The Kumaraswamy Inverse Weibull family

Usage

KumIW(mu.link = "log", sigma.link = "log", nu.link = "log")

Arguments

mu.link: defines the mu.link, with "log" link as the default for the mu parameter.
sigma.link: defines the sigma.link, with "log" link as the default for the sigma.
nu.link: defines the nu.link, with "log" link as the default for the nu parameter.

Value

Returns a gamlss.family object which can be used to fit a KumIW distribution in the gamlss() function.

Details

The Kumaraswamy Inverse Weibull Distribution with parameters mu, sigma and nu has density given by

\(f(x)= \mu \sigma \nu x^{-\sigma - 1} \exp{- \mu x^{-\sigma}} (1 - \exp{- \mu x^{-\sigma}})^{\nu - 1},\)

for \(x > 0\), \(\mu > 0\), \(\sigma > 0\) and \(\nu > 0\).

The KumIW distribution with \(\nu=1\) corresponds with the IW distribution.

References

almalki2014modificationsRelDists

shahbaz2012kumaraswamyRelDists

Author

Freddy Hernandez, fhernanb@unal.edu.co

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rKumIW(n=1000, mu=1.5, sigma=0.3, nu=1)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family=KumIW,
              control=gamlss.control(n.cyc=5000, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 12836.61 
#> GAMLSS-RS iteration 2: Global Deviance = 12494.11 
#> GAMLSS-RS iteration 3: Global Deviance = 12491.52 
#> GAMLSS-RS iteration 4: Global Deviance = 12489.82 
#> GAMLSS-RS iteration 5: Global Deviance = 12488.33 
#> GAMLSS-RS iteration 6: Global Deviance = 12486.99 
#> GAMLSS-RS iteration 7: Global Deviance = 12485.78 
#> GAMLSS-RS iteration 8: Global Deviance = 12484.69 
#> GAMLSS-RS iteration 9: Global Deviance = 12483.71 
#> GAMLSS-RS iteration 10: Global Deviance = 12482.83 
#> GAMLSS-RS iteration 11: Global Deviance = 12482.04 
#> GAMLSS-RS iteration 12: Global Deviance = 12481.33 
#> GAMLSS-RS iteration 13: Global Deviance = 12480.69 
#> GAMLSS-RS iteration 14: Global Deviance = 12480.12 
#> GAMLSS-RS iteration 15: Global Deviance = 12479.61 
#> GAMLSS-RS iteration 16: Global Deviance = 12479.15 
#> GAMLSS-RS iteration 17: Global Deviance = 12478.73 
#> GAMLSS-RS iteration 18: Global Deviance = 12478.36 
#> GAMLSS-RS iteration 19: Global Deviance = 12478.02 
#> GAMLSS-RS iteration 20: Global Deviance = 12477.72 
#> GAMLSS-RS iteration 21: Global Deviance = 12477.44 
#> GAMLSS-RS iteration 22: Global Deviance = 12477.2 
#> GAMLSS-RS iteration 23: Global Deviance = 12476.97 
#> GAMLSS-RS iteration 24: Global Deviance = 12476.77 
#> GAMLSS-RS iteration 25: Global Deviance = 12476.59 
#> GAMLSS-RS iteration 26: Global Deviance = 12476.43 
#> GAMLSS-RS iteration 27: Global Deviance = 12476.28 
#> GAMLSS-RS iteration 28: Global Deviance = 12476.15 
#> GAMLSS-RS iteration 29: Global Deviance = 12476.03 
#> GAMLSS-RS iteration 30: Global Deviance = 12475.92 
#> GAMLSS-RS iteration 31: Global Deviance = 12475.82 
#> GAMLSS-RS iteration 32: Global Deviance = 12475.74 
#> GAMLSS-RS iteration 33: Global Deviance = 12475.66 
#> GAMLSS-RS iteration 34: Global Deviance = 12475.58 
#> GAMLSS-RS iteration 35: Global Deviance = 12475.52 
#> GAMLSS-RS iteration 36: Global Deviance = 12475.46 
#> GAMLSS-RS iteration 37: Global Deviance = 12475.4 
#> GAMLSS-RS iteration 38: Global Deviance = 12475.35 
#> GAMLSS-RS iteration 39: Global Deviance = 12475.31 
#> GAMLSS-RS iteration 40: Global Deviance = 12475.27 
#> GAMLSS-RS iteration 41: Global Deviance = 12475.23 
#> GAMLSS-RS iteration 42: Global Deviance = 12475.2 
#> GAMLSS-RS iteration 43: Global Deviance = 12475.17 
#> GAMLSS-RS iteration 44: Global Deviance = 12475.14 
#> GAMLSS-RS iteration 45: Global Deviance = 12475.12 
#> GAMLSS-RS iteration 46: Global Deviance = 12475.09 
#> GAMLSS-RS iteration 47: Global Deviance = 12475.07 
#> GAMLSS-RS iteration 48: Global Deviance = 12475.06 
#> GAMLSS-RS iteration 49: Global Deviance = 12475.04 
#> GAMLSS-RS iteration 50: Global Deviance = 12475.02 
#> GAMLSS-RS iteration 51: Global Deviance = 12475.01 
#> GAMLSS-RS iteration 52: Global Deviance = 12475 
#> GAMLSS-RS iteration 53: Global Deviance = 12474.99 
#> GAMLSS-RS iteration 54: Global Deviance = 12474.98 
#> GAMLSS-RS iteration 55: Global Deviance = 12474.97 
#> GAMLSS-RS iteration 56: Global Deviance = 12474.96 
#> GAMLSS-RS iteration 57: Global Deviance = 12474.95 
#> GAMLSS-RS iteration 58: Global Deviance = 12474.94 
#> GAMLSS-RS iteration 59: Global Deviance = 12474.94 
#> GAMLSS-RS iteration 60: Global Deviance = 12474.93 
#> GAMLSS-RS iteration 61: Global Deviance = 12474.93 
#> GAMLSS-RS iteration 62: Global Deviance = 12474.92 
#> GAMLSS-RS iteration 63: Global Deviance = 12474.92 
#> GAMLSS-RS iteration 64: Global Deviance = 12474.91 
#> GAMLSS-RS iteration 65: Global Deviance = 12474.91 
#> GAMLSS-RS iteration 66: Global Deviance = 12474.91 
#> GAMLSS-RS iteration 67: Global Deviance = 12474.9 
#> GAMLSS-RS iteration 68: Global Deviance = 12474.9 
#> GAMLSS-RS iteration 69: Global Deviance = 12474.9 
#> GAMLSS-RS iteration 70: Global Deviance = 12474.9 
#> GAMLSS-RS iteration 71: Global Deviance = 12474.89 
#> GAMLSS-RS iteration 72: Global Deviance = 12474.89 
#> GAMLSS-RS iteration 73: Global Deviance = 12474.89 
#> GAMLSS-RS iteration 74: Global Deviance = 12474.89 
#> GAMLSS-RS iteration 75: Global Deviance = 12474.89 
#> GAMLSS-RS iteration 76: Global Deviance = 12474.89 
#> GAMLSS-RS iteration 77: Global Deviance = 12474.89 
#> GAMLSS-RS iteration 78: Global Deviance = 12474.88 
#> GAMLSS-RS iteration 79: Global Deviance = 12474.88 

# Extracting the fitted values for mu, sigma and nu 
# using the inverse link function
exp(coef(mod, what="mu"))
#> (Intercept) 
#>    1.615627 
exp(coef(mod, what="sigma"))
#> (Intercept) 
#>   0.2796471 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>    1.082394 

# Example 2
# Generating random values under some model
n <- 200
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(1 +  -1 * x1)
sigma <- exp(1 + -1 * x2)
nu <- 5
y <- rKumIW(n=n, mu=mu, sigma=sigma, nu=nu)

mod <- gamlss(y~x1, sigma.fo=~x2, nu.fo=~1, family=KumIW,
              control=gamlss.control(n.cyc=5000, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 104.6923 
#> GAMLSS-RS iteration 2: Global Deviance = 92.5663 
#> GAMLSS-RS iteration 3: Global Deviance = 92.3988 
#> GAMLSS-RS iteration 4: Global Deviance = 92.2324 
#> GAMLSS-RS iteration 5: Global Deviance = 92.0726 
#> GAMLSS-RS iteration 6: Global Deviance = 91.9191 
#> GAMLSS-RS iteration 7: Global Deviance = 91.7717 
#> GAMLSS-RS iteration 8: Global Deviance = 91.6318 
#> GAMLSS-RS iteration 9: Global Deviance = 91.4971 
#> GAMLSS-RS iteration 10: Global Deviance = 91.3675 
#> GAMLSS-RS iteration 11: Global Deviance = 91.2428 
#> GAMLSS-RS iteration 12: Global Deviance = 91.1227 
#> GAMLSS-RS iteration 13: Global Deviance = 91.0071 
#> GAMLSS-RS iteration 14: Global Deviance = 90.8959 
#> GAMLSS-RS iteration 15: Global Deviance = 90.789 
#> GAMLSS-RS iteration 16: Global Deviance = 90.6875 
#> GAMLSS-RS iteration 17: Global Deviance = 90.5897 
#> GAMLSS-RS iteration 18: Global Deviance = 90.4954 
#> GAMLSS-RS iteration 19: Global Deviance = 90.4045 
#> GAMLSS-RS iteration 20: Global Deviance = 90.3168 
#> GAMLSS-RS iteration 21: Global Deviance = 90.2322 
#> GAMLSS-RS iteration 22: Global Deviance = 90.1506 
#> GAMLSS-RS iteration 23: Global Deviance = 90.0718 
#> GAMLSS-RS iteration 24: Global Deviance = 89.9958 
#> GAMLSS-RS iteration 25: Global Deviance = 89.9225 
#> GAMLSS-RS iteration 26: Global Deviance = 89.8516 
#> GAMLSS-RS iteration 27: Global Deviance = 89.7833 
#> GAMLSS-RS iteration 28: Global Deviance = 89.7173 
#> GAMLSS-RS iteration 29: Global Deviance = 89.6535 
#> GAMLSS-RS iteration 30: Global Deviance = 89.592 
#> GAMLSS-RS iteration 31: Global Deviance = 89.5325 
#> GAMLSS-RS iteration 32: Global Deviance = 89.4751 
#> GAMLSS-RS iteration 33: Global Deviance = 89.4217 
#> GAMLSS-RS iteration 34: Global Deviance = 89.3701 
#> GAMLSS-RS iteration 35: Global Deviance = 89.3201 
#> GAMLSS-RS iteration 36: Global Deviance = 89.2717 
#> GAMLSS-RS iteration 37: Global Deviance = 89.2247 
#> GAMLSS-RS iteration 38: Global Deviance = 89.1793 
#> GAMLSS-RS iteration 39: Global Deviance = 89.1352 
#> GAMLSS-RS iteration 40: Global Deviance = 89.0925 
#> GAMLSS-RS iteration 41: Global Deviance = 89.0511 
#> GAMLSS-RS iteration 42: Global Deviance = 89.011 
#> GAMLSS-RS iteration 43: Global Deviance = 88.9721 
#> GAMLSS-RS iteration 44: Global Deviance = 88.9344 
#> GAMLSS-RS iteration 45: Global Deviance = 88.8978 
#> GAMLSS-RS iteration 46: Global Deviance = 88.8624 
#> GAMLSS-RS iteration 47: Global Deviance = 88.828 
#> GAMLSS-RS iteration 48: Global Deviance = 88.7947 
#> GAMLSS-RS iteration 49: Global Deviance = 88.7624 
#> GAMLSS-RS iteration 50: Global Deviance = 88.731 
#> GAMLSS-RS iteration 51: Global Deviance = 88.7006 
#> GAMLSS-RS iteration 52: Global Deviance = 88.671 
#> GAMLSS-RS iteration 53: Global Deviance = 88.6424 
#> GAMLSS-RS iteration 54: Global Deviance = 88.6146 
#> GAMLSS-RS iteration 55: Global Deviance = 88.5876 
#> GAMLSS-RS iteration 56: Global Deviance = 88.5614 
#> GAMLSS-RS iteration 57: Global Deviance = 88.536 
#> GAMLSS-RS iteration 58: Global Deviance = 88.5114 
#> GAMLSS-RS iteration 59: Global Deviance = 88.4874 
#> GAMLSS-RS iteration 60: Global Deviance = 88.4642 
#> GAMLSS-RS iteration 61: Global Deviance = 88.4416 
#> GAMLSS-RS iteration 62: Global Deviance = 88.4196 
#> GAMLSS-RS iteration 63: Global Deviance = 88.3983 
#> GAMLSS-RS iteration 64: Global Deviance = 88.3777 
#> GAMLSS-RS iteration 65: Global Deviance = 88.3576 
#> GAMLSS-RS iteration 66: Global Deviance = 88.338 
#> GAMLSS-RS iteration 67: Global Deviance = 88.3191 
#> GAMLSS-RS iteration 68: Global Deviance = 88.3007 
#> GAMLSS-RS iteration 69: Global Deviance = 88.2827 
#> GAMLSS-RS iteration 70: Global Deviance = 88.2653 
#> GAMLSS-RS iteration 71: Global Deviance = 88.2484 
#> GAMLSS-RS iteration 72: Global Deviance = 88.232 
#> GAMLSS-RS iteration 73: Global Deviance = 88.216 
#> GAMLSS-RS iteration 74: Global Deviance = 88.2005 
#> GAMLSS-RS iteration 75: Global Deviance = 88.1854 
#> GAMLSS-RS iteration 76: Global Deviance = 88.1707 
#> GAMLSS-RS iteration 77: Global Deviance = 88.1564 
#> GAMLSS-RS iteration 78: Global Deviance = 88.1425 
#> GAMLSS-RS iteration 79: Global Deviance = 88.129 
#> GAMLSS-RS iteration 80: Global Deviance = 88.1159 
#> GAMLSS-RS iteration 81: Global Deviance = 88.1031 
#> GAMLSS-RS iteration 82: Global Deviance = 88.0907 
#> GAMLSS-RS iteration 83: Global Deviance = 88.0786 
#> GAMLSS-RS iteration 84: Global Deviance = 88.0669 
#> GAMLSS-RS iteration 85: Global Deviance = 88.0554 
#> GAMLSS-RS iteration 86: Global Deviance = 88.0443 
#> GAMLSS-RS iteration 87: Global Deviance = 88.0334 
#> GAMLSS-RS iteration 88: Global Deviance = 88.0229 
#> GAMLSS-RS iteration 89: Global Deviance = 88.0126 
#> GAMLSS-RS iteration 90: Global Deviance = 88.0026 
#> GAMLSS-RS iteration 91: Global Deviance = 87.9929 
#> GAMLSS-RS iteration 92: Global Deviance = 87.9835 
#> GAMLSS-RS iteration 93: Global Deviance = 87.9742 
#> GAMLSS-RS iteration 94: Global Deviance = 87.9653 
#> GAMLSS-RS iteration 95: Global Deviance = 87.9565 
#> GAMLSS-RS iteration 96: Global Deviance = 87.948 
#> GAMLSS-RS iteration 97: Global Deviance = 87.9397 
#> GAMLSS-RS iteration 98: Global Deviance = 87.9317 
#> GAMLSS-RS iteration 99: Global Deviance = 87.9238 
#> GAMLSS-RS iteration 100: Global Deviance = 87.9162 
#> GAMLSS-RS iteration 101: Global Deviance = 87.9087 
#> GAMLSS-RS iteration 102: Global Deviance = 87.9015 
#> GAMLSS-RS iteration 103: Global Deviance = 87.8944 
#> GAMLSS-RS iteration 104: Global Deviance = 87.8875 
#> GAMLSS-RS iteration 105: Global Deviance = 87.8808 
#> GAMLSS-RS iteration 106: Global Deviance = 87.8742 
#> GAMLSS-RS iteration 107: Global Deviance = 87.8676 
#> GAMLSS-RS iteration 108: Global Deviance = 87.8612 
#> GAMLSS-RS iteration 109: Global Deviance = 87.855 
#> GAMLSS-RS iteration 110: Global Deviance = 87.849 
#> GAMLSS-RS iteration 111: Global Deviance = 87.8431 
#> GAMLSS-RS iteration 112: Global Deviance = 87.8373 
#> GAMLSS-RS iteration 113: Global Deviance = 87.8317 
#> GAMLSS-RS iteration 114: Global Deviance = 87.8263 
#> GAMLSS-RS iteration 115: Global Deviance = 87.821 
#> GAMLSS-RS iteration 116: Global Deviance = 87.8158 
#> GAMLSS-RS iteration 117: Global Deviance = 87.8108 
#> GAMLSS-RS iteration 118: Global Deviance = 87.8058 
#> GAMLSS-RS iteration 119: Global Deviance = 87.8011 
#> GAMLSS-RS iteration 120: Global Deviance = 87.7964 
#> GAMLSS-RS iteration 121: Global Deviance = 87.7918 
#> GAMLSS-RS iteration 122: Global Deviance = 87.7874 
#> GAMLSS-RS iteration 123: Global Deviance = 87.7831 
#> GAMLSS-RS iteration 124: Global Deviance = 87.7789 
#> GAMLSS-RS iteration 125: Global Deviance = 87.7748 
#> GAMLSS-RS iteration 126: Global Deviance = 87.7708 
#> GAMLSS-RS iteration 127: Global Deviance = 87.7669 
#> GAMLSS-RS iteration 128: Global Deviance = 87.7631 
#> GAMLSS-RS iteration 129: Global Deviance = 87.7594 
#> GAMLSS-RS iteration 130: Global Deviance = 87.7558 
#> GAMLSS-RS iteration 131: Global Deviance = 87.7522 
#> GAMLSS-RS iteration 132: Global Deviance = 87.7488 
#> GAMLSS-RS iteration 133: Global Deviance = 87.7455 
#> GAMLSS-RS iteration 134: Global Deviance = 87.7422 
#> GAMLSS-RS iteration 135: Global Deviance = 87.739 
#> GAMLSS-RS iteration 136: Global Deviance = 87.7359 
#> GAMLSS-RS iteration 137: Global Deviance = 87.7329 
#> GAMLSS-RS iteration 138: Global Deviance = 87.7299 
#> GAMLSS-RS iteration 139: Global Deviance = 87.727 
#> GAMLSS-RS iteration 140: Global Deviance = 87.7242 
#> GAMLSS-RS iteration 141: Global Deviance = 87.7215 
#> GAMLSS-RS iteration 142: Global Deviance = 87.7188 
#> GAMLSS-RS iteration 143: Global Deviance = 87.7162 
#> GAMLSS-RS iteration 144: Global Deviance = 87.7136 
#> GAMLSS-RS iteration 145: Global Deviance = 87.7112 
#> GAMLSS-RS iteration 146: Global Deviance = 87.7085 
#> GAMLSS-RS iteration 147: Global Deviance = 87.706 
#> GAMLSS-RS iteration 148: Global Deviance = 87.7034 
#> GAMLSS-RS iteration 149: Global Deviance = 87.701 
#> GAMLSS-RS iteration 150: Global Deviance = 87.6986 
#> GAMLSS-RS iteration 151: Global Deviance = 87.6963 
#> GAMLSS-RS iteration 152: Global Deviance = 87.6941 
#> GAMLSS-RS iteration 153: Global Deviance = 87.6919 
#> GAMLSS-RS iteration 154: Global Deviance = 87.6898 
#> GAMLSS-RS iteration 155: Global Deviance = 87.6877 
#> GAMLSS-RS iteration 156: Global Deviance = 87.6857 
#> GAMLSS-RS iteration 157: Global Deviance = 87.6837 
#> GAMLSS-RS iteration 158: Global Deviance = 87.6818 
#> GAMLSS-RS iteration 159: Global Deviance = 87.6799 
#> GAMLSS-RS iteration 160: Global Deviance = 87.6781 
#> GAMLSS-RS iteration 161: Global Deviance = 87.6763 
#> GAMLSS-RS iteration 162: Global Deviance = 87.6746 
#> GAMLSS-RS iteration 163: Global Deviance = 87.6729 
#> GAMLSS-RS iteration 164: Global Deviance = 87.6712 
#> GAMLSS-RS iteration 165: Global Deviance = 87.6696 
#> GAMLSS-RS iteration 166: Global Deviance = 87.6681 
#> GAMLSS-RS iteration 167: Global Deviance = 87.6666 
#> GAMLSS-RS iteration 168: Global Deviance = 87.6651 
#> GAMLSS-RS iteration 169: Global Deviance = 87.6637 
#> GAMLSS-RS iteration 170: Global Deviance = 87.6623 
#> GAMLSS-RS iteration 171: Global Deviance = 87.6609 
#> GAMLSS-RS iteration 172: Global Deviance = 87.6596 
#> GAMLSS-RS iteration 173: Global Deviance = 87.6583 
#> GAMLSS-RS iteration 174: Global Deviance = 87.657 
#> GAMLSS-RS iteration 175: Global Deviance = 87.6558 
#> GAMLSS-RS iteration 176: Global Deviance = 87.6546 
#> GAMLSS-RS iteration 177: Global Deviance = 87.6534 
#> GAMLSS-RS iteration 178: Global Deviance = 87.6523 
#> GAMLSS-RS iteration 179: Global Deviance = 87.6512 
#> GAMLSS-RS iteration 180: Global Deviance = 87.6501 
#> GAMLSS-RS iteration 181: Global Deviance = 87.649 
#> GAMLSS-RS iteration 182: Global Deviance = 87.648 
#> GAMLSS-RS iteration 183: Global Deviance = 87.647 

coef(mod, what="mu")
#> (Intercept)          x1 
#>   0.9748666  -0.7999194 
coef(mod, what="sigma")
#> (Intercept)          x2 
#>    1.002897   -1.088306 
exp(coef(mod, what="nu"))
#> (Intercept) 
#>     5.32747