Estimate Zero Truncated Poisson Parameters

This function will attempt to estimate the Zero Truncated Poisson lambda parameter given some vector of values .x. The function will return a tibble output, and if the parameter .auto_gen_empirical is set to TRUE then the empirical data given to the parameter .x will be run through the tidy_empirical() function and combined with the estimated Zero Truncated Poisson data.

Usage

util_zero_truncated_poisson_param_estimate(.x, .auto_gen_empirical = TRUE)

Arguments

.x

The vector of data to be passed to the function. Must be non-negative integers.

.auto_gen_empirical

This is a boolean value of TRUE/FALSE with default set to TRUE. This will automatically create the tidy_empirical() output for the .x parameter and use the tidy_combine_distributions(). The user can then plot out the data using $combined_data_tbl from the function output.

Value

A tibble/list

Details

This function estimates the parameter lambda of a Zero-Truncated Poisson distribution based on a vector of non-negative integer values .x. The Zero-Truncated Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time, given that at least one event has occurred.

The estimation is performed by minimizing the negative log-likelihood of the observed data .x under the Zero-Truncated Poisson model. The negative log-likelihood function used for optimization is defined as:

$$-\sum_{i=1}^{n} \log(P(X_i = x_i \mid X_i > 0, \lambda))$$

where \( X_i \) are the observed values in .x and lambda is the parameter of the Zero-Truncated Poisson distribution.

The optimization process uses the optim function to find the value of lambda that minimizes this negative log-likelihood. The chosen optimization method is Brent's method (method = "Brent") within a specified interval [0, max(.x)].

If .auto_gen_empirical is set to TRUE, the function will generate empirical data statistics using tidy_empirical() for the input data .x and then combine this empirical data with the estimated Zero-Truncated Poisson distribution using tidy_combine_distributions(). This combined data can be accessed via the $combined_data_tbl element of the function output.

The function returns a tibble containing the estimated parameter lambda along with other summary statistics of the input data (sample size, minimum, maximum).

See also

Other Parameter Estimation: util_bernoulli_param_estimate(), util_beta_param_estimate(), util_binomial_param_estimate(), util_burr_param_estimate(), util_cauchy_param_estimate(), util_chisquare_param_estimate(), util_exponential_param_estimate(), util_f_param_estimate(), util_gamma_param_estimate(), util_generalized_beta_param_estimate(), util_generalized_pareto_param_estimate(), util_geometric_param_estimate(), util_hypergeometric_param_estimate(), util_inverse_burr_param_estimate(), util_inverse_pareto_param_estimate(), util_inverse_weibull_param_estimate(), util_logistic_param_estimate(), util_lognormal_param_estimate(), util_negative_binomial_param_estimate(), util_normal_param_estimate(), util_paralogistic_param_estimate(), util_pareto1_param_estimate(), util_pareto_param_estimate(), util_poisson_param_estimate(), util_t_param_estimate(), util_triangular_param_estimate(), util_uniform_param_estimate(), util_weibull_param_estimate(), util_zero_truncated_binomial_param_estimate(), util_zero_truncated_geometric_param_estimate(), util_zero_truncated_negative_binomial_param_estimate()

Other Poisson: tidy_poisson(), tidy_zero_truncated_poisson(), util_poisson_param_estimate(), util_poisson_stats_tbl(), util_zero_truncated_poisson_stats_tbl()

Author

Steven P. Sanderson II, MPH

Examples

library(dplyr)
library(ggplot2)

tc <- tidy_zero_truncated_poisson() |> pull(y)
output <- util_zero_truncated_poisson_param_estimate(tc)

output$parameter_tbl
#> # A tibble: 1 × 5
#>   dist_type              samp_size   min   max lambda
#>   <chr>                      <int> <dbl> <dbl>  <dbl>
#> 1 Zero Truncated Poisson        50     1     4  0.997

output$combined_data_tbl |>
  tidy_combined_autoplot()