Question

We have been talking a lot about specific sample statistics such as (sample) mean or (sample) variance. In principle, any quantity that characterizes (is derived from) a sample is a legitimate sample statistics (and since it depends on the sample, or in other word on a particular realization/sample drawing, this quantity is going to be a random variable itself). A sum of squares of the values in the sample, or a product of their logarithms, or simply the maximum value in the sample - these are all "sample statistics". Of course some statistics are "better" and/or more useful than the others. Think about how we can (or would like) to characterize a sample, and what would make a "good" statistic? Why maximum value in a sample is not such a good statistics in most cases? Why mean is better? What problems can mean still suffer from? What other "good" sample statistics can you think of?

In this context, I'd like to stress that "estimator" can be viewed simply as a particular sample statistics that approximates (i.e. "estimates form the available sample") some meaningful parameter/property of the underlying distribution. This statistic better be "good" in any of the senses you came up with above; but what additional properties we would like the estimator to have? What would make a better or a worse estimator, how could we quantify their "quality"?

Answer 1

Why maximum value in a sample is not such a good statistics in most cases?

because Maximum value talking about a specific value not full sample or population that's

why maximum value is not good statistics

Why mean is better?

Bacause mean is talking about all the sample or population , which is based on all the value of the sample or population that's why mean is better statistcis than maximum value of the data .

What problems can mean still suffer from?

mean is not fair representation of data when data skewed ther may some outlier in the data

in this case very large value and Very Small value will affect on the mean.

What other "good" sample statistics can you think of?

Median is good statisctis of because if your data is skewed in this case median will give middle value of the data so this is better representation of the data .