Question

Let \(X\) be a random variable with cdf \(F(x)\) and let \(T(F)\) be a functional. We say that \(T(F)\) is a scale functional if it satisfies the three properties $$ \text { (i) } T\left(F_{a X}\right)=a T\left(F_{X}\right), \text { for } a>0 $$ (ii) \(T\left(F_{X+b}\right)=T\left(F_{X}\right), \quad\) for all \(b\) $$ \text { (iii) } T\left(F_{-X}\right)=T\left(F_{X}\right) \text { . } $$ Show that the following functionals are scale functionals. (a) The standard deviation, \(T\left(F_{X}\right)=(\operatorname{Var}(X))^{1 / 2}\). (b) The interquartile range, \(T\left(F_{X}\right)=F_{X}^{-1}(3 / 4)-F_{X}^{-1}(1 / 4)\).

Short answer

Both the standard deviation and the interquartile range are scale functionals. This is because they both satisfy the three properties outlined, i.e., scaling the argument of the functional by a factor of a or -a results in the functional value also being scaled by that factor, adding a constant to the random variable does not change the value of the functional and the value of the functional remains unchanged when negating the random variable. The key principles involved in proving this are the properties of the variance and the cumulative distribution function (CDF) and its inverse.

Step-by-step solution

Prove standard deviation is a scale functional

The standard deviation of a random variable X is described by \(T(F_X) = (\operatorname{Var}(X))^{1 /2}\). This can be rewritten in terms of the expected value as\( T(F_X) = (E[X^2] - (E[X])^2)^{1/2}\). Thus proving \(\text { (i) } T\left(F_{aX}\right)=a T\left(F_{X}\right), \text { for } a>0 \) involves plugging \(aX\) instead of \(X\) in the expected value: \( T(F_{aX}) = (E[(aX)^2] - (E[aX])^2)^{1/2} = a(E[X^2] - (E[X])^2)^{1/2} = aT(F_X)\).For \(\text { (ii) }\), we substitute \( F_{X+b}\) for \( F_{X}\) and realize that adding a constant \( b\) to a random variable \( X\) does not change the standard deviation as \( b\) affects neither the spread of the data nor its 'scale', hence \( T(F_{X+b}) = T(F_{X})\). For the last property \(\text{ (iii)}, we know that the standard deviation is a measure of dispersion or 'spread' of a random variable, so taking the negative of \(X\) does not affect this spread, hence \( T(F_{-X}) = T(F_{X}).\)

Prove the interquartile range is a scale functional

The interquartile range is given by \(T(F_X) = F_X^{-1}(3/4) - F_X^{-1}(1/4)\). For \(\text { (i) }\), when the random variable \( X\) is scaled by a factor of \( a > 0\), the difference between the third and the first quartiles also scales by a factor of a resulting to \( T\left(F_{aX}\right) = a T\left(F_{X}\right)\). For \(\text { (ii) }\), when a constant \( b\) is added to a random variable \( X\), the interquartile range stays the same hence \( T(F_{X+b}) = T(F_{X}).\) Finally, for \(\text { (iii),}\) if \( X\) is negated, this does not affects its interquartile range hence \( T(F_{-X}) = T(F_{X}).\)

Key concepts

Statistical Scale Functionals

Statistical scale functionals are tools used by statisticians to measure the variability or spread of a data set. They are a subset of a broader category known as statistical functionals, which map functions, such as cumulative distribution functions (CDFs), to real numbers, summarizing certain aspects of the distribution.
Scale functionals have unique properties: they are invariant to shifts in location and they scale in a predictable manner when all data points are scaled by a constant factor. Additionally, they are not affected by taking the negative of all data values. These properties make scale functionals excellent measures for comparing the variability of different distributions regardless of their location or sign.

Standard Deviation

The standard deviation is a widely-used statistical scale functional which quantifies the amount of variation or dispersion of a set of values. It is defined as the square root of the variance. In mathematical terms, for a random variable X, standard deviation is represented by \(T(F_X) = (\operatorname{Var}(X))^{1/2}\), where \(\operatorname{Var}(X)\) is the variance of X.
It satisfies the properties required for a scale functional, making it a key indicator of consistency within a dataset. For example, a smaller standard deviation implies that the data points tend to be closer to the mean of the dataset.

Interquartile Range

The interquartile range (IQR) represents the middle 50% of a dataset and is calculated as the difference between the 75th percentile (third quartile) and the 25th percentile (first quartile). In the context of scale functionals, the IQR can be expressed as \(T(F_X) = F_X^{-1}(3/4) - F_X^{-1}(1/4)\).

Because the IQR focuses on the central portion of the data, it is less influenced by outliers or extreme values than other measures such as the range or standard deviation. As a scale functional, the IQR meets the three properties: it scales with the data, it is not influenced by shifts in location (adding a constant), and does not change when the data is reflected (multiplying by -1).

Cumulative Distribution Function

The cumulative distribution function (CDF) is a function associated with a random variable and indicates the probability that the variable will take a value less than or equal to a specific value. The CDF is integral in defining and analyzing properties of statistical scale functionals since it encapsulates all the information about the distribution of a random variable.
The CDF for a random variable X is denoted as \(F(x)\), and it ranges from 0 to 1 with \(F(-\infty) = 0\) and \(F(+\infty) = 1\). When analyzing scale functionals, the CDF forms the basis to establish whether the functional remains consistent under transformation of the data, such as scaling.

Random Variable Properties

Random variables are basic building blocks in probability and statistics. They are quantities that assume different values depending on the outcome of a stochastic process. Properties of random variables guide the behavior and interpretation of statistical scale functionals. Some key properties include the expected value (or mean), which is the average outcome if an experiment is repeated many times; variance, which measures how much the values of the random variable vary from the mean; and skewness and kurtosis, which describe the shape of the random variable's distribution.
Understanding these properties is crucial when working with scale functionals as they influence the measure of variability and play a pivotal role in analyzing the spread of data.