 Meanpreserving spread

In probability and statistics, a meanpreserving spread (MPS)^{[1]} is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A's probability density function while leaving the mean (the expected value) unchanged. As such, the concept of meanpreserving spreads provides a stochastic ordering of equalmean gambles (probability distributions) according to their degree of risk; this ordering is partial, meaning that of two equalmean gambles, it is not necessarily true that either is a meanpreserving spread of the other. Definitionally, if B is a meanpreserving spread of A then A is said to be a meanpreserving contraction of B.
Ranking gambles by meanpreserving spreads is a special case of ranking gambles by secondorder stochastic dominance – namely, the special case of equal means: If B is a meanpreserving spread of A, then A is secondorder stochastically dominant over B; and the converse holds if A and B have equal means.
If B is a meanpreserving spread of A, then B has a higher variance than A; but the converse is not in general true, because the variance is a complete ordering while ordering by meanpreserving spreads is only partial.
Contents
Example
This example from ^{[2]} shows that to have a meanpreserving spread does not require that all or most of the probability mass move away from the mean. Let A have equal probabilities 1 / 100 on each outcome x_{Ai} , with x_{Ai} = 198 for and x_{Ai} = 202 for ; and let B have equal probabilities 1 / 100 on each outcome x_{Bi}, with x_{B1} = 100, x_{Bi} = 200 for , and x_{B100} = 300. Here B has been constructed from A by moving one chunk of 1% probability from 198 to 100 and moving 49 probability chunks from 198 to 200, and then moving one probability chunk from 202 to 300 and moving 49 probability chunks from 202 to 200. This sequence of two meanpreserving spreads is itself a meanpreserving spread, despite the fact that 98% of the probability mass has moved to the mean (200).
Mathematical definitions
Let x_{A} and x_{B} be the random variables associated with gambles A and B. Then B is a meanpreserving spread of A if and only if for some random variable z having E(z  x_{A}) = 0 for all values of x_{A}. Here means "is equal in distribution to" (that is, "has the same distribution as").
Meanpreserving spreads can also be defined in terms of the cumulative distribution functions F_{A} and F_{B} of A and B. If A and B have equal means, B is a meanpreserving spread of A if and only if the area under F_{A} from minus infinity to x is less than or equal to that under F_{B} from minus infinity to x for all real numbers x, with strict inequality at some x.
Both of these mathematical definitions replicate those of secondorder stochastic dominance for the case of equal means.
Relation to expected utility theory
If B is a meanpreserving spread of A then A will be preferred by all expected utility maximizers having concave utility. The converse also holds: if A and B have equal means and A is preferred by all expected utility maximizers having concave utility, then B is a meanpreserving spread of A.
See also
References
 ^ Rothschild, Michael, and Stiglitz, Joseph, "Increasing risk I: A definition," Journal of Economic Theory, 1970, 225–243.
 ^ Landsberger, M., and Meilijson, I., "Meanpreserving portfolio dominance," Review of Economic Studies 60, April 1993, 479–485.
Categories: Theory of probability distributions
 Decision theory
Wikimedia Foundation. 2010.
Look at other dictionaries:
Statistical dispersion — In statistics, statistical dispersion (also called statistical variability or variation) is variability or spread in a variable or a probability distribution. Common examples of measures of statistical dispersion are the variance, standard… … Wikipedia
MPC — or Mpc or mpc stands for: Contents 1 Astronomy 2 Computers and electronics 3 Mathematics and statistics … Wikipedia
Credit rationing — refers to the situation where lenders limit the supply of additional credit to borrowers who demand funds, even if the latter are willing to pay higher interest rates. It is an example of market imperfection, or market failure, as the price… … Wikipedia
MPS — may refer to: Robinson List, aka Mail Preference Service, direct mail opt out system Malmin Palloseura, association football club from Helsinki, Finland. Marginal propensity to save Master Production Schedule Ministry of Public Security of the… … Wikipedia
Michael Rothschild — For other uses, see Michael Rothschild (disambiguation). Michael Rothschild Born August 2, 1942 (1942 08 02) (age 69) Chicago, Illinois Nationality United States Institution … Wikipedia
Risk aversion — is a concept in psychology, economics, and finance, based on the behavior of humans (especially consumers and investors) while exposed to uncertainty. Risk aversion is the reluctance of a person to accept a bargain with an uncertain payoff rather … Wikipedia
Europe, history of — Introduction history of European peoples and cultures from prehistoric times to the present. Europe is a more ambiguous term than most geographic expressions. Its etymology is doubtful, as is the physical extent of the area it designates.… … Universalium
biblical literature — Introduction four bodies of written works: the Old Testament writings according to the Hebrew canon; intertestamental works, including the Old Testament Apocrypha; the New Testament writings; and the New Testament Apocrypha. The Old… … Universalium
china — /chuy neuh/, n. 1. a translucent ceramic material, biscuit fired at a high temperature, its glaze fired at a low temperature. 2. any porcelain ware. 3. plates, cups, saucers, etc., collectively. 4. figurines made of porcelain or ceramic material … Universalium
China — /chuy neuh/, n. 1. People s Republic of, a country in E Asia. 1,221,591,778; 3,691,502 sq. mi. (9,560,990 sq. km). Cap.: Beijing. 2. Republic of. Also called Nationalist China. a republic consisting mainly of the island of Taiwan off the SE coast … Universalium