Probability density for the re-parameterized two-component normal mixture distribution.
dist2(x, prob = 0.5, d = 0, sd_ratio = 1, overallmean = 0, overallsd = 1)The evaluated probability density value(s).
A numeric vector. The location to evaluate the density function.
A numeric value of \(\pi = \frac{n_1}{N}\) parameter of two-component Gaussian mixture distribution, where \(n_1\) is the estimated number of examinees belonging to the first Gaussian component and \(N\) is the total number of examinees (Li, 2021).
A numeric value of \(\delta = \frac{\mu_2 - \mu_1}{\bar{\sigma}}\) parameter of two-component Gaussian mixture distribution, where \(\mu_1\) and \(\mu_2\) are the estimated mean of the first and second Gaussian component, respectively. And \(\bar{\sigma}\) is the overall standard deviation of the latent distribution (Li, 2021). Without loss of generality, \(\mu_2 \ge \mu_1\) is assumed, thus \(\delta \ge 0\).
A numeric value of \(\zeta = \frac{\sigma_2}{\sigma_1}\) parameter of two-component Gaussian mixture distribution, where \(\sigma_1\) and \(\sigma_2\) are the estimated standard deviation of the first and second Gaussian component, respectively (Li, 2021).
A numeric value of \(\bar{\mu}\) that determines the overall mean of two-component Gaussian mixture distribution.
A numeric value of \(\bar{\sigma}\) that determines the overall standard deviation of two-component Gaussian mixture distribution.
Seewoo Li cu@yonsei.ac.kr
1) Overall mean (\(\bar{\mu}\)) $$\bar{\mu}=\pi\mu_1 + (1-\pi)\mu_2$$
2) Overall standard deviation (\(\bar{\sigma}\)) $$\bar{\sigma}=\sqrt{\pi\sigma_{1}^{2}+(1-\pi)\sigma_{2}^{2}+\pi(1-\pi)(\mu_2-\mu_1)^2}$$
Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. Journal of Educational Evaluation, 34(4), 759-789.
# Evaluated density
dnst <- dist2(seq(-6,6,.1), prob = 0.3, d = 1, sd_ratio=0.5)
# Plot of the density
plot(seq(-6,6,.1), dnst)
Run the code above in your browser using DataLab