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Michael Lugo
Michael Lugo
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What m minimizes E(|m-EX|^3X|^3) for a random variable X?

Let X be a random variable. Then E(|m-EX|^1X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-EX|^2X|^2) is minimized when m is the mean of x.

A couple weeks ago in a technical stretch of a proof involving the Lyapunov condition for the central limit theorem I ended up with the expression E(|m-EX|^3X|^3). Does this statistic have a name, or any nice properties?

Edit: Earlier versions of this question had |m-EX| where |m-X| was; this isn't what I meant.

What m minimizes E(|m-EX|^3) for a random variable X?

Let X be a random variable. Then E(|m-EX|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-EX|^2) is minimized when m is the mean of x.

A couple weeks ago in a technical stretch of a proof involving the Lyapunov condition for the central limit theorem I ended up with the expression E(|m-EX|^3). Does this statistic have a name, or any nice properties?

What m minimizes E(|m-X|^3) for a random variable X?

Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x.

A couple weeks ago in a technical stretch of a proof involving the Lyapunov condition for the central limit theorem I ended up with the expression E(|m-X|^3). Does this statistic have a name, or any nice properties?

Edit: Earlier versions of this question had |m-EX| where |m-X| was; this isn't what I meant.

Source Link
Michael Lugo
Michael Lugo
  • 14k
  • 7
  • 58
  • 80

What m minimizes E(|m-EX|^3) for a random variable X?

Let X be a random variable. Then E(|m-EX|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-EX|^2) is minimized when m is the mean of x.

A couple weeks ago in a technical stretch of a proof involving the Lyapunov condition for the central limit theorem I ended up with the expression E(|m-EX|^3). Does this statistic have a name, or any nice properties?