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This work is motivated by analysis of protein expression levels quantified using immunofluorescence immunohistochemistry assays of human tissues. The distributions of cellular protein expression levels in a tissue often exhibit multimodality, skewness and heavy tails, but there is a substantial variability between distributions in different tissues from different subjects, while some of these mixture distributions include components consistent with the assumption of a normal distribution. To accommodate such diversity, we propose a mixture of 4-parameter Tukey\u2019s <jats:italic>g<\/jats:italic>- &amp;-<jats:italic>h<\/jats:italic> distributions for fitting finite mixtures with both Gaussian and non-Gaussian components. Tukey\u2019s <jats:italic>g<\/jats:italic>- &amp;-<jats:italic>h<\/jats:italic> distribution is a flexible model that allows variable degree of skewness and kurtosis in mixture components, including normal distribution as a particular case. Since the likelihood of the Tukey\u2019s <jats:italic>g<\/jats:italic>- &amp;-<jats:italic>h<\/jats:italic> mixtures does not have a closed analytical form, we propose a quantile least Mahalanobis distance (QLMD) estimator for parameters of such mixtures. QLMD is an indirect estimator minimizing the Mahalanobis distance between the sample and model-based quantiles, and its asymptotic properties follow from the general theory of indirect estimation. We have developed a stepwise algorithm to select a parsimonious Tukey\u2019s <jats:italic>g<\/jats:italic>- &amp;-<jats:italic>h<\/jats:italic> mixture model and implemented all proposed methods in the R package QuantileGH available on CRAN. A simulation study was conducted to evaluate performance of the Tukey\u2019s <jats:italic>g<\/jats:italic>- &amp;-<jats:italic>h<\/jats:italic> mixtures and compare to performance of mixtures of skew-normal or skew-<jats:italic>t<\/jats:italic> distributions. The Tukey\u2019s <jats:italic>g<\/jats:italic>- &amp;-<jats:italic>h<\/jats:italic> mixtures were applied to model cellular expressions of Cyclin D1 protein in breast cancer tissues, and resulting parameter estimates evaluated as predictors of progression-free survival.<\/jats:p>","DOI":"10.1007\/s11222-025-10596-9","type":"journal-article","created":{"date-parts":[[2025,3,15]],"date-time":"2025-03-15T05:13:07Z","timestamp":1742015587000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Estimation and model selection for finite mixtures of Tukey\u2019s g- &amp;-h distributions"],"prefix":"10.1007","volume":"35","author":[{"given":"Tingting","family":"Zhan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Misung","family":"Yi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Amy R.","family":"Peck","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hallgeir","family":"Rui","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Inna","family":"Chervoneva","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,3,15]]},"reference":[{"issue":"6","key":"10596_CR1","doi-asserted-by":"publisher","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","volume":"19","author":"H Akaike","year":"1974","unstructured":"Akaike, H.: A new look at the statistical model identification. 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