Question

Let \(Y_{1}

Short answer

The problem shows how one can demonstrate the independence of a complete sufficient statistic from other statistics in order statistics. This problem demonstrates the use of transformations and the properties of order statistics in the field of statistical inference. Overall, the complete sufficient statistic, \(Y_{4}\) (which corresponds to the parameter \(\theta\)), is independent of each \(Y_{1} / Y_{4}\) and \(\left(Y_{1}+Y_{2}\right) / \left(Y_{3}+Y_{4}\right)\) as they all follow a uniform distribution on interval (0, 1).

Step-by-step solution

Understanding of the Given Number of Random Sample

The exercise presents \(Y_{1}<Y_{2}<Y_{3}<Y_{4}\) as the order statistics of a random sample of size 4 extracted from a certain distribution. This implies \(Y_{4}\) is the maximum value of the sample, which is the complete sufficient statistic for \(\theta\). Under this condition, we need to show that \(Y_{4}\) is independent of the ratios \(Y_{1} / Y_{4}\) and \(\left(Y_{1}+Y_{2}\right) / \left(Y_{3}+Y_{4}\right)\).

Verifying the Probability Density Function (pdf)

The pdf of the distribution given in the problem is \(f(x ; \theta)=1 / \theta\), defined on the interval \(0<x<\theta\). Therefore, the function is uniform on this interval.

Redefining the pdf

To argue the required independence, we need to represent the given pdf in the hinted form. This means transforming \(f(x ; \theta)\) into \((1 / \theta) f(x / \theta)\) where \(f(w)=1,0<w<1\). For the given pdf, this reformulation represents \(x/\theta\) as \(w\), so the new pdf is \(f(w) = 1\) for \(0 < w < 1\). Hence, \(f(w)\) proves to be a uniform distribution on the interval (0,1).

Arguing the Independence of Statistics

The complete sufficient statistic \(Y_{4}\) is chosen for the parameter \(\theta\). Since the distribution of the sample elements is now uniform on (0,1), \(Y_{4}\) also has a uniform distribution on this interval. Since each \(Y_{i}\) for \(i = 1,2,3\) is less than \(Y_{4}\), each \(Y_{i} / Y_{4}\) and \(\left(Y_{1}+Y_{2}\right) / \left(Y_{3}+Y_{4}\right)\) have values within the range (0,1) and are also characterized by a uniform distribution. As such, \(Y_{4}\) can be viewed as independent of each of these statistics.

Key concepts

Probability Density Function

A probability density function (PDF) lays the groundwork for understanding various types of distributions. In simple terms, a PDF, which is a function of a continuous random variable, describes the relative likelihood for this random variable to occur at a given point. For the uniform distribution featured in our exercise, the PDF as provided is \( f(x ; \theta)=1 / \theta \) for the interval \( 0
To put it another way, if we think about picking a random point along a straight line of length \( \theta \) where any segment of the line has an equal chance of being chosen, that's what the PDF for a uniform distribution describes. In the context of the exercise improvement advice, it is crucial to note that the understanding and manipulation of the PDF are needed for transforming the random variable and proving statistical independence.

Complete Sufficient Statistic

The concept of a complete sufficient statistic is at the heart of statistical estimation. A statistic is sufficient for parameter \( \theta \) if it captures all available information about \( \theta \) contained in a sample. This means after computing the statistic, you don’t need the actual sample itself to estimate the parameter. Completeness adds another layer, indicating that no other statistic that can be calculated from the same sample provides any additional information about \( \theta \) that the sufficient statistic didn't already provide.

In the given exercise, \( Y_{4} \) is presented as the complete sufficient statistic for parameter \( \theta \) because it encompasses all the necessary information from the sample about the upper bound of the uniform distribution. Essentially, \( Y_{4} \) represents the maximum sample value and thus directly relates to \( \theta \) in the uniform distribution context. In terms of improvements, to help students understand this concept better, it's beneficial to explain the reasoning behind why \( Y_{4} \) is sufficient and complete in the context of the problem at hand.

Statistical Independence

When discussing statistical independence, we're referring to a scenario in which knowing the value of one variable tells us nothing about the value of another. In probability terms, two random variables are independent if the occurrence of one does not affect the probability distribution of the other.

In our exercise, we are particularly concerned with showing that the complete sufficient statistic \( Y_{4} \) is independent of the other given statistics. For the uniform distribution from which the sample is drawn, the PDF transformation showed that each sample element is equally affected by the scaling factor \( \theta \) and therefore \( Y_{4} \) remains the maximum regardless of the specific values of the other order statistics. This underpins the idea that \( Y_{4} \) is independent of ratios involving other order statistics because those ratios are effectively dimensionless and the scaling factor washes out in the comparison.

Improving comprehension on this topic could involve presenting examples of dependent and independent variables in various statistical contexts, as well as explicitly demonstrating the mathematical rationale behind the independence in this specific exercise.