To determine which statements about estimators are correct, let's analyze each option in relation to statistical estimation, specifically for a normal distribution with mean \(\mu\) and variance \(\sigma^2\).
- Understanding the Sample Variance and Estimators:
- Given that \(X_1, X_2, \ldots, X_n\) are independent and identically distributed random variables from a normal distribution \(N(\mu, \sigma^2)\).
- The sample mean is denoted as \(\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i\).
- The sample variance (unbiased estimator) is:
- Minimum Variance Unbiased Estimator (MVUE) of \(\sigma^2\):
- The sample variance \(\frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2\) is indeed the Minimum Variance Unbiased Estimator of \(\sigma^2\). Thus, the first option is correct.
- Maximum Likelihood Estimator (MLE):
- The Maximum Likelihood Estimator of \(\sigma^2\) is given by:
- This is because the MLE does not correct for bias, unlike the unbiased estimator, hence the divisor is \(n\) rather than \(n-1\).
- Maximum Likelihood Estimator of \(\sigma\):
- The MLE of \(\sigma\) can be derived from the MLE of \(\sigma^2\):
- This option correctly describes the MLE for the standard deviation, \(\sigma\).
Based on this analysis, the correct options are:
- The first option: \(\frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2\) is the Minimum Variance Unbiased Estimate of \(\sigma^2\)
- The third option: \(\frac{1}{n}\sum_{i=1}^n (X_i - \bar{X})^2\) is the Maximum Likelihood Estimate of \(\sigma^2\)
- The fourth option: \(\sqrt{\frac{1}{n}\sum_{i=1}^n (X_i - \bar{X})^2}\) is the Maximum Likelihood Estimate of \(\sigma\)
Important Questions from Elementary Bayesian Inference
- Let $X|\theta \sim \text{Uniform}[0, \theta]$ and $\theta$ has an Exponential distribution with mean $\lambda$, where $\lambda > 3$ is known. If the realized value of $X$ is $2025$, then the posterior mode equals
Let $X_1, X_2, \dots, X_7$ be a random sample from $N(\mu, \sigma^2)$ where $\mu$ and $\sigma^2$ are unknown. Consider the problem of testing $H_0: \mu = 2$ against $H_1: \mu > 2$. Suppose the observed values of $x_1, x_2, \dots, x_7$ are $1.2, 1.3, 1.7, 1.8, 2.1, 2.3, 2.7$. If we use the Uniformly Most Powerful test, which of the following is true?
Suppose $X_i \mid \theta_i \sim N(\theta_i, \sigma^2), i = 1, 2$ are independently distributed. Under the prior distribution, $\theta_1$ and $\theta_2$ are i.i.d $N(\mu, \tau^2)$, where $\sigma^2, \mu$ and $\tau^2$ are known. Then which of the following is true about the marginal distributions of $X_1$ and $X_2$?
Let $(X,Y)$ have the joint discrete distribution such that $X \mid Y=y \sim \text{Binomial} \quad (y, 0.5)$ and $Y \sim \text{Poisson}(\lambda), \lambda > 0$, where $\lambda$ is an unknown parameter. Let $T = T(X,Y)$ be any unbiased estimator of $\lambda$. Then
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?
