• Title/Summary/Keyword: conditional expectations

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On the maximum and minimum in a bivariate uniform distribution

  • Lee, Changsoo;Shin, Hyejung;Moon, Yeung-Gil
    • Journal of the Korean Data and Information Science Society
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    • v.26 no.6
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    • pp.1495-1500
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    • 2015
  • We obtain means and variances of max {X, Y} and min {X, Y} in the underlying Morgenstern type bivariate uniform variables X and Y with same scale parameters and different scale parameters respectively. And we obtain the conditional expectations in the underlying Morgenstern type bivariate uniform variables. Here, we shall consider the conditional expectations to know the dependence of one variable on the other variable and we consider the behaviors of means and variances of max {X, Y} and min {X, Y} with respect to changes in means, variances, and the correlation coeffcient of the underlying Morgenstern type bivariate uniform variables.

CHARACTERIZATIONS OF BETA DISTRIBUTION OF THE FIRST KIND BY CONDITIONAL EXPECTATIONS OF RECORD VALUES

  • Lee, Min-Young;Chang, Se-Kyung
    • Journal of applied mathematics & informatics
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    • v.13 no.1_2
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    • pp.441-446
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    • 2003
  • Let { $X_{n}$ , n $\geq$ 1} be a sequence of independent and identically distributed random variables with a common continuous distribution function F(x) and probability density function f(x). Let $Y_{n}$ = max{ $X_1$, $X_2$, …, $X_{n}$ } for n $\geq$ 1. We say $X_{j}$ is an upper record value of { $X_{n}$ , n$\geq$1} if $Y_{j}$ > $Y_{j-1}$, j > 1. The indices at which the upper record values occur are given by the record times {u(n)}, n$\geq$1, where u(n) = min{j|j>u(n-1), $X_{j}$ > $X_{u}$ (n-1), n$\geq$2} and u(1) = 1. We call the random variable X $\in$ Beta (1, c) if the corresponding probability cumulative function F(x) of x is of the form F(x) = 1-(1-x)$^{c}$ , c>0, 0$\leq$x$\leq$1. In this paper, we will give a characterization of the beta distribution of the first kind by considering conditional expectations of record values.s.

Conditional Integral Transforms on a Function Space

  • Cho, Dong Hyun
    • Kyungpook Mathematical Journal
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    • v.52 no.4
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    • pp.413-431
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    • 2012
  • Let $C^r[0,t]$ be the function space of the vector-valued continuous paths $x:[0,t]{\rightarrow}\mathbb{R}^r$ and define $X_t:C^r[0,t]{\rightarrow}\mathbb{R}^{(n+1)r}$ and $Y_t:C^r[0,t]{\rightarrow}\mathbb{R}^{nr}$ by $X_t(x)=(x(t_0),\;x(t_1),\;{\cdots},\;x(t_{n-1}),\;x(t_n))$ and $Y_t(x)=(x(t_0),\;x(t_1),\;{\cdots},\;x(t_{n-1}))$, respectively, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n=t$. In the present paper, using two simple formulas for the conditional expectations over $C^r[0,t]$ with the conditioning functions $X_t$ and $Y_t$, we establish evaluation formulas for the analogue of the conditional analytic Fourier-Feynman transform for the function of the form $${\exp}\{{\int_o}^t{\theta}(s,\;x(s))\;d{\eta}(s)\}{\psi}(x(t)),\;x{\in}C^r[0,t]$$ where ${\eta}$ is a complex Borel measure on [0, t] and both ${\theta}(s,{\cdot})$ and ${\psi}$ are the Fourier-Stieltjes transforms of the complex Borel measures on $\mathbb{R}^r$.

Limit Theorems for Fuzzy Martingales

  • Joo, Sang-Yeol;Kim, Gwan-Young;Kim, Yun-Kyong
    • Journal of the Korean Statistical Society
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    • v.28 no.1
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    • pp.21-34
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    • 1999
  • In this paper, conditional expectation of a fuzzy random variable is introduced and its properties are investigated. Using this, we introduce the concept of fuzzy martingales and prove some convergence theorems which generalize te corresponding results for the classical martingales.

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EVALUATION FORMULAS FOR AN ANALOGUE OF CONDITIONAL ANALYTIC FEYNMAN INTEGRALS OVER A FUNCTION SPACE

  • Cho, Dong-Hyun
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.655-672
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    • 2011
  • Let $C^r$[0,t] be the function space of the vector-valued continuous paths x : [0,t] ${\rightarrow}$ $R^r$ and define $X_t$ : $C^r$[0,t] ${\rightarrow}$ $R^{(n+1)r}$ and $Y_t$ : $C^r$[0,t] ${\rightarrow}$ $R^{nr}$ by $X_t(x)$ = (x($t_0$), x($t_1$), ..., x($t_{n-1}$), x($t_n$)) and $Y_t$(x) = (x($t_0$), x($t_1$), ..., x($t_{n-1}$)), respectively, where 0 = $t_0$ < $t_1$ < ... < $t_n$ = t. In the present paper, with the conditioning functions $X_t$ and $Y_t$, we introduce two simple formulas for the conditional expectations over $C^r$[0,t], an analogue of the r-dimensional Wiener space. We establish evaluation formulas for the analogues of the analytic Wiener and Feynman integrals for the function $G(x)=\exp{{\int}_0^t{\theta}(s,x(s))d{\eta}(s)}{\psi}(x(t))$, where ${\theta}(s,{\cdot})$ and are the Fourier-Stieltjes transforms of the complex Borel measures on ${\mathbb{R}}^r$. Using the simple formulas, we evaluate the analogues of the conditional analytic Wiener and Feynman integrals of the functional G.

CONDITIONAL FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS OF UNBOUNDED FUNCTIONS ON A GENERALIZED WIENER SPACE

  • Cho, Dong Hyun
    • Journal of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1105-1127
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    • 2013
  • Let C[0, $t$] denote the function space of real-valued continuous paths on [0, $t$]. Define $X_n\;:\;C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ and $X_{n+1}\;:\;C[0,t]{\rightarrow}\mathbb{R}^{n+2}$ by $X_n(x)=(x(t_0),x(t_1),{\ldots},x(t_n))$ and $X_{n+1}(x)=(x(t_0),x(t_1),{\ldots},x(t_n),x(t_{n+1}))$, respectively, where $0=t_0 <; t_1 <{\ldots} < t_n < t_{n+1}=t$. In the present paper, using simple formulas for the conditional expectations with the conditioning functions $X_n$ and $X_{n+1}$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions, which have the form $fr((v_1,x),{\ldots},(v_r,x)){\int}_{L_2}_{[0,t]}\exp\{i(v,x)\}d{\sigma}(v)$ for $x{\in}C[0,t]$, where $\{v_1,{\ldots},v_r\}$ is an orthonormal subset of $L_2[0,t]$, $f_r{\in}L_p(\mathbb{R}^r)$, and ${\sigma}$ is the complex Borel measure of bounded variation on $L_2[0,t]$. We then investigate the inverse conditional Fourier-Feynman transforms of the function and prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions can be expressed by the products of the analytic conditional Fourier-Feynman transform of each function.

CONDITIONAL INTEGRAL TRANSFORMS AND CONVOLUTIONS OF BOUNDED FUNCTIONS ON AN ANALOGUE OF WIENER SPACE

  • Cho, Dong Hyun
    • Journal of the Chungcheong Mathematical Society
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    • v.26 no.2
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    • pp.323-342
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    • 2013
  • Let $C[0,t]$ denote the function space of all real-valued continuous paths on $[0,t]$. Define $Xn:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ and $X_{n+1}:C[0,t]{\rightarrow}\mathbb{R}^{n+2}$ by $X_n(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$ and $X_{n+1}(x)=(x(t_0),x(t_1),{\cdots},x(t_n),x(t_{n+1}))$, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n$ < $t_{n+1}=t$. In the present paper, using simple formulas for the conditional expectations with the conditioning functions $X_n$ and $X_{n+1}$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions which have the form $${\int}_{L_2[0,t]}{{\exp}\{i(v,x)\}d{\sigma}(v)}{{\int}_{\mathbb{R}^r}}\;{\exp}\{i{\sum_{j=1}^{r}z_j(v_j,x)\}dp(z_1,{\cdots},z_r)$$ for $x{\in}C[0,t]$, where $\{v_1,{\cdots},v_r\}$ is an orthonormal subset of $L_2[0,t]$ and ${\sigma}$ and ${\rho}$ are the complex Borel measures of bounded variations on $L_2[0,t]$ and $\mathbb{R}^r$, respectively. We then investigate the inverse transforms of the function with their relationships and finally prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the products of the conditional Fourier-Feynman transforms of each function.

DERIVATION OF A PRICE PROCESS FOR MULTITYPE MULTIPLE DEFAULTABLE BONDS

  • Park Heung-Sik
    • Journal of the Korean Statistical Society
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    • v.35 no.2
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    • pp.193-199
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    • 2006
  • We consider a zero coupon bond that is at the risk of multitype multiple defaults. Assuming defaults occur according to k Cox processes, we find a price process for zero coupon bonds. To derive this process we follow the Lando (1998)'s method which uses conditional expectations instead of the traditional methods.

A Consistent Test for Linearity for a Class of General First order Nonlinear Time Series

  • Hwang, Sun Y.
    • Journal of the Korean Statistical Society
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    • v.27 no.4
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    • pp.451-458
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    • 1998
  • Problem of testing linearity among general class of first order nonlinear time series models is discussed. The null hypotheses of linearity is identified via conditional expectations. A consistent test is then suggested and relevant limiting results are derived. It is worth indicating that any specific alternatives are not specified.

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Stochastic interpolation of earthquake ground motions under spectral uncertainties

  • Morikawa, Hitoshi;Kameda, Hiroyuki
    • Structural Engineering and Mechanics
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    • v.5 no.6
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    • pp.839-851
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    • 1997
  • Closed-form solutions are analytically derived for stochastic properties of earthquake ground motion fields, which are conditioned by an observed time series at certain observation sites and are characterized by spectra with uncertainties. The theoretical framework presented here can estimate not only the expectations of such simulated earthquake ground motions, but also the prediction errors which offer important information for the field of engineering. Before these derivations are made, the theory of conditional random fields is summarized for convenience in this study. Furthermore, a method for stochastic interpolation of power spectra is explained.