An Outlier Detection Methodology with Consideration for an Inefficient Frontier

1
An Outlier Detection Methodology with
Consideration for an Inefficient Frontier

• By
• Andy Johnson
• Leon McGinnis

2
Outline

• Background and motivation
• Current two-stage DEA method
• Proposed changes to two-stage DEA
• Definitions Relative to an Inefficient Frontier
• Leave-one-out method / threshold value
• Iterative Outlier Detection
• Second Stage Bootstrap
• Summary

3
Background

• Outlier detection in a non-parametric framework
  is important because many of these methods do not
  consider measurement error or random fluctuations
  when constructing a frontier
• Thus allowing over stated data to be included in
  the reference set can bias not only one
  efficiency estimate, but several efficiency
  estimates if the over stated observation is used
  to construct the frontier

4
Motivation

• The iDEAs project for warehouse performance
  benchmarking
• On-line data collection requires more scrutiny
  than data collected and analyzed by a single
  person
• What could be the/a cause of a negatively skewed
  efficiency distribution

5
Motivation

• To investigate the impact of environmental
  characteristics on efficiency the two-stage DEA
  method has been developed
• A data set is identified as using a similar
  technology
• In the first stage efficiency estimates are
  calculated
• In the second stage the estimates are regressed
  against environmental characteristics
• In this setting over stated observations are a
  problem in the first stage, but both overstated
  and understated observations are a problem in the
  second stage

6
Current Two-stage DEA Method when Outlier
Detection is Considered

• Consider the problem of understanding sources of
  inefficiency
• Identify outliers relative to the efficient
  frontier and remove them
• Calculate efficiency estimates
• Regress efficiency estimates against
  environmental variables

7
An Outlier Detection Methodology with
Consideration for an Inefficient Frontier
Output
input
8
An Outlier Detection Methodology with
Consideration for an Inefficient Frontier

• A proposed improvement on the current method
  requires an outlier methodology
• First, identify outliers relative to both the
  efficient and inefficient frontier
• Use a two-stage DEA method where DEA estimates
  are calculated in the first stage and regressed
  against environmental variables in the second
  stage
• Use bootstrapping in the second stage to deal
  with the problem of correlation among the error
  terms

9
An Outlier Detection Methodology with
Consideration for an Inefficient Frontier
output
input
10
Definitions Relative to an Inefficient Frontier

• The production possibility set when the
  inefficient frontier is included
• Shephards input inefficient distance function
  can be defined as
• Shephards output inefficient distance function
  can be defined as

11
Inefficiency Measures Relative to an Inefficient
Frontier

• The Single-Output Inefficient Production
  Frontiers and the Measure of Technical Efficiency
• An input oriented inefficiency measure relative
  to an inefficient frontier as
• An output-oriented measure of inefficiency
  relative to an inefficient frontier is given by
  the function

12
Definition of an Inefficient Frontier

• The Multiple-Output Inefficient Production
  Frontiers and the Measure of Technical Efficiency
  
• The inefficient frontier with respect to the
  subset X(y) can be found by

13
Linear Program for Calculating Inefficiency

• The Multiple-Output Inefficient Production
  Frontiers and the Measure of Technical Efficiency
  
• The inefficiency estimate calculated from the
  input perspective can be found by solving the
  following linear program

14
Definition of an Inefficient Frontier

• The Multiple-Output Inefficient Production
  Frontiers and the Measure of Technical Efficiency
  
• The inefficient frontier with respect to the
  subset Y(x) can be found by

15
Linear Program for Calculating Inefficiency

• The Multiple-Output Inefficient Production
  Frontiers and the Measure of Technical Efficiency
  
• The inefficiency estimate calculated from the
  output perspective can be found by solving the
  following linear program

16
Outlier Detection

• As suggested by Simar 2003 an outlier needs to be
  identified by both an input and an output
  oriented detection method

17
Leave-One-Out DEA Linear Program

• The leave-one-out DEA inefficiency estimate is

18
Threshold Value for Identifying Outliers
Input oriented Threshold Value
the worst observation or convex combination of
bad observations in the reference set excluding
the observation under evaluation can produce the
same level of output as the given observation
using half the inputs
Output oriented Threshold Value
the worst observation or convex combination of
bad observations in the reference set excluding
the observation under evaluation can use the same
level of input as the given observation and
produce twice the output
The reciprocal of the input threshold
19
Iterative Outlier Detection

• Identify outliers based on agreement between both
  the input and output orientation detection method
• Remove identified outliers
• Rerun outlier detection method
• Continue for the number of outlier detected is
  below some limit or for a set number of iterations

20
Bootstrapping method for the second stage of the
two-stage DEA method

• Necessary because of the correlation among error
  terms in the second stage regression
• Sample n observations, call this set b, with
  replacement from the set of input/output data
• Calculate efficiency estimates for each of the
  original n observations relative to the set b
• Repeat these two steps 2000 times
• Construct confidence intervals and bias estimates
  for each of the original n efficiency estimates

21
Second Stage Bootstrap

• Bootstrapping method for the second stage of the
  two-stage DEA method
• Because of bias present, the corrected efficiency
  estimate
• Confidence interval estimates

22
Banker and Morey Data

• When outliers are only determined based on the
  efficient frontier, the finding is population has
  no impact on efficiency
• When outliers are determined based on an
  efficient and an inefficient frontier, the
  finding is population is negative correlated with
  efficiency at the 95 confidence level

23
Summary

• By not considering an inefficient frontier the
  second stage results of the two-stage method are
  biased. This bias is not corrected for by the
  bootstrapping method.
• Developed a more comprehensive outlier detection
  methodology for non-parametric efficiency methods

24
Thank You