Asymmetric Loss Functions for Forecasting in Criminal Justice Settings

The statistical procedures typically used for forecasting in criminal justice settings rest on symmetric loss functions. For quantitative response variables, overestimates are treated the same as underestimates. For categorical response variables, it does not matter in which class a case is inaccurately placed. In many criminal justice settings, symmetric costs are not responsive to the needs of stakeholders. It can follow that the forecasts are not responsive either. In this paper, we consider asymmetric loss functions that can lead to forecasting procedures far more sensitive to the real consequences of forecasting errors. Theoretical points are illustrated with examples using criminal justice data of the kind that might be used for “predictive policing.”

[1]  Howard G. Borden.  Factors for Predicting Parole Success , 1928 .

[2]  O. D. Duncan,et al.  The Efficiency of Prediction in Criminology , 1949, American Journal of Sociology.

[3]  Sheldon Glueck,et al.  Unraveling Juvenile Delinquency , 1951, Mental Health.

[4]  Generalizing the Problem of Prediction , 1952 .

[5]  The Use and Validity of a Prediction Instrument. I. a Reformulation of the Use of a Prediction Instrument , 1953, American Journal of Sociology.

[6]  II. The Validation of Prediction , 1953, American Journal of Sociology.

[7]  Richard Larson,et al.  Models of a Total Criminal Justice System , 1969, Oper. Res..

[8]  P. Lachenbruch.  Mathematical Statistics, 2nd Edition , 1972 .

[9]  E. Parzen.  Nonparametric Statistical Data Modeling , 1979 .

[10]  Oliver D. Anderson,et al.  Forecasting in Business and Economics , 1981 .

[11]  Walter Oberhofer,et al.  The Consistency of Nonlinear Regression Minimizing the $L_1$-Norm , 1982 .

[12]  Isobel Clark.  Regression revisited , 1983 .

[13]  Richard A. Berk,et al.  Prisons as Self-Regulating Systems: A Comparison of Historical Patterns in California for Male and Female Offenders , 1983 .

[14]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[15]  Kenneth C. Land,et al.  Age structure and crime: symmetry versus asymmetry and the projection of crime rates through the 1990s , 1987 .

[16]  David P. Farrington,et al.  Predicting Individual Crime Rates , 1987, Crime and Justice.

[17]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[18]  R. Tibshirani,et al.  Generalized Additive Models , 1991 .

[19]  Z. Ying,et al.  A resampling method based on pivotal estimating functions , 1994 .

[20]  Pin T. Ng,et al.  Quantile smoothing splines , 1994 .

[21]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[22]  Roger Koenker,et al.  Conditional Quantile Estimation and Inference for Arch Models , 1996, Econometric Theory.

[23]  Norman R. Swanson,et al.  Forecasting economic time series using flexible versus fixed specification and linear versus nonlinear econometric models , 1997 .

[24]  Kenneth F. Wallis,et al.  Density Forecasting: A Survey , 2000 .

[25]  Feifang Hu,et al.  Markov Chain Marginal Bootstrap , 2002 .

[26]  A. Piquero,et al.  Deadly Demographics: Population Characteristics and Forecasting Homicide Trends , 2003 .

[27]  Snigdhansu Chatterjee,et al.  Generalized bootstrap for estimators of minimizers of convex functions , 2003 .

[28]  Eric R. Ziegel,et al.  The Elements of Statistical Learning , 2003, Technometrics.

[29]  Leo Breiman,et al.  Random Forests , 2001, Machine Learning.

[30]  R. Koenker,et al.  Penalized triograms: total variation regularization for bivariate smoothing , 2004 .

[31]  R. Berk,et al.  Developing a Practical Forecasting Screener for Domestic Violence Incidents , 2004, Evaluation review.

[32]  Avishai Mandelbaum,et al.  Statistical Analysis of a Telephone Call Center , 2005 .

[33]  Xuming He,et al.  Practical Confidence Intervals for Regression Quantiles , 2005 .

[34]  R. Koenker.  Quantile Regression: Quantile Regression in R: A Vignette , 2005 .

[35]  R. Koenker.  Quantile Regression: Name Index , 2005 .

[36]  Wilpen L. Gorr,et al.  Development of Crime Forecasting and Mapping Systems for Use by Police , 2005 .

[37]  Nicolai Meinshausen,et al.  Quantile Regression Forests , 2006, J. Mach. Learn. Res..

[38]  S. D. Gottfredson,et al.  Statistical Risk Assessment: Old Problems and New Applications , 2006 .

[39]  Richard A. Berk,et al.  Cost-sensitive stochastic gradient boosting within a quantitative regression framework , 2007 .

[40]  R. Berk,et al.  Estimating the Homeless Population in Los Angeles: An Application of Cost-Sensitive Stochastic Gradient Boosting , 2007 .

[41]  R. Koenker,et al.  Regression Quantiles , 2007 .

[42]  William J. Bratton,et al.  Police Performance Management in Practice: Taking COMPSTAT to the Next Level , 2008 .

[43]  J. Maindonald.  Statistical Learning from a Regression Perspective , 2008 .

[44]  Shalabh.  Statistical Learning from a Regression Perspective , 2009 .

[45]  R. Berk.  The Role of Race in Forecasts of Violent Crime , 2009 .

[46]  Klaus Nordhausen,et al.  The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition by Trevor Hastie, Robert Tibshirani, Jerome Friedman , 2009 .

[47]  C. Britt.  Modeling the Distribution of Sentence Length Decisions Under a Guidelines System: An Application of Quantile Regression Models , 2009 .

[48]  R. Berk,et al.  Forecasting murder within a population of probationers and parolees: a high stakes application of statistical learning , 2009 .

[49]  R. Berk,et al.  Small Area Estimation of the Homeless in Los Angeles: An Application of Cost-Sensitive stochastic Gradient Boosting , 2010, 1011.2890.

[50]  Wesley G. Jennings,et al.  Studying the costs of crime across offender trajectories , 2010 .

[51]  D. Eckberg.  Estimates of early twentieth-century U.S. homicide rates: An econometric forecasting approach , 1995, Demography.

[52]  Wei-Yin Loh,et al.  Classification and regression trees , 2011, WIREs Data Mining Knowl. Discov..