Test for homogeneity in gamma mixture models using likelihood ratio

Wong, Tony Siu Tung; Li, Wai Keung

doi:10.1016/j.csda.2013.09.001

Cited by 9 publications

(7 citation statements)

References 25 publications

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“…For the single gamma distribution, parameters were estimated with the maximum-likelihood (ML) method, while for the gamma mixture distribution they were estimated with the expectation maximization (EM) method (Dempster et al, 1977 ). A penalized likelihood ratio test was implemented to determine the goodness of fit of the mixture gamma compared to the single gamma model (Wong and Li, 2014 ) and choose between the two.…”

Section: Methodsmentioning

confidence: 99%

MRI Analysis of White Matter Myelin Water Content in Multiple Sclerosis: A Novel Approach Applied to Finding Correlates of Cortical Thinning

et al. 2017

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A novel lesion-mask free method based on a gamma mixture model was applied to myelin water fraction (MWF) maps to estimate the association between cortical thickness and myelin content, and how it differs between relapsing-remitting (RRMS) and secondary-progressive multiple sclerosis (SPMS) groups (135 and 23 patients, respectively). It was compared to an approach based on lesion masks. The gamma mixture distribution of whole brain, white matter (WM) MWF was characterized with three variables: the mode (most frequent value) m1 of the gamma component shown to relate to lesion, the mode m2 of the component shown to be associated with normal appearing (NA) WM, and the mixing ratio (λ) between the two distributions. The lesion-mask approach relied on the mean MWF within lesion and within NAWM. A multivariate regression analysis was carried out to find the best predictors of cortical thickness for each group and for each approach. The gamma-mixture method was shown to outperform the lesion-mask approach in terms of adjusted R2, both for the RRMS and SPMS groups. The predictors of the final gamma-mixture models were found to be m1 (β = 1.56, p < 0.005), λ (β = −0.30, p < 0.0005) and age (β = −0.0031, p < 0.005) for the RRMS group (adjusted R2 = 0.16), and m2 (β = 4.72, p < 0.0005) for the SPMS group (adjusted R2 = 0.45). Further, a DICE coefficient analysis demonstrated that the lesion mask had more overlap to an ROI associated with m1, than to an ROI associated with m2 (p < 0.00001), and vice versa for the NAWM mask (p < 0.00001). These results suggest that during the relapsing phase, focal WM damage is associated with cortical thinning, yet in SPMS patients, global WM deterioration has a much stronger influence on secondary degeneration. Through these findings, we demonstrate the potential contribution of myelin loss on neuronal degeneration at different disease stages and the usefulness of our statistical reduction technique which is not affected by the typical bias associated with approaches based on lesion masks.

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Section: Methodsmentioning

confidence: 99%

MRI Analysis of White Matter Myelin Water Content in Multiple Sclerosis: A Novel Approach Applied to Finding Correlates of Cortical Thinning

et al. 2017

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“…Define the modified likelihood ratio statistic by

M L R T_{m} = 2 log L^{p} ({\overset{Ψ}{true}}_{m + 1}^{p}; y, c) - 2 log L^{p} ({\overset{Ψ}{true}}_{m}^{p}; y, 0) .

Wong & Li () showed that when the true model is a homogeneous model, the statistic

M L R T_{1}

is distributed as a mixture of a point mass at zero and a chi‐squared distribution. This null distribution is bounded below the aforementioned chi‐square distribution.…”

Section: Residuals Analysismentioning

confidence: 99%

“…The modified likelihood ratio statistic (Chen, Chen & Kalbfleisch 2001) overcomes the difficulty of obtaining consistent estimators under the breakdown of regularity conditions to which the ordinary likelihood ratio statistic (Wilks 1938) is subject. A new problem arises however, in that the limiting distribution of this modified statistic is a mixture of chi-squared distributions (Lindsay 1995;Wong & Li 2014;Zeng & Wong 2014). Estimation of the unknown weights in the mixture distribution is not very precise (Wong & Li 2014).…”

Section: Introductionmentioning

confidence: 99%

“…A new problem arises however, in that the limiting distribution of this modified statistic is a mixture of chi‐squared distributions (Lindsay ; Wong & Li ; Zeng & Wong ). Estimation of the unknown weights in the mixture distribution is not very precise (Wong & Li ). The research presented in this paper provides a decision criterion on the basis of an upper bound obtained from the simple chi‐squared distribution.…”

Section: Introductionmentioning

confidence: 99%

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Statistical Analysis of Heat Waves in the State of Victoria in Australia

Wong

2015

Aus NZ J of Statistics

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Summary The recent blistering heat waves of 2009 in the state of Victoria in Australia were so unprecedented in terms of duration and intensity that society was largely unprepared. These heat waves caused serious health, social and economic problems. In this paper, the daily maximum temperatures at ten selected stations are studied. Auto‐regressive integrated moving‐average models are used to prewhiten the time series. Uncorrelated, non‐normal and heavy‐tailed residuals are analyzed by means of a new skew t‐mixture distribution. The number of mixture components is effectively determined by an innovative penalisation procedure. It is shown that the resulting skew t‐mixture models provide an acceptable fit in all cases. Possible future temperature patterns are obtained through simulation. It is forecast that the average duration of high temperature episodes will increase by two to three days per year and a new eight‐year high temperature level is very likely in the coming few years. The relationship between heavy tail behaviour of the fitted distribution and heat waves is noteworthy.

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“…Verbelen et al (2015) developed numerical strategies for fitting Erlang mixtures with structural scale parameter when data also contain censored and truncated observations. More application examples of finite Gamma mixtures can be found in Liu et al (2003), Wong and Li (2014), Willmot and Lin (2011), and He and Chen (2020).…”

Section: Introductionmentioning

confidence: 99%

Strong consistency of the MLE under two-parameter Gamma mixture models with a structural scale parameter

Chen

2021

Adv Data Anal Classif

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We study the strong consistency of the maximum likelihood estimator under a special finite mixture of two-parameter Gamma distributions. Somewhat surprisingly, the likelihood function under Gamma mixture with a set of independent and identically distributed observations is unbounded. There exist many sets of nonsensical parameter values at which the likelihood value is arbitrarily large. This leads to an inconsistent, or arguably undefined, maximum likelihood estimator. Interestingly, when the scale or shape parameter in the finite Gamma mixture model is structural, the maximum likelihood estimator of the mixing distribution is well defined and strongly consistent. Establishing the consistency when the shape parameter is structural is technically less challenging and already given in the literature. In this paper, we prove the consistency when the scale parameter is structural and provide some illustrative simulation experiments. We further include an application example of the model with a structural scale parameter to salary potential data. We conclude that the Gamma mixture distribution with a structural scale parameter provides another flexible yet relatively parsimonious model for observations with intrinsic positive values.

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Test for homogeneity in gamma mixture models using likelihood ratio

Cited by 9 publications

References 25 publications

MRI Analysis of White Matter Myelin Water Content in Multiple Sclerosis: A Novel Approach Applied to Finding Correlates of Cortical Thinning

MRI Analysis of White Matter Myelin Water Content in Multiple Sclerosis: A Novel Approach Applied to Finding Correlates of Cortical Thinning

Statistical Analysis of Heat Waves in the State of Victoria in Australia

Strong consistency of the MLE under two-parameter Gamma mixture models with a structural scale parameter

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