| Type: | Package |
| Title: | Generalized Exponentiated Composite Distributions |
| Version: | 0.1.0 |
| Author: | Bowen Liu [aut, cre], Malwane M.A. Ananda [aut], Dwaipayan Mukhopadhyay [aut] |
| Maintainer: | Bowen Liu <bowen.liu@umkc.edu> |
| Description: | Contains the framework of the estimation, sampling, and hypotheses testing for two special distributions (Exponentiated Exponential-Pareto and Exponentiated Inverse Gamma-Pareto) within the family of Generalized Exponentiated Composite distributions. The detailed explanation and the applications of these two distributions were introduced in Bowen Liu, Malwane M.A. Ananda (2022) <doi:10.1080/03610926.2022.2050399>, Bowen Liu, Malwane M.A. Ananda (2022) <doi:10.3390/math10111895>, and Bowen Liu, Malwane M.A. Ananda (2022) <doi:10.3390/app13010645>. |
| License: | GPL-3 |
| Encoding: | UTF-8 |
| Imports: | stats, mistr |
| RoxygenNote: | 7.2.3 |
| NeedsCompilation: | no |
| Packaged: | 2023-10-21 14:30:24 UTC; liuab |
| Repository: | CRAN |
| Date/Publication: | 2023-10-24 16:40:02 UTC |
Likelihood Ratio Test (LRT) for the exponent parameter in EEP model.
Description
This function computes the test statistic and the p-value of LRT for the exponent parameter in EEP model.
Usage
LRT_eep(data, theta0, theta1, eta1)
Arguments
data |
Observations. |
theta0 |
The MLE of theta when eta = 1. |
theta1 |
The unrestricted MLE of theta. |
eta1 |
The unrestricted MLE of eta. |
Details
LRT_eep
Value
This function returns the test statistic and the p-value of the LRT test
Examples
sample1 = eep_sampling(200,eta = 1.1,theta = 6)
eta1 = mle_search_eep(data = sample1)$eta
theta1 = mle_search_eep(data = sample1)$theta
theta0 = mle_iter_eep(data = sample1,eta = 1)
LRT_eep(sample1,theta0,theta1,eta1)
Likelihood Ratio Test (LRT) for the exponent parameter in EIGP model.
Description
This function computes the test statistic and the p-value for LRT for the exponent parameter in EIGP model.
Usage
LRT_eigp(data, theta0, theta1, eta1)
Arguments
data |
Observations. |
theta0 |
The MLE of theta when eta = 1. |
theta1 |
The unrestricted MLE of theta. |
eta1 |
The unrestricted MLE of eta. |
Details
LRT_eigp
Value
This function returns the test statistic and the p-value from the LRT test
Examples
sample1 = eigp_sampling(200,eta = 1.1,theta = 3)
eta1 = mle_search_eigp(data = sample1)$eta
theta1 = mle_search_eigp(data = sample1)$theta
theta0 = mle_iter_eigp(data = sample1,eta = 1)
LRT_eigp(sample1,theta0,theta1,eta1)
Asymptotic Wald's test for testing the exponent in a EEP model.
Description
This function computes the test statistic and the p-value of Wald's test for the exponent parameter in EEP model.
Usage
asymptotic_eep(data, eta0, theta1, eta1)
Arguments
data |
Observations. |
eta0 |
To test if the exponent equals 1, the default for eta0 is et to be 1. |
theta1 |
The unrestricted MLE of theta. |
eta1 |
The unrestricted MLE of eta. |
Details
asymptotic_eep
Value
This function returns the test statistic and the p-value of the Wald's test.
Examples
sample1 = eep_sampling(200,eta = 1.1,theta = 3)
theta1 = mle_search_eep(data = sample1)$theta
eta1 = mle_search_eep(data = sample1)$eta
asymptotic_eep(sample1,eta0 = 1,theta1,eta1)
Asymptotic Wald's test for testing the exponent in a EIGP model.
Description
This function computes the test statistic and the p-value of Wald's test for the exponent parameter in EIGP model.
Usage
asymptotic_eigp(data, eta0 = 1, theta1, eta1)
Arguments
data |
Observations. |
eta0 |
To test if the exponent equals 1, the default for eta0 is et to be 1. |
theta1 |
The unrestricted MLE of theta. |
eta1 |
The unrestricted MLE of eta. |
Details
asymptotic_eigp
Value
This function returns the test statistic and the p-value of the Wald's test.
Examples
sample1 = eigp_sampling(200,eta = 1.1,theta = 3)
theta1 = mle_search_eigp(data = sample1)$theta
eta1 = mle_search_eigp(data = sample1)$eta
asymptotic_eigp(sample1,eta0 = 1,theta1,eta1)
The cumulative distribution function of EEP.
Description
cdf_eep
Usage
cdf_eep(theta, eta, data)
Arguments
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
data |
Observations. |
Value
Return the cumulative probability of EEP at the specific location.
Examples
cdf_eep(1,2,5)
The cumulative distribution function of EIGP.
Description
cdf_eigp
Usage
cdf_eigp(theta, eta, data)
Arguments
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
data |
Observations. |
Value
Return the cumulative probability of EIGP at the specific location.
Examples
cdf_eigp(1,2,5)
The EEP Negative Log-likelihood Function.
Description
This function serves as the objective function for the Maximum Likelihood Estimation procedure for EEP.
Usage
eep_nll(x, m, data)
Arguments
x |
Vector of parameters. |
m |
The number of data items less than the density change point. |
data |
Observations. |
Details
eep_nll
Value
A scalar that represents the negative loglikelihood of a EEP sample given the model parameter and the data.
Examples
eep_nll(c(2,2),50,seq(1:100))
The Wrapper Function that Returns the Final Estimates from Maximum Likelihood Estimation for EEP.
Description
This function serves as a wrapper that returns the final estimates of theta, eta, and the corresponding density change point
Usage
eep_optim(data, init = c(1, 1), lower_bound = c(0.01, 0.01))
Arguments
data |
Observations. |
init |
The vector of initial values of the model parameters. The default is c(1,1). |
lower_bound |
The vector of the lower bound for the parameters. The default is c(0.01,0.01). |
Details
eep_optim
Value
A data frame with 1 row and 3 columns that contains the MLE of theta, eta, and the predicted density change point.
Examples
eep_optim(seq(1:100))
The Random Number Generation Function for EIGP
Description
Create a EEP random sample.
Usage
eep_sampling(n, theta, eta)
Arguments
n |
Number of observations. (n>=1) |
theta |
The location parameter for the parent EP distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The parameter should be positive. |
Details
eep_sampling
Value
returns a numerical vector of size n.
Examples
eep_sampling(100,1,1)
The EIGP Negative Log-likelihood Function.
Description
This function serves as the objective function for the Maximum Likelihood Estimation procedure for EIGP.
Usage
eigp_nll(x, m, data)
Arguments
x |
Vector of parameters. |
m |
The number of data items less than the density change point. |
data |
Observations. |
Details
eigp_nll
Value
A scalar that represents the negative loglikelihood of a EIGP sample given the model parameter and the data.
Examples
eigp_nll(c(2,2),50,seq(1:100))
The Wrapper Function that Returns the Final Estimates from Maximum Likelihood Estimation for EIGP.
Description
This function serves as a wrapper that returns the final estimates of theta, eta, and the corresponding density change point
Usage
eigp_optim(data, init = c(1, 1), lower_bound = c(0.01, 0.01))
Arguments
data |
Observations. |
init |
The vector of initial values of the model parameters. The default is c(1,1). |
lower_bound |
The vector of the lower bound for the parameters. The default is c(0.01,0.01). |
Details
eigp_optim
Value
A data frame with 1 row and 3 columns that contains the MLE of theta, eta, and the predicted density change point.
Examples
eigp_optim(seq(1:100))
The Random Number Generation Function for EIGP
Description
Create a EIGP random sample.
Usage
eigp_sampling(n, theta, eta)
Arguments
n |
Number of observations. (n>=1) |
theta |
The location parameter for the parent IGP distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The parameter should be positive. |
Details
eigp_sampling
Value
This function returns a numerical vector of size n.
Examples
eigp_sampling(100,1,1)
The negative log density of a sample item if it follows exponential in a EEP model
Description
This function return the negative log density of a sample item if if it follows exponential in a EEP model.
Usage
exp_eep(x, theta, eta)
Arguments
x |
The value of a sample item. |
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
Details
exp_exp
Value
This function return the negative log density of a sample item if if it follows exponential in a EEP model.
Examples
exp_eep(1,5,2)
The hazard function of EEP.
Description
hazard_eep
Usage
hazard_eep(theta, eta, data)
Arguments
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
data |
Observations. |
Value
Return the hazard of EEP at the specific location.
Examples
hazard_eep(2,1,5)
plot(hazard_eep(2,1,seq(0.01,100,by=0.01)))
The hazard function of EIGP.
Description
hazard_eigp
Usage
hazard_eigp(theta, eta, data)
Arguments
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
data |
Observations. |
Value
Return the hazard of EIGP at the specific location.
Examples
hazard_eigp(1,2,5)
plot(hazard_eep(2,1,seq(0.01,100,by=0.01)))
The negative log density of a sample item if it follows inverse gamma in a EIGP model
Description
This function return the negative log density of a sample item if if it follows inverse gamma in a EIGP model.
Usage
inv_gamma_eigp(x, theta, eta)
Arguments
x |
The value of a sample item. |
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
Details
inv_gamma_eigp
Value
This function return the negative log density of a sample item if if it follows inverse gamma in a EIGP model.
Examples
inv_gamma_eigp(1,5,2)
Analytical solution of theta given eta in EEP model.
Description
This function provides the analytical solution of theta for given eta EEP model.
Usage
mle_eep(s, m, n)
Arguments
s |
A numeric value the sum of log(1/x_i^eta), where i is from 1 to m. |
m |
m is the number of data items less than the density change point. |
n |
n is the sample size, n has to be greater than m. |
Details
mle_eep
Value
This function returns the Maximum Likelihood Estimate of theta for a given eta
Examples
mle_eep(5,2,5)
Analytical solution of theta given eta in EIGP model.
Description
This function provides the analytical solution of theta for given eta EIGP model.
Usage
mle_eigp(s, m, n)
Arguments
s |
a numeric value the sum of log(1/x_i^eta), where i is from 1 to m. |
m |
m is the number of data items less than the density change point. |
n |
n is the sample size, n has to be greater than m. |
Details
mle_eigp
Value
This function returns the Maximum Likelihood Estimate of theta for a given eta.
Examples
mle_eigp(5,2,5)
Iteration function to find the analytical solution of theta given eta and data in EEP model.
Description
This function finds the analytical solution of theta given eta and data in EEP model.
Usage
mle_iter_eep(data, eta)
Arguments
data |
Observations. |
eta |
The exponent parameter. This value is greater than 0. |
Details
mle_iter_eep
Value
This function returns the Maximum Likelihood Estimate of theta for a given eta with data.
Examples
mle_iter_eep(seq(1:100),2)
Iteration function to find the analytical solution of theta given eta and data in EIGP model.
Description
This function finds the analytical solution of theta given eta and data in EIGP model.
Usage
mle_iter_eigp(data, eta)
Arguments
data |
Observations. |
eta |
The exponent parameter. This value is greater than 0. |
Details
mle_iter_eigp
Value
This function returns the Maximum Likelihood Estimate of theta for a given eta with data.
Examples
mle_iter_eigp(seq(1:100),2)
The grid search procedure for parameter estimation of EEP.
Description
This function find the parameter estimates of EEP throgh a grid search procedure.
Usage
mle_search_eep(eta_seq = seq(0.5, 10, by = 0.01), data)
Arguments
eta_seq |
A predefined range for eta values. The default is c(0.5,10,by = 0.01) |
data |
Observations. |
Details
mle_search_eep
Value
This function returns a data frame as the parameter estimates for EEP from grid search methods.
Examples
sample1 = eep_sampling(200,eta = 2,theta = 3)
mle_search_eep(data = sample1)
The grid search procedure for parameter estimation of EIGP.
Description
This function find the parameter estimates of EIGP through a grid search procedure.
Usage
mle_search_eigp(eta_seq = seq(0.5, 10, by = 0.01), data)
Arguments
eta_seq |
A predefined range for eta values. The default is c(0.5,10,by = 0.01) |
data |
n by 1 vector with all positive entries. |
Details
mle_search_eigp
Value
This function returns data frame as the parameter estimates for EIGP from grid search methods.
Examples
sample1 = eigp_sampling(200,eta = 2,theta = 3)
mle_search_eigp(data = sample1)
The negative log likelihood function for EEP distribution.
Description
This function computes the negative log-likelihood for EEP distribution.
Usage
neg_log_eep(y, theta, eta)
Arguments
y |
n by 1 vector with all positive entries. |
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
Details
neg_log_eigp
Value
This function return the negative log density of a sample item if if it follows Pareto in a EEP model.
Examples
neg_log_eep(seq(1:100),2,2)
The negative log likelihood function for EIGP distribution.
Description
This function computes the negative log-likelihood for EIGP distribution.
Usage
neg_log_eigp(y, theta, eta)
Arguments
y |
n by 1 vector with all positive entries. |
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
Details
neg_log_eigp
Value
This function return the negative log density of a sample item if if it follows Pareto in a EIGP model.
Examples
neg_log_eigp(seq(1:100),2,2)
The negative log density of a sample item if it follows Pareto in a EEP model
Description
This function return the negative log density of a sample item if if it follows Pareto in a EEP model.
Usage
pareto_eep(x, theta, eta)
Arguments
x |
The value of a sample item. |
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
Details
pareto_eep
Value
This function return the negative log density of a sample item if if it follows Pareto in a EEP model.
Examples
pareto_eep(10,5,2)
The negative log density of a sample item if it follows Pareto in a EIGP model
Description
This function return the negative log density of a sample item if if it follows Pareto in a EIGP model.
Usage
pareto_eigp(x, theta, eta)
Arguments
x |
The value of a sample item. |
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
Details
pareto_eigp
Value
This function return the negative log density of a sample item if if it follows Pareto in a EIGP model.
Examples
pareto_eigp(10,5,2)
The probability function of EEP.
Description
pdf_eep
Usage
pdf_eep(theta, eta, data)
Arguments
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
data |
Observations. |
Value
Return the density of EEP
Examples
pdf_eep(1,2,5)
The probability density function of EIGP.
Description
pdf_eigp
Usage
pdf_eigp(theta, eta, data)
Arguments
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
data |
Observations. |
Value
Return the density of EIGP
Examples
pdf_eigp(1,2,5)
The quantile function of EEP.
Description
q_eep
Usage
q_eep(theta, eta, p)
Arguments
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
p |
This indicates the p-th percentile. p is greater than 0 and less than 100. |
Value
Return the p-th percentile of EEP.
Examples
q_eigp(1,2,5)
The quantile function of EIGP.
Description
q_eigp
Usage
q_eigp(theta, eta, p)
Arguments
theta |
The location parameter for the base distribution (eta = 1). The value needs to be positive. |
eta |
The exponent parameter. The value provided needs to be positive. |
p |
This indicates the p-th percentile. p is greater than 0 and less than 100. |
Value
Return the p-th percentile of EIGP.
Examples
q_eigp(1,2,5)
The Optimization Function for EEP Maximum Likelihood Estimation.
Description
This function serves as the optimization function for EEP at different locations of density change points.
Usage
raw_est_eep(data, init = c(1, 1), lower_bound = c(0.01, 0.01))
Arguments
data |
Observations. |
init |
The vector of initial values of the model parameters. The default is c(1,1). |
lower_bound |
The vector of the lower bound for the parameters. The default is c(0.01,0.01). |
Details
raw_est_eep
Value
The matrix with estimates of theta and eta for n-1 different locations of density change points (1st column for theta, 2nd column for eta).
Examples
raw_est_eep(seq(1:100))
The Optimization Function for EIGP Maximum Likelihood Estimation.
Description
This function serves as the optimization function for EIGP at different locations of density change points.
Usage
raw_est_eigp(data, init = c(1, 1), lower_bound = c(0.01, 0.01))
Arguments
data |
Observations. |
init |
The vector of initial values of the model parameters. The default is c(1,1). |
lower_bound |
The vector of the lower bound for the parameters. The default is c(0.01,0.01). |
Details
raw_est_eigp
Value
The matrix with estimates of theta and eta for n-1 different locations of density change points (1st column for theta, 2nd column for eta).
Examples
raw_est_eigp(seq(1:100))
The function for calculating the standard errors of the parameters of EEP model.
Description
This function find the parameter estimates of EEP through a grid search procedure.
Usage
se_eep(data, theta, eta)
Arguments
data |
Observations. |
theta |
The MLE of theta |
eta |
The MLE of eta |
Details
se_eep
Value
The estimate of SE for theta and eta
Examples
sample1 = eep_sampling(200,eta = 2,theta = 3)
theta = mle_search_eep(data = sample1)$theta
eta = mle_search_eep(data = sample1)$eta
se_eep(sample1,theta,eta)
The function for calculating the standard errors of the parameters of EIGP model.
Description
This function find the parameter estimates of EIGP through a grid search procedure.
Usage
se_eigp(data, theta, eta)
Arguments
data |
Observations. |
theta |
The MLE of theta |
eta |
The MLE of eta |
Details
se_eigp
Value
The estimate of SE for theta and eta
Examples
sample1 = eigp_sampling(200,eta = 2,theta = 3)
theta = mle_search_eigp(data = sample1)$theta
eta = mle_search_eigp(data = sample1)$eta
se_eigp(sample1,theta,eta)
The validation Function for Model Parameters.
Description
This function checks if the estimates from raw_est_eigp or raw_est_eep satisfy the pre-defined conditions for the parameters.
Usage
validation(data, estimate)
Arguments
data |
Observations. |
estimate |
The data frame with 2 columns named 'theta' and 'eta'. |
Details
validation
Value
A Boolean vector.
Examples
estimate = raw_est_eigp(seq(1:100),init = c(1,1),lower_bound = c(0.01,0.01))
estimate = data.frame(estimate)
colnames(estimate) = c('theta','eta')
validation(seq(1:100),estimate)