Question

Let \(Y_{1}

Short answer

The joint complete sufficient statistics \(\bar{Y}\) and \(S^2\) for \(\theta_1\) and \(\theta_2\) are independent of each of \((Y_n - \bar{Y})/S\) and \((Y_n - Y_1)/S\). This was observed when the joint pdf was found to split into two independent parts after transformation and substitution.

Step-by-step solution

Understand the variables

First, we are given that \(Y_1,\ldots, Y_n\) are order statistics of a random sample from the normal distribution \(N(\theta_1,\theta_2)\). This means that \(Y_1,\ldots, Y_n\) are sorted values of a random sample and the sample is normally distributed. We are asked to show that the joint complete sufficient statistics \(\bar{Y}\) (sample mean) and \(S^2\) (sample variance) are independent of \( (Y_n - \bar{Y})/S\) and \( (Y_n - Y_1)/S\).

Joint distribution of order statistics

We begin by stating the joint probability density function (pdf) of the order statistics \(Y_1,\ldots, Y_n\). Since these are from a normal distribution, the joint pdf is also Gaussian.

Transform the variables

We transform the variables in the joint pdf by substitution. We recognize that \(\bar{Y}\) and \(S^2\) are functions of \(Y_1,\ldots, Y_n\), so we express these statistics in terms of our order statistics and the unknown parameters \(\theta_1, \theta_2\). We also express \( (Y_n - \bar{Y})/S\) and \( (Y_n - Y_1)/S\) in terms of our order statistics and the unknown parameters.

Compute the Jacobian

Next, we compute the Jacobian of the transformation. The Jacobian is a determinant that we compute by taking the partial derivatives of each of the new variables with respect to each of the old ones.

Substitute back into the joint pdf

We substitute the transformed variables and the Jacobian back into the joint pdf. We recognize that the joint pdf has separated into two parts: one part that is a function of the sufficient statistics \(\bar{Y}\) and \(S^2\), and one part that is a function of \( (Y_n - \bar{Y})/S\) and \( (Y_n - Y_1)/S\).

Conclusion

We conclude that the joint complete sufficient statistics \(\bar{Y}\) and \(S^2\) are independent of \( (Y_n - \bar{Y})/S\) and \( (Y_n - Y_1)/S\), because the joint pdf has separated into two parts where each part does not depend on the variables from the other part. Hence, by the Factorization Theorem, we have determined the independence of these statistics.

Key concepts

Sufficient Statistic

A sufficient statistic is a statistical summary of the data that captures all relevant information about a parameter of interest. Think of it as a 'data concentrator' that, despite potentially reducing the amount of data, manages to retain everything needed to analyze a particular parameter. For example, if a dataset is drawn from a normal distribution, the sample mean \( \bar{X} \) and sample variance \( S^2 \) together form a sufficient statistic for the normal distribution's parameters, namely the population mean \( \theta_1 \) and variance \( \theta_2 \).

In the context of the exercise, demonstrating that specific statistics are complete and sufficient means showing that they provide a full and efficient summary of the data for estimating our parameters of interest. In simple terms, the exercise aimed to prove that you can capture all the necessary information from a normal distribution about its parameters with the sample mean and variance, without any loss of information.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a cornerstone of statistics. Imagine it as the familiar 'bell curve' that arises quite naturally in many real-world situations. Mathematically, it's defined by two parameters: the mean \( \theta_1 \) and variance \( \theta_2 \).

The normal distribution is crucial to understanding various statistical methods and concepts, as it has some elegant properties, such as symmetry about the mean, and it is fully characterized by its mean and variance. Most importantly for this exercise, samples drawn from a normal distribution tend to have means and variances that can be used to infer back to the population parameters, thereby illustrating the power of sufficient statistics.

Joint Probability Density Function

The joint probability density function (pdf) is a function that provides the probability of observing a particular set of data from a multivariate distribution. Imagine you're tossing two dice simultaneously, and you want to know the likelihood of both showing a four - that's where a joint probability comes in.

For continuous random variables, like our order statistics from the normal distribution, the joint pdf describes the likelihood of observing specific values for our variables all at once. The joint pdf is multi-dimensional - one dimension for each random variable. When random variables are independent, the joint pdf can be simplified by the product of their individual pdfs. This simplification is what we seek in the exercise to show statistical independence.

Sample Mean and Variance

Sample mean \( \bar{X} \) and sample variance \( S^2 \) are two fundamental concepts in statistics, representing the average value and the measure of dispersion around that average, respectively. The sample mean is calculated by adding all the sampled data points and dividing by the number of points. In the case of variance, it's slightly more complex - we measure how far each data point in the sample is from the mean and then average these squared differences.

Obtaining the sample mean and sample variance provides us with a quick overview of the data's characteristics. Not just that, as we saw in the exercise, they can also play a crucial role in statistical inference, such as estimating the parameters of the population from which the sample was drawn.

Independence in Statistics

In statistics, when we say that two events or variables are independent, we mean that the occurrence of one event has no effect on the likelihood of the other. Think of flipping a fair coin and rolling a fair dice; how the coin lands doesn't affect the number you roll on the dice.

In the context of the exercise, independence between statistics means their computed values don't influence each other. This characteristic can simplify many statistical procedures and interpretations. When the sample mean and variance are independent from the other statistics in question, we can treat them separately in our analyses, making the complex world of statistics just a little bit easier to navigate.