• Journal of the Korean Statistical Society
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• 제30권2호
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• pp.179-192
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• 2001

When a statistical model has a hierarchical structure such as multilayer perceptrons in neural networks or Gaussian mixture density representation, the model includes distribution with unidentifiable parameters when the structure becomes redundant. Since the exact structure is unknown, we need to carry out statistical estimation or learning of parameters in such a model. From the geometrical point of view, distributions specified by unidentifiable parameters become a singular point in the parameter space. The problem has been remarked in many statistical models, and strange behaviors of the likelihood ratio statistics, when the null hypothesis is at a singular point, have been analyzed so far. The present paper studies asymptotic behaviors of the maximum likelihood estimator and the Bayesian predictive estimator, by using a simple cone model, and show that they are completely different from regular statistical models where the Cramer-Rao paradigm holds. At singularities, the Fisher information metric degenerates, implying that the cramer-Rao paradigm does no more hold, and that he classical model selection theory such as AIC and MDL cannot be applied. This paper is a first step to establish a new theory for analyzing the accuracy of estimation or learning at around singularities.

• 응용통계연구
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• 제8권2호
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• pp.109-119
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• 1995

두 모집단에서 임의로 관측중단도니 두 표본을 얻었을 때, 두 모집단의 분포가 같다는 가설을 검정하기 위한 카이제곱 검정방법이 제안되었다. 여기서 제안된 통계량은 대립가설이 두 모집단의 분포가 같지 않다는 양측가설일 때 쓰일 수 있다. 귀무가설이 사실일 때 제안된 통계량의 극한분포는 카이제곱 분포가 된다. 두 가지 형태의 카이제곱 검정통계량이 제안되었는데, 하나는 product-limit 추정치로부터 얻은 관측된 칸(cell) 확률의 차이들의 벡터의 이차형식으로 표현된 것이고, 다른 하나는 간단한 합의 모양으로 표현된 것이다. 두 형태의 검정통계량을 사용하여 암치료를 위한 화학요법 실험으로부터 얻은 자료를 분석하여 보았다.

• Journal of the Korean Statistical Society
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• 제33권1호
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• pp.79-97
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• 2004

We propose a permutation approach to the classic Behrens-Fisher problem of comparing two means in the presence of unequal variances. It is motivated by the observation that a paired test is valid whether or not the variances are equal. Rather than using a single arbitrary pairing of the data, we average over all possible pairings. We do this in both a parametric and nonparametric setting. When the sample sizes are equal, the parametric version is equivalent to referral of the unpaired t-statistic to a t-table with half the usual degrees of freedom. The derivation provides an interesting representation of the unpaired t-statistic in terms of all possible pairwise t-statistics. The nonparametric version uses the same idea of considering all different pairings of data from the two groups, but applies it to a permutation test setting. Each pairing gives rise to a permutation distribution obtained by relabeling treatment and control within pairs. The totality of different mean differences across all possible pairings and relabelings forms the null distribution upon which the p-value is based. The conservatism of this procedure diminishes as the disparity in variances increases, disappearing completely when the ratio of the smaller to larger variance approaches 0. The nonparametric procedure behaves increasingly like a paired t-test as the sample sizes increase.

• Journal of the Korean Statistical Society
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• 제30권3호
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• pp.347-366
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• 2001

Robust analogues of the likelihood ratio test are considered for testing of hypotheses involving multiple discrete distributions. The test statistics are generalizations of the Hellinger deviance test of Simpson(1989) and disparity tests of Lindsay(1994), obtained by looking at a 'penalized' version of the distances; harris and Basu (1994) suggest that the penalty be based on reweighting the empty cells. The results show that often the tests based on the ordinary and penalized distances enjoy better robustness properties than the likelihood ratio test. Also, the tests based on the penalized distances are improvements over those based on the ordinary distances in that they are much closer to the likelihood ratio tests at the null and their convergence to the x$^2$ distribution appears to be dramatically faster; extensive simulation results show that the improvement in performance of the tests due to the penalty is often substantial in small samples.

• Journal of the Korean Data and Information Science Society
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• 제4권
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• pp.137-146
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• 1993

The Proposed goodness-of-fit test Statistic $\hat{\lambda}_{\alpha}$ derived from the test Statistc in Kim (1992) is itself a smoothing parameter which is selected to minimize an estimated MISE for a truncated series estimator, $d_{\hat{\lambda}{n}}$, of the comparison density function. Therefore, this test statistic leads immediately to a point estimate of the density function in the event that $H_{0}$ is ejected. The limiting distribution of $\hat{\lambda}_{\alpha}$ was obtained under the null hypothesis. It is also shown that this test is consistent against fixed alternatives.

• Communications for Statistical Applications and Methods
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• 제9권3호
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• pp.765-774
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• 2002

Suppose one is given a vector X of a finite set of quantities $X_i$ which are independent Poisson random variables. A null hypothesis $H_0$ about E(X) is to be tested against an alternative hypothesis $H_1$. A quantity $\sum\limits_{i}w_ix_i$ is to be computed and used for the test. The optimal values of $W_i$ are calculated for three cases: (1) signal to noise ratio is used in the test, (2) normal approximations with unequal variances to the Poisson distributions are used in the test, and (3) the Poisson distribution itself is used. The above three cases are considered to the situations that are without background noise and with background noise. A comparison is made of the optimal values of $W_i$ in the three cases for both situations.

• Journal of the Korean Data and Information Science Society
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• 제13권2호
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• pp.307-315
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• 2002

Suppose one is given a vector X of a finite set of quantities $X_i$ which are independent Poisson signals. A null hypothesis $H_0$ about E(X) is to be tested against an alternative hypothesis $H_1$. A quantity $$\sum\limits_{i}\omega_ix_i$$ is to be computed and used for the test. The optimal values of $\omega_i$ are calculated for three cases : (1) signal to noise ratio is used in the test, (2) normal approximations with unequal variances to the Poisson distributions are used in the test, and (3) the Poisson distribution it self is used. A comparison is made of the optimal values of $\omega_i$ in the three cases as parameter goes to infinity.

• Communications for Statistical Applications and Methods
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• 제8권2호
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• pp.337-346
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• 2001

In this paper we consider the problem of identifying and testing the outliers in linear regression. first we consider the problem for testing the null hypothesis of no outliers. The test based on the ratio of two scale estimates is proposed. We show the asymptotic distribution of the test statistic by Monte Carlo simulation and investigate its properties. Next we consider the problem of identifying the outliers. A forward sequential procedure based on the suggested test is proposed and shown to perform fairly well. The forward sequential procedure is unaffected by masking and swamping effects because the test statistic is based on robust estimate.

• 대한수학회지
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• 제47권2호
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• pp.385-407
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• 2010

To study the relationship between the levels of chemical pollutants and the number of daily total hospital admissions for respiratory diseases and to find the effect of temperature/relative humidity on the admission number, Wong et al. [17] introduced the varying-coefficient single-index model (VCSIM). As pointed out, it is a popular multivariate nonparametric fitting technique. However, the tests of the model have not been very well developed. In this paper, based on the estimators obtained by the local linear technique, the average method and the one-step back-fitting technique in the VCSIM, the generalized likelihood ratio (GLR) tests for varying-coefficient parts on the VCSIM are established. Under the null hypotheses the new proposed GLR tests follow the $\chi^2$-distribution asymptotically with scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Simulations are conducted to evaluate the test procedure empirically. A real example is used to illustrate the performance of the testing approach.

• International Journal of Reliability and Applications
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• 제15권2호
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• pp.99-110
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• 2014

Based on the kernel function, a new test is presented, testing $H_0:\bar{F}$ is exponential against $H_1:\bar{F}$ is UBACT and not exponential is given in section 2. Monte Carlos null distribution critical points for sample sizes n = 5(5)100 is investigated in section 3. The Pitman asymptotic efficiency for common alternatives is obtained in section 4. In section 5 we propose a test statistic for censored data. Finally, a numerical examples in medical science for complete and censored data using real data is presented in section 6.