Question

. Let \(X\) and \(Y\) have a joint bivariate normal distribution. An observation \((x, y)\) arises from the joint distribution with parameters equal to either $$\mu_{1}^{\prime}=\mu_{2}^{\prime}=0,\quad\left(\sigma_{1}^{2}\right)^{\prime}=\left(\sigma_{2}^{2}\right)^{\prime}=1, \quad \rho^{\prime}=\frac{1}{2}$$ or $$\mu_{1}^{\prime \prime}=\mu_{2}^{\prime \prime}=1,\left(\sigma_{1}^{2}\right)^{\prime \prime}=4, \quad\left(\sigma_{2}^{2}\right)^{\prime \prime}=9, \rho^{\prime \prime}=\frac{1}{2}$$ Show that the classification rule involves a second-degree polynomial in \(x\) and \(y\).

Short answer

The required classification rule obtained involves a second-degree polynomial in \(x\) and \(y\): \((-3/4)x^{2}+(-5/9)y^{2}+x+y-1/4-1/9 < 0\). This results from simplifying the likelihood ratio of the two joint normal distributions and setting it to a condition for separation.

Step-by-step solution

Specify the Joint Distributions

The joint distributions for \(X\) and \(Y\) are given as follows: \[f_{1}(x, y)=\frac{1}{2 \pi \sqrt{1-\rho^{2}}}\exp \left\{-\frac{1}{2(1-\rho^{2})}[(x-\mu_{1})^{2}+(y-\mu_{2})^{2}-2\rho(x-\mu_{1})(y-\mu_{2})]\right\}\]\[f_{2}(x, y)=\frac{1}{2 \pi \sqrt{1-\rho^{2}}}\exp \left\{-\frac{1}{2(1-\rho^{2})}[(x-\mu_{1}')^{2}+(y-\mu_{2}')^{2}-2\rho(x-\mu_{1}')(y-\mu_{2}')]\right\}\]

Derive the Likelihood Ratio

The likelihood ratio can be obtained from the two distributions. The classification rule based on maximum likelihood will decide between \(f_1\) and \(f_2\) and test whether the likelihood ratio \(f_2 / f_1\) is greater than or less than 1.Substituting the parameters from the given problem, the likelihood ratio can be simplified as:\[\frac{f_{2}(x, y)}{f_{1}(x, y)}=\exp \left\{-\frac{1}{2}[(x-1)^{2}/4+(y-1)^{2}/9-x^{2}-y^{2}+2x+2y-x+4y/9]\right\}\]

Obtain Second-Degree Polynomial Classification Rule

By simplifying and isolating the elements of the exponent in the likelihood ratio, a second-degree polynomial classification rule can be found:\[(x-1)^{2}/4+(y-1)^{2}/9-x^{2}-y^{2}+2x+2y-x+4y/9 < 0\]Solving the above inequality, the equation comes as:\[(-3/4)x^{2}+(-5/9)y^{2}+x+y-1/4-1/9 < 0\]This is a second-degree polynomial in \(x\) and \(y\), which constitutes the required classification rule.

Key concepts

Joint Distribution

In statistics, the concept of joint distribution refers to the probability distribution of two or more random variables at the same time. When we think of these variables as having a bivariate normal distribution, it means they are considering two variables, namely \(X\) and \(Y\), which are normally distributed and have a specified correlation coefficient \(\rho\).
Such distributions are often described by their means (\(\mu_1\) and \(\mu_2\)), variances (\(\sigma_1^2\) and \(\sigma_2^2\)), and the correlation \(\rho\). The joint distribution helps us understand how these two variables are related and how their probabilities are distributed across potential combinations.
For example, if \(X\) and \(Y\) are independent, \(\rho\) would be zero, indicating no linear relationship. Otherwise, if \(\rho\) is close to 1 or -1, it indicates a strong positive or negative linear relationship, respectively. This joint distribution forms the basis for calculating probabilities and understanding the relationship between the randomness of these variables.

Likelihood Ratio

The likelihood ratio is a vital concept in the realm of statistics, often used for hypothesis testing. In the context of a bivariate normal distribution, it helps determine which of the two potential distributions an observation most likely originates from.
The likelihood ratio compares two probability distributions: one under the null hypothesis \(H_0\) and another under the alternative hypothesis \(H_1\). In problem solving, the ratio \(\frac{f_2(x, y)}{f_1(x, y)}\) is calculated. If this ratio is greater than one, it suggests that the observation is more likely under \(H_1\).
In the exercise given, the likelihood ratio was simplified using given parameters, turning into a crucial tool for deciding between the two specified distributions. It essentially assesses the 'weight of evidence' supporting one hypothesis over another, shaping the choice made by any classification rule applied thereafter.

Polynomial Inequality

When analyzing statistical data, polynomial inequalities often arise in scenarios involving classification rules. A polynomial inequality is an inequality where a polynomial expression is set against zero or another polynomial.
In this exercise context, the polynomial inequality derived was \(-\frac{3}{4}x^2 - \frac{5}{9}y^2 + x + y - \frac{1}{4} - \frac{1}{9} < 0\).
This inequality emerges from manipulating the given likelihood ratio expression to isolate terms in \(x\) and \(y\). Solving this inequality helps define regions in the \((x, y)\) space that belong to one distribution versus another, effectively guiding the decision-making process in classification.
Polynomial inequalities are powerful as they allow us to map non-linear relationships and thresholds within data, which can be crucial in more complex decision boundaries in statistics and machine learning applications.

Classification Rule

Classification rules in statistics are fundamental for deciding between different data categories based on given criteria or data. They consist of a set of guidelines generated from data to distinguish between classes efficiently.
In this bivariate normal distribution problem, the classification rule arose from the polynomial inequality.
The process involves taking the polynomial inequality derived from the likelihood ratio and using it to separate the space into different classes, such as that the rule becomes a decision boundary.
For example, the problem's classification rule involved a second-degree polynomial, which practically splits the \((x, y)\) space into regions that belong to either \(\mu'\) or \(\mu''\) distribution. This kind of second-degree polynomial provides a more flexible boundary than linear separators in the classification, allowing for curvier decision boundaries which can adequately capture complex data patterns. In statistical terms, this helps improve the accuracy of the classification process.