Question

Let $X_1,X_2,\dots ,X_n$ be a random sample from the normal distribution $N(\theta ,1)$. Show that the likelihood ratio principle for testing $H_0:\theta ={\theta }^{\prime }$, where ${\theta }^{\prime }$ is specified, against $H_1:\theta \ne {\theta }^{\prime }$ leads to the inequality $|\overline{x}-{\theta }^{\prime }|\ge c$ (a) Is this a uniformly most powerful test of $H_0$ against $H_1?$ (b) Is this a uniformly most powerful unbiased test of $H_0$ against $H_1?$

Short answer

(a) Yes, this is a uniformly most powerful (UMP) test. (b) Yes, it is also a uniformly most powerful unbiased (UMPU) test.

Step-by-step solution

Derive the Likelihood Ratio Test

For this problem, the likelihood ratio test is $\mathrm{\Lambda }=\frac{L\left({\theta }^{\prime }\right)}{L\left(\hat{\theta }\right)}$ where $\hat{\theta }$ is the max likelihood estimator, equal to the sample mean $\overline{x}$. That simplifies to $\mathrm{\Lambda }=\frac{e^{-n{(\overline{x}-{\theta }^{\prime })}^2/2}}{e^{-n{(\overline{x}-\overline{x})}^2/2}}=e^{-n{(\overline{x}-{\theta }^{\prime })}^2/2}$ . The inequality $|\overline{x}-{\theta }^{\prime }|\ge c$ is the rejection region.

Determine if this is a uniformly most powerful (UMP) test

This test is actually UMP. It was derived from the Neyman-Pearson Lemma which indicates that the LRT is the most powerful for simple hypotheses.

Determine if this is a UMP unbiased test

To be a uniformly most powerful unbiased (UMPU) test, the test needs to not only be most powerful for every value under the alternative hypothesis (UMP), but also unbiased in the sense that the probability of rejecting the null hypothesis when it's true is not greater than the significance level $\alpha$. With this inequality as the rejection region, this likelihood ratio test fulfills all conditions, so it is UMPU.

Key concepts

Uniformly Most Powerful Test

A Uniformly Most Powerful (UMP) test is one that maximizes the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true, for all possible values under the alternative hypothesis. In simpler terms, it’s the best test we can create for detecting whether the null hypothesis is false.

To understand this better, let's consider the Neyman-Pearson Lemma. It tells us that for testing simple hypotheses, the likelihood ratio test is the most powerful. This means the test is designed to detect if the hypothesis is wrong more effectively than any other test of the same size (or significance level).

• The simple hypothesis specifies a fixed value for the parameter under the null hypothesis.
• The UMP test is ideal because it works optimally for all parameters under the alternative hypothesis.

In the exercise, the test for the normal distribution is UMP because it was derived using this principle. This makes it the best choice for determining whether there's a difference from the hypothesis $\theta ={\theta }^{\prime }$.

Normal Distribution

The normal distribution is one of the most important concepts in statistics. It is a bell-shaped curve that describes how data is spread out around a mean. In a normal distribution, most data points fall close to the mean, and the probabilities for data points decrease as you move away from the center.

Here's a quick breakdown of its properties:

• The mean, median, and mode of a normal distribution are all equal, located at the peak of the curve.
• It's symmetric around the mean, meaning the left and right sides are mirror images.
• The area under the curve represents the total probability, which is equal to 1.

The normal distribution is crucial in hypothesis testing and in the exercise, it forms the basis of the data points $X_1,X_2,\dots ,X_n$. When working with a normal distribution, we often deal with variables like the sample mean $\overline{x}$, which itself follows a normal distribution when the sample size is large enough according to the Central Limit Theorem.

Hypothesis Testing

Hypothesis testing is a method used in statistical analysis to determine if there is enough evidence to reject a null hypothesis $H_0$ in favor of an alternative hypothesis $H_1$. Here’s a basic rundown of how it works:

1. **Formulate Hypotheses**
We start with two hypotheses:

• **Null Hypothesis ($H_0$)**: Assumes no effect or no difference. It’s the hypothesis we seek to test against.
• **Alternative Hypothesis ($H_1$)**: Suggests an effect or a difference. It's what we hope to support.

2. **Select a Significance Level $\alpha$**

• This is the probability of rejecting the null hypothesis when it is true. Common values are 0.05 or 0.01.

3. **Calculate the Test Statistic**

• This involves determining how far the sample mean is from the hypothesized population mean under $H_0$.

4. **Make a Decision**

• Based on the test statistic and significance level, decide to either reject $H_0$ or not.

In our example, the rejection region is $|\overline{x}-{\theta }^{\prime }|\ge c$. If this condition holds, it indicates enough evidence to reject the null hypothesis, confirming an effect or difference exists.