Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be i.i.d from the uniform distribution \(U(a, b)\), \(-\infty

Short answer

The estimators of 'a' and 'b' in a uniform distribution using method of moments are \( \hat{a} = \bar{X} - \sqrt{\frac{3(n-1)}{n} S^{2}} \) and \( \hat{b} = \bar{X} + \sqrt{\frac{3(n-1)}{n} S^{2}} \), which are found by setting sample moments (sample mean and variance) equal to the theoretical moments (mean and variance of uniform distribution), and solving for 'a' and 'b'.

Step-by-step solution

Understanding Method of Moments

The method of moments is a way of estimating the parameters of a distribution by matching the sample moments (like the sample mean and variances) to the theoretical moments of the distribution.

Find the Theoretical Moments

The theoretical moments of a uniform distribution \( U(a, b) \) are: \[ \text{Mean} = \frac{a+b}{2} \] and \[ \text{Variance} = \frac{(b-a)^{2}}{12} \].

Find the Sample Moments

The sample moments are based on the numerical data given, and they are: \[ \bar{X} \] (the sample mean) and \[ S^{2} = \frac{1}{n-1}\sum_{i=1}^{n} (X_i - \bar{X})^{2} \] (the sample variance).

Set up the Equations

Based on the method of moments, we set the sample moments equal to the theoretical moments, which gives us the following two equations: \[ \bar{X} = \frac{a+b}{2} \] and \[ S^{2} = \frac{(b-a)^{2}}{12} \].

Solve the Equations

Solving these equations for 'a' and 'b', we get: \[ a = \bar{X} - \sqrt{\frac{3(n-1)}{n} S^{2}} \] and \[ b = \bar{X} + \sqrt{\frac{3(n-1)}{n} S^{2}} \], which are the desired estimators.

Key concepts

Uniform Distribution

The uniform distribution is a type of probability distribution where all outcomes are equally likely. When discussing continuous random variables, the uniform distribution over an interval from a to b, denoted as U(a, b), has a flat probability density function because every value between a and b has the same chance of occurring. This property results in specific formulas for the mean and variance of the distribution.

The mean, or expected value, of a uniform distribution can be calculated using the midpoint between a and b, expressed mathematically as \( \frac{a+b}{2} \). Similarly, the variance, which measures the spread of the distribution, is given by \( \frac{(b-a)^{2}}{12} \). This variance formula stems from the dispersion of values across the interval. Having a fundamental understanding of these properties is crucial for parameter estimation in statistics.

Parameter Estimation

Parameter estimation involves determining the values of the parameters of a statistical model that best fit the data. When working with a uniform distribution, we aim to estimate the lower and upper bounds, a and b. The method of moments is one such technique for parameter estimation. This method involves equating the theoretical moments (mean, variance) derived from the probability distribution to the corresponding sample moments calculated from the data. By solving these equations, we find the parameter estimates that make the sample moments as close as possible to the theoretical moments.

For the uniform distribution U(a, b), after finding expressions for a and b using the sample mean and variance, we gain estimators \( \hat{a} \) and \( \hat{b} \) that specify our best guesses for the actual parameter values based on the data we have.

Sample Mean

In statistics, the sample mean is one of the most fundamental measures of central tendency. It represents the average of a set of observations from a sample and is denoted by \( \bar{X} \). When we say 'sample mean', we're talking about the sum of all sample values divided by the total number of observations in the sample. This measure serves as a central marker for the data and is used in various statistical methods, including the method of moments, to provide an estimate for the theoretical mean of a distribution. In the context of the uniform distribution, the sample mean is used in conjunction with the sample variance to estimate the parameters a and b.

Sample Variance

The sample variance, denoted as \( S^2 \), measures how spread out the sample values are around the sample mean. It is calculated by taking the sum of the squared differences between each sample point and the sample mean, then dividing by the number of observations minus one — this subtraction known as Bessel's correction, accounts for the sample estimation to avoid bias. In symbol terms, it looks like this: \( S^{2} = \frac{1}{n-1}\sum_{i=1}^{n} (X_i - \bar{X})^{2} \).

The sample variance is a critical component in the method of moments, as it is set equal to the theoretical variance of the distribution to solve for the distribution's parameters. Understanding how to calculate and interpret the sample variance is essential for reliable parameter estimation in statistical analysis.