Linear Moments

█ OVERVIEW
The Linear Moments indicator, also known as L-moments, is a statistical tool used to estimate the properties of a probability distribution. It is an alternative to conventional moments and is more robust to outliers and extreme values.

█ CONCEPTS

█ Four moments of a distribution

We have mentioned the concept of the Moments of a distribution in one of our previous posts. The method of Linear Moments allows us to calculate more robust measures that describe the shape features of a distribution and are anallougous to those of conventional moments. L-moments therefore provide estimates of the location, scale, skewness, and kurtosis of a probability distribution.

The first L-moment, λ₁, is equivalent to the sample mean and represents the location of the distribution. The second L-moment, λ₂, is a measure of the dispersion of the distribution, similar to the sample standard deviation. The third and fourth L-moments, λ₃ and λ₄, respectively, are the measures of skewness and kurtosis of the distribution. Higher order L-moments can also be calculated to provide more detailed information about the shape of the distribution.

One advantage of using L-moments over conventional moments is that they are less affected by outliers and extreme values. This is because L-moments are based on order statistics, which are more resistant to the influence of outliers. By contrast, conventional moments are based on the deviations of each data point from the sample mean, and outliers can have a disproportionate effect on these deviations, leading to skewed or biased estimates of the distribution parameters.

█ Order Statistics

L-moments are statistical measures that are based on linear combinations of order statistics, which are the sorted values in a dataset. This approach makes L-moments more resistant to the influence of outliers and extreme values. However, the computation of L-moments requires sorting the order statistics, which can lead to a higher computational complexity.

To address this issue, we have implemented an Online Sorting Algorithm that efficiently obtains the sorted dataset of order statistics, reducing the time complexity of the indicator. The Online Sorting Algorithm is an efficient method for sorting large datasets that can be updated incrementally, making it well-suited for use in trading applications where data is often streamed in real-time. By using this algorithm to compute L-moments, we can obtain robust estimates of distribution parameters while minimizing the computational resources required.

█ Bias and efficiency of an estimator

One of the key advantages of L-moments over conventional moments is that they approach their asymptotic normal closer than conventional moments. This means that as the sample size increases, the L-moments provide more accurate estimates of the distribution parameters.

Asymptotic normality is a statistical property that describes the behavior of an estimator as the sample size increases. As the sample size gets larger, the distribution of the estimator approaches a normal distribution, which is a bell-shaped curve. The mean and variance of the estimator are also related to the true mean and variance of the population, and these relationships become more accurate as the sample size increases.

The concept of asymptotic normality is important because it allows us to make inferences about the population based on the properties of the sample. If an estimator is asymptotically normal, we can use the properties of the normal distribution to calculate the probability of observing a particular value of the estimator, given the sample size and other relevant parameters.

In the case of L-moments, the fact that they approach their asymptotic normal more closely than conventional moments means that they provide more accurate estimates of the distribution parameters as the sample size increases. This is especially useful in situations where the sample size is small, such as when working with financial data. By using L-moments to estimate the properties of a distribution, traders can make more informed decisions about their investments and manage their risk more effectively.

Below we can see the empirical dsitributions of the Variance and L-scale estimators. We ran 10000 simulations with a sample size of 100. Here we can clearly see how the L-moment estimator approaches the normal distribution more closely and how such an estimator can be more representative of the underlying population.


█ WAYS TO USE THIS INDICATOR

The Linear Moments indicator can be used to estimate the L-moments of a dataset and provide insights into the underlying probability distribution. By analyzing the L-moments, traders can make inferences about the shape of the distribution, such as whether it is symmetric or skewed, and the degree of its spread and peakedness. This information can be useful in predicting future market movements and developing trading strategies.

One can also compare the L-moments of the dataset at hand with the L-moments of certain commonly used probability distributions. Finance is especially known for the use of certain fat tailed distributions such as Laplace or Student-t. We have built in the theoretical values of L-kurtosis for certain common distributions. In this way a person can compare our observed L-kurtosis with the one of the selected theoretical distribution.

█ FEATURES

Source Settings

Source - Select the source you wish the indicator to calculate on

Source Selection - Selec whether you wish to calculate on the source value or its log return


Moments Settings

Moments Selection - Select the L-moment you wish to be displayed

Lookback - Determine the sample size you wish the L-moments to be calculated with

Theoretical Distribution - This setting is only for investingating the kurtosis of our dataset. One can compare our observed kurtosis with the kurtosis of a selected theoretical distribution.