• Title/Summary/Keyword: Beta function in n variables

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ON THE STABILITY OF FUNCTIONAL EQUATIONS IN n-VARIABLES AND ITS APPLICATIONS

  • KIM, GWANG-HUI
    • Communications of the Korean Mathematical Society
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    • v.20 no.2
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    • pp.321-338
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    • 2005
  • In this paper we investigate a generalization of the Hyers-Ulam-Rassias stability for a functional equation of the form $f(\varphi(X))\;=\;\phi(X)f(X)$, where X lie in n-variables. As a consequence, we obtain a stability result in the sense of Hyers, Ulam, Rassias, and Gavruta for many other equations such as the gamma, beta, Schroder, iterative, and G-function type's equations.

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.

Sequential confidence intervals for the mean with $\beta$-protection in a certain parameter space

  • Kim, Sung-Lai
    • Journal of the Korean Statistical Society
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    • v.19 no.2
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    • pp.113-121
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    • 1990
  • Let ${X_n : n=1,2,\cdots}$ be iid random variables with distribution $P_{\theta}, \theta \in H$ where $H$ is some abstract parameter space. We consider a sequential confidence interval I for the mean $\mu = \mu(\theta)$ of $P_{\theta}$ satisfying $P_{\theta}(\mu \in I) \geq 1-\alpha$ and $P_{\theta}(\mu-\delta(\mu) \in I) \leq \beta$ for all $\theta \in H$ for any given an imprecision real valued function $\delta(\mu) > 0$ and error probabilities $0 < \alpha, \beta < 1$. A one-sided sequential confidence interval is constructed under some restriction of the family {P_{\theta} : \theta \in H}$ and the imprecision function $\delta$. This is extended to the two-sided cases.

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RECURRENCE RELATIONS FOR QUOTIENT MOMENTS OF THE PARETO DISTRIBUTION BY RECORD VALUES

  • Lee, Min-Young;Chang, Se-Kyung
    • The Pure and Applied Mathematics
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    • v.11 no.1
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    • pp.97-102
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    • 2004
  • In this paper we establish some recurrence relations satisfied by quotient moments of upper record values from the Pareto distribution. Let {$X_n,n\qeq1$}be a sequence of independent and identically distributed random variables with a common continuous distribution function(cdf) F($chi$) and probability density function(pdf) f($chi$). Let $Y_n\;=\;mas{X_1,X_2,...,X_n}$ for $ngeq1$. We say $X_{j}$ is an upper record value of {$X_{n},n\geq1$}, if $Y_{j}$$Y_{j-1}$,j>1. The indices at which the upper record values occur are given by the record times ${u( n)}n,\geq1$, where u(n) = min{j|j >u(n-l), $X_{j}$$X_{u(n-1)}$,n\qeq2$ and u(l) = 1. Suppose $X{\epsilon}PAR(\frac{1}{\beta},\frac{1}{\beta}$ then E$(\frac{{X^\tau}}_{u(m)}}{{X^{s+1}}_{u(n)})\;=\;\frac{1}{s}E$ E$(\frac{{X^\tau}}_{u(m)}{{X^s}_{u(n-1)}})$ - $\frac{(1+\betas)}{s}E(\frac{{X^\tau}_{u(m)}}{{X^s}_{u(n)}}$ and E$(\frac{{X^{\tau+1}}_{u(m)}}{{X^s}_{u(n)}})$ = $\frac{1}{(r+1)\beta}$ [E$(\frac{{X^{\tau+1}}}_u(m)}{{X^s}_{u(n-1)}})$ - E$(\frac{{X^{\tau+1}}_u(m)}}{{X^s}_{u(n-1)}})$ - (r+1)E$(\frac{{X^\tau}_{u(m)}}{{X^s}_{u(n)}})$]

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Asymptotic Properties of the Stopping Times in a Certain Sequential Procedure

  • Kim, Sung-Lai
    • Journal of the Korean Statistical Society
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    • v.24 no.2
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    • pp.337-347
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    • 1995
  • In the problem of some sequential estimation, the stopping times may be written in the form $N(c) = inf{n \geq n_0; n \geq c^2 S^2_n/\delta^2 (\bar{X}_n)}$ where ${s^2_n}$ and ${\bar{X}_n}$ are the sequences of sample variance and sample mean of the independently and identically distributed (i.i.d.) random variables with distribution $F_{\theta}(x), \theta \in \Theta$, respectively, and $\delta$ is either constant or any given positive real valued function. We obtain some asymptotic normality and asymptotic expectation of the N(c) in various limiting situations. Specially, uniform asymptotic normality and uniform asymptotic expectation of the N(c) are given.

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An Alternative Model for Determining the Optimal Fertilizer Level (수도(水稻) 적정시비량(適正施肥量) 결정(決定)에 대한 대체모형(代替模型))

  • Chang, Suk-Hwan
    • Korean Journal of Soil Science and Fertilizer
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    • v.13 no.1
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    • pp.21-32
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    • 1980
  • Linear models, with and without site variables, have been investigated in order to develop an alternative methodology for determining optimal fertilizer levels. The resultant models are : (1) Model I is an ordinary quadratic response function formed by combining the simple response function estimated at each site in block diagonal form, and has parameters [${\gamma}^{(1)}_{m{\ell}}$], for m=1, 2, ${\cdots}$, n sites and degrees of polynomial, ${\ell}$=0, 1, 2. (2) Mode II is a multiple regression model with a set of site variables (including an intercept) repeated for each fertilizer level and the linear and quadratic terms of the fertilizer variables arranged in block diagonal form as in Model I. The parameters are equal to [${\beta}_h\;{\gamma}^{(2)}_{m{\ell}}$] for h=0, 1, 2, ${\cdots}$, k site variable, m=1, 2, ${\cdots}$ and ${\ell}$=1, 2. (3) Model III is a classical response surface model, I. e., a common quadratic polynomial model for the fertilizer variables augmented with site variables and interactions between site variables and the linear fertilizer terms. The parameters are equal to [${\beta}_h\;{\gamma}_{\ell}\;{\theta}_h$], for h=0, 1, ${\cdots}$, k, ${\ell}$=1, 2, and h'=1, 2, ${\cdots}$, k. (4) Model IV has the same basic structure as Mode I, but estimation procedure involves two stages. In stage 1, yields for each fertilizer level are regressed on the site variables and the resulting predicted yields for each site are then regressed on the fertilizer variables in stage 2. Each model has been evaluated under the assumption that Model III is the postulated true response function. Under this assumption, Models I, II and IV give biased estimators of the linear fertilizer response parameter which depend on the interaction between site variables and applied fertilizer variables. When the interaction is significant, Model III is the most efficient for calculation of optimal fertilizer level. It has been found that Model IV is always more efficient than Models I and II, with efficiency depending on the magnitude of ${\lambda}m$, the mth diagonal element of X (X' X)' X' where X is the site variable matrix. When the site variable by linear fertilizer interaction parameters are zero or when the estimated interactions are not important, it is demonstrated that Model IV can be a reasonable alternative model for calculation of optimal fertilizer level. The efficiencies of the models are compared us ing data from 256 fertilizer trials on rice conducted in Korea. Although Model III is usually preferred, the empirical results from the data analysis support the feasibility of using Model IV in practice when the estimated interaction term between measured soil organic matter and applied nitrogen is not important.

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Analysis of urine β2-microglobulin in pediatric renal disease (소아 신장질환에서 요 β2-microglobulin검사의 분석)

  • Kim, Dong Woon;Lim, In Seok
    • Clinical and Experimental Pediatrics
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    • v.50 no.4
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    • pp.369-375
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    • 2007
  • Purpose : There have been numerous researches on urine ${\beta}_2$-microglobulin (${\beta}_2$-M) concerned with primary nephrotic syndrome and other glomerular diseases, but not much has been done in relation to pediatric age groups. Thus, our hospital decided to study the relations between the analysis of the test results we have conducted on pediatric patients and renal functions. Methods : Retrospective data analysis was done to 102 patients of ages 0 to 4 with renal diseases with symptoms such as hematuria, edema, and proteinuria who were admitted to Chung-Ang Yongsan Hospital and who participated in 24-hour urine and urine ${\beta}_2$-M excretion test between January of 2003 and January of 2006. Each disease was differentiated as independent variables, and the statistical difference of the results of urine ${\beta}_2$-M excretion of several groups of renal diseases was analyzed with student T-test by using test results as dependent variables. Results : Levels of urine ${\beta}_2$-M excretion of the 102 patients were as follows : 52 had primary nephrotic syndrome [MCNS (n=45, $72{\pm}45{\mu}g/g$ creatinine, ${\mu}g/g-Cr$), MPGN (n=3, $154{\pm}415{\mu}g/g-Cr$), FSGS (n=4, $188{\pm}46{\mu}g/-Cr$], six had APSGN ($93{\pm}404{\mu}g/g-Cr$), seven had IgA nephropathy ($3,414{\pm}106{\mu}g/g-Cr$), 9 had APN ($742{\pm}160{\mu}g/g-Cr$), 16 had cystitis ($179{\pm}168{\mu}g/g-Cr$), and 12 had HSP nephritis ($109{\pm}898{\mu}g/g-Cr$). IgA nephropathy (P<0.05) and APN (P<0.05) were significantly higher than in other renal diseases. Among primary nephrotic syndrome, FSGS with higher results of ${\beta}_2$-microglobulin test had longer treatment period (P<0.01) when compared to the lower groups, but no significant differences in Ccr, BUN, or Cr were observed. Conclusion : IgA nephropathy and APN groups showed significantly higher level of ${\beta}_2$-M excretion value than other groups. Although ${\beta}_2$-microglobulin value is not appropriate as an indicator of general renal function and pathology, it seems to be sufficient in the differential diagnosis of the UTI and in the prediction of the treat-ment period of nephrotic syndrome patients.

Related Factors of Quality of Life in Male Patients with Chronic Obstructive Pulmonary Disease (남성 만성폐쇄성폐질환자의 삶의 질 관련요인)

  • Lee, Hae-Jung;Jee, Young-Ju
    • Korean Journal of Adult Nursing
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    • v.23 no.4
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    • pp.309-320
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    • 2011
  • Purpose: The purpose of the study was to examine the related factors of quality of life (QOL) among patients with Chronic Obstructive Pulmonary Disease (COPD). Methods: Patients diagnosed with COPD (N=230) were recruited from four hospitals in Kyeong-Nam province, from March 2 to November 30, 2010. The data collection instruments were the Short Form 36, perceived dyspnea measure by Modified Medical Research Council, COPD and Asthma Sleep Impact Scale, COPD Self-efficacy Scale, and Center for Epidemiologic Studies Depression Scale were used. Following the completion of the data collection instruments Pulmonary function was tested. Data were analyzed with descriptive statistics, Pearson correlation and simultaneous multiple regression using SPSS/WIN. Results: The mean QOL of this study was 68.24. Using a multivariate approach, the significant correlates of QOL were depression (${\beta}$=-.37), dyspnea (${\beta}$=-.28), self-efficacy (${\beta}$=.20), and a sufficient degree of household income (${\beta}$=.16). These variables explained 49% of variance in QOL. Conclusion: The study suggests that psychological aspects are an important factor in explaining QOL of the patients. Screening and minimizing depression could be effective strategies in enhancing QOL of patients with COPD and further investigation to reduce depression could warrant the improvement of QOL in patients with COPD.

A Kernel-function-based Approach to Sequential Estimation with $\beta$-protection of Quantiles

  • 김성래;김성균
    • Proceedings of the Korean Society of Computational and Applied Mathematics Conference
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    • 2003.09a
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    • pp.14-14
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    • 2003
  • Given a sequence { $X_{n}$} of independent and identically distributed random variables with F, a sequential procedure for the p-th quantile ξ$_{P}$= $F^{-1}$ (P), 0$\beta$-protection. Some asymptotic properties for the proposed procedure and of an involved stopping time are proved: asymptotic consistency, asymptotic efficiency and asymptotic normality. From one of the results an effect of smoothing based on kernel functions is discussed. The results are also extended to the contaminated case.e.e.

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