Criminal Careers and "Career Criminals," Volume II (1986) / Chapter Skim
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10. Random Parameter Stochastic-Process Models of Criminal Careers
10. Random Parameter Stochastic-Process Models of Criminal Careers
Pages 380-404
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From page 380... ...
The approaches developed in this paper are the outgrowth of a long series of discussions concerning appropriate models for criminal behavior and the empirical evidence supporting those models. In addition, I wish to thank Donald Gaver for his many discussions concerning hierarchical models and corrections to biases in criminal justice data sets.
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Generally, criminal justice data sets JO not provide random samples from the offender population. Rather, inclivicluals are part of a sample because they meet a specific criterion that may be directly or inclirectly related to their parameter values.
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For multiple crime types, a competing-risks formulation is used. The models offer tractability, can include covariates, provide periods of high and low activity, and introduce some behavorial parameters.
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HIERARCHICAL MODELS . This section describes the use of hierarchical stochastic models for studying criminal careers.
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A NEW STOCHASTIC MODEL This section introduces a new family of stochastic moclels of the crime process and arrest process associated with a single criminal career. These new models are intended!
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are inclepenclent random variables with an exponential distribution having some mean, say 1/A. Associated with a crime process is an arrest process.
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. CRIMINAL CAREERS AND CAREER CRIMINALS cess models for the crime process (see, for example, Holden, 19831.
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Let us definc · a crime process that is a renewal process with the time between crimes being a phase distribution F with mean 1/A; · an arrest probability q; · a state-switching probability a; · a phase-type distribution G governing the amount of time the indiviclual spends in the Tow state, during which no crimes are committed; and · a probability ,B giving the probability that the career ends with the start of the current Tow period. We can describe the process intuitively, as follows.
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and let its Laplace-StieTtJes transform be given by AH. The arrest CRIMINAL CAREERS AND CAREER CRIMINALS process is also a terminating renewal process.
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, where cij is the probability that the offender, having last committed a crime of type i, will next commit a crime of type _i 389 The Markov crime-switch approach is much used in stochastic moclels of the crime process. Moreover, it can be made more general by allowing arrest probabilities to clepend on crime type.
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CRIMINAL CAREERS AND CAREER CRIMINALS ~ an arrest probability for crime type t, qt.
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that assumes that the distributions governing the individual crime and arrest processes contain random parameters. Both approaches will be used here.
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b'. The probability that an individual who commits a crime of type t at age a will drop this crime from his active set is given by CRIMINAL CAREERS AND CAREER CRIMINALS ,B~eZ(a~b3.
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~ 7 7 ~ ~ PARAMETER ESTIMATION This section addresses the problem of parameter estimation for the hierarchical models clevelopect in this paper. The statistical literature on the estimation of hierarchical models and the empirical Bayes approach is quite large and rapidly growing.
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Again a choice of methods is possi CRIMINAL CAREERS AND CAREER CRIMINALS ble, but the Bayes approach using the posterior distribution of ~ given ¢, and xi is preferable. The Simultaneous-Likelihood Approach A third approach is the simultaneouslikelihood approach.
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The most convenient choice is to introduce the conjugate prior distribution for the negative binomial distribution, the beta distribution.
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method should be avoided. CRIMINAL CAREERS AND CAREER CRIMINALS several properties of this class.
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of the times between crimes for multiple crime types, as was used above. The class of phase distributions is very large and explicitly contains a number of important parametric families.
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Window Arrest Data Sets CRIMINAL CAREERS AND CAREER CRIMINALS ration. The sampling plan thus is biased in favor of offenders with higher crime anct arrest rates.
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We can also achieve equilibrium by using a delayed renewal process formula tion with G,(t)
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Biases in Samples of Prisoners Two other biases can arise in sampling and analyzing criminal justice data sets involving prisoners. First, individuals are generally sentenced to prison as a result of a high frequency of offenses.
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Assume that contacts occur according to a Poisson process. We imagine that sentencing occurs at the time of the first contact having the property that there are also k other contacts within ~ units of time.
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SUMMARY AND SUGGESTIONS FOR FURTHER RESEARCH This paper has introduced two innovations to the quantitative modeling of criminal justice problems: a general structure of hierarchical models and a new stochastic model of a criminal career. These models allow one to distinguish
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be seen in the general population. The new stochastic model of a criminal career offers three advantages over the standard renewalprocess models in common use.
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Journal of the American Statistical Association 78:47~5. CRIMINAL CAREERS AND CAREER CRIMINALS Murray, C
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