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Multivariate mixtures of Erlangs for density estimation under censoring. (English) Zbl 1422.62194

Summary: Multivariate mixtures of Erlang distributions form a versatile, yet analytically tractable, class of distributions making them suitable for multivariate density estimation. We present a flexible and effective fitting procedure for multivariate mixtures of Erlangs, which iteratively uses the EM algorithm, by introducing a computationally efficient initialization and adjustment strategy for the shape parameter vectors. We furthermore extend the EM algorithm for multivariate mixtures of Erlangs to be able to deal with randomly censored and fixed truncated data. The effectiveness of the proposed algorithm is demonstrated on simulated as well as real data sets.

MSC:

62H12 Estimation in multivariate analysis
62N01 Censored data models

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References:

[1] Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716-723 · Zbl 0314.62039
[2] Ampe B, Goethals K, Laevens H, Duchateau L (2012) Investigating clustering in interval-censored udder quarter infection times in dairy cows using a gamma frailty model. Prev Vet Med 106(3):251-257
[3] Asmussen S, Nerman O, Olsson M (1996) Fitting phase-type distributions via the EM algorithm. Scand J Stat 23:419-441 · Zbl 0898.62104
[4] Assaf D, Langberg NA, Savits TH, Shaked M (1984) Multivariate phase-type distributions. Oper Res 32(3):688-702 · Zbl 0558.60070
[5] Azzalini A, Bowman A (1990) A look at some data on the Old Faithful geyser. Appl Stat 39:357-365 · Zbl 0707.62186
[6] Cossette H, Côté M-P, Marceau E, Moutanabbir K (2013a) Multivariate distribution defined with Farlie-Gumbel-Morgenstern copula and mixed Erlang marginals: aggregation and capital allocation. Insur Math Econ 52(3):560-572 · Zbl 1284.60027
[7] Cossette H, Mailhot M, Marceau E, Mesfioui M (2013b) Bivariate lower and upper orthant value-at-risk. Eur Actuar J 3(2):321-357 · Zbl 1304.91097
[8] Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B (Method) 39(1):1-38 · Zbl 0364.62022
[9] Dhaene J, Tsanakas A, Valdez EA, Vanduffel S (2012) Optimal capital allocation principles. J Risk Insur 79(1):1-28
[10] Efron B, Tibshirani R (1994) An introduction to the bootstrap. Monographs on statistics & applied probability. Chapman & Hall, London
[11] Eisele K-T (2005) EM algorithm for bivariate phase distributions. In: ASTIN Colloquium, Zurich, Switzerland. http://www.actuaries.org/ASTIN/Colloquia/Zurich/Eisele.pdf · Zbl 1284.60027
[12] Frees EW, Carriere J, Valdez E (1996) Annuity valuation with dependent mortality. J Risk Insur 63:229-261
[13] Georges P, Lamy A-G, Nicolas E, Quibel G, Roncalli T (2001) Multivariate survival modelling: a unified approach with copulas. Unpublished paper, Groupe de Recherche Operationnelle, Credit Lyonnais, France · Zbl 1325.62107
[14] Goethals K, Ampe B, Berkvens D, Laevens H, Janssen P, Duchateau L (2009) Modeling interval-censored, clustered cow udder quarter infection times through the shared gamma frailty model. J Agric Biol Environ Stat 14(1):1-14 · Zbl 1306.62230
[15] Härdle W (1991) Smoothing techniques: with implementation in S. Springer, New York · Zbl 0716.62040
[16] Joe H (1997) Multivariate models and multivariate dependence concepts, vol 73. CRC Press, Boca Raton · Zbl 0990.62517
[17] Klugman SA, Panjer HH, Willmot GE (2013) Loss models: further topics. Wiley, Hoboken · Zbl 1273.62008
[18] Laevens H, Deluyker H, Schukken Y, De Meulemeester L, Vandermeersch R, De Muelenaere E, De Kruif A (1997) Influence of parity and stage of lactation on the somatic cell count in bacteriologically negative dairy cows. J Dairy Sci 80(12):3219-3226
[19] Lee G, Scott C (2012) EM algorithms for multivariate Gaussian mixture models with truncated and censored data. Comput Stat Data Anal 56(9):2816-2829 · Zbl 1255.62308
[20] Lee S, McLachlan GJ (2014) Finite mixtures of multivariate skew \[t\] t-distributions: some recent and new results. Stat Comput 24(2):181-202 · Zbl 1325.62107
[21] Lee SC, Lin XS (2010) Modeling and evaluating insurance losses via mixtures of Erlang distributions. N Am Actuar J 14(1):107-130
[22] Lee SC, Lin XS (2012) Modeling dependent risks with multivariate Erlang mixtures. ASTIN Bull 42(1):153-180 · Zbl 1277.62255
[23] Leung K-M, Elashoff RM, Afifi AA (1997) Censoring issues in survival analysis. Annu Rev Public Health 18(1):83-104
[24] Li Y, Gillespie BW, Shedden K, Gillespie JA (2015) Calculating profile likelihood estimates of the correlation coefficient in the presence of left, right or interval censoring and missing data. Working paper
[25] Lin TI (2009) Maximum likelihood estimation for multivariate skew normal mixture models. J Multivar Anal 100(2):257-265 · Zbl 1152.62034
[26] Mailhot M (2012) Mesures de risque et dépendance. Ph.D. thesis, Université Laval
[27] McLachlan G, Jones P (1988) Fitting mixture models to grouped and truncated data via the EM algorithm. Biometrics 44:571-578 · Zbl 0707.62214
[28] McLachlan G, Peel D (2001) Finite mixture models. Wiley, New York · Zbl 0963.62061
[29] McLachlan GJ, Krishnan T (2007) The EM algorithm and extensions, vol 382. Wiley, New York · Zbl 0882.62012
[30] Nair VN (1984) Confidence bands for survival functions with censored data: a comparative study. Technometrics 26(3):265-275
[31] Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New York · Zbl 1152.62030
[32] Olsson M (1996) Estimation of phase-type distributions from censored data. Scand J Stat 23:443-460 · Zbl 0898.62105
[33] Peel D, McLachlan G (2000) Robust mixture modelling using the t distribution. Stat Comput 10(4):339-348
[34] Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461-464 · Zbl 0379.62005
[35] Silverman BW (1986) Density estimation for statistics and data analysis, vol 26. Chapman & Hall, London · Zbl 0617.62042
[36] Tijms HC (1994) Stochastic models: an algorithmic approach. Wiley, New York · Zbl 0838.60075
[37] Turnbull BW (1976) The empirical distribution function with arbitrarily grouped, censored and truncated data. J R Stat Soc Ser B (Method) 38:290-295 · Zbl 0343.62033
[38] Verbelen R, Gong L, Antonio K, Badescu A, Lin XS (2015) Fitting mixtures of Erlangs to censored and truncated data using the EM algorithm. ASTIN Bull 45(3):729-758 · Zbl 1390.62227
[39] Willmot GE, Woo J-K (2007) On the class of Erlang mixtures with risk theoretic applications. N Am Actuar J 11(2):99-115 · Zbl 1480.91253
[40] Willmot GE, Woo J-K (2015) On some properties of a class of multivariate Erlang mixtures with insurance applications. ASTIN Bull 45(01):151-173 · Zbl 1390.62092
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