Consider Part (a) of Exercise 6.4 .6. (a) Using the data of...

Consider Part (a) of Exercise $6.4 .6$. (a) Using the data of Exercise $6.4 .7$, compute the mle of $b$. Also obtain the estimate based on 90 th percentile of the data. (b) Edit the R function bootstrapcis64.R to compute a bootstrap confidence interval for $b$. Then run your $\mathrm{R}$ function on the data of Exercise $6.4 .7$ to compute a $95 \%$ confidence interval for $b$.

Consider Part (a) of Exercise $6.4 .6$.
(a) Using the data of Exercise $6.4 .7$, compute the mle of $b$. Also obtain the estimate based on 90 th percentile of the data.
(b) Edit the R function bootstrapcis64.R to compute a bootstrap confidence interval for $b$. Then run your $\mathrm{R}$ function on the data of Exercise $6.4 .7$ to compute a $95 \%$ confidence interval for $b$.

Key Concepts

Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a model by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. It provides a framework for parameter estimation that often yields estimators with desirable properties such as consistency and asymptotic normality, making it a fundamental tool in inferential statistics.

Percentile Estimation

Percentile estimation involves using sample quantiles to estimate parameters of a distribution. This approach can be useful when certain percentiles have specific significance in the model, such as the 90th percentile used to derive an estimate for a parameter. It leverages order statistics and is sometimes preferred for its robustness to model assumptions compared to methods that rely solely on central moments.

Bootstrap Confidence Intervals

Bootstrap confidence intervals are a resampling-based method for estimating the uncertainty of parameter estimators. By repeatedly drawing samples with replacement from the observed data and recalculating the estimate, one can approximate the sampling distribution and hence construct confidence intervals, such as a 95% interval. This technique is particularly useful when the theoretical distribution of the estimator is complex or unknown, making it a versatile tool in modern statistical inference.

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