Families of Multivariate Distributions Involving “Triangular” Transformations

Filus, Jerzy K.; Arnold, Barry C.

doi:10.1080/03610920802687793

Cited by 15 publications

(6 citation statements)

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“…We begin by reviewing a family of models called triangular transformation models which were introduced in Filus, Filus and Arnold [2] as follows:…”

Section: Triangular Transformation Modelsmentioning

confidence: 99%

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Statistical Inference for distributions with one Poisson conditional

Arnold

Manjunath

2020

Preprint

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It will be recalled that the classical bivariate normal distributions have normal marginals and normal conditionals. It is natural to ask whether a similar phenomenon can be encountered involving Poisson marginals and conditionals. Reference to Arnold, Castillo and Sarabia's (1999) book on conditionally specified models will confirm that Poisson marginals will be encountered, together with both conditionals being of the Poisson form, only in the case in which the variables are independent. Instead, in the present article we will be focusing on bivariate distributions with one marginal and the other family of conditionals being of the Poisson form. Such distributions are called Pseudo-Poisson distributions. We discuss distributional features of such models, explore inferential aspects and include an examples of applications of the Pseudo-Poisson model to sets of over-dispersed data.

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“…We begin by reviewing a family of models called triangular transformation models which were introduced in Filus, Filus and Arnold [2] as follows:…”

Section: Triangular Transformation Modelsmentioning

confidence: 99%

“…Now suppose that we have data of the form X (1) , X (2) , ..., X (n) which are i.i.d. with common distribution (5.1)-(5.2).…”

Section: Moments and Moment Estimatorsmentioning

confidence: 99%

Statistical Inference for distributions with one Poisson conditional

Arnold

Manjunath

2020

Preprint

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“…Returning to the main subject, it is noticed that the sequence of the random vector transformations 1, is the pseudoaffine version of sequence of triangular transformations R T R T (Filus et al, 2010).…”

Section: Examplementioning

confidence: 99%

“…One of the methods employs triangular transformations (J. K. Filus, L. Z. Filus, & Arnold, 2010), as the defining tool and may therefore be more useful in a further statistical analysis and possible simulation studies. This method is described in section two and three.…”

Section: Introductionmentioning

confidence: 99%

An Alternative for Time Series Models

2014

CUBR

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Two methods for construction of new stochastic processes with discrete time are presented. One of the methods employs as the defining tool "triangular (more specifically 'pseudoaffine') transformations" which are extended from the Euclidean R n to infinite dimension space. They transform any well-known discrete time stochastic process into the constructed one. The other more flexible method is the "method of parameter dependence", extended to infinite dimension. Properties of the obtained stochastic processes (by either method) indicate the possibility to apply them for financial analysis, as an alternative for the classical time series models. The advantage of the presented models over the existing ones first of all relies on expected better accuracy. This follows from the fact that the typically held assumption on Markovianity in the existing models can easily be relaxed. The defined processes may incorporate a quite long memory including, among others, the k-Markovian cases for k ≥ 2. Regardless the non-Markovianity of the models they still are tractable in an analytical or numerical way. The stochastic processes defined in this paper provide more flexible and more general tools than the existing time series models for modeling financial problems. Among others, they make it possible to incorporate the influence of environmental (explanatory) random variables on the underlying stochastic models' behavior. These additional features turn out to be describable by the method of parameter dependence. Some suggestions for an associated preliminary statistical analysis are included.

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“…Some similar but different methods of constructing models, under the name "parameter dependence method", can be found in [13] [14]. As it turned out many of the multivariate distributions obtained by that method can also be obtained by the "method of triangular transformations" (especially by pseudoaffine and pseudopower transformations), see for example [15].…”

Section: Introductionmentioning

confidence: 99%

Construction of <i>k</i>-Variate Survival Functions with Emphasis on the Case <i>k</i> = 3

Filus¹

2020

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The purpose of this paper is to present a general universal formula for k-variate survival functions for arbitrary k = 2, 3, •••, given all the univariate marginal survival functions. This universal form of k-variate probability distributions was obtained by means of "dependence functions" named "joiners" in the text. These joiners determine all the involved stochastic dependencies between the underlying random variables. However, in order that the presented formula (the form) represents a legitimate survival function, some necessary and sufficient conditions for the joiners had to be found. Basically, finding those conditions is the main task of this paper. This task was successfully performed for the case k = 2 and the main results for the case k = 3 were formulated as Theorem 1 and Theorem 2 in Section 4. Nevertheless, the hypothetical conditions valid for the general k ≥ 4 case were also formulated in Section 3 as the (very convincing) Hypothesis. As for the sufficient conditions for both the k = 3 and k ≥ 4 cases, the full generality was not achieved since two restrictions were imposed. Firstly, we limited ourselves to the, defined in the text, "continuous cases" (when the corresponding joint density exists and is continuous), and secondly we consider positive stochastic dependencies only. Nevertheless, the class of the k-variate distributions which can be constructed is very wide. The presented method of construction by means of joiners can be considered competitive to the copula methodology. As it is suggested in the paper the possibility of building a common theory of both copulae and joiners is quite possible, and the joiners may play the role of tools within the theory of copulae, and vice versa copulae may, for example, be used for finding proper joiners. Another independent feature of the joiners methodology is the possibility of constructing many new stochastic processes including stationary and Markovian.

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Families of Multivariate Distributions Involving “Triangular” Transformations

Cited by 15 publications

References 1 publication

Statistical Inference for distributions with one Poisson conditional

Statistical Inference for distributions with one Poisson conditional

An Alternative for Time Series Models

Construction of <i>k</i>-Variate Survival Functions with Emphasis on the Case <i>k</i> = 3

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Families of Multivariate Distributions Involving “Triangular” Transformations

Cited by 15 publications

References 1 publication

Statistical Inference for distributions with one Poisson conditional

Statistical Inference for distributions with one Poisson conditional

An Alternative for Time Series Models

Construction of &lt;i&gt;k&lt;/i&gt;-Variate Survival Functions with Emphasis on the Case &lt;i&gt;k&lt;/i&gt; = 3

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Construction of <i>k</i>-Variate Survival Functions with Emphasis on the Case <i>k</i> = 3