Suppose that Y1, Y2, Y3 denote a random sample from an...

Suppose that $Y_{1}, Y_{2}, Y_{3}$ denote a random sample from an exponential distribution with density function $$f(y)=\left\{\begin{array}{ll} \left(\frac{1}{\theta}\right) e^{-y / \theta}, & y>0 \\ 0, & \text { elsewhere } \end{array}\right.$$ Consider the following five estimators of $\theta$ : $$\hat{\theta}=Y_{1}, \quad \hat{\theta}_{2}=\frac{Y_{1}+Y_{2}}{2}, \quad \hat{\theta}_{3}=\frac{Y_{1}+2 Y_{2}}{3}, \quad \hat{\theta}_{4}=\min \left(Y_{1}, Y_{2}, Y_{3}\right), \quad \hat{\theta}_{5}=\bar{Y}$$ a. Which of these estimators are unbiased? b. Among the unbiased estimators, which has the smallest variance?

Suppose that $Y_{1}, Y_{2}, Y_{3}$ denote a random sample from an exponential distribution with density function
$$f(y)=\left\{\begin{array}{ll}
\left(\frac{1}{\theta}\right) e^{-y / \theta}, & y>0 \\
0, & \text { elsewhere }
\end{array}\right.$$
Consider the following five estimators of $\theta$ :
$$\hat{\theta}=Y_{1}, \quad \hat{\theta}_{2}=\frac{Y_{1}+Y_{2}}{2}, \quad \hat{\theta}_{3}=\frac{Y_{1}+2 Y_{2}}{3}, \quad \hat{\theta}_{4}=\min \left(Y_{1}, Y_{2}, Y_{3}\right), \quad \hat{\theta}_{5}=\bar{Y}$$
a. Which of these estimators are unbiased?
b. Among the unbiased estimators, which has the smallest variance?

Key Concepts

Exponential Distribution

This distribution is characterized by its memoryless property and a single parameter that determines both its mean and variance. It is often used to model the time between events in a Poisson process. Understanding the probability density function, expected value, and variance of the exponential distribution is essential when working with estimators derived from samples from this distribution.

Unbiased Estimator

An unbiased estimator is a statistic that, on average, equals the parameter it is estimating. This means that its expected value is equal to the true parameter value. In estimation theory, determining whether an estimator is unbiased helps in assessing its accuracy and fairness over repeated sampling.

Variance of Estimators

The variance of an estimator measures the spread of its sampling distribution around the expected value. An estimator with a lower variance is generally more desirable since it implies the estimator will yield values closer to the true parameter. Comparing variances among unbiased estimators is a common step when seeking the most efficient estimator.

Order Statistics

Order statistics, such as the minimum or maximum of a sample, have specific probability distributions derived from the parent distribution. In the context of the exponential distribution, the minimum of a sample is itself exponentially distributed with a different rate parameter. Understanding this concept is crucial when estimating parameters using sample extremes.

Linear Combination Estimators

Estimators that are constructed as linear combinations of sample observations are commonly used due to their simplicity and tractable properties. Analyzing such estimators involves calculating their expected values and variances, which typically depend on the properties of the individual random variables and the weights used in the combination.

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