• Communications for Statistical Applications and Methods
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• v.21 no.2
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• pp.183-191
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• 2014

We define a multivariate Cauchy distribution using a probability density function; subsequently, a Ferguson's definition of a multivariate Cauchy distribution can be viewed as a characterization theorem using the characteristic function approach. To clarify this characterization theorem, we construct two dependent Cauchy random variables, but their sum is not Cauchy distributed. In doing so the proofs depend on the characteristic function, but we use the cumulative distribution function to obtain the exact density of their sum. The derivation methods are relatively straightforward and appropriate for graduate level statistics theory courses.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.33 no.3
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• pp.130-136
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• 2010

In this study, we propose a new method for generating candidate solutions based on both the Cauchy and the Gaussian probability distributions in order to use the merit of the solutions generated by these distributions. The Cauchy probability distribution has larger probability in the tail region than the Gaussian distribution. Thus, the Cauchy distribution can yield higher probabilities of generating candidate solutions of large-varied variables, which in turn has an advantage of searching wider area of variable space. On the contrary, the Gaussian distribution can yield higher probabilities of generating candidate solutions of small-varied variables, which in turn has an advantage of searching deeply smaller area of variable space. In order to compare and analyze the performance of the proposed method against the conventional method, we carried out experiments using benchmarking problems of real valued functions. From the result of the experiment, we found that the proposed method based on the Cauchy and the Gaussian distributions outperformed the conventional one for most of benchmarking problems, and verified its superiority by the statistical hypothesis test.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.9 no.1
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• pp.1-3
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• 2009

Recently, Carlsson, Full\acute{e}$r and Majlender [1] presented the concept of possibilitic correlation representing an average degree of interaction between marginal distribution of a joint possibility distribution as compared to their respective dispersions. They also formulated the weak and strong forms of the possibilistic Cauchy-Schwarz inequality. In this paper, we define a new probability measure. Then the weak and strong forms of the Cauchy-Schwarz inequality are immediate consequence of probabilistic Cauchy-Schwarz inequality with respect to the new probability measure.

• Journal of KIISE:Software and Applications
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• v.37 no.8
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• pp.591-598
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• 2010

In this study, we propose a new method for generating candidate solutions based on the Cauchy probability distribution in order to complement the shortcoming of the solutions generated by the normal distribution. The Cauchy probability distribution has infinite mean and variance, and it has rather large probability in the tail region relative to the normal distribution. Thus, the Cauchy distribution can yield higher probabilities of generating candidate solutions of large-varied variables, which in turn has an advantage of searching wider area of variable space. In order to compare and analyze the performance of the proposed method against the conventional method, we carried out an experiment using benchmarking problems of real valued function. From the result of the experiment, we found that the proposed method based on the Cauchy distribution outperformed the conventional one for all benchmarking problems, and verified its superiority by the statistical hypothesis test.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.29-34
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• 1993

J. Leray [7] proposed a sufficient condition ofr the solvability of the Cauchy problem on the initial hyperplane x$_{1}$=0 with Cauchy data which are holomorphic with respect to the variables parallel to some analytic subvariety S of the initial hyperplane. Limiting the problem to the case of operators with constant coefficients, A. Kaneko [2] proposed a new sharper sufficient condition. Later we generalized this condition and showed that it is necessary and sufficient for the solvability of the Cauchy problem for the hyperfunction Cauchy data and the distribution Cauchy data which contain variables parallel to S as holomorphic parameters in [5, 6]. In this paper, we extend the results in [6] to the case of operators with variable coefficients and show that it is sufficient for the solvability of the Cauchy problem for the hyperfunction Cauchy data. Our main theorem can be considered as an example of a deep theorem on micro-hyperbolic systems by Kashiwara-Schapira [4] and we give a direct proof based on an elementary sweeping out procedure developed in Kaneko [3].

• Journal of the Korean Statistical Society
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• v.34 no.1
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• pp.35-38
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• 2005

A single, tractable, special case of the problem of continuous mixtures of beta distributions over their parameters is considered. This is the uniform mixture of generalized arc-sine distributions which, curiously, turns out to be linked by transformation to the Cauchy distribution.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.361-370
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• 2008

We consider the Hyers-Ulam type stability of the Cauchy, Jensen, Pexider, Pexider-Jensen differences: $$(0.1){\hspace{55}}C(u):=u{\circ}A-u{\circ}P_1-u{\circ}P_2,\\(0.2){\hspace{55}}J(u):=2u{\circ}\frac{A}{2}-u{\circ}P_1-u{\circ}P_2,\\(0.3){\hspace{18}}P(u,v,w):=u{\circ}A-v{\circ}P_1-w{\circ}P_2,\\(0.4)\;JP(u,v,w):=2u{\circ}\frac{A}{2}-v{\circ}P_1-w{\circ}P_2$$, with respect to bounded distributions.

• Journal of KIISE:Software and Applications
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• v.37 no.9
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• pp.694-705
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• 2010

The mutation operation is the main operation in the evolutionary programming which has been widely used for the optimization of real valued function. In general, the mutation operation utilizes both a probability distribution and its parameter to change values of variables, and the parameter itself is subject to its own mutation operation which requires other parameters. However, since the optimal values of the parameters entirely depend on a given problem, it is rather hard to find an optimal combination of values of parameters when there are many parameters in a problem. To solve this shortcoming at least partly, if not entirely, in this paper, we propose a new mutation operation in which the parameter for the variable mutation is theoretically estimated from the self-adaptive perspective. Since the proposed algorithm estimates the scale parameter of the Cauchy probability distribution for the mutation operation, it has an advantage in that it does not require another mutation operation for the scale parameter. The proposed algorithm was tested against the benchmarking problems. It turned out that, although the relative superiority of the proposed algorithm from the optimal value perspective depended on benchmarking problems, the proposed algorithm outperformed for all benchmarking problems from the perspective of the computational time.

• The Korean Journal of Applied Statistics
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• v.16 no.2
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• pp.335-349
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• 2003

In this paper, we use Bayesian method for model selection of poisson vs. negative binomial distribution, and normal, double exponential vs. cauchy distribution. The fractional Bayes factor of O'Hagan (1995) was applied to Bayesian model selection under the assumption of noninformative improper priors for all parameters in the models. Through the analyses of real data and simulation data, we examine the usefulness of the fractional Bayes factor in comparison with intrinsic Bayes factors of Berger and Pericchi (1996, 1998).

• Communications for Statistical Applications and Methods
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• v.9 no.1
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• pp.101-113
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• 2002

In this paper we introduce a new score generating (unction for the rank regression in the linear regression model. The score function compares the $\gamma$'th and s\`th power of the tail probabilities of the underlying probability distribution. We show that the rank estimate asymptotically converges to a multivariate normal. further we derive the asymptotic Pitman relative efficiencies and the most efficient values of $\gamma$ and s under the symmetric distribution such as uniform, normal, cauchy and double exponential distributions and the asymmetric distribution such as exponential and lognormal distributions respectively.