Unit 5: Relation Between Moments - Subjective Questions

The th raw moment (moment about the origin) of a random variable is

Thus,

The th central moment (moment about the mean) is

where . The first four central moments are:

• , the variance
• , which measures asymmetry
• , which is used to measure kurtosis.

Since is the deviation from the mean,

Using the binomial expansion,

where .

The required relations are:




These formulas convert moments measured about the origin into moments measured about the mean.

Write and expand using the binomial theorem. Since , we obtain:




Here, is the arithmetic mean and denotes the th central moment. These are the inverse relations to those expressing central moments in terms of raw moments.

Let the th moments about and be

Since

raising both sides to the th power and taking expectations gives

where .

For example,

If is the mean, then and , so the relation connects moments about an arbitrary origin directly with central moments.

For

the th raw moment of is

In particular,

Since

the central moments satisfy

Therefore:

• A change of origin does not affect central moments.
• A change of scale divides the th central moment by .
• If , odd central moments change sign, while even central moments remain nonnegative.

The Pearsonian moment coefficients are

For , the transformed central moments are

Hence,

and

Thus, and are dimensionless and remain unchanged by origin and scale transformations. The signed coefficient changes sign if .

Given

The central moments are




Therefore,


The excess kurtosis is

Thus, the distribution is positively skewed and platykurtic.

The transformed mean is

Under the transformation ,

Thus,



The coefficient of skewness is

The coefficient of kurtosis is

Therefore, is slightly negatively skewed and mesokurtic.

The Pearsonian moment coefficients are:

• Squared coefficient of skewness:
  
• Signed coefficient of skewness:
  
• Coefficient of kurtosis:
  
• Excess kurtosis:
  

Interpretation:

• indicates zero moment skewness.
• indicates positive skewness.
• indicates negative skewness.
• indicates mesokurtosis.
• indicates leptokurtosis.
• indicates platykurtosis.

The signed skewness coefficient is

Therefore,

The coefficient of kurtosis is

Hence, the excess kurtosis is

Interpretation:

• Since , the distribution is slightly negatively skewed.
• Since and , the distribution is leptokurtic.

Skewness measures the degree and direction of asymmetry of a distribution about its central value.

• Symmetric distribution: The two sides are mirror images. Generally, and .
• Positively skewed distribution: The right tail is longer. Usually,
  
  and .
• Negatively skewed distribution: The left tail is longer. Usually,
  
  and .

These mean-median-mode orders are common empirical patterns, not universal mathematical identities.

The principal measures are:

• Karl Pearson's coefficient:
  
  or, when the mode is ill-defined,
  
  It uses the mean and standard deviation and is affected by extreme values.

• Bowley's coefficient:
  
  It is based on quartiles and is suitable for open-ended distributions.

• Kelly's coefficient:
  
  It uses a larger portion of the data than Bowley's measure.

• Moment coefficient:
  
  It uses all observations, has a firm algebraic basis, and gives both the direction and magnitude of skewness.

Using the mode-based formula,

Using the median-based formula,

Both coefficients are positive, so both indicate a positively skewed distribution. Their magnitudes differ because the empirical relation among mean, median, and mode is only approximate. If the mode is clearly defined, the first coefficient may be used; otherwise, the median-based coefficient is generally preferred.

Bowley's coefficient is

Substituting the values,

Kelly's coefficient is

where . Therefore,

Both coefficients are positive and indicate slight positive skewness.

The third central moment is

Because cubing preserves the sign of deviations:

• Observations above the mean contribute positive terms.
• Observations below the mean contribute negative terms.
• A long or heavy right tail generally gives .
• A long or heavy left tail generally gives .

Since depends on the unit of measurement, it is standardized as

Every symmetric distribution with a finite third moment has . However, the converse is not necessarily true: an asymmetric distribution can have positive and negative cubic deviations that cancel. Thus, indicates zero moment skewness but does not by itself prove symmetry.

Kurtosis is measured using the standardized fourth central moment

The excess kurtosis is

because a normal distribution has .

• Mesokurtic: or ; the normal distribution is the standard example.
• Leptokurtic: or ; the distribution has greater tail weight and more extreme observations relative to its variance.
• Platykurtic: or ; the distribution has lower tail weight relative to a normal distribution.

Kurtosis is best interpreted as a measure of tail weight or extremity rather than merely the sharpness of the peak.

Given

the coefficient of kurtosis is

The excess kurtosis is

Since , or equivalently , the distribution is leptokurtic.

For ,

Therefore, its central moments satisfy

Hence,

If , this reduces to

Also,


Thus, translation and positive rescaling do not affect standardized skewness or kurtosis. If , the distribution is reflected, so , while , , and remain unchanged.

Let

so that , , , and .

For every real number ,

Expanding,

Using the standardized moments,

The left side is minimized when . Therefore,

Since ,

This inequality gives a necessary consistency condition between the skewness and kurtosis of any distribution having finite fourth moment.

For , the raw moments are

Its central moments are



Since ,




The moment coefficients are unchanged by positive scaling:



Thus, the distribution is positively skewed and mesokurtic.