Upper Bound of a Uniform Distribution

Solution & Explanation

To determine the upper limit $d$ of a uniform distribution $[0, d]$ given $n$ samples, we can use several statistical approaches. Here are some methods and their explanations:

1. Sample Mean Method

• Concept: The expected value $E[X]$ of a uniform distribution $[0, d]$ is $d/2$. Thus, an estimator for $d$ can be derived from the sample mean.
• Formula: $d = 2 \times \bar{x}$ where $\bar{x}$ is the sample mean of the $n$ samples.
• Explanation: Since the expected value is $d/2$, multiplying the sample mean by 2 gives an estimate for $d$.

2. Maximum Likelihood Estimation (MLE)

• Concept: The maximum likelihood estimate for $d$ is the maximum observed value $M$ from the samples.
• Formula: $d = \frac{n+1}{n} \times M$
• Explanation: The MLE approach considers the largest sample value as the best estimate for $d$. The adjustment factor $(n+1)/n$ corrects for bias in small sample sizes.

3. Variance Method

• Concept: The variance of a uniform distribution $[0, d]$ is $d^2/12$. This property can be used to estimate $d$.
• Formula: $d = \sqrt{12 \times \text{Var}(X)}$ where $\text{Var}(X)$ is the sample variance.
• Explanation: By rearranging the variance formula, we can solve for $d$ using the sample variance.

4. Combination of Methods

• Concept: Use a combination of the above methods to find the most robust estimate for $d$.
• Approach:
  1. Calculate $d$ using the sample mean method.
  2. Calculate $d$ using the MLE method.
  3. Calculate $d$ using the variance method.
  4. Choose the maximum of these estimates as the final estimate for $d$.

Conclusion

Each method provides a different perspective on estimating $d$. The choice between them depends on the context:

• Sample Mean Method: Simple and intuitive, suitable for large sample sizes.
• MLE: Effective for small sample sizes, but may require bias correction.
• Variance Method: Useful when variance is accurately estimated.
• Combination Approach: Provides a balanced estimate by leveraging multiple statistical properties.