Question

Show that \(Y=|X|\) is a complete sufficient statistic for \(\theta>0\), where \(X\) has the pdf \(f_{X}(x ; \theta)=1 /(2 \theta)\), for \(-\theta

Short answer

It can be proved that \(Y=|X|\) is a complete sufficient statistic for \(\theta>0\), and that \(Y=|X|\) and \(Z=\operatorname{sgn}(X)\) are independent by analyzing the given pdf and using properties of the sufficient and complete statistics, and definition of independent events in probability theory.

Step-by-step solution

Show Y is sufficient

Start with the joint pdf for \(n\) observations. Here observations are \(x_1,\dots,x_n\). The joint pdf is given by \[f(x_1,\dots,x_n;\theta) = \prod_{i=1}^{n}f_X(x_i;\theta) = \frac{1}{(2\theta)^n}\] where \(-\theta<x_i<\theta\) for all \(i\) and zero otherwise. This can be equally written as \[f(x_1,\dots,x_n;\theta) = \frac{1}{(2\theta)^n}\] where \(-\theta<|x_i|<\theta\) for all \(i\) and zero otherwise. This shows that we can replace \(x_i\) with \(|x_i|\), meaning \(Y=|X|\) is a sufficient statistic.

Show Y is complete

For \(Y=|X|\) to be a complete statistic, if \(E(g(Y)) = 0\) for all \(\theta\), then \(P(g(Y) = 0) = 1\) for all \(\theta\). For a function \(g(Y)\) where \(Y=|X|\), if its expectation \(E(g(Y))\) is \(0\) for all \(\theta\), then it must be \(0\) everywhere. That is \(P(g(Y) = 0) = 1\), for all \(\theta\). Hence, \(Y=|X|\) is also a complete statistic.

Prove Y and Z are independent

To check if \(Y=|X|\) and \(Z=\operatorname{sgn}(X)\) are independent, we begin by noting that \(Z=\operatorname{sgn}(X)\) can take only two values: \(+1\) or \(-1\). \(Y= |X|\) is nonnegative. We note that the occurrences of \(Z\) are equally likely, irrespective of the value of \(Y\). That is, \(P(Z=1)\) is the same whether \(Y\) is small or large, and the same holds for \(P(Z=-1)\). Since \(Z\) does not depend on \(Y\), we establish that \(Y\) and \(Z\) are statistically independent.

Key concepts

Mathematical Statistics

Mathematical statistics is a branch of applied mathematics that focuses on the collection, analysis, interpretation, and presentation of empirical data. It applies probability theory to random events, considering the randomness and uncertainty of data. In the context of the given exercise, mathematical statistics is utilized to determine if a statistic, such as the absolute value of a random variable, serves as a 'complete sufficient statistic' for an unknown parameter.

In assessing sufficiency, statisticians consider whether the statistic captures all necessary information from the data to make inferences about the parameter. Completeness adds a layer of assurance; a complete statistic means that no other statistic can provide additional information about the parameter. If a statistic is both sufficient and complete for a parameter, it is considered the best statistic for estimating that parameter, allowing for the most precise inferences.

Probability Density Function (PDF)

The probability density function (PDF) is a fundamental concept in statistics, used to specify the probability of a continuous random variable falling within a particular range of values. The PDF, represented by a mathematical formula, shows the shape of the distribution of the random variable.

For example, consider the PDF given in the exercise \[f_{X}(x ; \theta)=1 /(2 \theta)\], for \(-\theta

Statistical Independence

Statistical independence is a key concept in probability that describes a situation where the occurrence of one event does not affect the probability of another event. Two variables are independent if the probability of one variable taking on any value is not influenced by the value of the other variable.

In the exercise, we look at independence between the variables \(Y=|X|\) and \(Z=\operatorname{sgn}(X)\). Statistically, \(Y\) and \(Z\) are independent if the joint probability distribution can be expressed as the product of their individual distributions. This means that knowing the value of \(Z\) gives no information about the value of \(Y\), and vice versa. In simpler terms, if flipping a coin (\(Z\)) to decide whether it lands heads or tails does not impact the measurement of a separate and unrelated quantity (\(Y\)), these two are independent. The exercise then concludes statistical independence since the chance of \(Z\) being \(+1\) or \(-1\) remains the same, regardless of the value of \(Y\).

Statistical Independence

Statistical independence is a key concept in probability that describes a situation where the occurrence of one event does not affect the probability of another event. Two variables are independent if the probability of one variable taking on any value is not influenced by the value of the other variable.

In the exercise, we look at independence between the variables \(Y=|X|\) and \(Z=\operatorname{sgn}(X)\). Statistically, \(Y\) and \(Z\) are independent if the joint probability distribution can be expressed as the product of their individual distributions. This means that knowing the value of \(Z\) gives no information about the value of \(Y\), and vice versa. In simpler terms, if flipping a coin (\(Z\)) to decide whether it lands heads or tails does not impact the measurement of a separate and unrelated quantity (\(Y\)), these two are independent. The exercise then concludes statistical independence since the chance of \(Z\) being \(+1\) or \(-1\) remains the same, regardless of the value of \(Y\).