Some of my student asked me how good is an estimate and how do we measure it up.

Let us first consider the difference between an estimate and estimator. We are all familiar with the following formula

s^{2} = \frac{1}{n-1}[\sum{x^{2}} -\frac{(\sum{x})^{2}}{n}]

The above formula is an estimator. It allows us to input our collected data and values to %latex s^{2}$ which is the estimate. So to put things in perceptive, we first collect samples and obtain a sample variance (biased), then input it into the estimator formula to obtain the estimate s^{2} which is unbiased. And let us not forget that the unbiased estimate is still not the same as the true value, that is, s^{2} \ne {\sigma}^{2}

So there are definitely more factors that attributes to a good estimate, unbiased is one. The next in line will be consistent. We question if the estimate is consistent, this means to consider if the estimate tends to the true value as n tends to infinity.

Credits: https://www.google.com.sg/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAYQjB1qFQoTCNv566vBmMcCFVNtjgod7mIJaA&url=https%3A%2F%2Fcommons.wikimedia.org%2Fwiki%2FFile%3AConsistentEstimationFunction.png
Credits: https://www.google.com.sg/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAYQjB1qFQoTCNv566vBmMcCFVNtjgod7mIJaA&url=https%3A%2F%2Fcommons.wikimedia.org%2Fwiki%2FFile%3AConsistentEstimationFunction.png

This is really intuitive as I explained in class just yesterday.Imagine you are capable of obtaining the weight of EVERY SINGLE person in Singapore, then you will definitely be able to find the TRUE mean weight (granted ladies do not underestimate their weight). Thus, the value found will be consistent with the true value.

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