Consider the pdf
$f(x;\theta,\sigma) = \frac{0.9}{\sigma}\phi\left(\frac{x-\theta}{\sigma}\right) + 0.1\ \phi(x-\theta)$, where $-\infty < \theta < \infty$ and $\sigma > 0$ are unknown parameters and $\phi$ denotes the pdf of $N(0,1)$. Let $X_1, X_2, \cdots, X_n$ be a random sample from this probability distribution. Then which of the following is (are) correct?
This problem involves analyzing a mixture probability density function (pdf) and determining properties related to its parameters and estimators.
Model Analysis
The given probability density function is:
$f(x;\theta,\sigma) = \frac{0.9}{\sigma}\phi\left(\frac{x-\theta}{\sigma}\right) + 0.1\ \phi(x-\theta)$
where $\phi(z)$ is the pdf of the standard normal distribution $N(0,1)$.
- The term $\frac{1}{\sigma}\phi\left(\frac{x-\theta}{\sigma}\right)$ is the pdf of a normal distribution $N(\theta, \sigma^2)$.
- The term $\phi(x-\theta)$ is the pdf of a normal distribution $N(\theta, 1)$.
Thus, the distribution is a mixture of two normal distributions: $N(\theta, \sigma^2)$ with probability 0.9 and $N(\theta, 1)$ with probability 0.1.
The model includes unknown parameters $\theta$ and $\sigma$, making it a parametric model.
Option 1 Analysis: Parametric Model
A model is parametric if it is defined by a finite number of parameters. This model depends on parameters $\theta$ and $\sigma$. Therefore, the model is indeed parametric.
Conclusion: Option 1 is incorrect.
Option 2 Analysis: Method of Moments Estimators
Method of Moments (MoM) estimators are found by equating population moments to sample moments and solving for the parameters.
Population Moments:
- First moment: $E[X] = 0.9 E[X|\text{comp 1}] + 0.1 E[X|\text{comp 2}] = 0.9(\theta) + 0.1(\theta) = \theta$.
- Second moment: $E[X^2] = 0.9 E[X^2|\text{comp 1}] + 0.1 E[X^2|\text{comp 2}]$.
- For $N(\mu, \nu^2)$, $E[Y^2] = \nu^2 + \mu^2$.
- $E[X^2|\text{comp 1}] = \sigma^2 + \theta^2$ (for $N(\theta, \sigma^2)$).
- $E[X^2|\text{comp 2}] = 1 + \theta^2$ (for $N(\theta, 1)$).
- So, $E[X^2] = 0.9(\sigma^2 + \theta^2) + 0.1(1 + \theta^2) = 0.9\sigma^2 + 0.9\theta^2 + 0.1 + 0.1\theta^2 = 0.9\sigma^2 + \theta^2 + 0.1$.
Sample Moments:
- First sample moment: $m_1 = \bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$.
- Second sample moment: $m_2 = \frac{1}{n}\sum_{i=1}^n X_i^2$.
Equating Moments:
- $E[X] = \theta = m_1 = \bar{X}$. This gives the MoM estimator $\hat{\theta} = \bar{X}$.
- $E[X^2] = 0.9\sigma^2 + \theta^2 + 0.1 = m_2$. Substituting $\hat{\theta} = \bar{X}$: $0.9\sigma^2 + (\bar{X})^2 + 0.1 = m_2$.
- Solving for $\sigma^2$: $0.9\sigma^2 = m_2 - (\bar{X})^2 - 0.1$.
- $\hat{\sigma}^2 = \frac{m_2 - (\bar{X})^2 - 0.1}{0.9}$.
- Since $\sigma > 0$, the estimator is $\hat{\sigma} = \sqrt{\frac{m_2 - (\bar{X})^2 - 0.1}{0.9}}$.
These equations yield estimators for both $\theta$ and $\sigma$, provided $m_2 - (\bar{X})^2 > 0.1$. Thus, MoM estimators exist.
Conclusion: Option 2 is correct.
Option 3 Analysis: Unbiased Estimator for $\theta$
Consider the MoM estimator for $\theta$, which is $\hat{\theta} = \bar{X}$.
We need to check if $E[\hat{\theta}] = \theta$.
$E[\hat{\theta}] = E[\bar{X}] = E\left[\frac{1}{n}\sum_{i=1}^n X_i\right] = \frac{1}{n}\sum_{i=1}^n E[X_i]$.
Since $E[X_i] = E[X] = \theta$ for all $i$,
$E[\bar{X}] = \frac{1}{n}\sum_{i=1}^n \theta = \frac{1}{n}(n\theta) = \theta$.
Therefore, $\bar{X}$ is an unbiased estimator for $\theta$.
Conclusion: Option 3 is correct.
Option 4 Analysis: Consistent Estimators
We found the estimator $\hat{\theta} = \bar{X}$. By the Law of Large Numbers, since $E[X] = \theta$ is finite, $\bar{X}$ converges in probability to $E[X]$.
$\bar{X} \xrightarrow{p} E[X] = \theta$.
This shows that $\hat{\theta} = \bar{X}$ is a consistent estimator for $\theta$.
Similarly, using the properties of converging sequences and continuous functions, the MoM estimator $\hat{\sigma}$ can also be shown to be consistent for $\sigma$.
Conclusion: Consistent estimators *do exist*. Option 4 is incorrect.
Final Summary
Based on the analysis:
- Option 1 is incorrect (the model is parametric).
- Option 2 is correct (MoM estimators can be derived).
- Option 3 is correct ($\bar{X}$ is an unbiased estimator for $\theta$).
- Option 4 is incorrect (consistent estimators exist).
The correct options are 2 and 3.
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