The problem asks to show that the given family of distributions is an exponential family (ΜΕΟΚ in Greek, short for "Μέγιστη Εκθετική Οικογένεια Κατανομών") and to find a sufficient statistic for this family. The given probability density function is:

$f_1(x; \theta) = \frac{3\theta x^2}{(1 + x^3)^{\theta + 1}} \mathbb{I}_{(0, \infty)}(x), \quad \theta \in \Theta = (0, \infty).$

Step 1: Show that the family of distributions belongs to the exponential family

To show that a given family belongs to the exponential family, we express the density function in the following general form:

$f(x; \theta) = h(x) \exp \left( \eta(\theta) T(x) - A(\theta) \right),$

where:

• $h(x)$ is a function of the data,
• $\eta(\theta)$ is the natural parameter,
• $T(x)$ is the sufficient statistic,
• $A(\theta)$ is the log-partition function (or cumulant function).

We start by rewriting the given density:

$f_1(x; \theta) = \frac{3 \theta x^2}{(1 + x^3)^{\theta + 1}}.$

We can separate terms involving $x$ and $\theta$:

$f_1(x; \theta) = 3 x^2 \cdot \theta \cdot \left( \frac{1}{(1 + x^3)} \right)^{\theta + 1}.$

Take the logarithm of the function to help express it in exponential form:

$\log f_1(x; \theta) = \log(3 x^2) + \log(\theta) - (\theta + 1)\log(1 + x^3).$

This can be rearranged as:

$\log f_1(x; \theta) = \log(3 x^2) - \log(1 + x^3) + \theta \left( -\log(1 + x^3) + \log(\theta) \right).$

Comparing this with the form of the exponential family:

• The term $\log(3 x^2) - \log(1 + x^3)$ is independent of $\theta$, so it corresponds to $h(x)$.
• The term $-\log(1 + x^3)$ can be treated as $T(x)$, which is the sufficient statistic.
• $\eta(\theta) = \log(\theta)$ is the natural parameter.
• $A(\theta)$ corresponds to the terms related to normalization.

Thus, the family is part of the exponential family.

Step 2: Find a sufficient statistic

For the exponential family, the sufficient statistic is typically the function of $x$ that multiplies the natural parameter. In this case, the sufficient statistic $T(x)$ is:

$T(x) = -\log(1 + x^3).$

Final Answer:

1. The family of distributions belongs to the exponential family.
2. A sufficient statistic for this family is $T(x) = -\log(1 + x^3)$.

Let me know if you'd like a deeper explanation or have more questions! Here are five related questions:

1. What are the properties of a sufficient statistic in the exponential family?
2. How do you compute the maximum likelihood estimator for $\theta$ given this family?
3. Can you derive the cumulant function $A(\theta)$ for this specific exponential family?
4. What are the advantages of recognizing a distribution as part of the exponential family?
5. How does the concept of conjugate priors apply to distributions from the exponential family?

Tip: Understanding how to express a probability density function in the exponential family form can simplify solving problems related to sufficient statistics and maximum likelihood estimation.