• Journal for History of Mathematics
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• v.13 no.1
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• pp.1-14
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• 2000

The first part of this thesis discusses the Pearson's Curve Family which gives $\beta$distribution, $\Gamma$-distribution, $X^2$-distribution and t-distribution. The second part of this thesis gives some brief process of calculations for normal distribution density and t-distribution density by the 7-th type Curve of Pearson's Curve Family. Finally, a conclusion arrives that Student(Gosset) could not find out his famous 'Student's t-distribution' without his attending of 'Pearson's Differential Equation' class taught by Pearson himself when he was a senior student. However, if he had got a professorship at the Pearson Statistics Laboratory, the University of London, then he could not have found 'Student's t-distribution' for small sampling technique of modern statistics.

• Journal of the Korean Statistical Society
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• v.7 no.1
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• pp.3-10
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• 1978

Let $X_i,...,X_n$ be a random sample from a distribution with cumulants $K_1, K_2,...$. The statistic $t = \frac{\sqrt{x}(\bar{X}-K_1)}{S}$ has the well-known 'student' distribution with $\nu = n-1$ degrees of freedom if the $X_i$ are normally distributed (i.e., $K_i = 0$ for $i \geq 3$). An Edgeworth series expansion for the distribution of t when the $X_i$ are not normally distributed is obtained. The form of this expansion is Prob $(t

• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.547-562
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• 2005

The distribution of products of random variables is of interest in many areas of the sciences, engineering and medicine. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this note, the distribution of the product | XY | is derived when X and Y are Student's t and Bessel function random variables distributed independently of each other.

• Bulletin of the Korean Mathematical Society
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• v.44 no.2
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• pp.265-270
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• 2007

The modern approach for wavelets imposes a Bayesian prior model on the wavelet coefficients to capture the sparseness of the wavelet expansion. The idea is to build flexible probability models for the marginal posterior densities of the wavelet coefficients. In this note, we derive exact expressions for a popular model for the marginal posterior density.

• Journal of the Korean Statistical Society
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• v.22 no.1
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• pp.13-25
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• 1993

The purpose of this paper is to investigate the properties of order statistics under various stochastic relations. We study the stochastic comparison of order statistics in a single sample. And we consider two sample case too. For example, F(t) > G9t) for t > 0 when X and Y are random variables symmetric about 0, with c.d.f.s F and G. Two examples are provided.

• Communications for Statistical Applications and Methods
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• v.30 no.2
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• pp.163-178
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• 2023

GARCH-X(1, 1) model specifies that conditional variance follows an AR(1) process and includes a past exogenous variable. This study proposes a new class from that model by allowing a more general (non-linear) variance function to follow an AR(1) process. The functions applied to the variance equation include exponential, Tukey's ladder, and Yeo-Johnson transformations. In the framework of normal and student-t distributions for return errors, the empirical analysis focuses on two stock indices data in developed countries (FTSE100 and SP500) over the daily period from January 2000 to December 2020. This study uses 10-minute realized volatility as the exogenous component. The parameters of considered models are estimated using the adaptive random walk metropolis method in the Monte Carlo Markov chain algorithm and implemented in the Matlab program. The 95% highest posterior density intervals show that the three transformations are significant for the GARCHX(1, 1) model. In general, based on the Akaike information criterion, the GARCH-X(1, 1) model that has return errors with student-t distribution and variance transformed by Tukey's ladder function provides the best data fit. In forecasting value-at-risk with the 95% confidence level, the Christoffersen's independence test suggest that non-linear models is the most suitable for modeling return data, especially model with the Tukey's ladder transformation.

• Communications for Statistical Applications and Methods
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• v.30 no.6
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• pp.605-629
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• 2023

This paper proposes some diagnostics procedures for the skew-t linear regression model with censored response. The skew-t distribution is an attractive family of asymmetrical heavy-tailed densities that includes the normal, skew-normal and student's-t distributions as special cases. Inspired by the power and wide applicability of the EM-type algorithm, local and global influence analysis, based on the conditional expectation of the complete-data log-likelihood function are developed, following Zhu and Lee's approach. For the local influence analysis, four specific perturbation schemes are discussed. Two real data sets, from education and economics, which are right and left censoring, respectively, are analyzed in order to illustrate the usefulness of the proposed methodology.

• Communications for Statistical Applications and Methods
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• v.29 no.3
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• pp.333-351
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• 2022

The skew-t distribution is an attractive family of asymmetrical heavy-tailed densities that includes the normal, skew-normal and Student's-t distributions as special cases. In this work, we propose an EM-type algorithm for computing the maximum likelihood estimates for skew-t linear regression models with censored response. In contrast with previous proposals, this algorithm uses analytical expressions at the E-step, as opposed to Monte Carlo simulations. These expressions rely on formulas for the mean and variance of a truncated skew-t distribution, and can be computed using the R library MomTrunc. The standard errors, the prediction of unobserved values of the response and the log-likelihood function are obtained as a by-product. The proposed methodology is illustrated through the analyses of simulated and a real data application on Letter-Name Fluency test in Peruvian students.

• The Journal of Industrial Distribution & Business
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• v.10 no.2
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• pp.25-37
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• 2019

Purpose - The goal of this study is to analyze the differences in education performances between students of the government's financial support program and those who do not receive support at a local university in Korea. Research design, data, and methodology - The questionnaire used was NASEL. NASEL is considered a highly suitable survey tool for professors, courses, and performances in Korean universities. The 290 students who participated and 44 students do not participate in the financial support program were surveyed for 10 days. The characteristics of students were investigated by frequency analysis and technical statistics. The analysis of student collective characteristics used independent t and f-tests,and one-way ANOVA with IBM SPSS Statistics 22.0 for statistical purposes. Results - The p-value of the group receiving financial support and the group without financial support in active-collaborative learning is 0.167. The p-value of the economically supported group and the non-supported group of the faculty-student interaction is 0.281. The confidence coefficient of the active-collaborative learning questionnaire is 0.861. The reliability coefficient of the questionnaire for the faculty-student interaction questionnaire is 0.871. Conclusions - There are no clear differences in active-collaborative learning and faculty-student interaction between participating and non-participating students in the economic program.

• Journal of the Korean Statistical Society
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• v.33 no.4
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• pp.411-433
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• 2004

The shrinkage preliminary test ridge regression estimators (SPTRRE) based on Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests for estimating the regression parameters of the multiple linear regression model with multivariate Student's t error distribution are considered in this paper. The quadratic biases and risks of the proposed estimators are compared under both null and alternative hypotheses. It is observed that there is conflict among the three estimators with respect to their risks because of certain inequalities that exist among the test statistics. In the neighborhood of the restriction, the SPTRRE based on LM test has the smallest risk followed by the estimators based on LR and W tests. However, the SPTRRE based on W test performs the best followed by the LR and LM based estimators when the parameters move away from the subspace of the restrictions. Some tables for the maximum and minimum guaranteed efficiency of the proposed estimators have been given, which allow us to determine the optimum level of significance corresponding to the optimum estimator among proposed estimators. It is evident that in the choice of the smallest significance level to yield the best estimator the SPTRRE based on Wald test dominates the other two estimators.