It can be shown that if Yn converges in probability to a constant τ, then h(Yn) converges to h(τ

step 1 We're told that it's provable that if the random variable YN converges in probability to a const

It can be shown that if Yn converges in probability to a...

It can be shown that if $Y_{n}$ converges in probability to a constant $\tau,$ then $h\left(Y_{n}\right)$ converges to $h(\tau)$ for any function $h$ that is continuous at $\tau$ . Use this to obtain a consistent estimator for the rate parameter $\lambda$ of an exponential distribution. [Hint: How does $\mu$ for an exponential distribution relate to the exponential parameter $\lambda ?$

Key Concepts

Method of Moment Estimation

The method of moments involves equating sample moments, such as the sample mean, to their corresponding theoretical moments to solve for unknown parameters. In the case of the exponential distribution, since the mean is 1/?, setting the sample mean equal to this value and solving for ? provides a simple and consistent estimator for the rate parameter.

Exponential Distribution Parameters

An exponential distribution is characterized by a rate parameter, ?, with its mean given by ? = 1/?. This relationship is essential when estimating ?, because a consistent estimator of ?, typically the sample mean, can be inverted (using the continuous mapping theorem) to yield a consistent estimator for ?.

Consistency of Estimators

An estimator is consistent if it converges in probability to the true parameter as the sample size tends to infinity. This property is fundamental in statistics as it assures that with increasing data, the estimator will yield values arbitrarily close to the parameter being estimated.

Continuous Mapping Theorem

The continuous mapping theorem states that if a sequence of random variables converges in probability to a constant, then any continuous function of these random variables will converge in probability to the function evaluated at that constant. This theorem is crucial for deriving consistent estimators when transformations of consistent estimators are applied.

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