[en] From a survival analysis perspective, bank failure data are often characterized by small default rates and heavy censoring. This empirical evidence can be explained by the existence of a subpopulation of banks likely immune from bankruptcy. In this regard, we use a mixture cure model to separate the factors with an influence on the susceptibility to default from the ones affecting the survival time of susceptible banks. In this paper, we extend a semi-parametric proportional hazards cure model to time-varying covariates and we propose a variable selection technique based on its penalized likelihood. By means of a simulation study, we show how this technique performs reasonably well. Finally, we illustrate an application to commercial bank failures in the United States over the period 2006-2016.
H., Akaike, Information theory and an extension of the maximum likelihood principle, 2nd International Symposium on Information Theory, Academiai Kiado, 1973.
J.W., Boag, Maximum likelihood estimates of the proportion of patients cured by cancer therapy, J. R. Stat. Soc. Ser. B. Stat. Methodol. 11 (1949), pp. 15–53.
L., Breiman, Heuristics of instability and stabilization in model selection, Ann. Statist. 24 (1996), pp. 2350–2383. doi: 10.1214/aos/1032181158
N.E., Breslow, Contribution to the discussion of the paper by D.R. Cox, J. R. Stat. Soc. Ser. B. Stat. Methodol. 34 (1972), pp. 216–217.
A.B., Cantor and J.J., Shuster, Parametric versus non-parametric methods for estimating cure rates based on censored survival data, Stat. Med. 11 (1992), pp. 931–937. doi: 10.1002/sim.4780110710
R.A., Cole and J.W., Gunther, Separating the likelihood and timing of bank failure, J. Bank. Financ. 19 (1995), pp. 1073–1089. doi: 10.1016/0378-4266(95)98952-M
D.R., Cox, Regression models and life-tables, J. R. Stat. Soc. Ser. B. Stat. Methodol. 34 (1972), pp. 187–220.
D., De Leonardis and R., Rocci, Default risk analysis via a discrete-time cure rate model, Appl. Stoch. Models Bus. Ind. 30 (2014), pp. 529–543. doi: 10.1002/asmb.1998
A., Demirgüç-Kunt, Deposit-institution failures: A review of empirical literature, Fed. Res. Bank Cleve. Econ. Rev. 25 (1989), pp. 2–19.
Y., Demyanyk and I., Hasan, Financial crises and bank failures: A review of prediction methods, Omega 38 (2010), pp. 315–324. doi: 10.1016/j.omega.2009.09.007
J.W., Denham, E., Denham, K.B., Dear, and G.V., Hudson, The follicular non-hodgkin's lymphomas–I. The possibility of cure, Eur. J. Cancer 32 (1996), pp. 470–479. doi: 10.1016/0959-8049(95)00607-9
L., Dirick, T., Bellotti, G., Claeskens, and B., Baesens, Macro-economic factors in credit risk calculations: Including time-varying covariates in mixture cure models, J. Bus. Econom. Statist. (2016), pp. 1–14. DOI:10.1080/07350015.2016.1260471.
L., Dirick, G., Claeskens, and B., Baesens, An akaike information criterion for multiple event mixture cure models, Eur. J. Oper. Res. 241 (2015), pp. 449–457. doi: 10.1016/j.ejor.2014.08.038
J., Fan and R., Li, Variable selection via nonconcave penalized likelihood and its oracle properties, J. Amer. Statist. Assoc. 96 (2001), pp. 1348–1360. doi: 10.1198/016214501753382273
J., Fan and R., Li, Variable selection for Cox's proportional hazards model and frailty model, Ann. Statist. 30 (2002), pp. 74–99. doi: 10.1214/aos/1015362185
J., Fan and R., Li, New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis, J. Amer. Statist. Assoc. 99 (2004), pp. 710–723. doi: 10.1198/016214504000001060
J., Fan and H., Peng, Nonconcave penalized likelihood with a diverging number of parameters, Ann. Statist. 32 (2004), pp. 928–961. doi: 10.1214/009053604000000256
V.T., Farewell, The use of mixture models for the analysis of survival data with long-term survivors, Biometrics 38 (1982), pp. 1041–1046. doi: 10.2307/2529885
V.T., Farewell, Mixture models in survival analysis: Are they worth the risk? Canad. J. Statist. 14 (1986), pp. 257–262. doi: 10.2307/3314804
M., Ghitany, R.A., Maller, and S., Zhou, Exponential mixture models with long-term survivors and covariates, J. Multivariate Anal. 49 (1994), pp. 218–241. doi: 10.1006/jmva.1994.1023
D.J., Hendry, Data generation for the Cox proportional hazards model with time-dependent covariates: A method for medical researchers, Stat. Med. 33 (2014), pp. 436–454. doi: 10.1002/sim.5945
A.E., Hoerl and R.W., Kennard, Ridge regression: Applications to nonorthogonal problems, Technometrics 12 (1970), pp. 69–82. doi: 10.1080/00401706.1970.10488635
D.R., Hunter and R., Li, Variable selection using MM algorithms, Ann. Statist. 33 (2005), pp. 1617–1642. doi: 10.1214/009053605000000200
D., Jones, R., Powles, D., Machin, and R., Sylvester, On estimating the proportion of cured patients in clinical studies, Biom. Praximetrie 21 (1981), pp. 1–11.
J.D., Kalbfleisch and R.L., Prentice, The Statistical Analysis of Failure Time Data, John Wiley & Sons, Hoboken, NJ, 2002.
A.Y., Kuk and C.H., Chen, A mixture model combining logistic regression with proportional hazards regression, Biometrika 79 (1992), pp. 531–541. doi: 10.1093/biomet/79.3.531
W.R., Lane, S.W., Looney, and J.W., Wansley, An application of the Cox proportional hazards model to bank failure, J. Bank. Financ. 10 (1986), pp. 511–531. doi: 10.1016/S0378-4266(86)80003-6
H., Leeb and B.M., Pötscher, On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding, J. Multivariate Anal. 100 (2009), pp. 2065–2082. doi: 10.1016/j.jmva.2009.06.010
F., Liu, Z., Hua, and A., Lim, Identifying future defaulters: A hierarchical Bayesian method, Eur. J. Oper. Res. 241 (2015), pp. 202–211. doi: 10.1016/j.ejor.2014.08.008
T.A., Louis, Finding the observed information matrix when using the EM algorithm, J. R. Stat. Soc. Ser. B. Stat. Methodol. 44 (1982), pp. 226–233.
C.L., Mallows, Some comments on Cp, Technometrics 15 (1973), pp. 661–675.
G., McLachlan and T., Krishnan, The EM Algorithm and Extensions, John Wiley & Sons, Hoboken, NJ, 2008.
Y., Peng and K.B., Dear, A nonparametric mixture model for cure rate estimation, Biometrics 56 (2000), pp. 237–243. doi: 10.1111/j.0006-341X.2000.00237.x
R.M., Pfeiffer, A., Redd, and R.J., Carroll, On the impact of model selection on predictor identification and parameter inference, Comput. Stat. 32 (2017), pp. 667–690. doi: 10.1007/s00180-016-0690-2
C., Sanderson and R., Curtin, Armadillo: A template-based c++ library for linear algebra, J. Open Source Softw. 1 (2016), pp. 26–32. doi: 10.21105/joss.00026
G., Schwarz, Estimating the dimension of a model, Ann. Statist. 6 (1978), pp. 461–464. doi: 10.1214/aos/1176344136
H., Shi and G., Yin, Landmark cure rate models with time-dependent covariates, Stat. Methods Med. Res. 26 (2017), pp. 2042–2054. doi: 10.1177/0962280217708681
J.P., Sy and J.M., Taylor, Estimation in a Cox proportional hazards cure model, Biometrics 56 (2000), pp. 227–236. doi: 10.1111/j.0006-341X.2000.00227.x
J.P., Sy and J.M., Taylor, Standard errors for the Cox proportional hazards cure model, Math. Comput. Model 33 (2001), pp. 1237–1251. doi: 10.1016/S0895-7177(00)00312-5
J.B., Thomson, Predicting bank failures in the 1980s, Fed. Res. Bank Cleve. Econ. Rev. 27 (1991), pp. 9–20.
R., Tibshirani, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B. Stat. Methodol. 58 (1996), pp. 267–288.
E.N., Tong, C., Mues, and L.C., Thomas, Mixture cure models in credit scoring: If and when borrowers default, Eur. J. Oper. Res. 218 (2012), pp. 132–139. doi: 10.1016/j.ejor.2011.10.007
G., Whalen, A proportional hazards model of bank failure: An examination of its usefulness as an early warning tool, Fed. Res. Bank Cleve. Econ. Rev. 27 (1991), pp. 21–30.
D.C., Wheelock and P.W., Wilson, Why do banks disappear? The determinants of US bank failures and acquisitions, Rev. Econ. Stat. 82 (2000), pp. 127–138. doi: 10.1162/003465300558560
Y., Zhang, R., Li, and C.L., Tsai, Regularization parameter selections via generalized information criterion, J. Amer. Statist. Assoc. 105 (2010), pp. 312–323. doi: 10.1198/jasa.2009.tm08013
H., Zou, The adaptive lasso and its oracle properties, J. Amer. Statist. Assoc. 101 (2006), pp. 1418–1429. doi: 10.1198/016214506000000735
