We consider a complex-valued linear mixture model, under discrete weakly stationary processes. We recover latent components of interest, which have undergone a linear mixing. We study asymptotic properties of a classical unmixing estimator, that is based on simultaneous diagonalization of the covariance matrix and an autocovariance matrix with lag $tau$. Our main contribution is that our asymptotic results can be applied to a large class of processes. In related literature, the processes are typically assumed to have weak correlations. We extend this class and consider the unmixing estimator under stronger dependency structures. In particular, we analyze the asymptotic behavior of the unmixing estimator under both, long- and short-range dependent complex-valued processes. Consequently, our theory covers unmixing estimators that converge slower than the usual $sqrt{T}$ and unmixing estimators that produce non-Gaussian asymptotic distributions. The presented methodology is a powerful prepossessing tool and highly applicable in several fields of statistics. Complex-valued processes are frequently encountered in, for example, biomedical applications and signal processing. In addition, our approach can be applied to model real-valued problems that involve temporally uncorrelated pairs. These are encountered in, for example, applications in finance.
{"title":"Modeling temporally uncorrelated components of complex-valued stationary processes","authors":"Niko Lietzén, L. Viitasaari, Pauliina Ilmonen","doi":"10.15559/21-vmsta190","DOIUrl":"https://doi.org/10.15559/21-vmsta190","url":null,"abstract":"We consider a complex-valued linear mixture model, under discrete weakly stationary processes. We recover latent components of interest, which have undergone a linear mixing. We study asymptotic properties of a classical unmixing estimator, that is based on simultaneous diagonalization of the covariance matrix and an autocovariance matrix with lag $tau$. Our main contribution is that our asymptotic results can be applied to a large class of processes. In related literature, the processes are typically assumed to have weak correlations. We extend this class and consider the unmixing estimator under stronger dependency structures. In particular, we analyze the asymptotic behavior of the unmixing estimator under both, long- and short-range dependent complex-valued processes. Consequently, our theory covers unmixing estimators that converge slower than the usual $sqrt{T}$ and unmixing estimators that produce non-Gaussian asymptotic distributions. The presented methodology is a powerful prepossessing tool and highly applicable in several fields of statistics. Complex-valued processes are frequently encountered in, for example, biomedical applications and signal processing. In addition, our approach can be applied to model real-valued problems that involve temporally uncorrelated pairs. These are encountered in, for example, applications in finance.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"69 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86646558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we investigate a problem of large deviations for continuous Volterra processes under the influence of model disturbances. More precisely, we study the behavior, in the near future after $T$, of a Volterra process driven by a Brownian motion in a case where the Brownian motion is not directly observable, but only a noisy version is observed or some linear functionals of the noisy version are observed. Some examples are discussed in both cases.
{"title":"Pathwise asymptotics for Volterra processes conditioned to a noisy version of the Brownian motion","authors":"B. Pacchiarotti","doi":"10.15559/20-VMSTA149","DOIUrl":"https://doi.org/10.15559/20-VMSTA149","url":null,"abstract":"In this paper we investigate a problem of large deviations for continuous Volterra processes under the influence of model disturbances. More precisely, we study the behavior, in the near future after $T$, of a Volterra process driven by a Brownian motion in a case where the Brownian motion is not directly observable, but only a noisy version is observed or some linear functionals of the noisy version are observed. Some examples are discussed in both cases.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"6 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75551535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the existence of an optimal hedging strategy for the shortfall risk measure in the game options setup. We consider the continuous time Black--Scholes (BS) model. Our first result says that in the case where the game contingent claim (GCC) can be exercised only on a finite set of times, there exists an optimal strategy. Our second and main result is an example which demonstrates that for the case where the GCC can be stopped on the all time interval, optimal portfolio strategies need not always exist.
{"title":"On shortfall risk minimization for game options","authors":"Y. Dolinsky","doi":"10.15559/20-vmsta164","DOIUrl":"https://doi.org/10.15559/20-vmsta164","url":null,"abstract":"In this paper we study the existence of an optimal hedging strategy for the shortfall risk measure in the game options setup. We consider the continuous time Black--Scholes (BS) model. Our first result says that in the case where the game contingent claim (GCC) can be exercised only on a finite set of times, there exists an optimal strategy. Our second and main result is an example which demonstrates that for the case where the GCC can be stopped on the all time interval, optimal portfolio strategies need not always exist.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"19 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88389374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tests for proportional hazards assumption concerning specified covariates or groups of covariates are proposed. The class of alternatives is wide: log-hazard rates under different values of covariates may cross, approach, go away. The data may be right censored. The limit distribution of the test statistic is derived. Power of the test against approaching alternatives is given. Real data examples are considered.
{"title":"Testing proportional hazards for specified covariates","authors":"Vilijandas Bagdonavivcius, Ruta Levulien.e","doi":"10.15559/19-VMSTA129","DOIUrl":"https://doi.org/10.15559/19-VMSTA129","url":null,"abstract":"Tests for proportional hazards assumption concerning specified covariates or groups of covariates are proposed. The class of alternatives is wide: log-hazard rates under different values of covariates may cross, approach, go away. The data may be right censored. The limit distribution of the test statistic is derived. Power of the test against approaching alternatives is given. Real data examples are considered.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"47 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80867849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Taylor's power law states that the variance function decays as a power law. It is observed for population densities of species in ecology. For random networks another power law, that is, the power law degree distribution is widely studied. In this paper the original Taylor's power law is considered for random networks. A precise mathematical proof is presented that Taylor's power law is asymptotically true for the $N$-stars network evolution model.
{"title":"Taylor’s power law for the N-stars network evolution model","authors":"I. Fazekas, Csaba Noszály, Noémi Uzonyi","doi":"10.15559/19-VMSTA137","DOIUrl":"https://doi.org/10.15559/19-VMSTA137","url":null,"abstract":"Taylor's power law states that the variance function decays as a power law. It is observed for population densities of species in ecology. For random networks another power law, that is, the power law degree distribution is widely studied. In this paper the original Taylor's power law is considered for random networks. A precise mathematical proof is presented that Taylor's power law is asymptotically true for the $N$-stars network evolution model.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"37 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88828431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main subject of the study in this paper is the simultaneous renewal time for two time-inhomogeneous Markov chains which start with arbitrary initial distributions. By a simultaneous renewal we mean the first time of joint hitting the specific set $C$ by both processes. Under the condition of existence a dominating sequence for both renewal sequences generated by the chains and non-lattice condition for renewal probabilities an upper bound for the expectation of the simultaneous renewal time is obtained.
{"title":"On estimation of expectation of simultaneous renewal time of time-inhomogeneous Markov chains using dominating sequence","authors":"V. Golomoziy","doi":"10.15559/19-VMSTA138","DOIUrl":"https://doi.org/10.15559/19-VMSTA138","url":null,"abstract":"The main subject of the study in this paper is the simultaneous renewal time for two time-inhomogeneous Markov chains which start with arbitrary initial distributions. By a simultaneous renewal we mean the first time of joint hitting the specific set $C$ by both processes. Under the condition of existence a dominating sequence for both renewal sequences generated by the chains and non-lattice condition for renewal probabilities an upper bound for the expectation of the simultaneous renewal time is obtained.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"64 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73513471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein--Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.
{"title":"Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator","authors":"M. Zili, Eya Zougar","doi":"10.15559/19-VMSTA139","DOIUrl":"https://doi.org/10.15559/19-VMSTA139","url":null,"abstract":"We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein--Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"32 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73977026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a mixture with varying concentrations in which each component is described by a nonlinear regression model. A modified least squares estimator is used to estimate the regressions parameters. Asymptotic normality of the derived estimators is demonstrated. This result is applied to confidence sets construction. Performance of the confidence sets is assessed by simulations.
{"title":"Asymptotic normality of modified LS estimator for mixture of nonlinear regressions","authors":"V. Miroshnichenko, R. Maiboroda","doi":"10.15559/20-vmsta167","DOIUrl":"https://doi.org/10.15559/20-vmsta167","url":null,"abstract":"We consider a mixture with varying concentrations in which each component is described by a nonlinear regression model. A modified least squares estimator is used to estimate the regressions parameters. Asymptotic normality of the derived estimators is demonstrated. This result is applied to confidence sets construction. Performance of the confidence sets is assessed by simulations.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"24 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86952933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define power variation estimators for the drift parameter of the stochastic heat equation with the fractional Laplacian and an additive Gaussian noise which is white in time and white or correlated in space. We prove that these estimators are consistent and asymptotically normal and we derive their rate of convergence under the Wasserstein metric.
{"title":"Estimation of the drift parameter for the fractional stochastic heat equation via power variation","authors":"Z. Khalil, C. Tudor","doi":"10.15559/19-VMSTA141","DOIUrl":"https://doi.org/10.15559/19-VMSTA141","url":null,"abstract":"We define power variation estimators for the drift parameter of the stochastic heat equation with the fractional Laplacian and an additive Gaussian noise which is white in time and white or correlated in space. We prove that these estimators are consistent and asymptotically normal and we derive their rate of convergence under the Wasserstein metric.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"13 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89049992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let ${L(t),tgeq 0}$ be a L'{e}vy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and L'{e}vy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic L'{e}vy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the L'{e}vy measure of the subordinated process $Z(t)$ is a shifted L'{e}vy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding L'{e}vy measures are equal.
{"title":"Logarithmic Lévy process directed by Poisson subordinator","authors":"Penka Mayster, Assen Tchorbadjieff","doi":"10.15559/19-VMSTA142","DOIUrl":"https://doi.org/10.15559/19-VMSTA142","url":null,"abstract":"Let ${L(t),tgeq 0}$ be a L'{e}vy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and L'{e}vy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic L'{e}vy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the L'{e}vy measure of the subordinated process $Z(t)$ is a shifted L'{e}vy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding L'{e}vy measures are equal.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"32 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80373424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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