This paper investigates the modified likelihood ratio test（LRT） for homogeneity in normal mixtures of two samples with mixing proportions unknown. It is proved that the limit distribution of the modified likelihood ...This paper investigates the modified likelihood ratio test（LRT） for homogeneity in normal mixtures of two samples with mixing proportions unknown. It is proved that the limit distribution of the modified likelihood ratio test is X^2（1）.展开更多
In this paper, we extend the generalized likelihood ratio test to the varying-coefficient models with censored data. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null d...In this paper, we extend the generalized likelihood ratio test to the varying-coefficient models with censored data. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distribution follows a distribution, with the scale constant and the number of degree of freedom being independent of nuisance parameters or functions, which is called the wilks phenomenon. Both simulated and real data examples are given to illustrate the performance of the testing approach.展开更多
This paper investigates the asymptotic properties of the modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models with an unknown structural parameter. It is shown that the modifi...This paper investigates the asymptotic properties of the modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models with an unknown structural parameter. It is shown that the modified likelihood ratio statistic has χ22 null limiting distribution.展开更多
This paper investigates the asymptotic properties of a modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models of two samples.The asymptotic null distribution of the modified lik...This paper investigates the asymptotic properties of a modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models of two samples.The asymptotic null distribution of the modified likelihood ratio statistic is found to be X2^2,where X2^2 is a chi-squared distribution with 2 degrees of freedom.展开更多
In this paper, asymptotic expansions of the distribution of the likelihood ratio statistic for testing sphericity in a crowth curve model have been derived in the null and nonnull cases when the alternatives are dose ...In this paper, asymptotic expansions of the distribution of the likelihood ratio statistic for testing sphericity in a crowth curve model have been derived in the null and nonnull cases when the alternatives are dose to the null hypothesis. These expansions are given in series form of beta distributions.展开更多
It is well-known that the power of Cochran’s Q test to assess the presence of heterogeneity among treatment effects in a clinical meta-analysis is low due to the small number of studies combined. Two modified tests (...It is well-known that the power of Cochran’s Q test to assess the presence of heterogeneity among treatment effects in a clinical meta-analysis is low due to the small number of studies combined. Two modified tests (PL1, PL2) were proposed by replacing the profile maximum likelihood estimator (PMLE) into the variance formula of logarithm of risk ratio in the standard chi-square test statistic for testing the null common risk ratios across all k studies (i = 1, L, k). The simply naive test (SIM) as another comparative candidate has considerably arisen. The performance of tests in terms of type I error rate under the null hypothesis and power of test under the random effects hypothesis was done via a simulation plan with various combinations of significance levels, numbers of studies, sample sizes in treatment and control arms, and true risk ratios as effect sizes of interest. The results indicated that for moderate to large study sizes (k?≥ 16)?in combination with moderate to large sample sizes?(?≥ 50), three tests (PL1, PL2, and Q) could control type I error rates in almost all situations. Two proposed tests (PL1, PL2) performed best with the highest power when?k?≥ 16?and moderate sample sizes (= 50,100);this finding was very useful to make a recommendation to use them in practical situations. Meanwhile, the standard Q test performed best when?k?≥ 16 and large sample sizes (≥ 500). Moreover, no tests were reasonable for small sample sizes (≤ 10), regardless of study size k. The simply naive test (SIM) is recommended to be adopted with high performance when k = 4 in combination with (≥ 500).展开更多
Hypothesis testing analysis and unknown parameter estimation of both the intermediate frequency(IF) and baseband GPS signal detection are given by using the generalized likelihood ratio test(GLRT) approach,applying th...Hypothesis testing analysis and unknown parameter estimation of both the intermediate frequency(IF) and baseband GPS signal detection are given by using the generalized likelihood ratio test(GLRT) approach,applying the model of GPS signal in white Gaussian noise,It is proved that the test statistic follows central or noncentral F distribution,It is also pointed out that the test statistic is nearly identical to central or noncentral chi-squared distribution because the processing samples are large enough to be considered as infinite in GPS acquisition problem.It is also proved that the probability of false alarm,the probability of detection and the threshold are affected largely when the hypothesis testing refers to the full pseudorandom noise(PRN) code phase and Doppler frequency search space cells instead of each individual cell.The performance of the test statistic is also given with combining the noncoherent integration.展开更多
Varying-coefficient models are a useful extension of classical linear model. They are widely applied to economics, biomedicine, epidemiology, and so on. There are extensive studies on them in the latest three decade y...Varying-coefficient models are a useful extension of classical linear model. They are widely applied to economics, biomedicine, epidemiology, and so on. There are extensive studies on them in the latest three decade years. In this paper, many of models related to varying-coefficient models are gathered up. All kinds of the estimation procedures and theory of hypothesis test on the varying-coefficients model are summarized. Prom my opinion, some aspects waiting to study are proposed.展开更多
In this papert we give an approach for detecting one or more outliers inrandomized linear model.The likelihood ratio test statistic and its distributions underthe null hypothesis and the alternative hypothesis are giv...In this papert we give an approach for detecting one or more outliers inrandomized linear model.The likelihood ratio test statistic and its distributions underthe null hypothesis and the alternative hypothesis are given. Furthermore,the robustnessof the test statistic in a certain sense is proved. Finally,the optimality properties of thetest are derived.展开更多
In this paper, we propose the test statistic to check whether the nonparametric function in partially linear models is linear or not. We estimate the nonparametric function in alternative by using the local linear met...In this paper, we propose the test statistic to check whether the nonparametric function in partially linear models is linear or not. We estimate the nonparametric function in alternative by using the local linear method, and then estimate the parameters by the two stage method. The test statistic under the null hypothesis is calculated, and it is shown to be asymptotically normal.展开更多
We propose the test statistic to check whether the nonpararnetric functions in two partially linear models are equality or not in this paper. We estimate the nonparametric function both in null hypothesis and the alte...We propose the test statistic to check whether the nonpararnetric functions in two partially linear models are equality or not in this paper. We estimate the nonparametric function both in null hypothesis and the alternative by the local linear method, where we ignore the parametric components, and then estimate the parameters by the two stage method. The test statistic is derived, and it is shown to be asymptotically normal under the null hypothesis.展开更多
Coutsourides derived an ad hoc nuisance paratmeter removal test for testing equality of two multiple correlation matrices of two independent p variate normal populations under the assumption that a sample of size ...Coutsourides derived an ad hoc nuisance paratmeter removal test for testing equality of two multiple correlation matrices of two independent p variate normal populations under the assumption that a sample of size n is available from each population. This paper presents a likelihood ratio test criterion for testing equality of K multiple correlation matrices and extends the results to the testing of equality of K partial correlation matrices.展开更多
Consider I pairs of independent binomial variates x0i and x1i with corresponding parameters P0i and p1i and sample sizes n0i and n1i for i=1, …,I. Let △i = P1i-P0i be the difference of the two binomial parameters, w...Consider I pairs of independent binomial variates x0i and x1i with corresponding parameters P0i and p1i and sample sizes n0i and n1i for i=1, …,I. Let △i = P1i-P0i be the difference of the two binomial parameters, where △i’s are to be of interest and P0i’s are nuisance parameters. The null hypothesis of homogeneity on the risk difference can be written as展开更多
The paper considers a high-dimensional likelihood ratio(LR)test on the intraclass correlation structure of the multivariate normal population.When the dimension p and sample size N satisfy N−1>p→∞,it is proved th...The paper considers a high-dimensional likelihood ratio(LR)test on the intraclass correlation structure of the multivariate normal population.When the dimension p and sample size N satisfy N−1>p→∞,it is proved that the logarithmic LR statistic asymptotically obeys Gaussian distribution,and the explicit expressions of the mean and the variance are also obtained.The simulations demonstrate that our high-dimensional LR test method outperforms the traditional Chi-square approximation method or F-approximation method,and performs as efficient as the accurate high-dimensional Edgeworth expansion method and the more accurate high-dimensional Edgeworth expansion method in analyzing the intraclass covariance structure of highdimensional data.展开更多
Count data is almost always over-dispersed where the variance exceeds the mean. Several count data models have been proposed by researchers but the problem of over-dispersion still remains unresolved, more so in the c...Count data is almost always over-dispersed where the variance exceeds the mean. Several count data models have been proposed by researchers but the problem of over-dispersion still remains unresolved, more so in the context of change point analysis. This study develops a likelihood-based algorithm that detects and estimates multiple change points in a set of count data assumed to follow the Negative Binomial distribution. Discrete change point procedures discussed in literature work well for equi-dispersed data. The new algorithm produces reliable estimates of change points in cases of both equi-dispersed and over-dispersed count data;hence its advantage over other count data change point techniques. The Negative Binomial Multiple Change Point Algorithm was tested using simulated data for different sample sizes and varying positions of change. Changes in the distribution parameters were detected and estimated by conducting a likelihood ratio test on several partitions of data obtained through step-wise recursive binary segmentation. Critical values for the likelihood ratio test were developed and used to check for significance of the maximum likelihood estimates of the change points. The change point algorithm was found to work best for large datasets, though it also works well for small and medium-sized datasets with little to no error in the location of change points. The algorithm correctly detects changes when present and fails to detect changes when change is absent in actual sense. Power analysis of the likelihood ratio test for change was performed through Monte-Carlo simulation in the single change point setting. Sensitivity analysis of the test power showed that likelihood ratio test is the most powerful when the simulated change points are located mid-way through the sample data as opposed to when changes were located in the periphery. Further, the test is more powerful when the change was located three-quarter-way through the sample data compared to when the change point is closer (quarter-way) to the first observation.展开更多
Ridge type estimators are used to estimate regression parameters in a multiple linear regression model when multicolinearity exists among predictor variables. When different estimators are available, preliminary test ...Ridge type estimators are used to estimate regression parameters in a multiple linear regression model when multicolinearity exists among predictor variables. When different estimators are available, preliminary test estimation procedure is adopted to select a suitable estimator. In this paper, two ridge estimators, the Stochastic Restricted Liu Estimator and Liu Estimator are combined to define a new preliminary test estimator, namely the Preliminary Test Stochastic Restricted Liu Estimator (PTSRLE). The stochastic properties of the proposed estimator are derived, and the performance of PTSRLE is compared with SRLE in the sense of mean square error matrix (MSEM) and scalar mean square error (SMSE) for the two cases in which the stochastic restrictions are correct and not correct. Moreover the SMSE of PTSRLE based on Wald (WA), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are derived, and the performance of PTSRLE is compared using WA, LR and LM tests as a function of the shrinkage parameter d with respect to the SMSE. Finally a numerical example is given to illustrate some of the theoretical findings.展开更多
Mixed models provide a wide range of applications including hierarchical modeling and longitudinal studies. The tests of variance component in mixed models have long been a methodological challenge because of its boun...Mixed models provide a wide range of applications including hierarchical modeling and longitudinal studies. The tests of variance component in mixed models have long been a methodological challenge because of its boundary conditions. It is well documented in literature that the traditional first-order methods: likelihood ratio statistic, Wald statistic and score statistic, provide an excessively conservative approximation to the null distribution. However, the magnitude of the conservativeness has not been thoroughly explored. In this paper, we propose a likelihood-based third-order method to the mixed models for testing the null hypothesis of zero and non-zero variance component. The proposed method dramatically improved the accuracy of the tests. Extensive simulations were carried out to demonstrate the accuracy of the proposed method in comparison with the standard first-order methods. The results show the conservativeness of the first order methods and the accuracy of the proposed method in approximating the p-values and confidence intervals even when the sample size is small.展开更多
Empirical estimates of power and Type I error can be misleading if a statistical test does not perform at the stated rejection level under the null hypothesis. We employed the permutation test to control the empirical...Empirical estimates of power and Type I error can be misleading if a statistical test does not perform at the stated rejection level under the null hypothesis. We employed the permutation test to control the empirical type I errors for zero-inflated exponential distributions. The simulation results indicated that the permutation test can be used effectively to control the type I errors near the nominal level even the sample sizes are small based on four statistical tests. Our results attest to the permutation test being a valuable adjunct to the current statistical methods for comparing distributions with underlying zero-inflated data structures.展开更多
基金Supported by the National Natural Science Foundation of China(10661003)the SRF for ROCS,SEM([2004]527)the NSF of Guangxi(0728092)
文摘This paper investigates the modified likelihood ratio test（LRT） for homogeneity in normal mixtures of two samples with mixing proportions unknown. It is proved that the limit distribution of the modified likelihood ratio test is X^2（1）.
文摘In this paper, we extend the generalized likelihood ratio test to the varying-coefficient models with censored data. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distribution follows a distribution, with the scale constant and the number of degree of freedom being independent of nuisance parameters or functions, which is called the wilks phenomenon. Both simulated and real data examples are given to illustrate the performance of the testing approach.
基金the National Natural Science Foundation of China (Grant No. 10661003)the Natural Science Foundation of Guangxi (Grant No. 0728092) SRF for ROCS, SEM (Grant No. [2004]527)
文摘This paper investigates the asymptotic properties of the modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models with an unknown structural parameter. It is shown that the modified likelihood ratio statistic has χ22 null limiting distribution.
基金supported by the National Natural Science Foundation of China under Grant No.10661003SRF for ROCS,SEM under Grant No.[2004]527the Natm'aI Science Foundation of Guangxi under Grant No.0728092
文摘This paper investigates the asymptotic properties of a modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models of two samples.The asymptotic null distribution of the modified likelihood ratio statistic is found to be X2^2,where X2^2 is a chi-squared distribution with 2 degrees of freedom.
文摘In this paper, asymptotic expansions of the distribution of the likelihood ratio statistic for testing sphericity in a crowth curve model have been derived in the null and nonnull cases when the alternatives are dose to the null hypothesis. These expansions are given in series form of beta distributions.
文摘It is well-known that the power of Cochran’s Q test to assess the presence of heterogeneity among treatment effects in a clinical meta-analysis is low due to the small number of studies combined. Two modified tests (PL1, PL2) were proposed by replacing the profile maximum likelihood estimator (PMLE) into the variance formula of logarithm of risk ratio in the standard chi-square test statistic for testing the null common risk ratios across all k studies (i = 1, L, k). The simply naive test (SIM) as another comparative candidate has considerably arisen. The performance of tests in terms of type I error rate under the null hypothesis and power of test under the random effects hypothesis was done via a simulation plan with various combinations of significance levels, numbers of studies, sample sizes in treatment and control arms, and true risk ratios as effect sizes of interest. The results indicated that for moderate to large study sizes (k?≥ 16)?in combination with moderate to large sample sizes?(?≥ 50), three tests (PL1, PL2, and Q) could control type I error rates in almost all situations. Two proposed tests (PL1, PL2) performed best with the highest power when?k?≥ 16?and moderate sample sizes (= 50,100);this finding was very useful to make a recommendation to use them in practical situations. Meanwhile, the standard Q test performed best when?k?≥ 16 and large sample sizes (≥ 500). Moreover, no tests were reasonable for small sample sizes (≤ 10), regardless of study size k. The simply naive test (SIM) is recommended to be adopted with high performance when k = 4 in combination with (≥ 500).
文摘Hypothesis testing analysis and unknown parameter estimation of both the intermediate frequency(IF) and baseband GPS signal detection are given by using the generalized likelihood ratio test(GLRT) approach,applying the model of GPS signal in white Gaussian noise,It is proved that the test statistic follows central or noncentral F distribution,It is also pointed out that the test statistic is nearly identical to central or noncentral chi-squared distribution because the processing samples are large enough to be considered as infinite in GPS acquisition problem.It is also proved that the probability of false alarm,the probability of detection and the threshold are affected largely when the hypothesis testing refers to the full pseudorandom noise(PRN) code phase and Doppler frequency search space cells instead of each individual cell.The performance of the test statistic is also given with combining the noncoherent integration.
基金Foundation item： Supported by the National Natural Science Foundation of China（10501053） Acknowledgement I would like to thank Henan Society of Applied Statistics for which give me a chance to declare my opinion about the varying-coefficient model.
文摘Varying-coefficient models are a useful extension of classical linear model. They are widely applied to economics, biomedicine, epidemiology, and so on. There are extensive studies on them in the latest three decade years. In this paper, many of models related to varying-coefficient models are gathered up. All kinds of the estimation procedures and theory of hypothesis test on the varying-coefficients model are summarized. Prom my opinion, some aspects waiting to study are proposed.
文摘In this papert we give an approach for detecting one or more outliers inrandomized linear model.The likelihood ratio test statistic and its distributions underthe null hypothesis and the alternative hypothesis are given. Furthermore,the robustnessof the test statistic in a certain sense is proved. Finally,the optimality properties of thetest are derived.
文摘In this paper, we propose the test statistic to check whether the nonparametric function in partially linear models is linear or not. We estimate the nonparametric function in alternative by using the local linear method, and then estimate the parameters by the two stage method. The test statistic under the null hypothesis is calculated, and it is shown to be asymptotically normal.
文摘We propose the test statistic to check whether the nonpararnetric functions in two partially linear models are equality or not in this paper. We estimate the nonparametric function both in null hypothesis and the alternative by the local linear method, where we ignore the parametric components, and then estimate the parameters by the two stage method. The test statistic is derived, and it is shown to be asymptotically normal under the null hypothesis.
文摘Coutsourides derived an ad hoc nuisance paratmeter removal test for testing equality of two multiple correlation matrices of two independent p variate normal populations under the assumption that a sample of size n is available from each population. This paper presents a likelihood ratio test criterion for testing equality of K multiple correlation matrices and extends the results to the testing of equality of K partial correlation matrices.
文摘Consider I pairs of independent binomial variates x0i and x1i with corresponding parameters P0i and p1i and sample sizes n0i and n1i for i=1, …,I. Let △i = P1i-P0i be the difference of the two binomial parameters, where △i’s are to be of interest and P0i’s are nuisance parameters. The null hypothesis of homogeneity on the risk difference can be written as
基金Supported by National Natural Science Foundation of China(Grant No.11401169)Natural Science Foundation of Henan Province of China(Grant No.202300410089).
文摘The paper considers a high-dimensional likelihood ratio(LR)test on the intraclass correlation structure of the multivariate normal population.When the dimension p and sample size N satisfy N−1>p→∞,it is proved that the logarithmic LR statistic asymptotically obeys Gaussian distribution,and the explicit expressions of the mean and the variance are also obtained.The simulations demonstrate that our high-dimensional LR test method outperforms the traditional Chi-square approximation method or F-approximation method,and performs as efficient as the accurate high-dimensional Edgeworth expansion method and the more accurate high-dimensional Edgeworth expansion method in analyzing the intraclass covariance structure of highdimensional data.
文摘Count data is almost always over-dispersed where the variance exceeds the mean. Several count data models have been proposed by researchers but the problem of over-dispersion still remains unresolved, more so in the context of change point analysis. This study develops a likelihood-based algorithm that detects and estimates multiple change points in a set of count data assumed to follow the Negative Binomial distribution. Discrete change point procedures discussed in literature work well for equi-dispersed data. The new algorithm produces reliable estimates of change points in cases of both equi-dispersed and over-dispersed count data;hence its advantage over other count data change point techniques. The Negative Binomial Multiple Change Point Algorithm was tested using simulated data for different sample sizes and varying positions of change. Changes in the distribution parameters were detected and estimated by conducting a likelihood ratio test on several partitions of data obtained through step-wise recursive binary segmentation. Critical values for the likelihood ratio test were developed and used to check for significance of the maximum likelihood estimates of the change points. The change point algorithm was found to work best for large datasets, though it also works well for small and medium-sized datasets with little to no error in the location of change points. The algorithm correctly detects changes when present and fails to detect changes when change is absent in actual sense. Power analysis of the likelihood ratio test for change was performed through Monte-Carlo simulation in the single change point setting. Sensitivity analysis of the test power showed that likelihood ratio test is the most powerful when the simulated change points are located mid-way through the sample data as opposed to when changes were located in the periphery. Further, the test is more powerful when the change was located three-quarter-way through the sample data compared to when the change point is closer (quarter-way) to the first observation.
文摘Ridge type estimators are used to estimate regression parameters in a multiple linear regression model when multicolinearity exists among predictor variables. When different estimators are available, preliminary test estimation procedure is adopted to select a suitable estimator. In this paper, two ridge estimators, the Stochastic Restricted Liu Estimator and Liu Estimator are combined to define a new preliminary test estimator, namely the Preliminary Test Stochastic Restricted Liu Estimator (PTSRLE). The stochastic properties of the proposed estimator are derived, and the performance of PTSRLE is compared with SRLE in the sense of mean square error matrix (MSEM) and scalar mean square error (SMSE) for the two cases in which the stochastic restrictions are correct and not correct. Moreover the SMSE of PTSRLE based on Wald (WA), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are derived, and the performance of PTSRLE is compared using WA, LR and LM tests as a function of the shrinkage parameter d with respect to the SMSE. Finally a numerical example is given to illustrate some of the theoretical findings.
文摘Mixed models provide a wide range of applications including hierarchical modeling and longitudinal studies. The tests of variance component in mixed models have long been a methodological challenge because of its boundary conditions. It is well documented in literature that the traditional first-order methods: likelihood ratio statistic, Wald statistic and score statistic, provide an excessively conservative approximation to the null distribution. However, the magnitude of the conservativeness has not been thoroughly explored. In this paper, we propose a likelihood-based third-order method to the mixed models for testing the null hypothesis of zero and non-zero variance component. The proposed method dramatically improved the accuracy of the tests. Extensive simulations were carried out to demonstrate the accuracy of the proposed method in comparison with the standard first-order methods. The results show the conservativeness of the first order methods and the accuracy of the proposed method in approximating the p-values and confidence intervals even when the sample size is small.
文摘Empirical estimates of power and Type I error can be misleading if a statistical test does not perform at the stated rejection level under the null hypothesis. We employed the permutation test to control the empirical type I errors for zero-inflated exponential distributions. The simulation results indicated that the permutation test can be used effectively to control the type I errors near the nominal level even the sample sizes are small based on four statistical tests. Our results attest to the permutation test being a valuable adjunct to the current statistical methods for comparing distributions with underlying zero-inflated data structures.
