#### Fire: A lognormal-Pareto distribution

This example is taken from Section 5 of

Scollnik, David P. M. , 'On composite lognormal-Pareto models', Scandinavian Actuarial Journal, 2007:1, pages 20 - 33

using the lognormal-pareto II model.

The data are fire insurance losses (in Danish Krone). The model is lognormal with mean mu and variance sigma when the losses (X) are less than an unknown parameter, theta, and pareto with parameter alpha when X is greater than theta.

The composite density is coded with BUGS functions and implemented using the 'logLike' distribution.

model{

for ( i in 1 : N){

dummy[i] <- 0
dummy[i] ~ dloglik(logLike[i])
logLike[i] <- log(r / phi(alpha * sigma)) * (1 - stepxtheta[i]) + log(1 - r) * stepxtheta[i] +
(-0.5 * log(2 * pi) - log(x[i]) - log(sigma) - 0.5 * pow((log(x[i]) - mu )/ sigma, 2) ) *
(1 - stepxtheta[i]) +
(log(alpha) + alpha * log(theta) - (alpha + 1)* log(x[i])) * stepxtheta[i]

stepxtheta[i] <- step(x[i] - theta)

}

theta ~ dgamma(0.001, 0.001) # dexp(0.5) #
alpha ~ dgamma(0.001, 0.001) # dexp(0.5) #
sigma ~ dgamma(0.001, 0.001) # dexp(0.5) #

r <- (sqrt(2 *pi) * alpha * sigma * phi(alpha * sigma))
/ (sqrt(2 * pi) * alpha * sigma * phi(alpha * sigma) + exp(-0.5 * pow(alpha * sigma, 2)))
mu <- log(theta) - alpha * pow(sigma, 2)
pi < -3.14159565

# xf prediction from fitted distribution
xf <- xa * delta + xb * (1 - delta )
xa ~ dlnorm(mu, tau )T( , theta )
xb ~ dpar(alpha, theta)

delta ~ dbern(r)
tau <- 1 / pow(sigma, 2)

}

##### Data
``` list(N = 2492, x=c(1.6837482, 2.0937042, 1.7325813, 1.7797540, 4.6120059,8.7252738, 7.8989751, 2.2080454, 1.4860908, 2.7961713,7.3206442, 3.3674963, 1.4641288, 1.7222225, 11.3748170,2.4827394, 26.2146413, 2.0024305, 4.5300146, 1.8417526,3.8067350, 14.1220761, 5.4242533, 11.7130307, 1.5153734,2.5385886, 2.0497804, 12.4655930, 1.7354451, 1.6837482,3.3238521, 1.8210249, 2.4158126, 1.4641288, 5.8759019,1.7924392, 3.3382138, 1.5844876, 1.9313206, 2.0497804,1.5737584, 1.7569546, 4.9311859, 5.4172767, 1.5373353,17.5695461, 1.6911230, 1.5816120, 1.4671157, 2.0497804,1.9287116, 3.2631537, 1.7042460, 2.0746076, 3.2942899,1.8399751, 7.3206442, 7.3206442, 1.5762152, 4.8560981,1.5422753, 13.6207906, 1.7569546, 1.6398243, 2.0363777,21.9619327, 1.7990264, 2.4524158, 2.6691069, 2.0241464,1.7247438, 1.6438638, 1.9619327, 4.6266471, 1.4641288,3.9632504, 5.5638521, 4.3923865, 3.6403133, 1.6778917,1.4494876, 263.2503660, 2.5007321, 1.8257687, 9.8828697,2.1466179, 2.8702592, 6.3199136, 2.1234612, 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1.1419142,1.6460396, 1.7706271, 0.8663366, 4.1254125, 1.9801980,1.3201320, 1.0066007, 8.2508251, 0.8910891, 1.1138614,0.9900990, 1.6501650, 0.9158416, 1.2665016, 1.5536304,4.0841584, 1.4488449, 0.8498350, 14.8514851, 0.8250825,0.8919142, 1.5965347, 0.8910891, 3.8778878, 0.8605611,2.0214521, 1.6501650, 0.8250825, 0.9653465, 0.9653465,1.0107261, 1.4851485, 0.8712871, 1.2788779, 0.9900990,1.0231023, 4.7029703, 3.1353135, 1.8646865, 2.0379538,2.1452145, 1.1551155, 1.0561056, 144.6575908, 0.8803630,0.9199670, 0.9282178, 28.6303630, 19.2656766, 5.5280528,1.5429043, 4.6204620, 1.0726073, 3.7128713, 1.0140264,0.8250825, 0.9579208, 1.1138614, 0.8828383, 0.8663366,0.9191419, 0.9983498, 2.3399340, 1.9389439, 0.9900990,1.2376238, 3.0528053, 1.4092409, 2.5536304, 0.8250825,0.9900990, 5.7755776, 1.1344884, 3.4075908, 1.0726073,1.1674917, 0.8745875, 0.9900990, 1.0726073, 1.2706271,0.9661716, 1.4727723, 1.0363036, 1.6501650, 1.6782178,2.6402640, 1.6014851, 17.7392739, 0.8663366, 4.3729373,1.3613861, 1.1839934, 0.9282178, 2.9702970, 1.0231023,0.9075908, 1.1303630, 3.0115512, 1.4026403, 0.9075908,0.9752475, 0.8745875, 2.3226073, 1.1155116, 1.6914191,1.2376238, 1.1146865, 0.9735974, 1.4026403, 4.8679868,1.0726073, 4.1254125)) ```
##### Inits for chain 1
``` list(alpha=1, sigma = 0.1, theta=1) # MLEs are 1.3234, 0.1927, 1.1969 ```

##### Inits for chain 2
``` list(alpha=2, sigma = 0.5, theta = 2) # MLEs are 1.3234, 0.1927, 1.1969 ```

Results