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The bias of an estimator helps us understand how accurately it can predict the true value of a parameter. It's defined simply as the difference between the expected value of the estimator and the actual parameter value. In mathematical terms, this is expressed as \( \text{Bias}(\widehat{\beta}) = E[\widehat{\beta}] - \beta \).

To derive the bias in our given scenario with the Pareto distribution, we need to calculate the expected value \( E[\widehat{\beta}] \). Using the proven steps, we discover that \( E[\widehat{\beta}] = \beta \frac{n}{n-1} \). This means the estimator's expected value inherently has some bias away from the true value \( \beta \).

So, by substituting this into our bias formula, we find that \( \text{Bias}(\widehat{\beta}) = \frac{\beta}{n-1} \). This indicates that as the sample size \( n \) increases, the bias decreases, meaning that larger samples provide more accurate estimates of \( \beta \).

Mean Squared Error (MSE) is a key measure of an estimator's quality. It combines both bias and variability of an estimator into a single number. Essentially, MSE provides a comprehensive picture of how concentrated or dispersed the estimator values are around the actual parameter.

The MSE is calculated as the sum of the variance of the estimator and its bias squared: \( \text{MSE}(\widehat{\beta}) = \text{Var}(\widehat{\beta}) + (\text{Bias}(\widehat{\beta}))^2 \).

For our Pareto distribution example, we already derived both components:\( \text{Var}(\widehat{\beta}) = \frac{\beta^2}{(n-1)(n-2)} \) and \( \text{Bias}(\widehat{\beta}) = \frac{\beta}{n-1} \).

Plugging these into the MSE formula, we obtain:

• \( \text{MSE}(\widehat{\beta}) = \frac{\beta^2}{(n-1)(n-2)} + \frac{\beta^2}{(n-1)^2} \)

This formula illustrates how MSE decreases as \( n \) increases, indicating more reliable estimates with larger samples.

A Probability Density Function (PDF) is crucial for understanding the likelihood of different outcomes for continuous random variables. It depicts how probabilities of values are distributed across different points in a random variable's scale.

In the given Pareto distribution, the PDF is defined as \( f(y) = 3 \beta^3 y^{-4} \) for \( y \geq \beta \). This function sharply declines as \( y \) increases, demonstrating less probability mass as you move farther from the minimum threshold \( \beta \).

To derive the PDF of the minimum estimator, we differentiate its CDF with respect to \( y \), resulting in \( f_{\widehat{\beta}}(y) = 3n\beta^3 y^{-(3n+1)} \) for \( y \geq \beta \). This reveals that the likelihood of observing the minimum value decreases significantly as the value moves away from \( \beta \).

Understanding and utilizing the PDF is key for calculating expectations and predicting behaviors within a given distribution.

The Cumulative Distribution Function (CDF) is fundamentally important in statistics. It sums up the probability that a random variable takes a value less than or equal to a particular point. The CDF increases stepwise or continuously from 0 to 1.

For a continuous distribution like the Pareto, the CDF can illustrate the spread of probabilities across a range of values. Particularly, for the minimum of a sample, the CDF formula is expressed as \( P(\widehat{\beta} \leq y) = 1 - (1 - P(Y_i \leq y))^n \), where \( P(Y_i \leq y) = 1 - (\beta/y)^3 \) for \( y \geq \beta \). This indicates how likely it is for the minimum observed sample to be less than or equal to a certain amount.

By understanding and utilizing the CDF effectively, we can determine probabilities and glean insights about the behavior and properties of our distribution.