Moment estimations for Bivariate Poisson Distribution - Geoffroy

This function obtains moment estimators for the Bivariate Poisson distribution under the parameterization of Geoffroy et. al (2021).

moments_estim_BP_Geoffroy(x)

Arguments

x

vector or matrix of quantiles. When x is a matrix, each row is taken to be a quantile and columns correspond to the number of dimensions p.

Value

Returns a vector with \(\hat{\lambda_1}\), \(\hat{\lambda_2}\) and \(\hat{\alpha}\).

References

Kouakou, K. J. G., Hili, O., & Dupuy, J. F. (2021). Estimation in the zero-inflated bivariate Poisson model with an application to health-care utilization data. Afrika Statistika, 16(2), 2767-2788.

Author

Freddy Hernandez-Barajas, fhernanb@unal.edu.co

Examples

# Example 1 ---------------------------------------------------------------
# Probability for single values of X1 and X2
dBP_Geoffroy(c(0, 0), l1=3, l2=4, l0=1)
#> Error in dBP_Geoffroy(c(0, 0), l1 = 3, l2 = 4, l0 = 1): unused argument (l0 = 1)
dBP_Geoffroy(c(1, 0), l1=3, l2=4, l0=1)
#> Error in dBP_Geoffroy(c(1, 0), l1 = 3, l2 = 4, l0 = 1): unused argument (l0 = 1)
dBP_Geoffroy(c(0, 1), l1=3, l2=4, l0=1)
#> Error in dBP_Geoffroy(c(0, 1), l1 = 3, l2 = 4, l0 = 1): unused argument (l0 = 1)

# Probability for a matrix the values of X1 and X2
x <- matrix(c(0, 0,
              1, 0,
              0, 1), ncol=2, byrow=TRUE)
x
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    1    0
#> [3,]    0    1
dBP_Geoffroy(x=x, l1=3, l2=4, l0=1)
#> Error in dBP_Geoffroy(x = x, l1 = 3, l2 = 4, l0 = 1): unused argument (l0 = 1)

# Checking if the probabilities sum 1
val_x1 <- val_x2 <- 0:50
space <- expand.grid(val_x1, val_x2)
space <- as.matrix(space)

l1 <- 3
l2 <- 4
l0 <- 5

probs <- dBP_Geoffroy(x=space, l1=l1, l2=l2, l0=l0)
#> Error in dBP_Geoffroy(x = space, l1 = l1, l2 = l2, l0 = l0): unused argument (l0 = l0)
sum(probs)
#> Error in eval(expr, envir, enclos): object 'probs' not found

# Example 2 ---------------------------------------------------------------
# Heat map for a BP_Geoffroy

l1 <- 1
l2 <- 2
l0 <- 4

X1 <- 0:10
X2 <- 0:10
data <- expand.grid(X1=X1, X2=X2)
data$Prob <- dBP_Geoffroy(x=data, l1=l1, l2=l2, l0=l0)
#> Error in dBP_Geoffroy(x = data, l1 = l1, l2 = l2, l0 = l0): unused argument (l0 = l0)
data$X1 <- factor(data$X1)
data$X2 <- factor(data$X2)

library(ggplot2)
ggplot(data, aes(X1, X2, fill=Prob)) +
  geom_tile() +
  scale_fill_gradient(low="darkgreen", high="pink")
#> Error in geom_tile(): Problem while computing aesthetics.
#> ℹ Error occurred in the 1st layer.
#> Caused by error:
#> ! object 'Prob' not found

# Example 3 ---------------------------------------------------------------
# Generating random values and moment estimations

l1 <- 1
l2 <- 2
l0 <- 4

x <- rBP_Geoffroy(n=500, l1, l2, l0)
moments_estim_BP_Geoffroy(x)
#> l1_hat l2_hat mu_hat 
#> 0.9927 2.0207 3.9113 

# Example 4 ---------------------------------------------------------------
# Estimating the parameters using the loglik function

# Loglik function
llBP_Geoffroy <- function(param, x) {
  l1 <- param[1]  # param: is the parameter vector
  l2 <- param[2]
  l0 <- param[3]
  sum(dBP_Geoffroy(x=x, l1=l1, l2=l2, l0=l0, log=TRUE))
}

# The known parameters
l1 <- 1
l2 <- 2
l0 <- 4

set.seed(12345)
x <- rBP_Geoffroy(n=500, l1=l1, l2=l2, l0=l0)
#> Error in rBP_Geoffroy(n = 500, l1 = l1, l2 = l2, l0 = l0): unused argument (l0 = l0)

# To obtain reasonable values for l0
start_param <- moments_estim_BP_Geoffroy(x)
start_param
#> l1_hat l2_hat mu_hat 
#> 0.9927 2.0207 3.9113 

# Estimating parameters
res1 <- optim(fn = llBP_Geoffroy,
              par = start_param,
              lower = c(0.001, 0.001, 0.001),
              upper = c(  Inf,   Inf,   Inf),
              method = "L-BFGS-B",
              control = list(maxit=100000, fnscale=-1),
              x=x)
#> Error in dBP_Geoffroy(x = x, l1 = l1, l2 = l2, l0 = l0, log = TRUE): unused argument (l0 = l0)

res1
#> Error in eval(expr, envir, enclos): object 'res1' not found