Estimator for Mean and Variance of Sampled Data

Estimator is a statistic, usually in a function of the data, that is used to infer the value of an unknown parameter in a statistical model. During this post, we will talk about estimator for mean and variance of sampled data. We can determine a good estimator by calculating the bias of it. A good estimator should give bias closed to zero. Let $\theta$ is parameter we want to estimate/observe, our estimator result will be $\hat{\theta}$. The bias of our estimator is defined as follows.

$bias = E[\hat{\theta}]-\theta$

We will use bias formula above to check whether our estimator is good or not. And during this post, we will check our estimator we already derived by MLE here, which are mean and variance. Let’s write them first.

$\mu_{MLE}=\hat{\mu}=\frac{1}{n}\sum_{i=1}^{n}x_i\\\\ \sigma^2_{MLE}=\hat{\sigma}^2=\frac{1}{n}\sum_{i=1}^{n}(x_i-\hat{x})^2$

where $\bar{x}=\mu$ and $\hat{x}=\hat{\mu}$ Continue reading “Estimator for Mean and Variance of Sampled Data”