied and by Theorem the asymptotically optimal (as e->0) test =(u,) is based on the statistic T(X,...,X,)=(ff-O)/x/n+e-2 and has the form u= min{n" T(-A(al), A(a0))}, =x{T>-A(ao)}, where A(p) is the (1-p)-quantile of the standard normal distribution.

Methodology and Computing in Applied Probability

Transactions of A. Razmadze Mathematical Institute, 2020

Consistent estimators of the mixing distribution in Poisson mixture models are constructed for bo... more Consistent estimators of the mixing distribution in Poisson mixture models are constructed for both the right censored and the uncensored case. The estimators are based on a kind of Laplace inversion via factorial moments. The rate of convergence of the mean integrated squared error of these estimators is (log n/ log log n) 2. It is also shown that there do not exist estimators for which this rate is better.

Statistics, 2012

An unknown moment-determinate cumulative distribution function or its density function can be rec... more An unknown moment-determinate cumulative distribution function or its density function can be recovered from corresponding moments and estimated from the empirical moments. This method of estimating an unknown density is natural in certain inverse estimation models like multiplicative censoring or biased sampling when the moments of unobserved distribution can be estimated via the transformed moments of the observed distribution. In this paper, we introduce a new nonparametric estimator of a probability density function defined on the positive real line, motivated by the above. Some fundamental properties of proposed estimator are studied. The comparison with traditional kernel density estimator is discussed.

Georgian Mathematical Journal, 1996

The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for ... more The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.

We present the first results from a new population study of the radio pulsar content in globular ... more We present the first results from a new population study of the radio pulsar content in globular clusters. Our goal is to develop a set of publicly available tools to constrain the underlying population distribution functions based on the sample of 140 pulsars in 26 clusters. In this work, we will present our main statistical techniques and apply them to

Applied Mathematics and Computation, 2015

The problem of recovering the ruin probability in the classical risk model based on the scaled La... more The problem of recovering the ruin probability in the classical risk model based on the scaled Laplace transform inversion is studied. It is shown how to overcome the problem of evaluating the ruin probability at large values of an initial surplus process. Comparisons of proposed approximations with the ones based on the Laplace transform inversions using a fixed Talbot algorithm as well as on the ones using the Trefethen-Weideman-Schmelzer and maximum entropy methods are presented via a simulation study.

Journal of Computational and Applied Mathematics

Abstract The problems of recovering a multivariate function f from the scaled values of its Lapla... more Abstract The problems of recovering a multivariate function f from the scaled values of its Laplace and Radon transforms are studied, and two novel methods for approximating and estimating the unknown function are proposed. Moreover, using the empirical counterparts of the Laplace transform of the underlying function, a new estimate of the Radon transform itself is obtained. Under smoothed conditions on the underlying function the uniform convergence of the proposed constructions are established, and their accuracy is illustrated graphically with several simple examples.

Statistics & Probability Letters, 2018

The problem of recovering quantiles and quantile density functions of a positive random variable ... more The problem of recovering quantiles and quantile density functions of a positive random variable via the values of frequency moments is studied. The uniform upper bounds of the proposed approximations are derived. Several simple examples and corresponding plots illustrate the behavior of the recovered approximations. Some applications of the constructions are discussed as well. Namely, using the empirical counterparts of the constructions yield the estimates of the quantiles, and the quantile density functions. By means of simulations, the average errors in terms of L 2-norm are evaluated to justify the consistency of the estimate of the quantile density function. As an application of the constructions, the question of estimating the so-called expected shortfall measure in risk models is also studied.

Journal of Mathematical Analysis and Applications, 2020

Abstract This article deals with characterizations of a function in terms of its circular mean Ra... more Abstract This article deals with characterizations of a function in terms of its circular mean Radon transform. We present a new approach (the consistency method) showing how to describe the class of real-valued, planar functions f which have the given circular mean Radon transform M f over circles centered on the unit circle. Also, expressions are derived for the geometric moments of an unknown function in terms of its circular mean Radon transform.

Transactions of A. Razmadze Mathematical Institute, 2018

New nonparametric procedure for estimating the probability density function of a positive random ... more New nonparametric procedure for estimating the probability density function of a positive random variable is suggested. Asymptotic expressions of the bias term and Mean Squared Error are derived. By means of graphical illustrations and evaluating the Average of L 2-errors we conducted comparisons of the finite sample performance of proposed estimate with the one based on kernel density method. c