A sufficient statistic for a parameter of a probability distribution is a statistic that encapsulates all the information needed about that parameter from the data. The factorization theorem is a tool used to find sufficient statistics. For a log-normal distribution, we'll identify sufficient statistics by applying the factorization theorem to its likelihood function.

1. Log-Normal Distribution Overview

If $X \sim \text{Log-Normal}(\mu, \sigma^2)$, then:

• $Y = \log(X) \sim N(\mu, \sigma^2)$, i.e., a normal distribution with mean $\mu$ and variance $\sigma^2$.
• The probability density function (pdf) of $X$ is: $f(x|\mu, \sigma^2) = \frac{1}{x \sigma \sqrt{2\pi}} \exp\left(-\frac{(\log(x) - \mu)^2}{2\sigma^2}\right)$ for $x > 0$.

2. Factorization Theorem

The factorization theorem states that a statistic $T(X)$ is sufficient for parameter $\theta$ if the likelihood function can be factorized as: $L(\theta; x) = g(T(x), \theta) \cdot h(x)$ where $g(T(x), \theta)$ depends on the data only through $T(x)$ and $\theta$, and $h(x)$ depends only on the data and not on $\theta$.

3. Likelihood Function of Log-Normal

The likelihood function for $n$ i.i.d. log-normal observations $X_1, X_2, \dots, X_n$ is: $L(\mu, \sigma^2; x_1, x_2, \dots, x_n) = \prod_{i=1}^{n} f(x_i | \mu, \sigma^2)$ Substituting the log-normal pdf: $L(\mu, \sigma^2; x_1, \dots, x_n) = \prod_{i=1}^{n} \frac{1}{x_i \sigma \sqrt{2\pi}} \exp\left(-\frac{(\log(x_i) - \mu)^2}{2\sigma^2}\right)$ This can be rewritten as: $L(\mu, \sigma^2; x_1, \dots, x_n) = \frac{1}{(2\pi)^{n/2} \sigma^n \prod_{i=1}^n x_i} \exp\left(-\frac{1}{2\sigma^2} \sum_{i=1}^{n} (\log(x_i) - \mu)^2\right)$ We aim to express this in the form of the factorization theorem.

4. Factorization and Sufficient Statistic

The likelihood function can be factorized as: $L(\mu, \sigma^2; x_1, \dots, x_n) = \left[\frac{1}{(2\pi)^{n/2} \sigma^n} \exp\left(-\frac{1}{2\sigma^2} \left(\sum_{i=1}^n (\log(x_i))^2 - 2\mu \sum_{i=1}^n \log(x_i) + n\mu^2 \right)\right)\right] \cdot \frac{1}{\prod_{i=1}^n x_i}$ From this, we observe that the likelihood depends on the data $x_1, x_2, \dots, x_n$ through:

1. $\sum_{i=1}^n \log(x_i)$ (sum of log-transformed values)
2. $\sum_{i=1}^n (\log(x_i))^2$ (sum of squared log-transformed values)

Thus, the sufficient statistics are:

• $T_1(X) = \sum_{i=1}^n \log(X_i)$
• $T_2(X) = \sum_{i=1}^n (\log(X_i))^2$

Conclusion: Sufficient Statistic

For the log-normal distribution with parameters $\mu$ and $\sigma^2$, the sufficient statistics, derived using the factorization theorem, are: $T(X) = \left( \sum_{i=1}^n \log(X_i), \sum_{i=1}^n (\log(X_i))^2 \right)$ These capture all the necessary information about $\mu$ and $\sigma^2$ from the data.

Would you like more details on the application of the factorization theorem? Here are 5 related questions for further exploration:

1. How does the log-normal distribution relate to the normal distribution?
2. What is the interpretation of the sufficient statistic in the context of parameter estimation?
3. Can you explain the importance of minimal sufficient statistics?
4. How would the Maximum Likelihood Estimators (MLE) be derived for the log-normal distribution?
5. What is the role of exponential families in the context of sufficient statistics?

Tip: Sufficient statistics allow for data reduction without losing information about the parameter, making them key for efficient estimation.