Question

If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a distribution having pdf of the form \(f(x ; \theta)=\theta x^{\theta-1}, 0

Short answer

The best critical region for testing the hypothesis \(H_{0}: \theta=1\) versus \(H_{1}: \theta=2\) is \(C=\left\{\left(x_{1}, x_{2}, \ldots, x_{n}\right): c \leq \prod_{i=1}^{n} x_{i}\right\}\)

Step-by-step solution

Set up the likelihood ratio

The likelihood ratio for the hypothesis \(H_{0}: \theta=1\) versus \(H_{1}: \theta=2\) is given by:\[ \Lambda = \frac{L(1)}{L(2)} \]In these expressions, \(L(1)\) and \(L(2)\) denote the likelihoods of the data under the respective hypotheses. Each one is the product of the probability density function \(f(x; \theta)\) evaluated at each datum under \(\theta=1\) and \(\theta=2\), respectively.

Compute likelihoods

Now, compute the likelihoods as follows:\[ L(1) = \prod_{i=1}^{n} f(x_i; 1) = \prod_{i=1}^{n} x_i^{1-1} = 1 \]\[ L(2) = \prod_{i=1}^{n} f(x_i; 2) = \prod_{i=1}^{n} 2 x_i^{2-1} = 2^n \prod_{i=1}^{n} x_i \]The likelihood ratio thus evaluates to \(\Lambda = \frac{1}{2^n \prod_{i=1}^{n} x_i}\).

Show that the likelihood ratio is a monotonic function

The likelihood ratio \(\Lambda\) is a decreasing function of the product \(\prod_{i=1}^{n} x_i\). It follows that the most powerful test rejects \(H_0\) for low values of \(\Lambda\), i.e., large values of \(\prod_{i=1}^{n} x_i\), and accepts \(H_0\) for high values of \(\Lambda\), i.e., small values of \(\prod_{i=1}^{n} x_i\). Therefore, the best critical region is of the form \(C = \{\left(x_{1}, x_{2}, \ldots, x_{n}\right): c \leq \prod_{i=1}^{n} x_{i}\}\), where the threshold \(c\) is chosen to satisfy the desired significance level.

Key concepts

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used by statisticians to make decisions about the properties of populations based on sample data. At its core, the method involves proposing two competing hypotheses: the null hypothesis, denoted as \(H_0\), which represents a default position, and an alternative hypothesis, \(H_1\), which we seek to demonstrate evidence for.

In a typical hypothesis test, the goal is to determine whether there is sufficient evidence from the sample data to reject \(H_0\) in favor of \(H_1\). This decision is made by calculating a test statistic from the sample data and determining whether the statistic falls into a critical region or not. The critical region represents a set of values for which the null hypothesis is considered to be unlikely and is therefore rejected.

To prevent incorrect conclusions, statisticians set a significance level \(\alpha\), often at 0.05 or 5%, which directly corresponds to the probability of rejecting the null hypothesis when it is actually true (a type I error). The lower the significance level, the more stringent and conservative the test.

Probability Density Function

The probability density function (PDF), denoted as \(f(x; \theta)\), describes how the probability is distributed over the values of a continuous random variable. For any given value of \(x\), the PDF gives the relative likelihood of that value occurring within a small interval around \(x\). It is important to note that the function itself is not a probability, but an integral of the PDF over a range of values yields a probability for the variable to lie within that range.

In the context of the example exercise, the PDF is defined as \(f(x ; \theta)=\theta x^{\theta-1}\) for \(0 < x < 1\), which highlights that the likelihood of observing a particular sample can be calculated as a function of \(\theta\) and values of \(x_{i}\). By establishing this relationship, the PDF serves as the fundamental building block for computing likelihoods and conducting hypothesis tests.

Critical Region

The critical region, also called the rejection region, is a key concept in hypothesis testing. It represents the set of all possible outcomes of a test statistic that would lead to the rejection of the null hypothesis \(H_0\). The boundaries of this region are defined and determined based on the desired significance level \(\alpha\) and the distribution of the test statistic under the null hypothesis.

In the step-by-step solution, the critical region is expressed as \(C = \{\left(x_{1}, x_{2}, \ldots, x_{n}\right): c \leq \prod_{i=1}^{n} x_{i}\}\), showing the best set of outcomes for which we reject \(H_0\) when comparing two values of \(\theta\). Essentially, if the calculated test statistic falls within this critical region, there is sufficient evidence to conclude that the sample data is significantly different from what would be expected under the null hypothesis, thus supporting the alternative hypothesis \(H_1\). It's imperative for the significance level and resultant region to be established prior to conducting the test to avoid bias.