Recovering original parameters of two-component Gaussian mixture distribution from re-parameterized values
original_par_2GM(
prob = 0.5,
d = 0,
sd_ratio = 1,
overallmean = 0,
overallsd = 1
)This function returns a vector of length 4: c(m1,m2,s1,s2).
The location parameter (mean) of the first Gaussian component.
The location parameter (mean) of the second Gaussian component.
The scale parameter (standard deviation) of the first Gaussian component.
The scale parameter (standard deviation) of the second Gaussian component.
The \(\pi = \frac{n_1}{N}\) 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).
The \(\delta = \frac{\mu_2 - \mu_1}{\bar{\sigma}}\) parameter of two-component Gaussian mixture distribution, where \(\mu_1\) and \(\mu_2\) are the estimated means of the first and second Gaussian components, 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 deviations of the first and second Gaussian components, 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
$$f(x)=\pi\times \phi(x | \mu_1, \sigma_1)+(1-\pi)\times \phi(x | \mu_2, \sigma_2)$$ , where \(\phi\) is a Gaussian component.
$$f(x)=2GM(x|\pi, \delta, \zeta, \bar{\mu}, \bar{\sigma})$$ , where \(\bar{\mu}\) is overall mean and \(\bar{\sigma}\) is overall standard deviation of the distribution.
1) Mean of the first Gaussian component (m1).
$$\mu_1=-(1-\pi)\delta\bar{\sigma}+\bar{\mu}$$
2) Mean of the second Gaussian component (m2).
$$\mu_2=\pi\delta\bar{\sigma}+\bar{\mu}$$
3) Standard deviation of the first Gaussian component (s1).
$$\sigma_1^2=\bar{\sigma}^2\left(\frac{1-\pi(1-\pi)\delta^2}{\pi+(1-\pi)\zeta^2}\right)$$
4) Standard deviation of the second Gaussian component (s2).
$$\sigma_2^2=\bar{\sigma}^2\left(\frac{1-\pi(1-\pi)\delta^2}{\frac{1}{\zeta^2}\pi+(1-\pi)}\right)=\zeta^2\sigma_1^2$$
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.