Relative asymptotic equivalence of dynamic equations on time scales

This paper aims to study the relative equivalence of the solutions of the following dynamic equations $y^{\Delta }(t)=A(t)y(t)$ y Δ ( t ) = A ( t ) y ( t ) and $x^{\Delta }(t)=A(t)x(t)+f(t,x(t))$ x Δ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) in the sense that if $y(t)$ y ( t ) is a given solution of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions $x(t)$ x ( t ) for the perturbed system such that $\Vert y(t)-x(t) \Vert =o( \Vert y(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ y ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ , and conversely, given a solution $x(t)$ x ( t ) of the perturbed system, we give sufficient conditions for the existence of a family of solutions $y(t)$ y ( t ) for the unperturbed system, and such that $\Vert y(t)-x(t) \Vert =o( \Vert x(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ x ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ ; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales.

Orthogonal decomposition of the sum-symmetry model for square contingency tables with ordinal categories: Use of the exponential sum-symmetry model

Summary In the existing decomposition theorem, the sum-symmetry model holds if and only if both the exponential sum-symmetry and global symmetry models hold. However, this decomposition theorem does not satisfy the asymptotic equivalence for the test statistic. To address the aforementioned gap, this study establishes a decomposition theorem in which the sum-symmetry model holds if and only if both the exponential sum-symmetry and weighted global-sum-symmetry models hold. The proposed decomposition theorem satisfies the asymptotic equivalence for the test statistic. We demonstrate the advantages of the proposed decomposition theorem by applying it to datasets comprising real data and artificial data.

Orthogonal decomposition of the sum-symmetry model using the two-parameters sum-symmetry model for ordinal square contingency tables

Summary Studies have been carried out on decomposing a model with symmetric structure using a model with asymmetric structure. In the existing decomposition theorem, the sum-symmetry model holds if and only if all of the two-parameters sum-symmetry, global symmetry and concordancediscordance models hold. However, this existing decomposition theorem does not satisfy the asymptotic equivalence for the test statistic, namely that the value of the likelihood ratio chi-squared statistic of the sum-symmetry model is asymptotically equivalent to the sum of those of the decomposed models. To address this issue, this study introduces a new decomposition theorem in which the sum-symmetry model holds if and only if all of the two-parameters sum-symmetry, global symmetry and weighted global-sum-symmetry models hold. The proposed decomposition theorem satisfies the asymptotic equivalence for the test statistic—the value of the likelihood ratio chi-squared statistic of the sum-symmetry model is asymptotically equivalent to the sum of those of the two-parameters sum-symmetry, global symmetry and weighted global-sum-symmetry models.

Divisible Statistics and Their Partial Sum Processes: Asymptotic Properties and Applications

<p><b>Divisible statistics have been widely used in many areas of statistical analysis. For example, Pearson's Chi-square statistic and the log-likelihood ratio statistic are frequently used in goodness of fit (GOF) and categorical analysis; the maximum likelihood (ML) estimators of the Shannon's and Simpson's diversity indices are often used as measure of diversity; and the spectral statistic plays a key role in the theory of large number of rare events. In the classical multinomial model, where the number of disjoint events N and their probabilities are all fixed, limit distributions of many divisible statistics have gradually been established. However, most of the results are based on the asymptotic equivalence of these statistics to Pearson's Chi-square statistic and the known limit distribution of the latter. In fact, with deeper analysis, one can conclude that the key point is not the asymptotic behavior of the Chi-square statistic, but that of the normalized frequencies. Based on the asymptotic normality of the normalized frequencies in the classical model, a unified approach to the limit theorems of more general divisible statistics can be established, of which the case of the Chi-square statistic is simply a natural corollary.</b></p> <p>In many applications, however, the classical multinomial model is not appropriate, and an extension to new models becomes necessary. This new type of model, called "non-classical" multinomial models, considers the case when N increases and the {Pni} change as sample size n increases. As we will see, in these non-classical models, both the asymptotic normality of the normalized frequencies and the asymptotic equivalence of many divisible statistics to the Chi-square statistic are lost, and the limit theorems established in classical model are no longer valid in non-classical models.</p> <p>The extension to non-classical models not only met the demands of many real world applications, but also opened a new research area in statistical analysis, which has not been thoroughly investigated so far. Although some results on the limit distributions of the divisible statistics in non-classical models have been acquired, e.g., Holst (1972); Morris (1975); Ivchenko and Levin (1976); Ivchenko and Medvedev (1979), they are far from complete. Though not yet attracting much attention by many applied statisticians, another advanced approach, introduced by Khmaladze (1984), makes use of modern martingale theory to establish functional limit theorems of the partial sum processes of divisible statistics successfully. In the main part of this thesis, we show that this martingale approach can be extended to more general situations where both Gaussian and Poissonian frequencies exist, and further discuss the properties and applications of the limiting processes, especially in constructing distribution-free statistics.</p> <p>The last part of the thesis is about the statistical analysis of large number of rare events (LNRE), which is an important class of non-classical multinomial models and presented in numerous applications. In LNRE models, most of the frequencies are very small and it is not immediately clear how consistent and reliable inference can be achieved. Based on the definitions and key concepts firstly introduced by Khmaladze (1988), we discuss a particular model with the context of diversity of questionnaires. The advanced statistical techniques such as large deviation, contiguity and Edgeworth expansion used in establishing limit theorems underpin the potential of LNRE theory to become a fruitful research area in future.</p>

Divisible Statistics and Their Partial Sum Processes: Asymptotic Properties and Applications

<p><b>Divisible statistics have been widely used in many areas of statistical analysis. For example, Pearson's Chi-square statistic and the log-likelihood ratio statistic are frequently used in goodness of fit (GOF) and categorical analysis; the maximum likelihood (ML) estimators of the Shannon's and Simpson's diversity indices are often used as measure of diversity; and the spectral statistic plays a key role in the theory of large number of rare events. In the classical multinomial model, where the number of disjoint events N and their probabilities are all fixed, limit distributions of many divisible statistics have gradually been established. However, most of the results are based on the asymptotic equivalence of these statistics to Pearson's Chi-square statistic and the known limit distribution of the latter. In fact, with deeper analysis, one can conclude that the key point is not the asymptotic behavior of the Chi-square statistic, but that of the normalized frequencies. Based on the asymptotic normality of the normalized frequencies in the classical model, a unified approach to the limit theorems of more general divisible statistics can be established, of which the case of the Chi-square statistic is simply a natural corollary.</b></p> <p>In many applications, however, the classical multinomial model is not appropriate, and an extension to new models becomes necessary. This new type of model, called "non-classical" multinomial models, considers the case when N increases and the {Pni} change as sample size n increases. As we will see, in these non-classical models, both the asymptotic normality of the normalized frequencies and the asymptotic equivalence of many divisible statistics to the Chi-square statistic are lost, and the limit theorems established in classical model are no longer valid in non-classical models.</p> <p>The extension to non-classical models not only met the demands of many real world applications, but also opened a new research area in statistical analysis, which has not been thoroughly investigated so far. Although some results on the limit distributions of the divisible statistics in non-classical models have been acquired, e.g., Holst (1972); Morris (1975); Ivchenko and Levin (1976); Ivchenko and Medvedev (1979), they are far from complete. Though not yet attracting much attention by many applied statisticians, another advanced approach, introduced by Khmaladze (1984), makes use of modern martingale theory to establish functional limit theorems of the partial sum processes of divisible statistics successfully. In the main part of this thesis, we show that this martingale approach can be extended to more general situations where both Gaussian and Poissonian frequencies exist, and further discuss the properties and applications of the limiting processes, especially in constructing distribution-free statistics.</p> <p>The last part of the thesis is about the statistical analysis of large number of rare events (LNRE), which is an important class of non-classical multinomial models and presented in numerous applications. In LNRE models, most of the frequencies are very small and it is not immediately clear how consistent and reliable inference can be achieved. Based on the definitions and key concepts firstly introduced by Khmaladze (1988), we discuss a particular model with the context of diversity of questionnaires. The advanced statistical techniques such as large deviation, contiguity and Edgeworth expansion used in establishing limit theorems underpin the potential of LNRE theory to become a fruitful research area in future.</p>

Multiscale Convergence of the Inverse Problem for Chemotaxis in the Bayesian Setting

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe the organism’s movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller–Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms’ population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions.