• Title, Summary, Keyword: Chi-Square

### A Rao-Robson Chi-Square Test for Multivariate Normality Based on the Mahalanobis Distances

• Park, Cheolyong
• Communications for Statistical Applications and Methods
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• v.7 no.2
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• pp.385-392
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• 2000
• Many tests for multivariate normality are based on the spherical coordinates of the scaled residuals of multivariate observations. Moore and Stubblebine's (1981) Pearson chi-square test is based on the radii of the scaled residuals, or equivalently the sample Mahalanobis distances of the observations from the sample mean vector. The chi-square statistic does not have a limiting chi-square distribution since the unknown parameters are estimated from ungrouped data. We will derive a simple closed form of the Rao-Robson chi-square test statistic and provide a self-contained proof that it has a limiting chi-square distribution. We then provide an illustrative example of application to a real data with a simulation study to show the accuracy in finite sample of the limiting distribution.

### Likelihood ratio in estimating Chi-square parameter

• Rahman, Mezbahur
• Journal of the Korean Data and Information Science Society
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• v.20 no.3
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• pp.587-592
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• 2009
• The most frequent use of the chi-square distribution is in the area of goodness-of-t of a distribution. The likelihood ratio test is a commonly used test statistic as the maximum likelihood estimate in statistical inferences. The recently revised versions of the likelihood ratio test statistics are used in estimating the parameter in the chi-square distribution. The estimates are compared with the commonly used method of moments and the maximum likelihood estimate.

### Distribution of a Sum of Weighted Noncentral Chi-Square Variables

• Heo, Sun-Yeong;Chang, Duk-Joon
• Communications for Statistical Applications and Methods
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• v.13 no.2
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• pp.429-440
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• 2006
• In statistical computing, it is often for researchers to need the distribution of a weighted sum of noncentral chi-square variables. In this case, it is very limited to know its exact distribution. There are many works to contribute to this topic, e.g. Imhof (1961) and Solomon-Stephens (1977). Imhof's method gives good approximation to the true distribution, but it is not easy to apply even though we consider the development of computer technology Solomon-Stephens's three moment chi-square approximation is relatively easy and accurate to apply. However, they skipped many details, and their simulation is limited to a weighed sum of central chi-square random variables. This paper gives details on Solomon-Stephens's method. We also extend their simulation to the weighted sum of non-central chi-square distribution. We evaluated approximated powers for homogeneous test and compared them with the true powers. Solomon-Stephens's method shows very good approximation for the case.

### Goodness-of-fit tests for a proportional odds model

• Lee, Hyun Yung
• Journal of the Korean Data and Information Science Society
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• v.24 no.6
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• pp.1465-1475
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• 2013
• The chi-square type test statistic is the most commonly used test in terms of measuring testing goodness-of-fit for multinomial logistic regression model, which has its grouped data (binomial data) and ungrouped (binary) data classified by a covariate pattern. Chi-square type statistic is not a satisfactory gauge, however, because the ungrouped Pearson chi-square statistic does not adhere well to the chi-square statistic and the ungrouped Pearson chi-square statistic is also not a satisfactory form of measurement in itself. Currently, goodness-of-fit in the ordinal setting is often assessed using the Pearson chi-square statistic and deviance tests. These tests involve creating a contingency table in which rows consist of all possible cross-classifications of the model covariates, and columns consist of the levels of the ordinal response. I examined goodness-of-fit tests for a proportional odds logistic regression model-the most commonly used regression model for an ordinal response variable. Using a simulation study, I investigated the distribution and power properties of this test and compared these with those of three other goodness-of-fit tests. The new test had lower power than the existing tests; however, it was able to detect a greater number of the different types of lack of fit considered in this study. I illustrated the ability of the tests to detect lack of fit using a study of aftercare decisions for psychiatrically hospitalized adolescents.

### On the Robustness of Chi-square Test Procedure for a Compounded Multivariate Normal Mean

• Kim, Hea-Jung
• Communications for Statistical Applications and Methods
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• v.2 no.2
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• pp.330-335
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• 1995
• The rebustness of one sample Chi-square test for multivariate normal mean vector is investigated when the multivariate normal population is mixed with another multivariate normal population with differing in the mean vector. Explicit expressions for the level of significance and power of the test are derived. Some numerical results indicate that the Chi-square test procedure is quite robust against slight mixtures of multivariate normal populations differing in location parameters.

### On an Approximation to the Distribution of Product of Independent Beta Variates

• Hea Jung Kim
• Communications for Statistical Applications and Methods
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• v.1 no.1
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• pp.81-86
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• 1994
• A Chi-square approximation to the distribution of product of independent Beta variates denoted by U is developed. The distribution is commonly used as a test criterion for the general linear hypothesis about the multivariate linear models. The approximation is obtained by fitting a logarithmic function of U to a Chi-square variate in terms of the first three moments. It is compared with the well known approximations due to Box(1949), Rao(1948), and Mudholkar and Trivedi(1980). It is found that the Chi-square approximation compares favorably with the other three approximations.

### SOME RESULTS RELATED TO DISTRIBUTION FUNCTIONS OF CHI-SQUARE TYPE RANDOM VARIABLES WITH RANDOM DEGREES OF FREEDOM

• Hung, Tran Loc;Thanh, Tran Thien;Vu, Bui Quang
• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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• pp.509-522
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• 2008
• The main aim of this paper is to present some results related to asymptotic behavior of distribution functions of random variables of chi-square type $X^2_N={\Sigma}^N_{i=1}\;X^2_i$ with degrees of freedom N, where N is a positive integer-valued random variable independent on all standard normally distributed random variables $X_i$. Two ways for computing the distribution functions of chi-square type random variables with random degrees of freedom are considered. Moreover, some tables concerning considered distribution functions are demonstrated in Appendix.

### Comparison of Rigorous Design Procedure with Approximate Design Procedure for Variable Sampling Plans Indexed by Quality Loss

• Ishii, Yoma;Arizono, Ikuo;Tomohiro, Ryosuke;Takemoto, Yasuhiko
• Industrial Engineering and Management Systems
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• v.15 no.3
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• pp.231-238
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• 2016
• Traditional acceptance sampling plans have focused on the proportion of nonconforming items as an attribute criterion for quality. In today's modern quality management under high quality production environments, the reduction of the deviation from a target value in a quality characteristic has become the most important purpose. In consequence, various inspection plans for the purpose of reducing the deviation from the target value in the quality characteristic have been investigated. In this case, a concept of the quality loss evaluated by the deviation from the target value has been accepted as the variable evaluation criterion of quality. Further, some quality measures based on the quality loss have been devised; e.g. the process loss and the process capability index. Then, as one of inspection plans based on the quality loss, the rigorous design procedure for the variable sampling plan having desired operating characteristics (VS-OC plan) indexed by the quality loss has been proposed by Yen and Chang in 2009. By the way, since the estimator of the quality loss obeys the non-central chi-square distribution, the rigorous design procedure for the VS-OC plan indexed by the quality loss is complicated. In particular, the rigorous design procedure for the VS-OC plan requires a large number of the repetitive and complicated numerical calculation about the non-central chi-square distribution. On the other hand, an approximate design procedure for the VS-OC plan has been proposed before the proposal of the above rigorous design procedure. The approximate design procedure for the VS-OC plan has been constructed by combining Patnaik approximation relating the non-central chi-square distribution to the central chi-square distribution and Wilson-Hilferty approximation relating the central chi-square distribution to the standard normal distribution. Then, the approximate design procedure has been devised as a convenient procedure without complicated and repetitive numerical calculations. In this study, through some comparisons between the rigorous and approximate design procedures, the applicability of the approximate design procedure has been confirmed.

### A Scene Change Detection Technique using the Weighted $\chi^2$-test and the Automated Threshold-Decision Algorithm (변형된 $\chi^2$- 테스트와 자동 임계치-결정 알고리즘을 이용한 장면전환 검출 기법)

• Ko, Kyong-Cheol;Rhee, Yang-Won
• Journal of the Institute of Electronics Engineers of Korea CI
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• v.42 no.4
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• pp.51-58
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• 2005
• This paper proposes a robust scene change detection technique that uses the weighted chi-square test and the automated threshold-decision algorithms. The weighted chi-square test can subdivide the difference values of individual color channels by calculating the color intensities according to NTSC standard, and it can detect the scene change by joining the weighted color intensities to the predefined chi-square test which emphasize the comparative color difference values. The automated threshold-decision at algorithm uses the difference values of frame-to-frame that was obtained by the weighted chi-square test. At first, The Average of total difference values is calculated and then, another average value is calculated using the previous average value from the difference values, finally the most appropriate mid-average value is searched and considered the threshold value. Experimental results show that the proposed algorithms are effective and outperform the previous approaches.

### Empirical Comparisons of Disparity Measures for Three Dimensional Log-Linear Models

• Park, Y.S.;Hong, C.S.;Jeong, D.B.
• Journal of the Korean Data and Information Science Society
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• v.17 no.2
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• pp.543-557
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• 2006
• This paper is concerned with the applicability of the chi-square approximation to the six disparity statistics: the Pearson chi-square, the generalized likelihood ratio, the power divergence, the blended weight chi-square, the blended weight Hellinger distance, and the negative exponential disparity statistic. Three dimensional contingency tables of small and moderate sample sizes are generated to be fitted to all possible hierarchical log-linear models: the completely independent model, the conditionally independent model, the partial association models, and the model with one variable independent of the other two. For models with direct solutions of expected cell counts, point estimates and confidence intervals of the 90 and 95 percentage points of six statistics are explored. For model without direct solutions, the empirical significant levels and the empirical powers of six statistics to test the significance of the three factor interaction are computed and compared.