Inferring hidden potentials in analytical regions: uncovering crime suspect communities in Medellín
IInferring hidden potentials in analytical regions:uncovering crime suspect communities in Medell´ın
Alejandro Puerta ∗ and Andr´es Ram´ırez–Hassan † This paper proposes a Bayesian approach to perform inferenceregarding the size of hidden populations at analytical region us-ing reported statistics. To do so, we propose a specification takinginto account one-sided error components and spatial effects withina panel data structure. Our simulation exercises suggest good finitesample performance. We analyze rates of crime suspects living perneighborhood in Medell´ın (Colombia) associated with four crimeactivities. Our proposal seems to identify hot spots or “crime com-munities”, potential neighborhoods where under-reporting is moresevere, and also drivers of crime schools. Statistical evidence sug-gests a high level of interaction between homicides and drug dealingin one hand, and motorcycle and car thefts on the other hand.JEL: C11, C21,K14Keywords: Bayesian econometrics, crime, hidden populations,neighbourhood, one-sided errors, spatial effects.
I. Introduction
We introduce in this paper an inferential framework within a Bayesian paradigmto incorporate that reported rates of individuals with unknown features in analyti-cal regions might be lower bounds. For instance, positive rate tests of individualshaving particular diseases, drug consumers, cheating behavior or police reportrates of criminal activity in analytical regions are under reports of the total pop-ulation size. All these are examples of what we called “hidden populations”. Ourproposal controls for spatial effects and unobserved heterogeneity on panel datasettings.We depart from the stochastic frontier analysis (composed error models) (Aigneret al., 1977; Meeusen and van Den Broeck, 1977), and consider inefficiency, whichis a one-sided error term from a statistical point of view, as the conditionalpercentage of the total population actually reported per analytical region. Ac-cordingly, we incorporate structural and transitory lower bounds as one-sidederror terms. In particular, we extend Tsionas and Kumbhakar (2014) proposalby including spatial effects; this component induces heteroscedasticity, and as a ∗ Department of Economics, School of Economics and Finance, Universidad EAFIT, Medell´ın, Colom-bia, E-mail: apuerta5@eafit.edu.co † Department of Economics, School of Economics and Finance, Universidad EAFIT, Medell´ın, Colom-bia. Grupo de estudios en econom´ıa y empresa. E-mail: aramir21@eafit.edu.co a r X i v : . [ ec on . E M ] S e p consequence, omitting one-sided error terms potentially causes biased estimators(Wang and Schmidt, 2002).Crime and arrest rates at a regional level based on reported statistics are lowerbounds (Kirk, 2006). Consistently, we apply our proposal to model the rate ofcrime suspects living per neighborhood in Medell´ın (Colombia). In particular, weinclude spatial effects as our cross-section units are neighborhoods, and perma-nent and transient one-sided errors to handle time-varying lower bound issues. Inthis way, we can perform inference about potential under reports of crime sus-pects living in a region; this would be valuable for policymakers. Particularly,it would suggest regional hot spots to implement structural interventions andefficient police operations to reduce criminality. This is especially relevant in acity like Medell´ın, which is a natural experiment field due to its history of vio-lence associated with drug trafficking and urban wars. Nevertheless, it is worthmentioning that Medell´ın is also an exceptional example of social transformationbased on public and private interventions.It seems that mainstream crime literature neither has taken into account po-tential bias due to omitting lower bound issues nor has implemented strategies toinfer the population of criminals. A remarkable exception for the latter is van derHeijden P. and Cruyff M. (2014), who based their strategy on count models, butrequires data at the individual level, whereas ours requires data at an aggregatedregional level. Andresen (2006); Kakamu et al. (2008); Kikuchi (2010); Arnio andBaumer (2012); Li et al. (2015) take into account spatial effects modeling crimedata. However, these authors do not take the lower bound issue into account,which may have consequences on the statistical properties of estimators. Observethat unlike most of the crime literature, which focuses on crime rates, we modelplace of residence of crime suspects, that is, we consider that neighborhoods havesocioeconomic conditions that might promote “schools of criminality”. So, fol-lowing the spirit of stochastic frontier analysis, we can consider these conditionsas production factors of potential criminals.On the other hand, Druska and Horrace (2003); Schmidt et al. (2009); Mastro-marco et al. (2016); Tsionas and Michaelides (2016); Glass et al. (2016); Gudeet al. (2018) propose stochastic frontier models including spatial effects, but focus-ing on “standard production functions” such as GDP at state level. Additionally,the tendency is to incorporate both effects in the same error: spatial dependenceand inefficiency, whereas we separate these components. Also, our proposal differsfrom previous stochastic frontier proposals with spatial effects due to including arandom conditional autoregressive spatial (CAR) effect. Schmidt et al. (2009) in-cludes CAR effects in the inefficiency component, whereas most of the stochasticfrontier literature includes spatial autoregressive (SAR) components. The formerproposal does not induce explicitly heteroscedasticity when inefficiency is omitteddue to being present precisely in this omitted part. On the other hand, the SARprocess is not Markovian, so it generates global spatial patterns, and it seemsthat criminal activity in Medell´ın is controlled by gangs with influence in specific areas (Collazos et al., 2020). Therefore, we use the CAR specification, which isa Markovian process in space (Ram´ırez Hassan and Montoya Bland´on, 2017), tocontrol for local spatial effects.We aim to contribute to crime and stochastic frontier literature, but mainly,considering one minus the (exponential) one-sided errors as the percentage ofcovered (uncaptured) criminals helps to build a link between these two well-developed areas of knowledge.It seems from our simulation exercises that the sampling performance of ourproposal is sound, allowing us to capture both the one-sided error terms and thespatial dependency. Additionally, it allows us to obtain good estimates for thelocation parameters in the presence of a five–way error component model. We alsofind that predictive inference regarding hidden populations has good predictiveinterval coverage.Our empirical analysis suggests that homicide and drug dealing are stronglylinked as both seem to generate local urban displacement, share some determi-nants and same hot spots (crime communities), which are mainly located in themost west analytical region in Medell´ın. On the other hand, there are commonlinks between car and motorcycle thefts. For instance, both crime activities areassociated with high local unemployment rates, and there is a crime communityin the central-west part of the city. Although, there are some isolated crimecommunities specialized in each crime activity with their own determinants.The remainder of this paper proceeds as follows: Section II outlines our econo-metric model, the conditional posterior distributions, and the results of simulationexercises. Section III presents our application. In particular, construction of theanalytical regions, unconditional spatial analysis based on hypothesis tests usingstandardized rates, and posterior inferential results. Section IV concludes. II. Econometric approach
A. The model
The point of departure is the observed under-reported ratio of the numberof target individuals per inhabitants ( Y it ) at analytical region i = 1 , , . . . , N and time period t = 1 , . . . , T , where target individuals belong to the “hiddenpopulation” ( P it ), which is also standardized, Y it = P it × R it , (1)where R it is the report rate, that is, the percentage of individuals belonging tothe target population that have been observed.We can think about P it as depending on environmental variables that pro-mote or discourage the number of individuals belonging to the target popula-tion as well as spatial effects reflecting spatial clusters, and also unobserved re-gional heterogeneity and idiosyncratic stochastic errors. So, we propose P it = f ( X it , β ) × exp { α i + v i + (cid:15) it } = (cid:81) Kk =1 X β k kit × exp { α i + v i + (cid:15) it } where X kit are k potential drivers which may include spatial lags, that is, given a set of controls( z it ), their spatial lags are (cid:80) Nj =1 w ij z ijt , where w ij is the ij -th element of thecontiguity matrix W N , x it = (cid:20) z (cid:48) it (cid:16)(cid:80) Nj =1 w ij z ijt (cid:17) (cid:48) (cid:21) (cid:48) . The location parametersare given by β k . In addition, α i is the unobserved stochastic heterogeneity, v i isthe spatial random effect, and (cid:15) it is the idiosyncratic stochastic error.On the other hand, we specify the report rate as R it = exp {− η + i − u + it } where η + i and u + it are one-sided positive stochastic errors to account for unobservedpersistent and transient lower bound issues, that is, if η + i = u + it = 0, then R it = 1,and Y it = P it , otherwise we observe just a lower bound of P it .Observe that this setting follows the statistical framework of stochastic frontieranalysis with permanent and transient one-sided components, unobserved hetero-geneity and spatial effects. Therefore, we extend Tsionas and Kumbhakar (2014)proposal including spatial effects, y it = x (cid:48) it β + α i + v i − η + i − u + it + (cid:15) it , (2)such that the reduce form equation 2 is in log-log form.Following Tsionas and Kumbhakar (2014) we assume that the i.i.d randomcomponents have the following distributions: α i ∼ N (0 , σ α ) , η + i ∼ N + (0 , σ η ) , u + it ∼ N + (0 , σ u ) , (cid:15) it ∼ N (0 , σ (cid:15) ) . (3)We assume that each v i has an improper (intrinsic) conditionally autoregressivestructure (Besag, 1991): v i | v i ∼ j ∼ N (cid:18) (cid:88) i ∼ j w ij v j (cid:80) i ∼ j w ij , σ v (cid:80) i ∼ j w ij (cid:19) , (4)where v i ∼ j is a vector of stochastic spatial errors for the neighbors j of i ( i ∼ j ).The joint distribution of the improper CAR is v ∼ N (¯ v , σ v ( D w − W N ) − ),where D w = diag( (cid:80) i ∼ j w ij ) (Banerjee et al., 2014). The ij -th element of W N is equal 1 if region i and j are neighbors, and 0 otherwise. By definition theelements of the main diagonal are set equal to zero. B. Likelihood and priors
Set τ it = α i + (cid:15) it such that taken assumptions in (3) into account, τ i ∼ N ( , Σ ), Σ = σ (cid:15) I T + σ α i T i (cid:48) T , where i T is a T -dimensional vector of 1’s and I T is a T -dimensional identity matrix. Given our model specification (equations (2), (3)and (4)), the joint conditional distribution function, given spatial random ef- fects, is the product over individuals of a T -variate closed skew normal distribu-tions (Dom´ınguez-Molina et al., 2003). Working directly with this distributionis demanding given that it is not readily available in closed form. So, we fol-low S´anchez-Gonz´alez et al. (2020) who in similar settings use data augmenting(Tanner and Wong, 1987). In particular, set θ = ( β (cid:48) , σ α , σ (cid:15) , σ v , σ u , σ η ) (cid:48) , and theaugmented vector Θ = ( θ (cid:48) , u + it , η + i , v i ), then taking into account equations (2),(3) and (4), the “augmented” likelihood is f ( y | X , Θ ) = N (cid:89) i =1 (2 π ) − T | Σ | − exp (cid:26) − (cid:0) y i − X i β − u + i − v i i T − η + i i T (cid:1) (cid:48) Σ − (cid:0) y i − X i β − u + i − v i i T − η + i i T (cid:1)(cid:27) × (2 π ) − T (cid:18) σ v (cid:80) i ∼ j w ij (cid:19) − T exp (cid:26) − (cid:80) i ∼ j w ij σ v (cid:18) v i i T − (cid:88) i ∼ j w ij v j (cid:80) i ∼ j w ij i T (cid:19) (cid:48) (cid:18) v i i T − (cid:88) i ∼ j w ij v j (cid:80) i ∼ j w ij i T (cid:19)(cid:27) I ( u + i > (cid:18) π (cid:19) − T ( σ u ) − T exp (cid:26) − σ u u + (cid:48) i u + i (cid:27) I ( η + i > (cid:18) π (cid:19) − ( σ η ) − exp (cid:26) − σ η η +2 i (cid:27) , where we stack information by individual such that X i is a T × dim { β } dimen-sional matrix with information of individual i .We follow standard practice in Bayesian econometrics with conditional indepen-dent priors such that β ∼ N ( β , B ), where β = and B = 1000 I which im-plies vague prior information. For the scale parameters, ¯ Q k σ k ∼ χ ( ¯ N k ), k = (cid:15), α, v ,where ¯ N k = 1 and ¯ Q k = 10 − (Tsionas and Kumbhakar, 2014). We follow Makie(cid:32)la(2017) for the priors of σ u and σ η , that is, we use σ u ∼ IG ( v u / , v u log ( r ∗ u ) / σ η ∼ IG ( v η / , v η log ( r ∗ η ) /
2) where the prior medians of the transientand persistent one-sided errors are equal to r ∗ u = 0 .
85 and r ∗ η = 0 .
70, and v u = v η = 10. Even though v has an improper distribution, Theorem 2 in Sunet al. (1999) guarantees that a proper posterior distribution exists if D w − W N is nonnegative definite, the precision parameters have gamma prior distributions,and the intercepts have diffuse prior distributions (we fulfill all these require-ments). C. Conditional posterior distributions
The conditional posterior distribution for the location parameters is β | Θ − β , y , X ∼ N ( ¯ β , ¯ B ) , where ¯ B = (cid:18) (cid:80) i X (cid:48) i Σ − X i + B − (cid:19) − , ¯ β = ¯ B (cid:18) (cid:80) i X (cid:48) i Σ − ˜ y i + B − β (cid:19) , and˜ y i = y i − u + i − v i i T − η + i i T . The notation Θ − ψ indicates all elements in Θ except We perform robustness analysis regarding these hyperparameters. Available upon request. ψ .The conditional posterior distribution for u + i is π ( u + i | Θ − u + i , y , X ) ∝ I ( u + i > × exp (cid:26) −
12 ( u + i − µ ) (cid:48) Ω − ( u + i − µ ) (cid:27) , where Ω = (cid:16) Σ − + I T σ u (cid:17) − and µ = Ω (cid:18) Σ − ( y i − X i β − v i i T − η + i i T ) (cid:19) . Giventhe multivariate condition I ( u + i > π ( u + it | Θ − u + it , y , X ) using the result from a con-ditional multivariate normal distribution (Eaton, 1983). Let u + i = ( u + i , . . . , u + iT ) (cid:48) , u + i = (cid:18) u +1 t u +2 t (cid:19) ∼ I ( u + i > × N (cid:20)(cid:18) µ µ (cid:19) , (cid:18) ω Ω Ω Ω (cid:19)(cid:21) , then, the conditional distribution of u +1 t given u +2 t is u +1 t | Θ − u +1 t , y , X , u +2 ,t ∼ I ( u +1 t > × N (¯ µ, ¯ ω ) , where ¯ µ = µ + Ω Ω − ( u +2 t − µ ) and ¯ ω = ω − Ω Ω − Ω .The conditional posterior distribution for η + i is: η + i ∼ N + ( m i , ψ )where ψ = σ η (cid:18) σ η i (cid:48) T Σ − i T (cid:19) − and m i = ψ i (cid:48) T Σ − ( y i − X i β − u + i − v i i T ).The conditional posterior distribution for v i is v i | Θ − v i , y , X ∼ N (¯ v i , ¯ σ vi ) , where ¯ σ vi = (cid:18) i (cid:48) T Σ − i T + (cid:80) i ∼ j w ij σ v (cid:19) − and ¯ v i = ¯ σ vi (cid:18) i (cid:48) T Σ − ( y i − X i β − u + i − η + i i T ) + (cid:80) i ∼ j w ij v j σ v (cid:19) . The conditional posterior distribution for σ v is¯ Q v + v (cid:48) (cid:0) D w − W N (cid:1) v σ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Θ − σ v , y , X ∼ χ ( N × T + ¯ N v ) . The conditional posterior distribution for σ u is σ u | Θ − σ u , y , X ∼ IG (cid:18) ( N × T ) + v u , u + (cid:48) u + + 2 v u log ( r ∗ u )2 (cid:19) . The conditional posterior distribution for σ η is σ η | Θ − σ η , y , X ∼ IG (cid:18) N + v η , η + (cid:48) η + + 2 v η log ( r ∗ η )2 (cid:19) . Up to this point we have standard conditional posterior distributions, so we canuse the Gibbs sampling algorithm. However, the conditional posterior distribu-tions for σ α and σ (cid:15) do not have standard form. We use the Metropolis-Hastingsalgorithm for these two parameters.In particular, the conditional posterior distribution for σ α is π ( σ α | Θ − σ α , y , X ) ∝ N (cid:89) i =1 | (cid:0) σ (cid:15) I t + σ α i T i (cid:48) T (cid:1) | − exp (cid:26) − (cid:0) y i − X i β − u + i − v i i T − η + i i T (cid:1) (cid:48) (cid:0) σ (cid:15) I T + σ α i T i (cid:48) T (cid:1) − (cid:0) y i − X i β − u + i − v i i T − η + i i T (cid:1)(cid:27) σ α ) ¯ N α / exp (cid:26) − ¯ Q α σ α (cid:27) . The conditional posterior distribution for σ (cid:15) is π ( σ (cid:15) | Θ − σ (cid:15) , y , X ) ∝ N (cid:89) i =1 | (cid:0) σ (cid:15) I t + σ α i T i (cid:48) T (cid:1) | − exp (cid:26) − (cid:0) y i − X i β − u + i − v i i T − η + i i T (cid:1) (cid:48) (cid:0) σ (cid:15) I T + σ α i T i (cid:48) T (cid:1) − (cid:0) y i − X i β − u + i − v i i T − η + i i T (cid:1)(cid:27) σ (cid:15) ) ¯ N (cid:15) / exp (cid:26) − ¯ Q (cid:15) σ (cid:15) (cid:27) . We use as proposal distributions scaled Chi-squared distributions with one degreeof freedom.
D. Simulations
We consider the following data generating process: y it = 0 . z it + 0 . N (cid:88) j =1 w ij z ij,t + α i + v i − η + i − u + it + (cid:15) it , (5)where the contiguity criterion is queen, and z it is drawn from a standard normaldistribution. Following Tsionas and Kumbhakar (2014), σ α = 0 . , σ η = 0 . , σ u =0 . , and σ (cid:15) = 0 .
1, and we set σ v = 0 .
4. We perform 20,000 iterations, a burn-inequal to 10,000, and a thinning parameter equal 5.
Population parameters: point estimate results. —
Table 1 displays samplingproperties regarding point estimates of our Bayesian proposal using different com-binations of N an T , where one of them closely matches our application. We cansee that our Bayesian proposal has good sampling properties as the highest den-sity intervals (HDI) are relatively narrow and contain the population scale andlocation parameters. Comparing the first scenario ( N = 49, T = 5) with the lastscenario ( N = 196, T = 10), we observe that the HDIs get narrower as the samplesize increases. This is particularly relevant for the scale parameters. In general,the expected value of the correlation of the posterior draws of one-sided errorsand spatial effects, and the Population values ( ˆ E ( ρ )) is higher than 0.5 except inone case. Also, the descriptive statistics (mean and median) of these unobservedstochastic components are similar.We produce another two sets of simulation results (see Appendix A, Tables A4and A5). The first shows the consequences of varying λ = σ η + σ u σ (cid:15) . This parameterhas attracted a lot of attention in the stochastic frontier community. Olson et al.(1980) identifies two main issues when λ → ∞ : two-step estimators have veryunstable empirical moments, and negative bias (constant term and ˆ σ u ). On theother hand, the wrong skew problem, λ →
0, implies opposite direction bias.Waldman (1982); Horrace and Wright (2019) prove the existence of a stationarypoint in this case, then the probability of the wrong skew problem convergesto zero when the sample size converges to infinity. However, Simar and Wilson(2009) show that finite sample problems remain.Robustness checks for λ are reported in Table A4 in the Appendix. The sam-pling properties of our proposal seems to follow previous studies, that is, goodperformance as far as λ is higher than one but bounded. Posterior estimates inour application suggests λ is between 2 and 9, which seems a safe ground.We also perform robustness checks regarding the presence of heavy tails in thestochastic error ( (cid:15) it ). In particular, we assume normality to perform statisticalinference, but simulating equation 5 using a Student’s t-distribution with fourdegrees of freedom. It seems from results in Table A5 in the Appendix that ourinferential procedure is robust to heavy tails presence. Hidden population: coverage results. —
One of the main purposes of ourproposal is being able to make inference about “hidden populations”. Given Y it = P it exp (cid:8) − ( η + i + u + it ) (cid:9) , then (cid:8) − ( η + i + u + it ) (cid:9) is the percentage of reportedcases of our target population, and as a consequence, 1-exp( − ( η + i + u + it )) rep-resents the percentage that is still covered. Therefore, we would expect thata sensible 1 − α predictive interval for the target population ( P it ) is P α = (cid:8) Y it × exp( η + i + u + it ) : π ( Y it × exp( η + i + u + it ) | Θ , Y , X ) ≥ k ( α ) (cid:9) , k ( α ) is the largestconstant such that P ( P α ) ≥ − α , that is, the (1 − α ) highest density (predictive)interval (HDI). T a b l e — : S a m p li n g p r o p e r t i e s o f B a y e s e s t i m a t o r s η + σ η u + σ u v σ v σ α σ (cid:15) β β M e a n M e d i a n ˆ E ( ρ ) M e a n H D I M e a n M e d i a n ˆ E ( ρ ) M e a n H D I M e d i a n ˆ E ( ρ ) M e a n H D I M e a n H D I M e a n H D I M e a n H D I M e a n H D I P o pu l a t i o n - . - . . - . - . . . . . . . - . E s t i m a t e - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = P o pu l a t i o n - . - . . - . - . . - . . . . . - . e s t i m a t e - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = P o pu l a t i o n - . - . . - . - . . . . . . . - . e s t i m a t e - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = P o pu l a t i o n - . - . . - . - . . . . . . . - . e s t i m a t e - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = P o pu l a t i o n - . - . . - . - . . . . . . . - . e s t i m a t e - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = N o t e : B y P op u l a t i o n v a l u e o f η + , u + a n d v w e m e a n t h e a ve r a ge a n d m e d i a n v a l u e s a s t h e y w e r ege n e r a t e d f r o m t h e M o n t e C a r l o ex p e r i m e n t . W e r e po r t po s t e r i o r m e a n s , m e d i a n s a n dh i g h e s t d e n s i t y i n t e r v a l s f r o m po s t e r i o r d r a w s u s i n g , i t e r a t i o n s , , b u r n - i n a n d t h i nn i n geq u a l t o5 . ˆ E ( ρ ) i s t h eex p ec t e d v a l u e o f t h ec o rr e l a - t i o n o f t h e po s t e r i o r d r a w s o f o n e - s i d e d e rr o r s a n d s pa t i a l e ff ec t s , a n d t h e P op u l a t i o n o n e - s i d e d e rr o r s a n d s pa t i a l e ff ec t s c o m i n g f r o m t h e M o n t e C a r l o ex p e r i m e n t , r e s p ec t i ve l y . We check the performance of this predictive interval calculating its coverage( cov ). In particular, we propose a Beta-Binomial model for this coverage usingas prior a non-informative Beta distribution, that is, π ( cov ) ∼ B (1 , cov | Θ , Y , X ∼ B ( a, b ) where a = 1 + (cid:80) Ni =1 (cid:80) Tt =1 I it , b = 1+ N × T − (cid:80) Ni =1 (cid:80) Tt =1 I it , I it = [ P it ∈ P α ]. Observe that E ( cov | Θ , Y , X ) ≈ (cid:80) Ni =1 (cid:80) Tt =1 I it N × T due to using a non-informative prior. Algorithm A1 shows details.It seems from Table 2 that our proposal has a good coverage as all means areclose to (1 − α ) credibility levels with narrow 95% HDIs, where most of themembrace the credibility levels.To have an idea of how close is our estimate of P it to the real value, we calculatethe mean absolute percentage error (MAPE) for each observation asMAPE it = 1 S S (cid:88) s =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P it − (cid:98) E [ P it ] P it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Table 3 reports summary statistics corresponding the MAPE’s for each of thesample sizes of Table 2. Results suggest that our point estimate for P it tends tobe very close to the real values. On average, our estimate diverges from P it in 23percentage points. Additionally, half of the estimates diverge in no more than 9percentage points from the real values. Algorithm A1
Coverage analysis
1: Simulate the dgp from Equation (2)2: Estimate the model by drawing samples from the posterior distributions in subsection II.C3: for i = 1 , . . . , N do for t = 1 , . . . , T do I it = [ P it ∈ P α ] where P α is the 1 − α highest density interval, that is, P α = (cid:110) Y it × exp( η + i + u + it ) : π ( Y it × exp( η + i + u + it ) | Θ , Y , X ) ≥ k ( α ) (cid:111) , k ( α ) is the largest constant suchthat P ( P α ) ≥ − α .6: end for end for
8: Draw S samples from B ( a, b ) where a = 1 + (cid:80) Ni =1 (cid:80) Tt =1 I it and b = 1 + N × T − (cid:80) Ni =1 (cid:80) Tt =1 I it .9: Obtain the point estimate of the coverage ( cov ) as the mean of these draws.10: Obtain the 95% highest density interval from these draws. III. Crime suspect communities: the Medell´ın case
Medell´ın is a natural experimental field to analyze crime. In particular, this cityhad a slightly increasing homicide rate from the mid-1960s to 1980, then there isa remarkable increase between 1981 to 1991 passing from 52 to 388 homicides perone hundred thousand inhabitants (Garc´ıa et al., 2012); this is mainly explainedby the drug trafficking war of the local cartel against its competitors and the state.Then, there is a significant decrease of the homicide rate until 2008, reaching 46, Table 2—: Hidden population: Coverage resultsCredibility level Mean coverage HDIN=49 T=5 0.90 0.911 (0.872,0.943)0.95 0.939 (0.905,0.966)0.99 0.992 (0.978,0.999)N=49 T=10 0.90 0.803 (0.767,0.836)0.95 0.884 (0.855,0.911)0.99 0.959 (0.939,0.975)N=100 T=5 0.90 0.9 (0.873,0.925)0.95 0.94 (0.918,0.959)0.99 0.98 (0.967,0.99)N=100 T=10 0.90 0.876 (0.856,0.896)0.95 0.943 (0.928,0.957)0.99 0.989 (0.982,0.994)N=196 T=10 0.90 0.881 (0.866,0.895)0.95 0.93 (0.918,0.941)0.99 0.982 (0.975,0.987)
Note: Coverage analysis under different sample sizes and credibility levels. It sug-gests that the − α highest density interval (HDI) for the hidden population, P α = (cid:110) Y it × exp( η + i + u + it ) : π ( Y it × exp( η + i + u + it ) | Θ , Y , X ) ≥ k ( α ) (cid:111) , k ( α ) is the largestconstant such that P ( P α ) ≥ − α , has good coverage as they are very close to thenominal credibility levels. that may be explained by the peace agreements with the guerrilla groups M-19 and EPL in 1990 and 1992, respectively, the dismantling of the local cartel,and death of its main figure in 1993, the intervention of the police and army(Orion operation) in Comuna 13 , a neighborhood characterized by no state lawenforcement, and the demobilization of the paramilitary group
Cacique Nutibara in 2003, and other groups until 2006 (Giraldo-Ram´ırez and Preciado-Restrepo,2015). However, there is a sudden upturn in 2009, the homicide rate was 94, thismay be explained by disputes to get control of the drug trafficking business afterthe extradition of former leaders. Since then, Medell´ın has experienced a steadydecrease in the homicide rate achieving 20 in 2015, which is a historical low recordfor the city in the last 50 years.We got confidential information from Colombian police reports about residen-tial address at capture moment of suspects associated with four criminal activi-ties: homicide, drug dealing, motorcycle and car thefts (see Appendix A, TableA1). We do not have the socioeconomic characteristics of these individuals. So,we calculate crime suspects rates per one hundred thousand inhabitants at theanalytical region level in Medell´ın between 2011-2014. Then, as we consider However, we use rates per one million inhabitants for regressors in our models for coefficient scalepurposes. Table 3—: Summary statistics: mean absolute percentage error posterior esti-mates of hidden populationsSample Size Average Median HDIN=49, T=5 0.2433 0.0886 0.0021 1.1326N=49, T=10 0.2055 0.0850 0.0018 0.7123N=100, T=5 0.2109 0.0872 0.0004 0.7125N=100, T=10 0.2195 0.0851 0.0013 0.8314N=196, T=10 0.2418 0.0855 0.0008 0.9275
Note: Mean absolute percentage error (MAPE),MAPE it = S (cid:80) Ss =1 (cid:12)(cid:12)(cid:12) P it − (cid:98) E [ P it ] P it (cid:12)(cid:12)(cid:12) associatedwith the posterior predictive interval P α = (cid:110) Y it × exp( η + i + u + it ) : π ( Y it × exp( η + i + u + it ) | Θ , Y , X ) ≥ k ( α ) (cid:111) , k ( α ) is the largest constant such that P ( P α ) ≥ − α . environmental factors at a neighborhood level as potential drivers of “crime com-munities”, we calculated for these analytical regions averages of socioeconomiccharacteristics that have been considered previously in crime literature (Bour-guignon et al., 2003; Andresen, 2006; Hipp, 2007; Kakamu et al., 2008; Kikuchi,2010; Arnio and Baumer, 2012; Li et al., 2015) using information from annualLiving Standards Surveys (See Appendix A, Table A2). However, these surveysare not representative at a neighborhood level (325 units) in Medell´ın. Therefore,we use the max-p-region algorithm (Duque et al., 2012) obtaining 175 analyticalregions which are more representative. This algorithm merges adjacent neighbor-hoods to create new analytical regions such that the algorithm minimizes withinattribute heterogeneity, but maximizes this heterogeneity between the new ana-lytical regions (see Duque et al. (2012) for details).Table A3 in Appendix A shows descriptive statistics. The main take away fromthis table is the high level of heterogeneity regarding control variables betweenthe analytical regions; this would suggest that Medell´ın is characterized by a veryhigh level of social inequality. We also observe in this table a mean homiciderate equal to 49 per one hundred thousand inhabitants; there are many analyticalregions without any homicide, whereas the central business district analyticalregion has a remarkable 2,062 rate, explained by relatively no many people livingin this area. Observe that this is a common flaw when modeling crime rates.On the other hand, we model crime suspects residence per region where we alsofound that there are many analytical regions without any, but others with veryhigh figures. Homicide suspects rate is the highest on average, followed by drugdealing, motorcycle thefts, and car thefts, respectively. A. Unconditional analysis
We perform unconditional analysis to identify potential “crime communities”using standardized incidence ratios (Banerjee et al., 2004),
SIR it = S it E it , where S it is number of crime suspects living in analytical region i at time t based oncapture police reports, and E it = n it (cid:80) Ni =1 S it (cid:80) Ni =1 n it is the expected number of crimesuspects, n it is the number of inhabitants in analytical region i at time t .It is possible to find a SIR ’s estimator by maximum likelihood; assuming S it | η it distributes poisson, that is, S it | η it ∼ P ( E it η it ). Then, the maximum likelihood(ML) estimator is ˆ η it = SIR it . Nonetheless, assuming equi-dispersion may benon-realistic (Clayton and Kaldor, 1987). To get a more flexible model we assumethat S it | η it ∼ P ( E it η it ) such that η it distributes gamma, η it ∼ G ( ν, α ); thisimplies that η it | S it = s it ∼ G ( s it + ν, E it + α ). So, at the end, we get smootherratios, through a prior distribution on η i , overcoming equi-dispersion.We are interested on the probability of a SIR being higher than an observedvalue, that is, H : S it = E it versus H : S it > E it . Then, if S it ∼ P ( E it η it ) underthe null hypothesis η it = 1, P ( η it > | s it , E it ) is the posterior probability againstthe null hypothesis of evenly distributed rates across space, P ( η it > | s it , E it ) =1 − P ( η it < | s it , E it ) = 1 − (cid:82) s it η sit + ν − it exp − ηi ( Eit + α ) ( E it + α ) sit + ν Γ( s it + ν ) dη it where Γ( . ) isthe gamma function, and ν = α = 0 .
01 to have non informative priors.Figure 1 shows probability maps, P ( η it > | s it , E it ) (Choynowski, 1959). Thishelp to easily identify “crime communities”. In particular, we observe that thecentral business district (map center) is a potential hot spot for homicide, drugdealing and car theft suspects. On the other hand, it seems that there are someother specialized “crime communities”. Homicide suspects are located in thewestern (see Figure 1a), drug dealing suspects at central-eastern (see Figure 1b),car thefts at north-western (see Figure 1c), and motorcycle thefts at north-eastern(see Figure 1d). B. Conditional posterior results
Table 4 shows posterior estimates of our econometric proposal for four criminalactivities: homicides, drug dealing, motorcycle and car thefts. Our dependentvariable is log(1 + Y it ) ≈ Y it where Y it = S it ( n it / , .There are some interesting results regarding home location of crime suspects.It seems that homicide and drug dealing suspect communities are positively as-sociated with high proportions of young males, and low population densities, butsurrounded by neighborhoods with a high population density. This would sug-gest local urban displacement associated with these crime communities, although,this displacement seems not to be explicitly forced. Observe that these charac-teristics do not play any statistical significant role in motorcycle and car theftssuspect communities, which on the other hand, are positively associated with Figure 1. : Standardized incidence ratios: Medell´ın 2011-2014 (a) Suspects: homicide rates (b) Suspects: drug dealing(c) Suspects: car thefts (d) Suspects: motorcycle thefts
Note: Probability of rejecting the null hypothesis of evenly distributed rates of crime suspects in Medell´ın.This map identifies potential captured “crime communities” at 90%, 95% and 99%. young unemployment rates. This suggests that local focused employment policiesmay reduce these criminal activities. Additionally, it seems that there is a kind of optimal location regarding these communities as they are located near middleincome neighborhoods.There are other specific statistical significant variables to each crime suspectcommunity. For instance, homicide communities are positively associated withless immigrants, low male education and household expenditures, high householdsizes and neighborhoods with a lower safety perception. Drug dealers communi-ties are associated with less proportion of Caucasians, but higher levels of safetyperception. The latter is also positively associated with motorcycle suspect com-munities, which in turn, is also positively associated with neighbors with highforced displacement and low safety perception. It seems that motorcycle thievestravel to close neighborhoods to commit their crimes (average travel time is 10minutes from crime location to home location). Finally, car thieves communitiesis positively associated with a higher proportion of divorced males, more immi-grants, less people per household locally and in surrounding neighborhoods.Table 5 reports posterior mean estimates of error components (one-sided andtwo-sided) associated with homicide, drug dealing, motorcycle and car thefts.We notice that λ is approximately between 2 and 8, which implies a safe groundfor inference in one-sided error models as shown from simulation exercises inTable A4 and previous studies (Olson et al., 1980; Simar and Wilson, 2009). Inaddition, the percentage of total variability due to spatial effects is between 20%(homicides) and 13% (car thefts), where σ v . ( (cid:80) i ∼ j w ij ) Ave is the marginal standarddeviation due to spatial effects (Ram´ırez Hassan and Montoya Bland´on, 2017), (cid:16)(cid:80) i ∼ j w ij (cid:17) Ave is the average number of neighbors. This highlights the relevanceof this effect. Finally, the posterior mean estimate of permanent percentage ofpotential covered (uncaptured) crime suspects (ˆ E (1 − exp( η + i ))) fluctuates between18% (car thefts) and 26% (drug dealing), and the total (permanent plus transient,ˆ E (1 − exp( η + i + u + it ))) is between 32% (car thefts) and 57% (homicides and drugdealing). This means that on average the highest transient effect were associatedwith homicides (33%).However, the former figures have a lot of heterogeneity through time and space.Figure 2 shows analytical regions specific total percentages (permanent and tran-sient) of potentially still covered suspects by crime and time. Regarding homicide(top-left panel), it seems that between 2011 and 2013 there was a hot spot of po-tential covered crime communities in the most western area (analytical region121) with percentages over 90%. However, this situation drastically changed in2014 for this area, it seems that these hot spots moved a little bit to east (an-alytical regions 121 and 177). In addition, there is a cluster of homicide crimecommunities from the central east (analytical region 80) to the north limit of thecentral business center (analytical region 163) for this last year.Drug dealing have similar pattern to homicides regarding the hot spot in themost western area between 2011 and 2013 (top-right panel). However, analyticalregion 121 still seems to be a drug dealers community in 2014. Observe that similar patters regarding statistically relevant variables was also found in estima-tion results, and coincides with the violent history of Medell´ın due to the drugtrafficking war of gangs for business control. Both crime activities (homicide anddrug dealing) have uncaptured percentage rates as high as 90%.Car thefts communities are located in the central-north area near the westriverside of the Medell´ın river (bottom-left panel). This river is a geographicalbarrier between the west and the east of the city, and plays an important roleregarding crime communities. It seems that there is a hot spot composed byanalytical regions 44 to 47. Another hot spot is composed by analytical regions95, 71 and 73 located on the central-west. It seems that this is also a communityof motorcycle thieves. Observe that the percentage of uncaptured car thieves isas high as 50%, whereas this figure is as high as 70% for motorcycle thieves.Table 4—: Bayesian posterior estimates: capture rates of crime suspects Homicides Drug dealing Motorcycle thefts Car theftsDivorced males 0.003 -0.056 -0.0596 0.0861**Caucasian -0.0174 -0.0753** 8.00E-04 -0.007Immigrants -0.2675*** -0.1181 -0.0348 0.162***Unemployment 24-15 0.0969 0.0305 0.0835** 0.0553*Male population 24-15 0.7973*** 0.3704** -0.0862 0.0722Male education -0.0649* -0.0271 0.0326 -0.0309Illiteracy 0.0379 0.0345 -0.0199 0.0524*Forced displacement 0.0201 0.0411 -0.0304 0.0131Safety perception -0.0676 0.2083* 0.1952** -0.0242Expenditure per capita -0.1493** -0.0261 0.0203 -0.0185Population density -0.0135*** -0.0113*** 0.0029 -0.0024People per household 0.1123** 0.0391 0.0562 -0.0763**Middle income -0.1624 -0.2189 -0.0676 -0.0864High income 0.2064 -0.3982 -0.2071 -0.1042Divorced males (spatial lag) 0.022 0.0215 -0.0051 -0.0041Caucasian (spatial lag) -0.0152 0.0131 0.0127 -3e-04Immigrants (spatial lag) 0.0506 0.0121 0.0286 -0.018Unemployment 24-15 (spatial lag) 0.026 -0.0078 0.011 -0.0016Male population 24-15 (spatial lag) -0.0309 -0.0196 0.0168 -0.0219Male education (spatial lag) 0.025 0.0096 -0.018 -0.0083Illiteracy (spatial lag) -0.0207 -8.00E-04 -0.0036 -0.0128Forced displacement (spatial lag) 0.0106 -0.0046 0.0186** 0.0036Safety perception (spatial lag) -0.085* -0.0417 -0.0634** 0.0199Expenditure per capita (spatial lag) 0.0209 0.0224 -0.0206 -0.01Population density (spatial lag) 0.0043*** 0.0022** 8.00E-04 -3e-04People per household (spatial lag) 0.0274 0.0144 0.0147 0.0236*Middle income (spatial lag) 0.0436 0.0473 0.0905** 0.065*High income (spatial lag) -0.0123 -0.0849 0.1249 0.0574
Notes: Posterior mean estimates. ***, ** and * are statistically significant variables at 1%, 5% and 10%.In particular, 99%, 95% and 90% highest density intervals do not embraces zero.7
Table 5—: Posterior estimates: error componentsHomicides Drug dealing Motorcycle thefts Car theftsˆ E (cid:2) η + i ) (cid:3) σ η E (cid:2) η + i + u + it )) (cid:3) σ u σ v σ α σ (cid:15) λ = σ η + σ u σ (cid:15) Notes: Posterior mean estimates. Standard deviations stochastic components (two-sided and one-sided), per-manent and total percentage of potential still covered crime suspects, and total one-sided variation to stochas-tic error variation ratio.
IV. Concluding remarks
We propose a Bayesian approach to perform inference regarding “hidden pop-ulations” at analytical region level such as criminal activity. We extend a gener-alized random effects model including spatial effects where “hidden populations”are taken into account using one-sided errors. Simulation exercises suggest thatour proposal has good sampling properties regarding point estimates and “hiddenpopulation” predictions.Our application based on home place of crime suspects suggest that there isassociation between homicide and drug dealing which has caused potential urbandisplacement to neighbourhoods near these crime communities. This is also sup-ported by historical facts and higher levels of uncaptured crime suspects, whichare as high as 90% in the hot spots of both activities. On the other hand, motor-cycle and car thefts have lower uncaptured rates (70% and 50%, respectively), andboth activities are associated with high local unemployment rates, which wouldsuggest that focused employment policies would mitigate these activities.Due to the elapsed time between crime moment and reported time using CCTVmonitoring, we suggest that a potentially good strategy to capture crime suspectsis to lock down the potential destination neighborhood of criminals. Our mod-elling strategy would help to predict this potential destination neighborhoods aswe identified potential “crime communities”. On the other hand, focused policyinterventions targeting these communities with specific education and employ-ment objectives would help to structurally reduce crime activity.Future research should take into account sensitivity of our proposal to spatialcontiguity criteria, where contiguity matrices can be selected based on Bayesfactors or performing Bayesian model average to take into account this uncertaintysource. Figure 2. : Percentage of potential uncaptured crime suspects: Medell´ın 2011-2014 (a) Suspects: homicide rates (b) Suspects: drug dealing(c) Suspects: car thefts (d) Suspects: motorcycle thefts
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A1. Descriptive statistics
Table A1—: Definition: dependent variablesVariable DescriptionHomicide Amount of captured homicide suspects living at analyticalregion i in year t per one hundred thousand inhabitants.Drug dealing Amount of captured drug dealing suspects living at analyticalregion i in year t per one hundred thousand inhabitants.Car theft Amount of captured car theft suspects living at analyticalregion i in year t per one hundred thousand inhabitants.Motorcycles theft Amount of captured motorcycle theft suspects living at an-alytical region i in year t per one hundred thousand inhabi-tants. Note: This information comes from police reports between 2011 and 2014.3
Table A2—: Definition: control variablesVariable DescriptionMale population 15-24 Proportion of males between 15 and 24 years oldDivorced males Proportion of divorced malesCaucasian Proportion of white race peopleImmigrants Proportion of people who migrated to regionUnemployment 15-24 Proportion of unemployed people between 15 and 24years oldLow income Proportion of low income householdsMiddle income Proportion of middle income householdsHigh income Proportion of high income householdsMale education Proportion of males head of household with college degreeor higherIlliteracy Proportion of population older than 15 years old thatdoes not know how to read or writeForced displacement Proportion of families that suffer forced displacementPeople per household Average number of people per householdPopulation density Average number of people per km Safety perception Proportion of population who feels unsafeExpenditure per capita Average per capita expenditure in COP$
Notes: Proportion variables are measured in terms of total per million inhabitants in analytical regions. In-formation comes from Medell´ın living standards survey between 2011 and 2014.4 T a b l e A — : D e s c r i p t i v e s t a t i s t i c s : a n a l y t i c a l r e g i o n s i n M e d e ll ´ ı n , - . M e a n O v e r a ll s . d . B e t w ee n s . d . W i t h i n s . d . M i n . M a x . M a l e p o pu l a t i o n - . . . . . . D i v o r ce d m a l e s . . . . . . C a u c a s i a n . . . . . . I mm i g r a n t s . . . . . . U n e m p l o y m e n t - + . . . . . . M a l ee du c a t i o n . . . . . . I lli t e r a c y . . . . . . F o r ce dd i s p l a ce m e n t . . . . . . S a f e t y p e r ce p t i o n . . . . . . E x p e nd i t u r e p e r c a p i t a1 , , , , , , , , P o pu l a t i o nd e n s i t y . . . . . . P e o p l e p e r h o u s e h o l d . . . . . . L o w i n c o m e . . . . . . M i dd l e i n c o m e . . . . . . H i g h i n c o m e . . . . . . H o m i c i d e r a t e . . . . . , . Su s p ec t s h o m i c i d e r a t e . . . . . . Su s p ec t s d r u g d e a li n g r a t e . . . . . . Su s p ec t s m o t o r c y c l e t h e f t s r a t e . . . . . . Su s p ec t s c a r t h e f t s r a t e . . . . . . N o t e s : + t h i s a r e p s e u do e m p l o y m e n t m e a s u r e s , s i n ce w ec o m p u t e d t h e m b y u s i n g t o t a l pop u l a t i o n a n d n o tt h eec o n o m i c a ll y a c t i ve pop u - l a t i o n . P r opo r t i o n v a r i a b l e s a r e m e a s u r e d i n t e r m s o f t o t a l p e r m i ll i o n i n ha b i t a n t s i n a n a l y t i c a l r eg i o n s . I n f o r m a t i o n c o m e s f r o m M e d e ll ´ ı n l i v i n g s t a n da r d ss u r ve y be t w ee n n d2014 . A2. Robustness checks T a b l e A — : S a m p li n g p r o p e r t i e s o f B a y e s e s t i m a t o r s : V a r y i n g λ = σ u + + σ η + σ (cid:15) λ η + σ η u + σ u v σ v σ α σ (cid:15) β β M e a n M e a n M e d i a n ˆ E ( ρ ) M e a n H D I M e a n M e d i a n ˆ E ( ρ ) M e a n H D I M e d i a n ˆ E ( ρ ) M e a n H D I M e a n H D I M e a n H D I M e a n H D I M e a n H D I T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . - . . . . - . - . - . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . - . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . - . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . - . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = T r u e . - . - . . - . - . . . . . . . - . E s t i m a t e d . - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = N o t e : B y P op u l a t i o n v a l u e o f η + , u + a n d v w e m e a n t h e a ve r a ge a n d m e d i a n v a l u e s a s t h e y w e r ege n e r a t e d f r o m t h e M o n t e C a r l o ex p e r i m e n t . W e r e po r t po s t e r i o r m e a n s , m e d i a n s a n dh i g h e s t d e n s i t y i n t e r v a l s f r o m po s t e r i o r d r a w s u s i n g , i t e r a t i o n s , , b u r n - i n a n d t h i nn i n geq u a l t o5 . ˆ E ( ρ ) i s t h eex p ec t e d v a l u e o f t h ec o rr e l a t i o n o f t h e po s t e r i o r d r a w s o f o n e - s i d e d e rr o r s a n d s pa t i a l e ff ec t s , a n d t h e P op u l a t i o n o n e - s i d e d e rr o r s a n d s pa t i a l e ff ec t s c o m i n g f r o m t h e M o n t e C a r l o ex p e r i m e n t , r e s p ec t i ve l y . T a b l e A — : S a m p li n g p r o p e r t i e s o f B a y e s e s t i m a t o r s : H e a vy t a il s η + σ η u + σ u v σ v σ α σ (cid:15) β β M e a n M e d i a n ˆ E ( ρ ) M e a n H D I M e a n M e d i a n ˆ E ( ρ ) M e a n H D I M e d i a n ˆ E ( ρ ) M e a n H D I M e a n H D I M e a n H D I M e a n H D I M e a n H D I P o pu l a t i o n - . - . . - . - . . - . . . . . - . e s t i m a t e - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = P o pu l a t i o n - . - . . - . - . . . . . . . - . e s t i m a t e - . - . . . . . - . - . . . . . . . . . . . . . . . . . . . - . - . - . N = , T = P o pu l a t i o n - . - . . - . - . . - . . . . . - . e s t i m a t e - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = P o pu l a t i o n - . - . . - . - . . - . . . . . - . e s t i m a t e - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = P o pu l a t i o n - . - . . - . - . . . . . . . - . e s t i m a t e - . - . . . . . - . - . . . . . - . . . . . . . . . . . . . . - . - . - . N = , T = N o t e : B y P op u l a t i o n v a l u e o f η + , u + a n d v w e m e a n t h e a ve r a ge a n d m e d i a n v a l u e s a s t h e y w e r ege n e r a t e d f r o m t h e M o n t e C a r l o ex p e r i m e n t . W e r e po r t po s t e r i o r m e a n s , m e d i a n s a n dh i g h e s t d e n s i t y i n t e r v a l s f r o m po s t e r i o r d r a w s u s i n g , i t e r a t i o n s , , b u r n - i n a n d t h i nn i n geq u a l t o5 . ˆ E ( ρ ) i s t h eex p ec t e d v a l u e o f t h ec o rr e l a t i o n o f t h e po s t e r i o r d r a w s o f o n e - s i d e d e rr o r s a n d s pa t i a l e ff ec t s , a n d t h e P op u l a t i o n o n e - s i d e d e rr o r s a n d s pa t i a l e ff ec t s c o m i n g f r o m t h e M o n t e C a r l o ex p e r i m e n t , r e s p ec t i ve l yy