Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N\left(\theta_{1}, \theta_{2}\right)\). (a) If the constant \(b\) is defined by the equation \(P(X \leq b)=0.90\), find the mle and the MVUE of \(b\). (b) If \(c\) is a given constant, find the mle and the MVUE of \(P(X \leq c)\).

Short answer

MLE of \(b\) is \(\overline{X} + k * S\), and MVUE of \(b\) is \(E(b) = \theta_{1} + k * \theta_{2}\). MLE of \(P(X \leq c)\) is \(P(Z \leq (c - \overline{X}) / S)\), and MVUE of \(P(X \leq c)\) is \(E[P(X \leq c)] = P(Z \leq (c - \theta_{1}) / \theta_{2})\).

Step-by-step solution

Identifying MLE and MVUE for \(b\)

Since \(X\) is normally distributed, note that \(b\) can be expressed as \(b=\theta_{1} + k * \theta_{2}\), where \(k\) is the z score for a 0.90 probability. The MLE of \(b\) is the same as the MLE of \(\theta_{1} + k * \theta_{2}\). Since the MLE of \(\theta_{1}\) is the sample mean \(\overline{X}\) and the MLE of \(\theta_{2}\) is the sample standard deviation \(S\) when \(S^2\) is the sample variance, then the MLE of \(b\) is \(\overline{X} + k * S\). For the MVUE of \(b\), it will simply be the expected value of \(b\), which is \(E(b) = \theta_{1} + k * \theta_{2}\)

Identifying MLE and MVUE for \(P(X \leq c)\)

The MLE and MVUE of \(P(X \leq c)\) are equivalent to identifying the probability that a standard normal variable \(Z\) is less than or equal to \(Z = (c - \overline{X}) / S\). This can be denoted as \(P(Z \leq (c - \overline{X}) / S)\), and the standard normal distribution table or computational software can be used to find this value. As for the MVUE, it will be the probability that a standard normal variable \(Z\) is less than or equal to \(Z = (c - \theta_{1}) / \theta_{2}\), which can be written as \(E[P(X \leq c)] = P(Z \leq (c - \theta_{1}) / \theta_{2})\). Similar to the MLE, a standard normal distribution table or computational software can be used to find this value

NOTE

Given that \(X_{1}, X_{2}, ..., X_{n}\) is a random sample from a normal distribution, these steps hold valid. However, if the distribution was different, the process might vary.

Key concepts

Minimum Variance Unbiased Estimator

A Minimum Variance Unbiased Estimator (MVUE) is a statistical method used to estimate a parameter in such a way that the estimation is unbiased and has the lowest possible variance among all unbiased estimators. This means that an MVUE provides the most accurate and consistent estimates of a parameter. Think of it this way: if you repeated an experiment or took multiple samples from a population, using an MVUE gives you estimates that are right on target on average. And among all the possible unbiased estimators, it spreads out or varies the least, meaning you can trust it more to be close to the true parameter value in repeated sampling. For example:

• When estimating the mean of a normally distributed population, the sample mean often acts as the MVUE. This is because it is unbiased (it correctly centers around the true mean) and tends to have minimum variance compared to other unbiased estimators of the mean.
• In the problem exercise, we look at finding MVUE for probabilities or specific statistics derived from normally distributed data. These properties help ensure our statistical models and conclusions are both reliable and predictable in practice.

Understanding and using MVUEs is a core aspect of ensuring your statistical analyses are robust.

Normal Distribution

In statistics, the Normal Distribution is one of the most important probability distributions. It is often referred to as a Gaussian distribution and is characterized by its bell-shaped curve.Key features include:

• The distribution is defined by two parameters: the mean (\(\theta_{1}\)) and the variance (\(\theta_{2}^{2}\)). These determine the center and the spread of the distribution, respectively.
• The curve is symmetric around the mean, implying that the values have an equal chance of falling on either side of the mean.
• Approximately 68% of the data falls within one standard deviation (\(\theta_{2}\)) from the mean. About 95% falls within two standard deviations, and nearly all (99.7%) lie within three standard deviations.

In many practical problems, including the textbook exercise, the normal distribution is used because of its unique properties. Many real-world phenomena such as test scores, heights, and errors in measurements naturally align with this distribution. This widely applies to inferential statistics where assumptions of normality are made to use statistical tests reliably.

Sample Mean

The Sample Mean is a fundamental concept in statistics, serving as a basic estimator of the population mean. It is calculated by summing up all the data points in a sample and dividing by the number of data points, essentially providing a measure of the central tendency of the data.When you have a normally distributed random sample, the sample mean (\(\overline{X}\)) is a crucial statistic. It is straightforward to compute and possesses a desirable property: it is an unbiased estimator of the population mean (\(\theta_{1}\)). That means, on average, the sample mean gives the exact value of the population mean when collected under random sampling conditions.Why is this important?

• The calculation of a sample mean is often the first step in statistical analysis, paving the way for more complex inferences like variance calculi, hypothesis testing, or confidence interval estimation.
• In Maximum Likelihood Estimation (MLE), the sample mean is often the preferred estimate for population characteristics due to its efficiency and simplicity.

Understanding the role of the sample mean allows you to make informed predictions and analyses from data, linking observation with theory in statistical practice.