91Ó°ÊÓ

A Probability Density Function (pdf) is a function used in statistics to describe the likelihood of a random variable taking on a given value. For a continuous random variable, the pdf provides the density of probability at each point in its range, though, in practice, it gives the shape of the distribution.

In the exercise, we are given the pdf of a random variable \( X \), specified as \( f_X(x; \theta) = \frac{1}{2\theta} \) for \(-\theta < x < \theta\) and zero elsewhere, indicating that the variable \( X \) has a uniform distribution over this interval. The parameter \( \theta \) is a scale parameter, adjusting the spread of the distribution.

Through a transformation, we obtain a new variable \( Y = |X| \), whose pdf is derived from \( X \). The transformation simplifies the original pdf to \( f_Y(y; \theta) = \frac{1}{\theta} \) for \( 0 < y < \theta \) and zero elsewhere. What this means is that \( Y \) also follows a uniform distribution but over the mirrored portion of \( X \) to reflect its absolute value constraint.

A statistic is considered sufficient if it captures all the information about a parameter of interest, allowing us to make likelihood-based inferences about the parameter without needing the original data.

In our exercise, we look at \( Y = |X| \) as a potential sufficient statistic for \( \theta \). For \( Y \) to be sufficient for \( \theta \), knowing \( Y \) should give us as much information about \( \theta \) as knowing \( X \) directly. From the transformation, we found that its pdf \( f_Y(y; \theta) = \frac{1}{\theta} \) relies directly on \( \theta \), similar to the initial pdf of \( X \).

Because \( Y \) retains the dependency on \( \theta \) in a way that fully encapsulates its influence, without requiring extra context from \( X \), \( Y \) is indeed a sufficient statistic for \( \theta \). This sufficiency is significant as it allows for a simplified model—often leading to easier data interpretations and computational benefits.

Completeness in statistics refers to a property of a family of distributions whereby no non-zero unbiased estimator exists for a function of the parameter unless the expected value of the estimator is zero. This property ensures that there is no redundant information in the sufficient statistic.

In the context of our exercise, we consider the family of distributions defined by \( f_Y(y; \theta) = \frac{1}{\theta} \) over the interval \( 0 < y < \theta \). To determine completeness, suppose there is a function \( h(y) \) such that \( E[h(Y)] = 0 \) for all \( \theta \). The problem leads us to the integral \( \int_{0}^{\theta} h(y) \cdot \frac{1}{\theta} \, dy = 0 \).

This integral equals zero only when \( h(y) \) itself is zero over \( 0 < y < \theta \). This outcome confirms that there is no information in the pdf of \( Y \) that could provide a non-trivial solution to this integral for all \( \theta \). Thus, the family \( \{f_Y(y; \theta) : \theta > 0\} \) is complete, reinforcing that \( Y \) is an optimal statistic for carrying all information about \( \theta \).