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This article explores the concept of extremes in statistical analysis, with a focus on economic forecasts by executives after 9/11 and CEO compensation. It discusses distribution theory, thresholds for extremes, and the dependence of extremes. Key lessons and techniques for identifying extremes are also covered.
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Identification and Analysis of Extremes with Two Examples:Executives’ Economic Forecasts after 9/11; CEO Compensation Ehsan S. Soofi, Paul C. Nystrom Sheldon B. Lubar School of Business University of Wisconsin-Milwaukee & Masoud Yasai-Ardekani School of Management George Mason University
Preamble: Extreme? • What is meant by extreme? • Semantically, extremes refer to cases farthest from the center. • Technically, extremes are plausible outcomes of the probability distributions of the farthest statistics from the sample center.
Preamble: Concepts of extreme & rare • Extremes(main focus of our study) • Farthest from the center of data • Rare events • Far from the mass of population (Heavy tail) • Relate to our study if have extreme impacts • Dot plots of simulated samples (n=100) Extreme Rare Extreme classification error rate, a • Horizontal scales are different
Preamble: Variations of sample central values and extremes • Example: Uniform distribution P(an X in an interval) = Area under the density Sample: n=15 Extreme Min Extreme Min Random order X1,X2, …X15 Random order X1,X2, …X15 Order statistics Y1<Y2< …<Y8<…<Y15 Order statistics Y1<Y2< …<Y8<…<Y15 Center Mean Center Mean Center Median Center Median Extreme Max Extreme Max P=.2 P=.2 P=.2 Mean .55 Mean .53
Outline • Why study extremes? • Distribution theory • Thresholds for extremes • Example: survey of executives after 9/11 • Bayesian inference, small number of observations • Example: Components of CEO compensation • Thresholds for low and high percentiles • Deciles, quartiles, thirds • Example: Edmondson et al (2001)
Learning • What has been learned? • From the best ones • Firm performance (Galunic & Eisenhardt 1996) • Product development cycle (Clark & Fujimoto 1991) • Inventory control (Wal-Mart) • Rapid organization crises and turnarounds (Hedberg, Nystrom & Starbuck 1976, Nystrom & Starbuck 1984) • From the worst ones • Decision errors (Starbuck & Milliken 1988) • What more can be learned? • Comparing relationship between a variable with • Extremes and non-extremes on another variable • Compare relationship between a pair of variables • For extremes of the two variables • Overall relationship
Order statistics • Standard assumption for statistical analysis of a sample of measurements x1 , x2 , …, xn • Are observations generated on random variables X1 ,…,Xn • Independent and have identical probability distribution FX(x) = Pr(Xi< x), for all i=1,…,n • Density functionfX(x)=derivative ofFX(x) • The order statistics of the sample • The sampled measurements arrayed in ascending order y1< y2< … < yn • Are observations from the set of random variables Y1< Y2< … < Yn
Distributions of order statistics • The probability distribution of Yi Gi(yi)= Pr(Yi< yi) • The probability density function is • A function of n, the order index i, and FX • FX is referred to as the parent distribution • gi(yi)are notidentical • Y1< Y2< … < Yn are not independent
Distributions of extremes • Minimum • Distribution function G1(y1) = Pr(Y1< y1)=1-[1- FX(y1)]n • Density function g1(y1) = [1- FX(y1)]n-1 fX(y1) • Maximum • Distribution function Gn(yn) = Pr(Yn< yn)=[FX(yn)]n • Density function gn(yn) = [FX(yn)]n-1 fX(yn)
Density functions of extremes of normal distribution • Normal (Mean=0, Standard deviation=1) • Parent is symmetric, extremes are mirror image • Three distributions are more separated for the larger n
Density functions of extremes of Weibull distribution • Weibull (Shape=2, Scale=1) • Parent is not symmetric, extremes are not mirror image • Three distributions are more separated for the larger n
Extreme value distribution • As the sample size increases, distribution of maximum, suitably standardized, approaches an extreme value distribution having a parameter, referred to as the extreme value index (EVI) • Three domains of attraction (set of distributions) • EVI>0 • EVI<0 • EVI=0 Includes Weibull
Discrepancy between distributions • As n increases, the distributions of extremes separate more from • the parent distribution • the median at a faster rate • each other at a much faster rate • The discrepancies do not depend on the parent distribution Extreme-Parent Extreme-median Extreme-Extreme
Dependence of extremes • As the sample size increases, dependence between • each extreme and its neighbor increases • the two extremes decreases • The dependence does not depend on the parent distribution
Thresholds for extremes • Definition: A measurement xi is an extreme if we can infer with high probability that it is a sample from the distribution of an extreme • Inference with probability .95 • 95% threshold for minimum is the solution to Pr(Y1< ymin)= G1(ymin) = 1-[1- FX(ymin)]n=.95 • 95% threshold for maximum is the solution to Pr(Yn > ymax)=1- Pr(Yn< ymax)=1- Gn(ymax) = 1-[FX(yn)]n=.95
Thresholds for extremes of normal distribution • Normal (Mean=0, SD=1), n= 93 • Pr(Y1< -1.89) = .95 • Threshold for minimum is -1.89, any observation less than 1.89 is an extreme • Pr(Yn > 1.89) = .95 • Threshold for maximum is 1.89, any observation greater than 1.89 is an extreme
Thresholds for extremes of Weibull distribution • Weibull (Shape=0, Scale=1), n= 100 • Pr(Y1< .17) = .95 • Threshold for minimum is .17, any observation less than .17 is an extreme • Pr(Yn > 1.88) = .95 • Threshold for maximum is 1.86, any observation greater than 1.86 is an extreme
Remarks on thresholds • Monotone transformation of variables • Threshold for log-normal is log of thresholds for normal • Threshold for Weibull is power of threshold for exponential • Thresholds for all distributions can be obtained by thresholds for uniform data • Probability integral transformation • Thresholds are functions of the parameters of the parent distribution FX(x;q) • In data analysis, the parameters must be estimated • Maximum likelihood method • Bayesian method
Variables chosen for studying extremes • Executives’ economic forecast accuracy • Executives’ perceived environmental uncertainty
Forecast accuracy • A paragraph described the U.S. economy's GDP (Gross Domestic Product) and its rates of change in recent years and quarters. • Likelihoods for three outcomes of the future economy (GDP growth by end of September 2002): • Full recovery (GDP>2%) • Modest recovery (0% < GDP<2%) • Further decline (GDP<0%) • Best estimates of GDP growth choosing from • GDP > 2% • 1% < GDP < 2% • 0% < GDP < 1% • -1% < GDP< 0% • -2% < GDP < -1% • GDP < -2%
Forecast distribution • Constructed piece-wise uniform distribution for the economic forecast (W) of each executive • Actual GDP growth by the end of September 2002 was 3.4% • For 87 of 93 respondents, the best estimate was assigned the highest probability (modal category) Most accurate Least accurate More accurate Less accurate
Forecast accuracy • The mean squared error (MSE) of each forecast distribution MSE(Wi)= E[(Wi - 3.4)2] =Variance(Wi) + [mean(Wi)– 3.4]2 • Quadratic loss function • Measures forecast inaccuracy • Our results are robust against variants of distribution for W and the loss function
Environmental uncertainty Probability P(S) • Type of uncertainty (Milliken 1987) • State (S): Outcome of the future economy • Effect (E): Impacts of each Son the organization • Highly positive • Positive • Neutral • Negative • Highly negative • Multiplicative rule of probability • Concept: Uniformity of the joint distribution (Argote 1982) • Measure: Shannon entropy (Leblebici and Salancik 1981) Conditional probability P(E | S) Bivariate distribution P(S & E) = P(E|S)P(S)
Distributions of the variables across the sample • Forecast inaccuracy • Weibull (shape=3.93, scale 10.14, location=-2.47) • Environmental uncertainty: Normal (mean 1.43, SD=0.54) • Probability plots and fit statistic (Anderson-Darling) AD=0.493 P=0.157 AD=0.466 P=0.247
Thresholds for extremes • Example: n= 93 (9/11 sample size) • Forecast inaccuracy: Weibull (shape=3.93, scale 10.14, location=-2.47) • Extreme high
Thresholds for extremes • n= 93 (9/11 sample size) • Maximum likelihood method
Forecast accuracy and strategic type • Strategic types (Miles and Snow 1978) • Prospector • Analyzer • Defender • Reactor • Each executive in the 9/11 study read four paragraphs that characterize four different strategic approaches • None of the paragraphs bore the type label • Each executive selected the paragraph that best described his/her organization
Hypotheses and data • Two hypotheses of interest: • H1: The probability of extreme accurate forecasting is higher for Prospector/Analyzer than for Defender/Reactor. • H2: The probability of extreme inaccurate forecasting is lower for Prospector/Analyzer than for Defender/Reactor. • Data: Cross-tabulation
Forecast accuracy: Bayesian analysis • Prior and posterior distributions for probability of being extremely accurate and extremely inaccurate (for each strategy type separately) • Prior: Uniform distribution (Bayes-Laplace) • Posterior Beta distributions for prospector • Posterior Beta distributions for reactors
Forecast accuracy: Bayesian analysis • Difference between the posterior distributions for the two strategy types • No analytical solution • Simulation results H1 holds H2 holds H2 Doesn’t hold H1 Doesn’t hold
Forecast accuracy: Bayesian analysis • Prior and posterior distributions for probability of being extremely accurate and extremely inaccurate (for each strategy type separately) • Prior: A highly skewed Beta distribution • Posterior Beta distributions for prospector/analyzer • Posterior Beta distributions for reactors/defender
Forecast accuracy: Bayesian analysis • Difference between the posterior distributions for the two strategy types • Results are robust H1 Doesn’t hold H2 Doesn’t hold
Forecast accuracy: Bayesian analysis • Prior probability of hypotheses: P(H true)=P(H false) • Posterior probability and posterior odds of hypothesis
Forecast accuracy & Environmentaluncertainty • Traditional: correlation and a regression analysis • Line of averages • Below on both variables • Above on both variables • Relationship between extremes on the two variables? Regression assumptions hold Thresholds for low Thresholds for high Positive association, mostly purple Regression to the mean
Bayesian analysis Not worth more than a bare mention
Data • The amounts of total compensation received by CEOs of America’s largest companies is controversial discussions • Data on 50 of Forbe’s 1991 US large firms (Frees 1996) • Two variables • Salary and bonuses • Additional compensations (mainly stock options exercised)
Distributions of variables • Dot plots of data • Models for the variables • Log-normal fits salary and bonus (AD test =.53, P=.16) • Weibull fits additional comp (AD test = .61, P=.12) Heavy tail
Relationship between extremes of two variables • Do CEOs with extreme salary and bonus get extreme additional compensation? • The traditional analysis • Positive association between salary and bonus and additional compensation • Correlation and a regression analysis • Regression assumptions do not hold • Transform data; assumptions hold • Line of averages • Both above average • Both below average • One above, one below • Positive Association Mostly blue kind 16 8 22 8
Relationship between extremes of two variables • Bonferroni type adjustment for two variables • 95% thresholds for two variable • 97.5% threshold for each variable • Positive association between extremes on the two variables? 3 0 • Extreme on both • Extreme on only one • Neither • Regression line 42 5
Bayesian analysis • Inference for probability of extreme on only one or both variables • Data: Cross-tabulation • Prior: Uniform distribution (or skewed Beta) • Posterior Beta distribution for extreme on both variables • Posterior Beta distribution for extreme on only one variable • Posterior odds in favor of being extreme in only one variable: 832to 1 Robust against the choice of prior
Generalization: Thresholds for Lower & Upper • Upper and lower observations in sample (e.g., Edmonton et al 2001) • n= 93 (9/11 sample size) • Raw count creates ties • Use distribution of respective order statistics Thresholds avoid tie
An example from the literature • Edmondson et al (2001) • Comparison of low and high performing hospitals (n = 16) • “The middle two scores are too close” • Normal model fits (Mean=21.5, SD=10.1, AD Test=.38, P-value=.37) Edmondson et al’s High Edmondson et al’s Low