3.5 Higher-order moments

Higher-order moments provide deeper insights into probability distributions beyond mean and variance. They capture nuanced aspects like skewness and kurtosis, enabling statisticians to analyze complex data patterns and make inferences about underlying populations.

Understanding these moments is crucial for various statistical applications. From parameter estimation to hypothesis testing, higher-order moments play a vital role in theoretical statistics, helping researchers develop robust methods for analyzing real-world data and making informed decisions.

Definition of higher-order moments

• Higher-order moments provide crucial insights into the shape and characteristics of probability distributions in theoretical statistics
• These moments extend beyond mean and variance to capture more nuanced aspects of data distributions
• Understanding higher-order moments enables statisticians to analyze complex data patterns and make inferences about underlying populations

Central vs raw moments

• Raw moments calculate expected values of powers of random variables from the origin
• Central moments measure deviations from the mean, offering insights into distribution spread and shape
• Raw moments defined as $E[X^k]$ where k is the moment order
• Central moments expressed as $E[(X - \mu)^k]$ where μ represents the mean
• Third and fourth central moments relate to skewness and kurtosis respectively

Standardized moments

• Standardized moments normalize central moments by dividing by the standard deviation raised to the moment's order
• Formula for standardized moments: $\frac{E[(X - \mu)^k]}{\sigma^k}$ where σ denotes standard deviation
• Dimensionless quantities allow for comparison across different scales and units
• Third standardized moment equates to skewness, fourth to kurtosis
• Higher standardized moments provide additional information about distribution tails and extreme values

Properties of moments

• Moments characterize probability distributions and provide insights into their shapes and behaviors
• Lower-order moments (mean, variance) often exist for a wider range of distributions than higher-order moments
• Theoretical statistics utilizes moment properties to develop estimation techniques and hypothesis tests

Existence and finiteness

• Not all probability distributions have finite moments for all orders
• Existence of kth moment requires the integral $\int_{-\infty}^{\infty} |x|^k f(x) dx$ to be finite
• Heavy-tailed distributions may have infinite higher-order moments (Cauchy distribution)
• Moment existence impacts the applicability of certain statistical methods and theorems
• Finite moments ensure the stability and convergence of various statistical estimators

Relationship to distribution shape

• First moment (mean) indicates central tendency
• Second central moment (variance) measures spread around the mean
• Third standardized moment (skewness) quantifies asymmetry
  • Positive skewness indicates a right-tailed distribution
  • Negative skewness suggests a left-tailed distribution
• Fourth standardized moment (kurtosis) describes tail behavior and peakedness
  • Higher kurtosis indicates heavier tails and a more peaked center
  • Lower kurtosis suggests lighter tails and a flatter distribution

Moment-generating functions

• Moment-generating functions (MGFs) serve as powerful tools in theoretical statistics for analyzing probability distributions
• MGFs encapsulate information about all moments of a distribution in a single function
• These functions facilitate the derivation of distribution properties and simplify calculations in probability theory

Definition and properties

• MGF defined as $M_X(t) = E[e^{tX}]$ where X is a random variable and t is a real number
• Exists only if the expectation is finite for some interval (-h, h) around zero
• Uniquely determines the probability distribution if it exists
• MGFs of independent random variables multiply: $M_{X+Y}(t) = M_X(t) \cdot M_Y(t)$
• Convolution of distributions simplifies to multiplication of their MGFs
• MGFs are always infinitely differentiable at t = 0

Relationship to moments

• kth derivative of MGF at t = 0 yields the kth raw moment: $M_X^{(k)}(0) = E[X^k]$
• Taylor series expansion of MGF around t = 0 expresses it in terms of moments: $M_X(t) = 1 + E[X]t + \frac{E[X^2]}{2!}t^2 + \frac{E[X^3]}{3!}t^3 + ...$
• Logarithm of MGF generates cumulants, which relate to central moments
• MGFs facilitate moment calculations for transformed random variables
• Provide a method for deriving moments of complex distributions through simpler MGF manipulations

Skewness

• Skewness quantifies the asymmetry of a probability distribution in theoretical statistics
• This third standardized moment plays a crucial role in understanding the shape and tail behavior of distributions
• Skewness analysis helps identify departures from normality and informs statistical modeling choices

Third standardized moment

• Defined as $\gamma_1 = E[\left(\frac{X-\mu}{\sigma}\right)^3] = \frac{\mu_3}{\sigma^3}$
• μ₃ represents the third central moment, σ the standard deviation
• Measures the degree and direction of asymmetry in a distribution
• Symmetric distributions (normal, uniform) have zero skewness
• Pearson's moment coefficient of skewness provides an alternative formulation

Interpretation and examples

• Positive skewness indicates a longer right tail (right-skewed)
  • Financial returns often exhibit positive skewness
  • Log-normal distribution has positive skewness
• Negative skewness suggests a longer left tail (left-skewed)
  • Beta distribution can exhibit negative skewness
  • Exam scores in a highly prepared class may show negative skewness
• Magnitude of skewness indicates the degree of asymmetry
• Skewness impacts risk assessment in finance and actuarial science
• Affects the choice of statistical tests and models in data analysis

Kurtosis

• Kurtosis measures the tailedness and peakedness of a probability distribution in theoretical statistics
• This fourth standardized moment provides insights into extreme values and outlier behavior
• Understanding kurtosis helps in assessing the appropriateness of statistical models and risk analysis

Fourth standardized moment

• Defined as $\gamma_2 = E[\left(\frac{X-\mu}{\sigma}\right)^4] = \frac{\mu_4}{\sigma^4}$
• μ₄ denotes the fourth central moment, σ the standard deviation
• Quantifies the combined weight of distribution tails relative to the center
• Higher kurtosis indicates heavier tails and more outlier-prone distributions
• Lower kurtosis suggests lighter tails and fewer extreme values

Excess kurtosis

• Excess kurtosis calculated as kurtosis minus 3 (kurtosis of normal distribution)
• Formula: $\gamma_2 - 3 = \frac{\mu_4}{\sigma^4} - 3$
• Provides a reference point for comparison with the normal distribution
• Positive excess kurtosis indicates leptokurtic distribution (heavier tails than normal)
• Negative excess kurtosis suggests platykurtic distribution (lighter tails than normal)

Interpretation and examples

• Normal distribution has kurtosis of 3 (excess kurtosis of 0)
• Student's t-distribution exhibits higher kurtosis than normal
  • Degrees of freedom influence the magnitude of kurtosis
• Uniform distribution has lower kurtosis (platykurtic)
• High kurtosis in financial returns suggests increased risk of extreme events
• Kurtosis impacts the performance of statistical estimators and tests
• Robust statistics often employed for high-kurtosis data to mitigate outlier effects

Applications in statistics

• Higher-order moments find extensive applications in various areas of theoretical statistics
• These moments contribute to parameter estimation, hypothesis testing, and model selection
• Understanding moment-based techniques enhances statistical inference and decision-making processes

Method of moments estimation

• Estimates population parameters by equating sample moments to theoretical moments
• Solves a system of equations to find parameter estimates
• Simple implementation for distributions with closed-form moment expressions
• Consistency of estimators depends on the existence of corresponding population moments
• Often used as initial estimates for more efficient methods (maximum likelihood estimation)
• Moment-based estimators may lack efficiency compared to other methods for some distributions

Hypothesis testing

• Moments used to construct test statistics for various statistical hypotheses
• Jarque-Bera test utilizes sample skewness and kurtosis to assess normality
• Anscombe-Glynn test focuses on kurtosis for detecting non-normality
• D'Agostino's K-squared test combines skewness and kurtosis for normality testing
• Higher-order moments contribute to power studies of statistical tests
• Moment-based tests often provide alternatives to more complex likelihood-based approaches

Moment problems

• Moment problems in theoretical statistics involve determining or characterizing probability distributions from their moments
• These problems play a crucial role in probability theory, functional analysis, and applied mathematics
• Understanding moment problems aids in distribution reconstruction and approximation techniques

Hamburger moment problem

• Addresses the existence and uniqueness of probability measures on the real line given a sequence of moments
• Seeks a measure μ such that $\int_{-\infty}^{\infty} x^n d\mu(x) = m_n$ for all n ≥ 0
• Solvability conditions involve positive definiteness of moment matrices
• Carleman's condition provides a sufficient criterion for uniqueness
• Applications in spectral analysis and quantum mechanics
• Relates to orthogonal polynomial theory and continued fractions

Stieltjes moment problem

• Focuses on probability measures supported on the non-negative real line [0, ∞)
• Aims to find a measure μ satisfying $\int_{0}^{\infty} x^n d\mu(x) = m_n$ for all n ≥ 0
• More restrictive than the Hamburger problem due to support constraint
• Solvability linked to complete monotonicity of moment sequence
• Hausdorff moment problem as a special case on [0, 1]
• Applications in analysis of positive definite functions and completely monotone functions

Moment inequalities

• Moment inequalities provide powerful tools for bounding expectations and probabilities in theoretical statistics
• These inequalities establish relationships between different moments or functions of random variables
• Understanding moment inequalities enhances statistical inference and helps derive concentration bounds

Jensen's inequality

• States that $f(E[X]) \leq E[f(X)]$ for convex functions f and random variable X
• Equality holds if and only if X is constant or f is linear
• Generalizes to $\varphi(E[X]) \leq E[\varphi(X)]$ for convex φ and $E[\varphi(X)] \leq \varphi(E[X])$ for concave φ
• Fundamental in deriving many other inequalities (Markov's, Chebyshev's)
• Applications in information theory (log-sum inequality)
• Used to prove AM-GM inequality and other mathematical relationships

Hölder's inequality

• Generalizes the Cauchy-Schwarz inequality to Lp spaces
• For p, q > 1 with 1/p + 1/q = 1, states $E[|XY|] \leq (E[|X|^p])^{1/p} (E[|Y|^q])^{1/q}$
• Special case p = q = 2 yields the Cauchy-Schwarz inequality
• Extends to more than two functions and different exponents
• Crucial in functional analysis and theory of Lp spaces
• Applications in proving other inequalities and bounding integrals

Multivariate moments

• Multivariate moments extend the concept of moments to multiple random variables in theoretical statistics
• These moments characterize joint distributions and capture dependencies between variables
• Understanding multivariate moments is crucial for analyzing complex systems and multivariate data

Joint moments

• Defined as expectations of products of powers of random variables
• For random variables X and Y, the (i,j)th joint moment is $E[X^i Y^j]$
• Central joint moments involve deviations from means: $E[(X-\mu_X)^i (Y-\mu_Y)^j]$
• Moment matrices organize joint moments for multivariate analysis
• Higher-order joint moments capture complex dependencies beyond linear relationships
• Applications in portfolio theory and risk management in finance

Covariance vs correlation

• Covariance measures linear dependence between two random variables
  • Defined as $Cov(X,Y) = E[(X-\mu_X)(Y-\mu_Y)]$
  • Equals the (1,1) central joint moment
• Correlation normalizes covariance to range [-1, 1]
  • Pearson correlation coefficient: $\rho = \frac{Cov(X,Y)}{\sigma_X \sigma_Y}$
• Covariance affected by scale, correlation dimensionless
• Zero correlation does not imply independence (except for multivariate normal)
• Higher-order analogues (coskewness, cokurtosis) capture non-linear dependencies
• Spearman's rank correlation based on ranks instead of raw values

Empirical moments

• Empirical moments estimate population moments using observed data in theoretical statistics
• These sample-based calculations provide insights into underlying distributions and inform statistical inference
• Understanding empirical moments is crucial for practical data analysis and model fitting

Sample moments

• Calculated from observed data to estimate population moments
• kth sample moment about the origin: $m_k = \frac{1}{n} \sum_{i=1}^n X_i^k$
• kth sample central moment: $\hat{\mu}_k = \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})^k$
• Sample mean (first moment): $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$
• Sample variance (second central moment): $s^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$
• Higher-order sample moments used to estimate skewness and kurtosis

Bias in moment estimation

• Sample moments may be biased estimators of population moments
• Bias generally decreases with increasing sample size
• Sample variance uses n-1 in denominator to achieve unbiasedness
• Higher-order moment estimates often have more substantial bias
• Corrections (G1 statistic for skewness, G2 for kurtosis) can reduce bias
• Bootstrap methods provide alternative approach to bias correction
• Understanding bias crucial for accurate inference and confidence interval construction

Robust alternatives

• Robust alternatives to traditional moments offer more stable and outlier-resistant measures in theoretical statistics
• These methods provide reliable estimates of distribution characteristics in the presence of extreme values or departures from normality
• Understanding robust alternatives enhances statistical analysis for real-world data with potential anomalies

Quantiles vs moments

• Quantiles (percentiles) provide robust alternatives to moments
• Median (50th percentile) offers robust measure of central tendency
• Interquartile range (IQR) provides robust measure of spread
• Quantile-based skewness: $\frac{(Q_3-Q_2)-(Q_2-Q_1)}{Q_3-Q_1}$
• Bowley's coefficient of skewness uses quartiles for robustness
• Quantile-based measures less sensitive to outliers than moment-based counterparts
• Applicable to distributions where moments may not exist (Cauchy distribution)

L-moments

• Linear combinations of order statistics that characterize probability distributions
• First L-moment (λ₁) equals the mean
• Second L-moment (λ₂) measures scale and dispersion
• L-moment ratios provide measures of shape:
  • L-CV (τ₂ = λ₂/λ₁) for relative dispersion
  • L-skewness (τ₃ = λ₃/λ₂) for asymmetry
  • L-kurtosis (τ₄ = λ₄/λ₂) for tail behavior
• More robust to outliers than conventional moments
• Exist for distributions with infinite variance (unlike standard moments)
• Useful in hydrological analysis and extreme value theory

Moments in probability theory

• Moments play a fundamental role in probability theory within theoretical statistics
• They provide alternative ways to characterize and analyze probability distributions
• Understanding moment-related concepts enhances the toolkit for theoretical and applied statistical analysis

Characteristic functions

• Fourier transform of the probability density function
• Defined as $\phi_X(t) = E[e^{itX}] = \int_{-\infty}^{\infty} e^{itx} f_X(x) dx$
• Always exist for any probability distribution
• Uniquely determine the distribution (unlike moment-generating functions)
• kth derivative at t=0 yields the kth raw moment multiplied by i^k
• Useful for proving limit theorems (Central Limit Theorem)
• Facilitate analysis of sums of independent random variables
• Relate to moment-generating functions through imaginary argument

Cumulants

• Alternative set of distribution descriptors related to moments
• Generated by the cumulant-generating function: $K_X(t) = \log(M_X(t))$
• First cumulant equals the mean, second equals the variance
• Higher-order cumulants relate to central moments but offer some advantages:
  • Additivity for independent random variables
  • Simpler expressions for some distributions
• Skewness expressed as $\gamma_1 = \kappa_3 / \kappa_2^{3/2}$
• Excess kurtosis as $\gamma_2 = \kappa_4 / \kappa_2^2$
• Used in statistical physics and for constructing distribution approximations