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, 2.9238653,
2.0363763, 1.7364568, 2.0497804, 1.9033675, 1.6105417,
1.7174231, 3.1294378, 2.0519575, 1.9675183, 2.1024890,
3.2591508, 1.9985359, 5.4172767, 2.8198360, 3.6970835,
2.0453880, 2.6793558, 3.4262240, 1.5885798, 4.2628712,
1.7558594, 1.9870059, 1.6619253, 7.6134700, 1.7948785,
6.6748170, 2.1859444, 3.8550512, 3.2210835, 2.1229868,
1.7602914, 2.0416530, 3.7559385, 2.2421113, 1.7211127,
1.4641288, 4.7118917, 2.1538243, 3.8748126, 19.0702782,
1.8872621, 1.4641288, 1.5361640, 2.5600659, 2.7997072,
2.0131772, 1.4641288, 6.3250366, 1.5673001, 2.1961933,
2.3352855, 2.1156662, 1.4695447, 5.4026354, 5.3147877,
2.8726208, 2.7818448, 2.5768214, 3.1010249, 1.4641288,
2.0497804, 1.9619327, 1.4641288, 2.6354319, 1.4641288,
5.8565154, 3.2942899, 2.5208565, 19.4729136, 1.8354627,
2.0497804, 2.7833089, 2.7818448, 1.7569546, 2.8404100,
2.3308931, 1.7562254, 1.3145478, 2.4039712, 6.9156304,
1.4918768, 1.5323958, 1.8738611, 1.8152031, 8.2568807,
1.4102045, 2.2066252, 34.1415465, 2.4433106, 3.5943696,
2.3722149, 3.6697248, 6.8884535, 1.8020970, 1.4416776,
9.1743119, 2.0969856, 2.8145478, 1.6849030, 2.0996068,
2.6212320, 1.3761468, 1.6644823, 2.3591088, 4.1364875,
1.3943958, 1.8020970, 1.8191350, 1.5137615, 3.0807733,
20.9698558, 4.1939712, 1.8616435, 2.4967235, 2.2137811,
5.2424640, 1.3222543, 8.5517693, 1.9659240, 12.8951507,
1.7889908, 2.1768270, 8.7776278, 1.8152031, 1.6831402,
3.6435125, 1.6133683, 1.4678899, 1.4056042, 3.1208034,
1.5981809, 3.8007864, 1.5100891, 1.5727392, 1.8300524,
2.6212320, 2.8064810, 1.3577982, 1.6186107, 1.9659240,
3.0222805, 56.2254260, 4.9803408, 1.9871625, 1.9659240,
2.7112923, 1.9003932, 1.7234600, 1.5727392, 1.7038008,
3.0196592, 1.8348624, 1.4193971, 4.3905636, 7.8636959,
1.5727392, 1.3366134, 1.7038008, 2.6356225, 1.8647182,
1.4577549, 7.5583224, 2.0313971, 1.3499345, 1.4748624,
5.1114024, 5.2424640, 1.4416776, 3.9361389, 4.0629096,
1.3761468, 1.6841415, 1.4406199, 1.3645491, 1.5727392,
2.1854220, 3.2813460, 1.3631009, 2.1231979, 2.6343381,
2.0982962, 3.4343526, 1.4787955, 2.6212320, 3.0013106,
1.4297339, 10.2228047, 2.5793958, 1.7553591, 1.4734050,
1.9528178, 1.7798689, 1.9868624, 1.9397117, 1.6759004,
1.5727392, 2.5249017, 14.6788991, 3.0340760, 1.9790301,
5.9377588, 1.4416776, 2.6212320, 2.0380079, 3.7549148,
2.4377457, 1.7020315, 7.1098702, 2.0196592, 1.8504902,
1.4159502, 7.6933159, 3.0144168, 2.3591088, 1.5923159,
1.3892529, 1.3329685, 1.4233290, 2.9775714, 1.3873552,
1.4049803, 2.6212320, 4.2791612, 1.6055046, 2.0117955,
2.2673657, 1.4646501, 3.3740655, 1.4907995, 2.0969856,
1.5701664, 1.3761468, 3.1454784, 8.4537353, 3.0332831,
2.0642202, 6.4220183, 1.8987877, 1.4901704, 50.0655308,
1.5596330, 1.9659240, 1.8000524, 2.4246396, 1.3106160,
1.6120577, 4.2211653, 1.2604043, 1.4036932, 1.2972652,
1.9274923, 1.9154578, 1.2074518, 5.6985826, 1.9024970,
1.1891344, 10.1780238, 1.6020797, 4.1617122, 2.3630499,
1.9650440, 1.9459822, 2.8537455, 1.2024400, 10.8204518,
1.5887634, 2.0963496, 5.4696790, 1.8929845, 2.0332937,
1.6052319, 1.4030916, 1.1950059, 3.2461356, 1.2631225,
2.9726516, 1.2049501, 1.8812735, 1.7835910, 3.9239001,
1.9024970, 1.6646849, 4.4528252, 1.3236207, 1.6052319,
24.9702735, 2.7348395, 1.7835910, 1.8109394, 2.1141498,
1.3912010, 1.3460166, 1.5073841, 1.9404875, 2.5697979,
1.5807658, 1.4875149, 1.5219976, 1.6908442, 1.7835910,
1.2588656, 1.5737479, 1.4673008, 2.7137051, 2.7360285,
1.4565993, 2.1374316, 2.4790904, 2.3781213, 4.6373365, # )) ,
1.1890606, 1.5457788, 2.3424495, 2.3376932, 1.3971463,
4.2211653, 1.6399524, 1.2187872, 3.6860880, 2.4970273,
2.4970273, 5.2318668, 1.3079667, 1.2681249, 2.1652794,
4.6401926, 2.9726516, 1.6646849, 1.1890606, 11.8906064,
2.8656361, 1.2443674, 1.7241379, 5.6956005, 1.3595696,
7.0749108, 1.4034756, 1.4024185, 1.3198573, 4.3205886,
1.8430440, 1.4149822, 1.4863258, 4.9460321, 1.9131986,
20.0499405, 1.4365077, 5.5077289, 3.2104637, 1.3993282,
5.0017348, 3.4482759, 3.5564804, 1.3575684, 2.1315113,
1.8418858, 1.4268728, 2.4175981, 1.2521546, 1.8997027,
1.5591831, 1.2492533, 1.9032414, 1.6052319, 2.2296587,
1.2128419, 3.4482759, 1.2432675, 1.6749310, 1.3531510,
1.7875446, 2.0368609, 1.9738407, 1.6409037, 2.9099322,
2.5509584, 1.8073722, 4.0025779, 1.4268728, 0.9512485,
2.3431118, 1.3888228, 4.1736029, 1.2785375, 3.5588585,
1.2001498, 1.5512342, 1.6996195, 65.7074911, 27.2625945,
15.9262782, 1.2794970, 3.0984411, 2.9469619, 1.2247325,
1.6646849, 1.3196017, 1.5165993, 1.2732307, 1.6646849,
1.3079667, 5.4696790, 1.2485137, 1.9868252, 1.4268728,
1.4268728, 2.1640904, 2.1403092, 1.3079667, 1.4983841,
1.6884661, 1.8399489, 4.3757432, 1.7866920, 1.6559441,
1.1890606, 3.8507598, 4.1617122, 2.1403092, 2.1403092,
2.6159334, 4.1617122, 22.2582259, 2.2592152, 4.2806183,
1.3062473, 5.5885850, 2.7942925, 1.4416018, 1.3496463,
1.5650723, 1.1123471, 2.8101635, 1.7834527, 2.1286986,
4.3492770, 2.7107898, 1.1345940, 2.6236863, 2.3937709,
6.2347052, 1.3781980, 1.2169077, 1.1635150, 2.4594071,
4.7274750, 1.5572859, 1.3070078, 3.1434928, 1.1971079,
1.1679644, 1.1123471, 1.7797553, 1.6474027, 1.7236919,
5.5617353, 1.3348165, 0.8620690, 1.4460512, 1.3348165,
1.5906563, 1.1123471, 3.0324360, 3.7864294, 2.2246941,
2.4471635, 10.0111235, 1.1635473, 1.1123471, 3.1528632,
3.7263626, 4.4493882, 2.2812636, 3.5595106, 4.4493882,
1.2091212, 1.1568409, 2.2246941, 1.2402670, 1.1592558,
1.6685206, 3.3370412, 10.0723026, 2.1412681, 1.4749722,
1.1123471, 1.6896552, 1.6685206, 1.6173526, 1.1123471,
2.1690768, 1.4938821, 1.4082314, 2.5027809, 2.2580645,
3.7263626, 1.8936997, 1.7041157, 2.3428543, 1.9466073,
1.2791991, 1.5106963, 2.4471635, 3.6751947, 2.3782414,
1.7741935, 1.5151279, 2.2347052, 1.8164627, 3.4482759,
4.6718576, 1.1568409, 1.2235818, 4.0789299, 1.1924360,
1.2040934, 2.5583982, 1.6184650, 1.8220245, 1.6105640,
1.8609566, 7.9920701, 1.7853170, 4.8943270, 2.5205784,
1.3948832, 1.3904338, 1.2971791, 3.6707453, 3.8765295,
3.5873192, 1.4379544, 1.1752748, 1.5016685, 1.5060033,
5.9254727, 12.6318131, 3.0611791, 1.4360400, 2.2791991,
1.4917664, 6.9165539, 1.6042291, 1.6685206, 1.1572113,
1.3203560, 3.4482759, 1.9757620, 1.8565072, 2.0022247,
5.3005039, 4.0044494, 1.1244683, 1.5464950, 2.5583982,
1.4791657, 1.2593993, 3.1979978, 3.1145717, 1.3125695,
5.6729700, 13.3481646, 11.4315907, 4.2591769, 3.3668265,
1.1123471, 1.1123471, 1.4460512, 2.0731146, 1.2235818,
1.6129032, 1.3192436, 1.1290323, 2.3359288, 1.7241379,
11.1234705, 1.6685206, 1.5572859, 1.5572859, 1.9333115,
1.7241379, 1.5016685, 2.2074346, 1.0471204, 1.4253613,
1.3612565, 9.3141361, 1.6448901, 2.6178010, 2.5411120,
1.5706806, 1.2565445, 1.4586387, 1.0471204, 2.2314136,
2.1989529, 5.0261780, 2.4083770, 1.6521780, 1.1518325,
1.3196838, 2.7015707, 2.5445026, 2.3560209, 1.2565445,
1.6753927, 1.3612565, 3.8743456, 1.7696335, 1.1183246,
2.2094241, 1.0984293, 1.1518325, 2.9637497, 11.6230366,
14.2931937, 1.2879581, 1.4659686, 13.6230366, 1.1298429,
2.4031414, 5.4450262, 1.0469110, 1.0591623, 2.0942408,
1.8554974, 1.5706806, 1.9371728, 2.2031414, 2.4607330,
1.4276168, 1.7350785, 4.0837696, 2.1570681, 1.4659686,
1.1685864, 1.2251309, 1.1518325, 3.7696335, 1.0950524,
1.2041885, 2.6178010, 1.8712042, 4.3979058, 4.6073298,
18.6464838, 1.0471204, 1.0471204, 1.0471204, 15.8115183,
1.9895288, 1.0471204, 1.2565445, 1.1518325, 1.0732984,
1.1944209, 1.0643613, 1.3842932, 1.1308901, 1.1518325,
1.7937173, 1.1413613, 1.6329058, 2.6439791, 1.2868963,
4.4502618, 1.2366492, 1.9591623, 1.1910503, 1.3144335,
1.9895288, 1.6495707, 2.7225131, 1.1324209, 1.7801047,
1.6073298, 0.9897749, 1.2041885, 1.0732984, 1.0994764,
1.4062827, 1.9654450, 3.3631906, 1.5706806, 1.2565445,
2.7624408, 1.0664859, 1.3779508, 1.3002534, 1.3193717,
3.4031414, 1.3612565, 1.1780105, 4.2931937, 4.8083770,
1.0994764, 2.2198953, 2.6178010, 1.4344963, 1.0471204,
2.5130890, 1.6214597, 19.1623037, 1.5706806, 2.3036649,
1.8455550, 1.0471204, 1.9895288, 2.8272251, 1.2565445,
3.9790576, 1.3612565, 1.6335079, 0.9424084, 18.8481675,
1.1256545, 5.3055780, 1.7277487, 1.1653424, 1.1518325,
1.2565445, 3.5685864, 7.2356021, 1.5706806, 7.6439791,
2.1220031, 2.9319372, 1.2879581, 1.0680073, 1.2756147,
1.6230366, 2.6701571, 1.2565445, 1.1030482, 1.2041885,
1.1518325, 1.8848168, 7.5392670, 1.0994764, 1.5706806,
2.6701571, 1.1518325, 3.8743456, 5.0261780, 1.7801047,
4.7643979, 1.1518325, 1.5000000, 1.2510000, 1.0300000,
1.0500000, 1.9000000, 1.1000000, 1.8817500, 1.0070000,
1.6300000, 1.0250000, 1.0072740, 3.5000000, 2.9000000,
2.4631370, 4.6250000, 1.0300000, 1.4000000, 1.0770750,
1.3200000, 2.6000000, 1.0800000, 1.5170000, 1.0700000,
2.2568230, 1.7884130, 1.2788960, 4.6090740, 3.1830000,
1.2000000, 2.7430000, 3.0000000, 1.0089380, 1.0040840,
1.8500000, 1.5209940, 1.2500890, 1.0000000, 1.9560000,
1.0500000, 2.7790000, 22.1375670, 1.4450000, 1.0051000,
1.0000000, 1.0000000, 16.3000000, 1.1100000, 1.4750000,
1.0498730, 1.7535420, 6.1433550, 1.2300000, 1.9390000,
46.5000000, 1.2414720, 3.4000000, 1.0000000, 6.2000000,
1.9662370, 2.1700000, 1.2960000, 1.5550000, 1.1500000,
1.3300000, 4.1242850, 1.3942390, 4.7612280, 4.8688520,
4.0000000, 1.6000000, 7.0850000, 1.7650540, 6.3066540,
1.2500000, 1.2000000, 1.2264120, 6.5630000, 4.1700000,
2.6800000, 1.0750000, 1.9329570, 1.3750000, 1.2500000,
3.3800000, 1.3801000, 2.2830000, 2.3500000, 3.1500000,
3.3500000, 4.0000000, 10.5000000, 2.8900000, 5.5000000,
2.8000000, 1.2600000, 1.1000000, 2.1267590, 2.2000000,
1.0700000, 9.2000000, 1.5000000, 1.6500000, 12.2250000,
1.4000000, 14.2390000, 1.0850000, 1.0500000, 2.6500000,
2.1230000, 5.2000000, 1.8250000, 1.1000000, 1.0000000,
2.0650000, 1.2774000, 1.1950000, 1.2000000, 1.1000000,
1.4000000, 1.5850000, 3.2650420, 1.2000000, 2.4330000,
1.6000000, 2.0000000, 4.7600000, 1.1650000, 1.3720000,
1.2000000, 1.6500000, 1.7000000, 1.3000000, 1.1000000,
1.1000000, 1.2410120, 1.0050000, 3.2820000, 1.0500000,
57.4106360, 1.0880000, 1.1180000, 2.6700000, 1.0749000,
2.1410000, 5.8500000, 14.3000000, 1.9710000, 1.7000000,
6.1670000, 1.0500000, 1.1560580, 3.5821500, 13.5000000,
1.2015000, 1.6500000, 1.3776000, 3.8000000, 1.3255000,
4.0500000, 6.7000000, 1.1600000, 1.5000000, 1.0600000,
1.4912870, 1.9818130, 1.5000000, 3.2143600, 4.1000000,
1.0000000, 10.7000000, 19.4000000, 1.6056030, 1.1320000,
2.2550000, 2.0200000, 1.2310000, 4.1000000, 7.2300000,
1.8500000, 1.0000000, 2.5600000, 1.5530000, 1.7000000,
1.0454720, 1.0540000, 1.0000000, 8.7100000, 1.5000000,
1.5000000, 1.2610600, 1.0550000, 1.2820000, 1.3000000,
1.3000000, 1.4220780, 1.0000000, 1.2000000, 1.8500000,
1.1800000, 1.0430000, 1.0000000, 1.6000000, 5.6000000,
1.8000000, 2.5000000, 1.7500000, 1.4500000, 1.4609450,
1.1571842, 1.4957570, 5.2073288, 3.6837030, 1.4464802,
1.0386577, 1.2536162, 1.0607522, 1.3982642, 1.3982642,
2.4108004, 1.4418515, 1.6393443, 1.0506268, 1.7550627,
0.9643202, 4.4840887, 0.9643202, 1.6779171, 1.0607522,
1.0000000, 3.0829315, 1.2777242, 1.2073288, 1.9286403,
1.0028930, 1.0896818, 1.7581292, 1.4464802, 1.5911283,
1.1089682, 1.0231099, 2.0009643, 1.3143105, 0.9726905,
2.0684667, 3.0858245, 6.7984571, 1.6393443, 1.6393443,
1.2295082, 1.7357763, 2.8234166, 2.5072324, 1.8667396,
4.4358727, 1.9286403, 1.0607522, 1.7414658, 3.0858245,
12.0540019, 1.1571842, 2.4108004, 1.5583414, 0.9643202,
1.3500482, 3.1822565, 1.9006635, 2.3143684, 3.7078110,
3.2063645, 1.0607522, 0.9643202, 1.4464802, 1.1571842,
5.2073288, 3.6644166, 1.6592285, 2.5072324, 0.9903568,
3.1918997, 16.4416586, 1.0829315, 2.4108004, 3.6692478,
1.1812922, 29.0260366, 4.9180328, 1.1485053, 1.5429122,
1.5670203, 1.2536162, 1.5429122, 1.8322083, 1.6634523,
1.2777242, 2.4435892, 2.8929605, 3.1340405, 2.6036644,
3.7897782, 1.9026037, 3.7704388, 1.1668274, 1.1890067,
0.9643202, 1.3211186, 1.7357763, 12.5361620, 1.3211186,
1.1571842, 5.7859209, 1.8322083, 1.3889797, 1.2054002,
1.1089682, 1.7293578, 0.3134040, 1.6393443, 1.0876480,
1.5429122, 1.4320154, 2.1697203, 5.4001929, 0.9739634,
5.7859209, 1.3500482, 6.0752170, 0.9643202, 18.3220829,
1.1571842, 0.9643202, 1.5911283, 1.9286403, 1.4464802,
1.6779171, 1.1571842, 1.0109981, 2.8447445, 1.7357763,
1.1504339, 2.5072324, 2.8929605, 2.0973963, 2.0864995,
0.9884282, 1.4464802, 0.9643202, 2.1502893, 1.1220926,
1.6586307, 1.6972035, 1.5911283, 5.7859209, 1.0559306,
1.4464802, 2.8929605, 1.1571842, 1.1205400, 1.1571842,
1.2033751, 5.5930569, 1.4329797, 1.2643202, 1.3992285,
0.9643202, 1.1477040, 1.2382343, 1.7598843, 1.5429122,
2.1215043, 1.5429122, 1.2536162, 1.9286403, 1.3307618,
0.9990357, 1.9286403, 1.1089682, 2.0250723, 8.6788814,
0.9836066, 2.2179364, 1.3018322, 1.1238968, 1.7275796,
1.3741562, 1.0069576, 8.1002893, 1.0028930, 1.1716490,
0.9643202, 10.2700096, 1.2713954, 2.8929605, 3.3452324,
1.0607522, 2.8929605, 1.1215043, 1.6972035, 1.7164899,
1.1024108, 17.0684667, 2.6036644, 1.2536162, 3.5679846,
1.4946962, 1.6393443, 1.7222758, 2.3510125, 1.6875603,
1.4705882, 2.3143684, 1.2536162, 1.1444291, 1.2054002,
2.6027001, 2.9417936, 2.1812922, 1.4464802, 4.8216008,
3.8379942, 1.3500482, 1.5429122, 1.1571842, 1.9286403,
2.2420444, 1.2632594, 1.0125362, 1.9286403, 0.9643202,
1.1378978, 1.0376085, 1.2536162, 2.1875603, 4.7672951,
1.5429122, 1.0607522, 2.7521697, 3.1822565, 1.4464802,
1.9286403, 1.5911283, 3.0858245, 4.0983607, 1.8804243,
2.4108004, 1.0125362, 1.9286403, 1.0607522, 0.9643202,
1.6393443, 1.1282546, 1.4464802, 1.4464802, 2.8929605,
1.3886210, 1.6393443, 2.0221794, 1.7051591, 1.0511090,
17.7434908, 1.1571842, 3.8572806, 1.7051591, 1.3500482,
5.4966249, 1.0231911, 1.3042672, 1.0667904, 1.8089054,
3.2467532, 1.4174397, 0.9740260, 2.4907236, 1.0111317,
2.9712430, 0.9276438, 3.0046382, 0.9461967, 0.9276438,
1.4202226, 2.2337662, 1.3979592, 2.3191095, 2.8525046,
1.9944341, 1.7421150, 1.6465677, 2.9730983, 1.1131725,
1.4842301, 1.1131725, 1.1493506, 3.1846011, 2.2263451,
3.8961039, 1.4378479, 2.3682746, 1.0213358, 1.0204082,
2.7365492, 23.2838590, 1.3914657, 1.3914657, 2.0408163,
1.0204082, 2.1530612, 0.9628942, 1.0667904, 1.3914657,
1.4591837, 4.9907236, 1.0575139, 4.1280148, 0.9833024,
1.0769944, 1.5027829, 1.0213358, 4.5000000, 3.1076067,
2.2541744, 1.7578850, 1.3988868, 0.9461967, 1.2523191,
4.4981447, 2.4118738, 2.1057514, 3.1883117, 3.6846011,
1.5769944, 1.6512059, 2.5231911, 1.6697588, 1.9044527,
1.8552876, 1.6233766, 0.9434137, 1.8552876, 2.3191095,
1.4378479, 0.9443414, 3.0380334, 1.2569573, 1.2448980,
3.8961039, 1.1131725, 1.2801484, 6.5862709, 0.9276438,
1.0853432, 1.6066790, 1.2987013, 1.7625232, 1.1131725,
1.2504638, 1.2356215, 1.1131725, 0.9322820, 0.9944341,
4.8237477, 2.4257885, 3.1474954, 1.6187384, 1.5213358,
1.8552876, 7.1428571, 0.9276438, 1.2170686, 1.1131725,
0.9740260, 1.5769944, 1.6233766, 1.7625232, 1.3914657,
1.1595547, 0.9740260, 11.1317254, 1.1131725, 1.8552876,
1.6697588, 12.5231911, 2.0408163, 0.9740260, 5.7513915,
0.9276438, 5.5658627, 5.7513915, 1.1131725, 4.1743970,
8.8126160, 3.7801484, 32.4675325, 0.9740260, 2.0871985,
3.2931354, 3.3395176, 1.3905380, 1.6233766, 4.4526902,
1.0538033, 1.3914657, 0.9276438, 1.3914657, 1.6697588,
1.6901670, 0.9573284, 1.3172542, 0.9276438, 4.4063080,
1.3914657, 2.1335807, 0.9424861, 1.2059369, 1.3636364,
2.3979592, 3.2467532, 1.0695733, 0.9564007, 2.8756957,
1.0204082, 1.1131725, 0.9406308, 0.9740260, 1.8163265,
1.3562152, 1.4842301, 2.7829314, 1.6141002, 0.9322820,
29.0371058, 18.5528757, 2.5046382, 3.0612245, 1.1641929,
1.1131725, 1.3636364, 0.9703154, 1.0760668, 1.5120594,
1.8478664, 16.8831169, 1.3914657, 0.9619666, 1.9016698,
1.4489796, 1.7680891, 1.0204082, 1.5797774, 0.9758813,
0.9276438, 1.2059369, 0.9833024, 0.9740260, 1.5166976,
1.0204082, 0.9276438, 1.8552876, 1.2059369, 1.0204082,
0.9350649, 1.2244898, 1.0046382, 7.7922078, 1.3497217,
0.9925789, 27.8293135, 4.1743970, 5.8998145, 1.0064935,
1.5074212, 1.2059369, 1.1131725, 12.0593692, 2.3191095,
3.3395176, 1.1595547, 2.3729128, 1.0964750, 1.1048237,
1.2523191, 3.6502783, 1.3914657, 1.2987013, 6.3079777,
9.4619666, 2.8756957, 1.6836735, 3.4814471, 7.6066790,
0.9276438, 1.3914657, 0.9276438, 3.8961039, 1.1252319,
11.5955473, 3.0769944, 2.1150278, 1.8552876, 1.7569573,
1.4842301, 3.0612245, 1.5769944, 0.9276438, 0.9740260,
1.7346939, 0.9879406, 1.7625232, 1.2894249, 1.4684601,
5.1948052, 1.9480519, 4.1187384, 4.9628942, 5.1020408,
0.9276438, 1.3450835, 1.0751391, 7.1011132, 2.0705009,
0.9276438, 2.4118738, 1.2374768, 2.0408163, 1.2987013,
2.8942486, 1.0667904, 0.9285714, 1.3914657, 1.5306122,
1.0482375, 1.1781076, 0.9740260, 1.3450835, 2.0408163,
0.9276438, 1.6697588, 1.6184561, 1.0275067, 0.9316770,
16.4152618, 1.5084295, 0.8873114, 0.9760426, 1.6858917,
1.3309672, 2.3070098, 1.9201420, 3.7710736, 0.8873114,
1.1535049, 5.0940550, 0.9387755, 2.9565217, 1.7985803,
1.5527950, 1.3309672, 1.8873114, 1.7746229, 1.3309672,
6.0159716, 3.2617569, 0.9405501, 1.1668146, 2.5208518,
0.9405501, 1.2937001, 18.4241349, 0.6876664, 1.0647737,
1.9556344, 2.0408163, 7.9858030, 3.4968944, 0.9716060,
6.5110914, 1.2599823, 1.8535936, 1.5527950, 0.9591837,
1.2670807, 2.1295475, 2.7062999, 1.5527950, 1.3309672,
1.4676131, 2.3070098, 0.9316770, 1.9077196, 0.9689441,
1.2422360, 0.9103815, 0.8961846, 1.2111801, 1.9964508,
38.1543922, 2.1295475, 2.4844720, 1.6903283, 1.7657498,
1.5385980, 1.6858917, 1.2422360, 0.8873114, 0.8873114,
1.3309672, 1.9822538, 1.8633540, 0.8873114, 1.8855368,
5.2102928, 1.1535049, 1.7888199, 0.9671695, 0.8961846,
5.7675244, 1.3309672, 0.8873114, 6.8322981, 1.3620231,
0.9458740, 0.9760426, 3.5492458, 1.3753327, 2.3070098,
0.9316770, 0.8873114, 2.6619343, 0.9316770, 7.5421473,
5.6814552, 9.2280390, 0.9316770, 1.0807453, 1.6122449,
1.2422360, 1.3566992, 1.5084295, 2.3070098, 1.3753327,
0.9263531, 1.5527950, 0.9724933, 27.3380657, 1.2377995,
0.8873114, 1.1925466, 0.9840284, 1.2777285, 0.8873114,
1.3309672, 7.5421473, 2.2236025, 1.1535049, 0.9405501,
1.7879326, 1.4374445, 6.1401952, 1.6166815, 2.0940550,
11.8012422, 0.9405501, 2.9724933, 1.9520852, 2.4880213,
3.9929015, 2.2227152, 25.2883762, 2.4401065, 3.7267081,
3.7524401, 0.8873114, 1.6415262, 1.2422360, 0.8873114,
0.8873114, 1.2422360, 4.4365572, 0.9432121, 0.9041704,
1.0647737, 1.1978705, 2.3957409, 10.2040816, 0.9316770,
0.9760426, 1.7746229, 2.3203194, 2.3070098, 1.0878438,
1.1428571, 1.1535049, 1.8669033, 2.4844720, 0.9157054,
0.9059450, 2.5732032, 0.8873114, 1.1215617, 5.9893523,
2.5093168, 0.9973381, 1.3283052, 1.0452529, 1.3877551,
0.9148181, 1.1535049, 1.2866016, 1.0470275, 4.6140195,
20.4525288, 2.2182786, 5.3238687, 0.9405501, 1.2165040,
7.0984916, 0.9316770, 1.3309672, 1.7746229, 0.8873114,
3.0124224, 0.9645075, 0.9813665, 47.0195209, 1.1002662,
2.3096717, 1.1455191, 1.3762201, 1.1428571, 2.8961846,
1.7746229, 0.8873114, 1.3540373, 24.5785271, 2.2617569,
4.4365572, 1.3487134, 0.9822538, 15.8828749, 1.0204082,
0.9405501, 1.2626442, 2.6619343, 0.9760426, 1.0514641,
1.9520852, 0.9050577, 1.3575865, 0.9538598, 1.6415262,
7.5421473, 2.2182786, 5.5013310, 2.2111801, 1.4773736,
0.9574091, 1.6415262, 1.0204082, 0.9503106, 3.2830524,
0.9139308, 25.9538598, 1.0647737, 0.9148181, 0.9937888,
4.0816327, 2.2883762, 0.9316770, 0.9503106, 0.9449867,
1.1002662, 2.8385093, 3.1943212, 0.9316770, 1.5971606,
0.8917480, 0.7054126, 1.7746229, 0.9840284, 1.7213842,
0.9094942, 2.3336291, 4.4942325, 8.8731145, 2.6761313,
0.8873114, 1.1472937, 1.3309672, 1.5084295, 1.2422360,
1.1091393, 0.9094942, 6.2111801, 1.7302573, 1.1535049,
0.9370009, 1.0559006, 1.8793256, 1.4019521, 0.7542147,
0.8873114, 0.9760426, 1.4196983, 2.7151730, 4.0816327,
1.7524401, 0.9316770, 1.0887311, 3.5492458, 1.2830524,
2.1739130, 2.8393966, 3.3717835, 1.3309672, 0.4436557,
4.6140195, 10.8251996, 5.7675244, 1.1091393, 31.0559006,
0.8917480, 3.1055901, 1.3309672, 0.9849157, 1.6858917,
3.9929015, 0.9937888, 0.8970719, 1.8408129, 1.2828112,
5.3767993, 0.8839966, 1.3615580, 1.0584251, 1.0033870,
2.9635902, 1.2709568, 1.9051651, 4.2337003, 1.9475021,
2.9110923, 0.8890771, 24.5554615, 1.2277731, 1.2193057,
3.2684166, 4.8264183, 1.6596105, 0.8467401, 1.2142252,
0.8704488, 1.0160881, 1.5232854, 2.6841660, 0.9314141,
1.5622354, 0.9390347, 1.1210838, 0.9424217, 1.0143946,
42.0914479, 1.6934801, 1.6401355, 1.9475021, 4.6570703,
1.0584251, 0.9339543, 1.1160034, 1.5241321, 1.3920406,
1.2277731, 0.8467401, 0.9314141, 14.3945809, 1.1303980,
1.7442845, 1.9475021, 0.9017782, 1.1854361, 0.9822185,
2.5656224, 1.1481795, 0.9822185, 2.1761219, 1.3310754,
20.8636749, 1.8941575, 2.1168501, 1.6511431, 1.0584251,
9.2294666, 0.8679086, 1.1430991, 0.9102456, 5.9271804,
1.1981372, 1.7747671, 1.8755292, 5.6646909, 1.1007621,
0.9923793, 4.1490262, 0.9314141, 1.2836579, 1.0584251,
2.2015241, 1.2895851, 1.3547841, 4.4165961, 0.8467401,
1.2701101, 14.3945809, 2.6867062, 1.6934801, 1.6240474,
3.3869602, 4.0643522, 1.3014395, 1.2701101, 1.5537680,
1.4767146, 1.1896698, 0.9906859, 1.0160881, 10.1371719,
2.6629975, 1.0457240, 0.8890771, 3.8103302, 1.2701101,
1.6934801, 1.4817951, 0.8975445, 1.5241321, 2.9635902,
1.1812024, 1.5656224, 3.0482642, 3.3378493, 1.4572396,
1.0160881, 4.3014395, 0.8890771, 0.3810330, 0.8763760,
0.9314141, 2.0321761, 4.0220152, 1.1219306, 0.9889924,
1.8797629, 0.9060119, 2.2015241, 0.8721423, 1.2701101,
0.8806097, 1.7341236, 0.9381880, 1.1685013, 6.7739204,
1.1769687, 4.5004234, 2.2133785, 1.0160881, 1.8628281,
2.2015241, 0.9187130, 0.8628281, 0.9356478, 1.2193057,
5.7154953, 6.4352244, 5.5038103, 2.5402202, 5.0804403,
5.0804403, 0.8467401, 0.9906859, 0.8975445, 2.0321761,
4.0914479, 0.9390347, 1.9017782, 0.8950042, 4.8264183,
2.1320914, 1.5698561, 0.9026249, 1.2531753, 0.8890771,
0.8467401, 0.8806097, 1.2701101, 3.0990686, 1.9102456,
0.9339543, 1.2955123, 2.5402202, 2.4877223, 2.2861981,
1.4055885, 1.1024555, 1.0160881, 0.9314141, 3.8526672,
12.8018628, 1.6088061, 1.5241321, 1.9729043, 1.2701101,
1.7849280, 16.0880610, 1.4394581, 152.4132091, 1.4394581,
3.1303980, 6.2870449, 2.7773074, 1.0330229, 1.0160881,
14.0135478, 0.8467401, 1.2701101, 0.9060119, 1.2150720,
0.9847587, 8.3674852, 1.0584251, 1.1176969, 3.8103302,
1.1007621, 0.9652837, 1.5207451, 0.8696020, 0.9263336,
1.4877223, 3.3869602, 4.6570703, 0.8467401, 2.2777307,
1.0414903, 1.1769687, 1.5241321, 1.2887384, 1.2701101,
3.7739204, 0.9314141, 1.1981372, 2.3522439, 1.0668925,
1.0668925, 9.3988146, 1.2701101, 0.8467401, 5.7705334,
1.0499577, 1.5368332, 0.8467401, 0.9695174, 1.0838273,
0.9314141, 1.8797629, 1.2785775, 1.3547841, 3.9796782,
5.2497883, 1.5656224, 12.7011008, 0.9314141, 4.5723963,
2.9212532, 1.7781541, 1.5241321, 11.6850127, 2.1168501,
0.9906859, 1.7332769, 2.5402202, 1.1761219, 0.8467401,
6.0118544, 32.3878069, 10.5842506, 1.6426757, 3.3869602,
0.8958510, 1.0160881, 4.8687553, 1.5283658, 1.1007621,
0.8467401, 1.0795936, 0.9314141, 1.6934801, 18.6282811,
1.7315834, 3.8103302, 1.1007621, 1.8374259, 1.0160881,
0.9822185, 4.6570703, 1.3149873, 1.5241321, 2.2861981,
0.8467401, 1.0160881, 1.8856901, 1.3116003, 1.2277731,
1.1007621, 3.3869602, 0.9017782, 1.6342083, 1.0135478,
1.7265030, 3.8103302, 2.4563929, 2.6867062, 0.8467401,
1.0033870, 1.0152413, 0.8679086, 1.8628281, 1.0838273,
2.1168501, 4.1490262, 1.0160881, 0.9314141, 1.0160881,
1.1769687, 0.9686469, 0.9447195, 1.8770627, 0.8539604,
0.9158416, 1.2681518, 1.3564356, 7.4257426, 1.1344884,
1.3292079, 0.9900990, 1.2871287, 1.5651815, 0.8457096,
1.0726073, 2.3102310, 1.0940594, 2.6171617, 1.9900990,
1.9876238, 0.8316832, 0.9075908, 0.6023102, 1.1559406,
0.8745875, 3.3787129, 1.2169967, 1.9513201, 4.1254125,
1.4026403, 1.0264026, 0.8465347, 1.7079208, 1.2376238,
3.3003300, 0.9067657, 0.8250825, 0.9240924, 1.2788779,
2.9810231, 4.0759076, 3.7128713, 0.8712871, 1.2376238,
0.8374587, 1.0313531, 1.4546205, 0.8622112, 0.8795380,
0.9653465, 0.9900990, 1.6897690, 2.3679868, 3.4653465,
1.2376238, 1.0643564, 1.4232673, 1.4851485, 2.4752475,
1.7367987, 0.9900990, 0.8250825, 10.9983498, 1.0849835,
0.9942244, 0.8547855, 1.1551155, 1.2582508, 1.8976898,
1.0066007, 1.1089109, 0.9727723, 1.0313531, 1.3201320,
1.9026403, 8.2508251, 1.0948845, 0.9282178, 0.9075908,
2.8877888, 2.3028053, 0.8663366, 1.7359736, 0.8927393,
3.2178218, 0.9570957, 0.8250825, 1.0849835, 0.8910891,
0.8844884, 0.9447195, 0.8910891, 1.4851485, 2.4991749,
4.8679868, 0.8622112, 1.1056106, 12.3762376, 1.7466997,
0.8663366, 0.9405941, 0.9356436, 3.5891089, 0.9117162,
1.0214521, 3.4653465, 4.5379538, 5.1980198, 2.6757426,
0.8919142, 0.8250825, 1.2755776, 1.6501650, 3.3003300,
1.3201320, 0.9900990, 0.8844884, 0.8333333, 1.4851485,
2.5330033, 3.8556106, 0.8415842, 0.8910891, 1.2087459,
0.8374587, 1.2904290, 2.8052805, 1.2656766, 5.6122112,
1.4438944, 2.1039604, 0.8745875, 0.9199670, 0.8787129,
2.4752475, 0.8663366, 1.8151815, 1.4950495, 2.0429043,
1.0709571, 1.2376238, 0.8333333, 0.9653465, 2.0627063,
1.0173267, 1.6501650, 15.2846535, 2.2937294, 1.0726073,
4.5379538, 0.8250825, 1.2376238, 0.8580858, 0.8250825,
10.1848185, 0.9727723, 1.6641914, 0.8993399, 2.3102310,
6.3729373, 0.8704620, 0.8498350, 1.4108911, 1.1146865,
1.8704620, 0.8250825, 2.0627063, 1.0684818, 1.6707921,
0.8663366, 1.3184818, 0.9282178, 1.1155116, 2.0627063,
1.0726073, 4.2904290, 0.8457096, 2.9826733, 1.0214521,
8.0858086, 3.1765677, 1.6542904, 1.2656766, 1.0816832,
1.0841584, 2.8052805, 2.1633663, 1.2788779, 0.9075908,
1.3209571, 1.3391089, 1.4834984, 1.8226073, 6.2004950,
0.9900990, 13.2013201, 4.3143564, 2.4075908, 1.3646865,
0.8663366, 0.8250825, 1.0057756, 0.8663366, 2.1452145,
1.9018152, 1.4612211, 20.8267327, 5.9405941, 2.7227723,
2.7623762, 0.9612211, 1.0272277, 0.9950495, 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

[fire1]