Question

In a scale family, \(Y=\tau \varepsilon\), where \(\varepsilon\) has a known density and \(\tau>0\). Consider testing the null hypothesis \(\tau=\tau_{0}\) against the alternative \(\tau \neq \tau_{0}\). Show that the appropriate group for constructing an invariant test has just one element (apart from permutations) and hence show that the test may be based on the maximal invariant \(Y_{(1)} / \tau_{0}, \ldots, Y_{(n)} / \tau_{0}\). When \(\varepsilon\) is exponential, show that the invariant test is based on \(\bar{Y} / \tau_{0}\).

Short answer

The invariant test is based on \( \bar{Y} / \tau_{0} \) when \( \varepsilon \) is exponential.

Step-by-step solution

Identifying the Problem

We need to construct an invariant test using the problem setup where \( Y = \tau \varepsilon \) with \( \varepsilon \) having a known density and \( \tau > 0 \). We are testing the hypothesis \( \tau = \tau_0 \) against \( \tau eq \tau_0 \).

Identifying the Group of Transformations

The suitable group of transformations for this problem will be a scaling group. We consider transformations \( g_c : Y \rightarrow cY \), where \( c > 0 \). Since scaling by any factor \( c \) simply modifies the scale of \( Y \), any invariant test must remain the same under such transformations.

Constructing the Maximal Invariant

Since the group acts by scaling, the ratio of observations, such as \( Y_{(i)}/\tau_0 \), remains invariant. Therefore, the maximal invariant is \( Y_{(1)} / \tau_{0}, \ldots, Y_{(n)} / \tau_{0} \). This set of ratios does not change when \( Y \) is scaled.

Special Case: Exponential Distribution

When \( \varepsilon \) follows an exponential distribution, the sample mean \( \bar{Y} \) is a sufficient statistic for \( \tau \). Thus, the invariant test can be based on \( \bar{Y} / \tau_{0} \), where the test uses the averages of the scaled observations as a simplifying factor.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. It involves setting up two competing hypotheses: the null hypothesis (H_0) and the alternative hypothesis (H_1). The null hypothesis is usually a statement of no effect or no difference, such as stating that a parameter equals a certain value. For example, in our case, the null hypothesis is \( \tau = \tau_0 \)\, meaning that the true value of the parameter \( \tau \) equals a specific value \( \tau_0 \). On the other hand, the alternative hypothesis represents the possibility of an effect or difference, such as \( \tau eq \tau_0 \). After formulating these hypotheses, the next step is to use statistical tests to determine which hypothesis is more likely given the data. The outcomes are often evaluated using a significance level \( \alpha \), which is the probability of rejecting the null hypothesis when it is true. Typically, \( \alpha \) is set to 0.05, but it can be adjusted based on the context of the study. The decision to reject or not reject \( H_0 \) is based on calculated statistics from the sample data, allowing us to assess the credibility of hypothesis \( H_0 \).

Invariant Tests

Invariant tests are crucial in statistical testing, as they maintain their form or decision rule unchanged under transformations applied to data. Consider a scenario involving scale transformations, where data is transformed by a multiplicative constant. An invariant test becomes valuable as it provides results that aren't affected by these changes. To construct such a test, we often rely on understanding the problem's symmetry and focus on statistics that remain unchanged under the specified transformations. In the example provided, we consider the transformation \( g_c : Y \rightarrow cY \), resulting in a scaling group where data is multiplied by a constant \( c \). With this knowledge, we seek a maximal invariant, which remains the same irrespective of how much the data is scaled. A simple way to construct a maximal invariant is to use ratios, such as \( Y_{(i)} / \tau_0 \). This approach effectively averages out the scaling factor \( \tau \), transforming or 'invariantizing' the data to focus on the underlying distribution properties rather than scaling differences.

Exponential Distribution

The exponential distribution is vital in scenarios involving the time between events in a Poisson process. It is continuous and characterized by a single parameter \( \lambda \), representing the rate of occurrence. This makes it popular in fields like reliability engineering and survival analysis. For a random variable following an exponential distribution, its probability density function is given by: \[ f(x | \lambda) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0 \]An interesting property of the exponential distribution is its memorylessness, meaning that the probability of an event occurring in the next instance is independent of how much time has already elapsed. In hypothesis testing, particularly for invariant tests, an exponential distribution leads to simplification since the sample mean \( \bar{Y} \) acts as a sufficient statistic for the scale parameter \( \tau \). This means that the test can focus on simple statistics without losing information about \( \tau \). For instance, if \( \varepsilon \) follows an exponential distribution, the invariant test can be effectively built using the ratio \( \bar{Y} / \tau_0 \), leveraging the simplicity provided by the exponential structure.