경험적 영향함수와 표본영향함수 간 차이 보정의 t통계량으로의 확장

강현석,김홍기 한국통계학회 2021 응용통계연구 Vol.34 No.6

This study is a follow-up study of Kang and Kim (2020). In this study, we derive the sample influence functions of the $t$-statistic which were not directly derived in previous researches. Throughout these results, we both mathematically examine the relationship between the empirical influence function and the sample influence function, and consider a method to approximate the sample influence function by the empirical influence function. Also, the validity of the relationship between an approximated sample influence function and the empirical influence function is verified by a simulation of a random sample of size 300 from normal distribution. As a result of the simulation, the relationship between the sample influence function which is derived from the $t$-statistic and the empirical influence function, and the method of approximating the sample influence function through the empirical influence function were verified. This research has significance in proposing both a method which reduces errors in approximation of the empirical influence function and an effective and practical method that evolves from previous research which approximates the sample influence function directly through the empirical influence function by constant revision. 본 연구는 Kang과 Kim (2020)의 후속 연구이다. 본 연구에서는 기존 연구에서 직접 유도하지 않았던 통계량의 표본영향함수를 유도한다. 그리고 이 결과를 바탕으로 경험적 영향함수와 표본영향함수는 어떠한 관계를 가지고 있는지 이론적으로 살펴보고, 경험적 영향함수를 통해 표본영향함수를 근사시켜 추정하는 방안에 대해 생각해 본다. 또한, 임의추출한 300개의 데이터를 바탕으로 모의실험을 통해 유도한 함수와 그 관계에 대한 그 타당성도 검증한다. 모의실험 결과 $t$통계량으로부터 유도한 표본영향함수와 경험적 영향함수와의 관계 및 경험적 영향함수를 통한 표본영향함수의 근사 방안에 대한 타당성도 검증해 냈다. 본 연구는 경험적 영향함수를 이용한 표본영향함수의 근사에서 오차를 줄이기 위한 방안을 제안하고 그 타당성을 검증하였으며, 이를 통해 기존의 연구에서 경험적 영향함수로 표본영향함수를 바로 근사시켰던 연구 방법에 효과적인 근사 방안을 제안한 점에서 의의를 갖는다.

경험적 영향함수와 표본영향함수의 차이 및 보정에 관한 연구

강현석,김홍기 한국통계학회 2020 응용통계연구 Vol.33 No.5

While analyzing data, researching outliers, which are out of the main tendency, is as important as researching data that follow the general tendency. In this study we discuss the influence function for outlier discrimination. We derive sample influence functions of sample mean, sample variance, and sample standard deviation, which were not directly derived in previous research. The results enable us to mathematically examine the relationship between the empirical influence function and sample influence function. We can also consider a method to approximate the sample influence function by the empirical influence function. Also, the validity of the relationship between the approximated sample influence function and the empirical influence function is also verified by the simulation of random sampled data in normal distribution. As the result of a simulation, both the relationship between the two influence functions, sample and empirical, and the method of approximating the sample influence function through the emperical influence function were verified. This research has significance in proposing a method that reduces errors in the approximation of the empirical influence function and in proposing an effective and practical method that proceeds from previous research that approximates the sample influence function directly through empirical influence function by constant revision. 이상치에 대한 적절한 선별과 배제없이 모든 데이터를 종합적으로 분석하게 되는 경우 데이터 분석을 통해 얻은 결과의 신뢰성과 해석의 일반성에 치명적인 위협을 받을 수 있다. 따라서 데이터의 분석 과정에서 이러한 이상치를 판별하고, 이상치가 통계량, 통계적 모형에 어떠한 영향을 주는 지에 대한 분석은 매우 중요한 일이라 할 수 있다. Hampel이 영향함수를 활용하여 이상치를 판별할 수 있는 방법을 소개한 이후, 이상치를 판별하기 위한 방법론으로 영향함수가 폭넓게 활용되어 왔다. 영향함수에는 경험적 영향함수와 표본영향함수가 있으며, 경험적 영향함수를 활용해 표본영향함수를 근사 추론하여 하나의 관측값이 제거되었을 때 통계량에 미치는 영향을 예측하는 방법론이 주로 활용되었다. 본 연구에서는 표본평균, 표본분산, 표본표준편차의 표본영향함수 유도를 통해 경험적 영향함수와 표본영향함수의 차이를 살펴 본다. 또한 경험적 영향함수로 표본영향함수를 근사하는 과정에서 발생하는 오차를 줄이기 위해 경험적 영향함수의 보정으로 표본영향함수를 근사 추론하는 방법을 제안하고, 모의실험을 통해 제안한 추론 방법의 타당성을 확인한다.

Influence in Fitting an Equicorrelation Model

Kim, Myung Geun,Jung, Kang-Mo The Korean Statistical Society 2001 Communications for statistical applications and me Vol.8 No.3

The influence in fitting an equicorrelation model is investigated using the influence function. The influence functions for the model parameters are derived and its sample versions are used for investigating the influence of observations on the estimators of the parameters. Some relationships among the sample versions are found. We will derive a measure for identifying observations that have a large influence on the test of fitting the equicorrelation model using the influence function method. An example is given for illustration.

Influence of an Observation on the t-statistic

Kim, Hong-Gie,Kim, Kyung-Hee 한국통계학회 2005 Communications for statistical applications and me Vol.12 No.2

We derive the influence function on t statistic and find its feature; the influence function on t statistic has two forms depending on the value of ${\mu}_0$. Sample influence functions are used to verify the validity of the derived influence function. We use random samples from normal distribution to show the validity of the function. The simulation study proves that the obtained influence function is very accurate to in estimating changes in t statistic when an observation is added or deleted.

한,중 아동 폐활량과 그 영향요인 비교 연구

김대선,차정훈,유승도,박경렬 한국환경보건학회 2003 한국환경보건학회지 Vol.29 No.5

To evaluate the health effects caused by air pollution, we conducted a preliminary study on pulmonary function and it's related factors as a first year's work of a cooperative research project between Korea and China. 200 schoolchildren, grade 3 to 6, from two schools on Beijing and Helong in China were recruited to perform the pulmonary function test(PFT). A questionnaire concerning medical history and potential influence factors(such as passive smoking, heating or cooking fuels, use a stove or keep a pet in indoor, crowding within a room, and family income) was filled out by children's parents. Regression analysis was utilized to determine which potential influence factors were significantly correlated with PFT measure(FVC and FEV1). We also compared the pulmonary functions of Chinese children with those of other studies of Korean children. The results of regression analysis between potential influence factors and PFT measure were not statistically significants. As the results of comparison of the pulmonary functions(controlled by height, weight, and age) of Chinese children with those of other studies of Korean children, FVC and FEV1 were lower in Korean children(FVC 0.015L, FEV1 0.026L in boys; FVC 0.128L, FEV1 0.136L in girls) compared with Chinese children. In this study, we could not present causation between air pollution and health effect because of some limitations(such as absent of air pollution data), but the results will be used usefully for design of next year's study which is to assess acute effect of air pollution(especially, PM10, PM2.5) on pulmonary function.

적응광학계 변형거울의 구동기 배열에 따른 성능 변화 연구

엄태경,이완술,윤성기,이준호 한국광학회 2002 한국광학회지 Vol.13 No.5

지상용 천문 망원경에서 대기의 영향으로 인해 발생하는 오차를 실시간으로 보상해주는 적응 광학을 이용하면 지상용 망원경으로도 막대한 비용이 드는 우주용 망원경에 버금가는 이미지를 얻을 수 있다. 일반적으로 적응 광학계에서는 대기에 의한 파면 오차를 제거하기 위하여, 변형 거울을 변형시켜 각 부분의 파면 오차를 보정하는 방법을 이용하므로, 변형 거울을 효과적으로 변형시키기 위한 구동기의 특성과 구동기의 배열에 대한 연구가 필수적이다. 하나의 구동기론 작동하여 거울을 변형시킬 때, 변형된 거울면의 형태를 영향 함수라고 정의하며, 이러한 영향 함수를 이용하여 변형 거울을 효과적으로 모형화하고 설계할 수 있다. 본 논문에서는 유한요소해석을 이용하여 계산된 변형 거울의 실제 영향 함수를 가우시안 함수 형태로 단순화하고, 추가로 구동기들 사이의 영향을 고려한 커플링 계수를 도입하여, 주어진 구동기 배열에 대한 영향 함수를 결정하였다. 또한 변형 거울에 사용되는 구동기들 사이의 적절한 커플링 계수를 결정하기 위하여, 커플링 계수 변화에 따른 변형거울의 성능 변화를 해석하였다. 이와 같이 구성된 영향 함수를 이용하석, 구동기가 삼각형과 사각형 형태와 같이 등간격으로 배치되어 있을 때의 구동기 간격에 따른 변형 거울의 성능을 해석하고 효과적인 배열을 제안하였다. In the earth telescope for space observation, the adaptive optical (AO) system that immediately compensates atmospheric turbulence is helpful to get high-resolution images. An adaptive optics for earth telescopes is very attractive, since the Earth telescopes can be made at lower costs and have larger optical apertures than space telescopes. Generally. in order to remove the wavefront error produced by atmospheric turbulence, a deformable mirror, whose surface shape changes in a controllable way in response to a drive signal, is used. The characteristics and patterns of actuators are very important for the effective control of a deformable mirror. The mirror surface shape deformed by one actuator is defined as an influence function and the deformable mirror can be effectively modeled and designed using this influence function. In this paper. by simplifying the actual influence function obtained by FEM analyses into the Gaussian function and introducing the coupling coefficient between actuators, the influence function is constructed. The proper coupling coefficient of the target system can be obtained by performance analyses of a deformable mirror for various coupling coefficients. Using the constructed influence function, the deformable mirror with equally spaced triangular and square actuator patterns is analyzed for various spacings and an effective actuator pattern is proposed.

Influence Measures for a Test Statistic on Independence of Two Random Vectors

Jung Kang-Mo The Korean Statistical Society 2005 Communications for statistical applications and me Vol.12 No.3

In statistical diagnostics a large number of influence measures have been proposed for identifying outliers and influential observations. However it seems to be few accounts of the influence diagnostics on test statistics. We study influence analysis on the likelihood ratio test statistic whether the two sets of variables are uncorrelated with one another or not. The influence of observations is measured using the case-deletion approach, the influence function. We compared the proposed influence measures through two illustrative examples.

Sensitivity Analysis in Variogram Estimation: Detecting Observations Influential to Cressie and Hawkins(1980) s Robust Variogram

Seung Bae Choi,Lee Jung Kang,Yutaka Tanaka 한국자료분석학회 2000 Journal of the Korean Data Analysis Society Vol.2 No.3

As a continuation to the study of the sensitivity analysis in the sample variograms (Choi and Tanaka, 1999), the present article proposes a method to detect observations which are influential to the robust variograms. To do this, we derive the influence function for Cressie and Hawkins (1980)s robust variograms assuming that the underlying process of the observed spatial data is second-order stationary. A real numerical example is analyzed to show the validity or usefulness of the proposed influence function.

Outlier detection based on a change of likelihood

Myung Geun Kim 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.5

A general method of detecting outliers based on a change of likelihood by using the influence function is suggested. It can be applied to all kinds of distributions that are specified by parameters. For the multivariate normal case, specific computations are made to get the corre- sponding conditional influence function. A numerical example is provided for illustration. A general method of detecting outliers based on a change of likelihood by using the influence function is suggested. It can be applied to all kinds of distributions that are specified by parameters. For the multivariate normal case, specific computations are made to get the corre- sponding conditional influence function. A numerical example is provided for illustration.

A Study on a One-step Pairwise GM-estimator in Linear Models

Song, Moon-Sup,Kim, Jin-Ho The Korean Statistical Society 1997 Journal of the Korean Statistical Society Vol.26 No.1

In the linear regression model $y_{i}$ = .alpha. $x_{i}$ $^{T}$ .beta. + .epsilon.$_{i}$ , i = 1,2,...,n, the weighted pairwise absolute deviation (WPAD) estimator was defined by minimizing the dispersion function D (.beta.) = .sum..sum.$_{{i<j}}$ $w_{{ij}}$$\mid$ $r_{j}$ (.beta.) $r_{i}$ (.beta.)$\mid$, where $r_{i}$ (.beta.)'s are residuals and $w_{{ij}}$'s are weights. This estimator can achive bounded total influence with positive breakdown by choice of weights $w_{{ij}}$. In this paper, we consider a more general type of dispersion function than that of D(.beta.) and propose a pairwise GM-estimator based on the dispersion function. Under some regularity conditions, the proposed estimator has a bounded influence function, a high breakdown point, and asymptotically a normal distribution. Results of a small-sample Monte Carlo study are also presented. presented.