Inferential Statistics 361
In our discussion, we will assume that individual draws are performed
independently and under identical conditions (i.e., the X 1 , X 2 ,... , Xn
are i.i.d.).
We know that the joint probability distribution of independent random
variables is obtained by multiplication of the marginal distributions.
Consider the daily stock returns of General Electric (GE) modeled by
the continuous random variable X. The returns on 10 different days (X 1 ,
X 2 ,... , X 10 ) can be considered a sample of i.i.d. draws. In reality, however,
stock returns are seldom independent. If, on the other hand, the observa-
tions are not made on 10 consecutive days but with larger gaps between
them, it is fairly reasonable to assume independence. Furthermore, the stock
returns are modeled as normal (or Gaussian) random variables.
Statistic What is the distinction between a statistic and a population
parameter? In the context of estimation, the population parameter is
inferred with the aid of the statistic. A statistic assumes some value that
holds for a specific sample only, while the parameter prevails in the entire
population.
The statistic, in most cases, provides a single number as an estimate of the
population parameter generated from the sample. If the true but unknown
parameter consists of, say, k components, the statistic will provide at least k
numbers, that is at least one for each component. We need to be aware of the
fact that the statistic will most likely not equal the population parameter due
to the random nature of the sample from which its value originates.
Technically, the statistic is a function of the sample (X 1 , X 2 ,... , Xn).
We denote this function by t. Since the sample is random, so is t and, conse-
quently, any quantity that is derived from it.
We need to postulate measurability so that we can assign a probability
to any values of the function t(x 1 , x 2 ,... , xn). Whenever it is necessary to
express the dependence of statistic t on the outcome of the sample (x), we
write the statistic as the function t(x). Otherwise, we simply refer to the
function t without explicit argument.
The statistic t as a random variable inherits its theoretical distribution
from the underlying random variables (i.e., the random draws X 1 , X 2 ,... ,
Xn). If we vary the sample size n, the distribution of the statistics will, in
most cases, change as well. This distribution expressing in particular the
dependence on n is called the sampling distribution of t. Naturally, the sam-
pling distribution exhibits features of the underlying population distribu-
tion of the random variable.
estimator The easiest way to obtain a number for the population parameter
would be to simply guess. But this method lacks any foundation since it is