Suppose that X1, X2, …, Xn are i.i.d. N(μ, σ^2). a. If μis known, what is the mle of σ? b. If σi

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Suppose that X1, X2, …, Xn are i.i.d. N(μ, σ^2). a. If μis...

Suppose that $X_{1}, X_{2}, \ldots, X_{n}$ are i.i.d. $N\left(\mu, \sigma^{2}\right)$. a. If $\mu$ is known, what is the mle of $\sigma ?$ b. If $\sigma$ is known, what is the mle of $\mu ?$ c. In the case above $(\sigma \text { known), does any other unbiased estimate of } \mu$ have smaller variance?

Suppose that $X_{1}, X_{2}, \ldots, X_{n}$ are i.i.d. $N\left(\mu, \sigma^{2}\right)$.
a. If $\mu$ is known, what is the mle of $\sigma ?$
b. If $\sigma$ is known, what is the mle of $\mu ?$
c. In the case above $(\sigma \text { known), does any other unbiased estimate of } \mu$ have smaller variance?

Key Concepts

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function, which quantifies the probability of the observed data given different parameter values. In the context of a normal distribution, MLE leads to explicit forms for estimators when either the mean or variance is known, and it leverages the sample information to produce consistent and often efficient estimates.

Normal Distribution Properties

The normal distribution is characterized by two parameters, the mean and the variance, and is symmetric about its mean. Its well-known density function and properties, such as the fact that linear combinations of normal random variables are also normal, allow for precise analytical derivations of estimators. This makes it a fundamental model in statistics for parametric inference, particularly when dealing with continuous data.

Unbiased Estimation

An unbiased estimator is one for which the expected value equals the true parameter value, meaning that on average, the estimator neither overestimates nor underestimates the parameter. In decision making, an unbiased estimator is desirable because it does not introduce systematic error, and establishing unbiasedness is a key step in assessing the quality of an estimator.

Efficiency and the Cramér-Rao Lower Bound

Efficiency of an estimator refers to the property of having the smallest possible variance among all unbiased estimators of a parameter. The Cramér-Rao Lower Bound (CRLB) provides a theoretical minimum variance that any unbiased estimator can achieve given the information in the data. An estimator that achieves this bound is deemed efficient, implying that no other unbiased estimator can have a lower variance.

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