Compound Poisson-Normal Regression

Fits a Compound Poisson-Normal (CPN) regression model to the response variable using maximum likelihood estimation via the Nelder-Mead method (Nelder and Mead 1965) .

Usage

cpn(formula, data = NULL, mu_init = NULL, sigma_init = NULL, k_max = 10)

Arguments

formula: A formula specifying the model, e.g., y ~ x1 + x2.
data: An optional data frame, list, or environment containing the variables in the model. If not found in data, the variables are taken from the environment from which cpn is called.
mu_init: Optional initial value for the Normal mean parameter. If NULL, it is set to the sample mean of y.
sigma_init: Optional initial value for the Normal standard deviation. If NULL, it is set to the sample standard deviation of y.
k_max: Upper limit of summation used in approximating the Poisson convolution. Should be between 10 and 100. Default is 10.

Value

An object of class "cpn" containing the following components:

coefficients: Estimated regression coefficients.
mu: Estimated mean of the Normal component.
sigma: Estimated standard deviation of the Normal component.
se: Standard errors of the estimated parameters.
model: The model frame used.
terms: The terms object from the model.
formula: The model formula.
data: The original data passed in.
deviance_residuals: Vector of deviance residuals.
fitted_values: Fitted values (Poisson mean times Normal mean).
neg_log_likelihood: Negative log-likelihood at the optimum.
null_deviance: Null model deviance.
residual_deviance: Residual deviance of the fitted model.
df_null: Degrees of freedom for the null model.
df_residual: Degrees of freedom for the fitted model.
k_max: Value used to truncate the Poisson convolution sum.
call: The matched function call.

Details

The function fits a regression model where the response is assumed to follow a Compound Poisson-Normal distribution. The model estimates regression coefficients for the Poisson intensity, and mean and standard deviation of the Normal component.

The optimization is performed via maximum likelihood using the Nelder-Mead method. Standard errors are computed via the observed information matrix using numerical Hessian. If the Hessian is singular or not positive definite, a warning is issued and standard errors are returned as NA.

References

Nelder JA, Mead R (1965). “A simplex method for function minimization.” The computer journal, 7(4), 308–313. doi:10.1093/comjnl/7.4.308 .

Examples

set.seed(123)
data <- simulate_cpn_data()

# Sequential analysis of deviance
fit <- cpn(y ~ x1 + x2, data = data)
summary(fit)
#> Call:
#> cpn(formula = y ~ x1 + x2, data = data)
#> 
#> Deviance Residuals:
#>    Min     1Q Median     3Q    Max 
#> -2.871 -2.059 -1.269  2.181  3.745 
#> 
#> Coefficients:
#>             Estimate Std.Error z.value      Pr.z    
#> (Intercept)  0.63754   0.15190  4.1969 2.705e-05 ***
#> x1B         -0.60713   0.22934 -2.6473  0.008114 ** 
#> x2           0.53609   0.10791  4.9679 6.767e-07 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Estimated mu parameter: 0.9600
#> Estimated sigma parameter: 1.7062
#> 
#> Null deviance: 478.20 on 99 degrees of freedom
#> Residual deviance: 449.74 on 95 degrees of freedom
#> AIC: 459.74