Moment (mathematics)


In mathematics, a moment is a specific quantitative measure of the shape of a function.
The concept is used in both mechanics and statistics. If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the zeroth moment is the total probability, the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.
For a distribution of mass or probability on a bounded interval, the collection of all the moments uniquely determines the distribution. The same is not true on unbounded intervals.

Significance of the moments

The -th moment of a real-valued continuous function f of a real variable about a value c is
It is possible to define moments for random variables in a more general fashion than moments for real values—see moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with c = 0.
For the second and higher moments, the central moment are usually used rather than the moments about zero, because they provide clearer information about the distribution's shape.
Other moments may also be defined. For example, the -th inverse moment about zero is and the -th logarithmic moment about zero is
The -th moment about zero of a probability density function f is the expected value of and is called a raw moment or crude moment. The moments about its mean are called central moments; these describe the shape of the function, independently of translation.
If f is a probability density function, then the value of the integral above is called the -th moment of the probability distribution. More generally, if F is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the -th moment of the probability distribution is given by the Riemann–Stieltjes integral
where X is a random variable that has this cumulative distribution F, and is the expectation operator or mean.
When
then the moment is said not to exist. If the -th moment about any point exists, so does the -th moment about every point.
The zeroth moment of any probability density function is 1, since the area under any probability density function must be equal to one.

Mean

The first raw moment is the mean, usually denoted

Variance

The second central moment is the variance. The positive square root of the variance is the standard deviation

Standardized moments

The normalised -th central moment or standardised moment is the -th central moment divided by ; the normalised -th central moment of the random variable is
These normalised central moments are dimensionless quantities, which represent the distribution independently of any linear change of scale.
For an electric signal, the first moment is its DC level, and the 2nd moment is proportional to its average power.

Skewness

The third central moment is the measure of the lopsidedness of the distribution; any symmetric distribution will have a third central moment, if defined, of zero. The normalised third central moment is called the skewness, often. A distribution that is skewed to the left will have a negative skewness. A distribution that is skewed to the right, will have a positive skewness.
For distributions that are not too different from the normal distribution, the median will be somewhere near ; the mode about.

Kurtosis

The fourth central moment is a measure of the heaviness of the tail of the distribution, compared to the normal distribution of the same variance. Since it is the expectation of a fourth power, the fourth central moment, where defined, is always nonnegative; and except for a point distribution, it is always strictly positive. The fourth central moment of a normal distribution is.
The kurtosis is defined to be the standardized fourth central moment If a distribution has heavy tails, the kurtosis will be high ; conversely, light-tailed distributions have low kurtosis.
The kurtosis can be positive without limit, but must be greater than or equal to ; equality only holds for binary distributions. For unbounded skew distributions not too far from normal, tends to be somewhere in the area of and.
The inequality can be proven by considering
where. This is the expectation of a square, so it is non-negative for all a; however it is also a quadratic polynomial in a. Its discriminant must be non-positive, which gives the required relationship.

Mixed moments

Mixed moments are moments involving multiple variables.
Some examples are covariance, coskewness and cokurtosis. While there is a unique covariance, there are multiple co-skewnesses and co-kurtoses.

Higher moments

High-order moments are moments beyond 4th-order moments. As with variance, skewness, and kurtosis, these are higher-order statistics, involving non-linear combinations of the data, and can be used for description or estimation of further shape parameters. The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality. This is due to the excess degrees of freedom consumed by the higher orders. Further, they can be subtle to interpret, often being most easily understood in terms of lower order moments – compare the higher derivatives of jerk and jounce in physics. For example, just as the 4th-order moment can be interpreted as "relative importance of tails versus shoulders in causing dispersion", the 5th-order moment can be interpreted as measuring "relative importance of tails versus center in causing skew".

Transformation of center

Since:
where is the binomial coefficient, it follows that the moments about b can be calculated from the moments about a by:

Cumulants

The first raw moment and the second and third unnormalized central moments are additive in the sense that if X and Y are independent random variables then
.
In fact, these are the first three cumulants and all cumulants share this additivity property.

Sample moments

For all k, the -th raw moment of a population can be estimated using the -th raw sample moment
applied to a sample drawn from the population.
It can be shown that the expected value of the raw sample moment is equal to the -th raw moment of the population, if that moment exists, for any sample size. It is thus an unbiased estimator. This contrasts with the situation for central moments, whose computation uses up a degree of freedom by using the sample mean. So for example an unbiased estimate of the population variance is given by
in which the previous denominator has been replaced by the degrees of freedom, and in which refers to the sample mean. This estimate of the population moment is greater than the unadjusted observed sample moment by a factor of and it is referred to as the "adjusted sample variance" or sometimes simply the "sample variance".

Problem of moments

The problem of moments seeks characterizations of sequences that are sequences of moments of some function f.

Partial moments

Partial moments are sometimes referred to as "one-sided moments." The -th order lower and upper partial moments with respect to a reference point r may be expressed as
Partial moments are normalized by being raised to the power 1/n. The upside potential ratio may be expressed as a ratio of a first-order upper partial moment to a normalized second-order lower partial moment. They have been used in the definition of some financial metrics, such as the Sortino ratio, as they focus purely on upside or downside.

Central moments in metric spaces

Let be a metric space, and let B be the Borel -algebra on M, the -algebra generated by the d-open subsets of M. Let.
The pth central moment of a measure on the measurable space about a given point is defined to be
μ is said to have finite -th central moment if the -th central moment of about x0 is finite for some.
This terminology for measures carries over to random variables in the usual way: if is a probability space and is a random variable, then the -th central moment of X about is defined to be
and X has finite -th central moment if the -th central moment of X about x0 is finite for some.