How does sampling scheme affect some statistics of interest?

Question

I have a sample of data sets A = {a1, a2, a3, ...aN}

We will change each data value following the same pattern, say a1’ =a1*10, a2’ = a2*10. Which statistics evaluated from this data set will be changed?

Assume that I increase the size of this data set simply by make copies for each data,

A’ = {a1, a1, a1, a2, a2, a2, a3, a3, a3,...}

How will this changing affect the different statistics, like mean, standard deviation, etc.?

Answer 1

Technical answers to these questions can be found in the (sub)discipline of robust statistics. It operates with the concept of influence function: $$ {\rm IF}(x;T,F) = \lim_{\epsilon\to +0} \frac{T[ (1-\epsilon) F + \epsilon \delta(x)] - T[F]}{{\rm d}\epsilon} $$ where $F(\cdot)$ is the distribution function of interest (typically, the sample cdf $F_n(x) = \sum_i 1\{ x \le x_i \}$, and $T(F)$ is a statistic that can be expressed as a functional of the distribution function. Sample mean is such a functional: ${\rm E}[X] = \int x \, {\rm d}F(x)$, and $\bar X = \int x \, {\rm d}F_n(x)$. Sample median is also a functional, although a more complicated one: the one that solves $\int {\rm sign}(x-m) \, {\rm d}F = 0$ with respect to $m$. Sample variance, $s^2 = \frac1{n-1} \sum_i (X_i - \bar X)^2$, does not belong to this class, as it has an ugly $n-1$ factor that cannot be related to $F_n(\cdot)$ function. Simply speaking, the influence function shows by how much your statistics changes when we add an observation with value $x$ to the data set.

If your statistic of interest belongs to the class of statistics that can be defined as functionals of the cdf, then it won't be affected by your second transformation (copies of the original sample). The answer to your first question is given by the IF: if you replace $x_1 \mapsto 10 x_1$, then the new statistic will change approximately by $T \mapsto T + \frac1n [{\rm IF}( 10 x_1, T, F_n)-{\rm IF}( x_1, T, F_n)]$. The IF of the sample mean is ${\rm IF}(x;\bar X, F) = \frac 1n x$, and of course it also easy to see algebraically that the change in the sample mean will be $\bar X \mapsto \bar X + 9x_1/n$.