The shape persistence of a crop yield probability density function (PDF) was studied by using two variants of the Visual Impact Method (VIM) to elicit subjective estimations by farmers. In one variant ten weights were used to describe the PDF and in the other variant the farmer chose the number of weights. Results were compared directly and by means of Weibull distributions fitting, with evidence being obtained in favor of methodological persistence and the equivalence of the two estimation methods.Subjective crop yield PDF elicitation, Visual impact method, Methodological persistence, Crop Production/Industries,

Clop-Gallart, M. Merce

Juarez-Rubio, Francisco

English

Research Papers in Economics

  1 
12
th Congress of the European Association of Agricultural Economists – EAAE 2008 
Shape Persistence in Elicited Subjective Crop Yield Probability Density Functions 
Clop-Gallart M.M.
1 and Juárez-Rubio F.
1 
1 Universitat de Lleida/Department of Business Administration and Economic Management of Natural  
Resources, Lleida, Catalonia, Spain  
 
Abstract—  The  shape  persistence  of  a  crop  yield 
probability density function (PDF) was studied by using 
two variants of the Visual Impact Method (VIM) to elicit 
subjective  estimations  by  farmers.  In  one  variant  ten 
weights were used to describe the PDF and in the other 
variant the farmer chose the number of weights. Results 
were  compared  directly  and  by  means  of  Weibull 
distributions  fitting,  with  evidence  being  obtained  in 
favor of methodological persistence and the equivalence 
of the two estimation methods. 
Keywords—  Subjective  crop  yield  PDF  elicitation, 
Visual impact method, Methodological persistence. 
I. INTRODUCTION  
In recent years, the debate over the shape of crop 
yield  probability  density  functions  (PDFs)  ([1],  [2], 
[3], [4], [5], [6], [7], [8], [9]), as well as the economic 
impact  of  crop  production  distributions  used  in  risk 
management  models,  has  moved  forward.  A 
significant contribution, and a summary of the state of 
the  art,  can  be  found  in  [10].  Nevertheless,  the 
objections raised by [11] cannot be considered to have 
been  resolved.  The  necessary  advances  in  the 
subjective estimation of variables in the area of risk 
programming have been defended by authors such as 
[12],  [13],  [14]  or  [15].  From  a  DSS  development 
perspective,  it  is  important  to  continue  moving 
forward  in  determining  the  best  PDF  estimation 
techniques for farmers. 
Previous  works  by  the  authors  ([16],  [17])  have 
shown  the  methodological  and  time  persistence  of 
subjective point parameter estimations (mean, median, 
mode, etc.) when farmers estimate crop yield PDFs. 
The  question  examined  in  this  work  is  whether  the 
functional  form  elicited  from  farmers  is  also 
maintained when variants are introduced to the method 
used to express the form (methodological persistence). 
Concretely,  the  farmers’  crop  yield  PDF  estimates 
were  examined  using  two  variants  of  the  Visual 
Impact  Method  (VIM)  with  data  from  rainfed  and 
irrigated crops. In one case ten weights were used to 
describe  the  PDF  and  in  the  other  case  the  farmer 
could freely select the number of weights. The results 
were  compared  to  determine  if  the  two  variants 
offered similar results.  
II. MATERIALS AND METHODS  
In the year 2000, a group of 44 farmers expressed 
their opinion on the form of a crop yield PDF with 
which  they  were  familiar  (34  responses  for  rainfed 
crops and 15 for irrigated crops). Two variants of the 
VIM ([18], [19]) were used to this end. 
Each farmer stated range (between maximum and 
minimum)  yield  values  for  the  crops  they  were 
familiar  with.  The  interviewers  (university  students) 
divided the range into five equal intervals and invited 
the  farmers  to  assign  a  frequency  to  each  of  the 
intervals. This assignment was done with the help of 
(a) ten weights that the farmer distributed amongst the 
five  intervals,  and  (b)  a  number  of  weights  chosen 
freely by the farmer. In the latter case all the farmers 
assigned an equal or greater number of weights than 
the original ten units. 
Histograms  estimated  using  the  ten-weight  VIM 
were  compared  with  histograms  estimated  using  a 
number  other  than  ten,  to  determine  the 
methodological  persistence  of  the  estimation.  This 
comparison was done in three ways:  
• Comparing  graphic  representations  of  the 
skewness and kurtosis moment-ratio diagrams of both 
distributions.  
• Comparing the PDFs obtained for both modalities 
of the VIM, using the Kolmogorov-Smirnov test.   2 
12
th Congress of the European Association of Agricultural Economists – EAAE 2008 
• Fitting to Weibull distributions and comparing the 
parameters  (shape  and  scale)  estimated  with  each 
method
1. 
III. RESULTS  
The  results  of  the  methodological  persistence 
evaluation  of  the  VIM,  for  the  density  functions 
estimated using ten weights or a distinct number of 
weights, are described here below. 
A. Skewness-kurtosis moment-ratio diagrams 
The  estimations  obtained  using  the  ten-weight  or 
ten-plus-weight  VIM  are  available.  Graphic 
representation  makes  it  possible  to  obtain  moment-
ratio  diagrams  in  which  skewness  and  kurtosis  are 
shown. The results can be seen in Figures 1 and 2 (for 
the rainfed crops) and 3 and 4 (for the irrigated crops). 
In the case of rainfed crops, farmers elicited largely 
asymmetric  functions,  with  positive  and  negative 
skewness in similar proportions for both cases. With 
respect  to  kurtosis,  a  greater  number  of  cases  with 
negative  values  are  seen,  especially  with  the  ten-
weight VIM. 
 
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Fig. 1 Skewness-Kurtosis diagram (ten-weight VIM, rainfed 
crops) 
In the case of irrigated crops, a greater quantity of 
cases with negative skewness are seen, both with the 
ten-weight and the ten-plus-weight VIM. With respect 
to kurtosis, negative values also predominate. 
                                                           
1 To avoid the bias mentioned by [20] and [21], if d1 and d2 are the values to be 
compared, relative differences are expressed as: (d1 – d2) / [(d1 + d2) / 2] 
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0
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Fig. 2 Skewness-Kurtosis diagram (ten-plus-weight VIM, 
rainfed crops) 
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-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6
Skewness
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Fig. 3 Skewness-Kurtosis diagram (ten-weight VIM, 
irrigated crops) 
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0
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-0,8 -0,6 -0,4 -0,2 0 0,2 0,4
Skewness
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Fig. 4 Skewness-Kurtosis diagram (ten-plus-weight VIM, 
irrigated crops)   3 
12
th Congress of the European Association of Agricultural Economists – EAAE 2008 
Which is to say, farmers tend to estimate platykurtic 
distributions for both for rainfed (ten-weight VIM) and 
irrigated crops using both variants of the VIM. 
B. Comparison of probability density functions 
The ten-weight and ten-plus-weight VIM estimated 
histograms  were  compared  for  each  farmer.  The 
proximity of the two distributions was estimated using 
the Kolmogorov-Smirnov test (0.05 significance). 
The results obtained indicate that the hypothesis of 
similarity  between  the  two  functions  cannot  be 
rejected  for  any  of  the  rainfed  and  irrigated  cases 
studied (for n=10, Massey 0.410), reinforcing the idea 
of persistence between the distinct estimations elicited 
from farmers. In the case of n=number of weights used 
by the farmer in the ten-plus-weight VIM, the result 
indicates  that  the  hypothesis  of  similarity  between 
functions cannot be rejected in 31 of 32 (97%) cases 
of rainfed crops and in all irrigated crops. 
C. Fitting to the Weibull distribution 
Weibull  distributions  were  fitted,  and  were 
evaluated using the Kolmogorov-Smirnov test (0.05), 
with the result that, for rainfed crops, the hypothesis of 
similarity between the two functions studied could not 
be rejected, in all the ten-weight cases and 29 of the 32 
ten-plus-weight cases. 
In  irrigated  crops,  it  is  not  possible  to  reject  the 
similarity hypothesis of the functions studied in all the 
ten-weight cases and in 10 of the 14 ten-plus-weight 
cases. 
The  shape  and  scale  parameters  for  the  Weibull 
distribution and the various PDFs elicited (using the 
ten-weight  and  ten-plus  weight  VIM  variations)  are 
available.  Table  1  shows  the  relative  differences 
obtained for these parameters. 
 
Table 1 Relative differences between shape and scale 
parameters of the Weibull distribution (%) 
Sh: Shape, Sc: Scale 
 
Mean  SD  Median  Max  Min  Crop  n 
Sh  Sc  Sh  Sc  Sh  Sc  Sh  Sc  Sh  Sc 
Rainfed  32  10  3  7  2  9  3  29  11  0  0 
Irrigated  14  14  2  21  3  6  1  75  8  0  0 
 
Given that the scale parameter has a much smaller 
variability than that of the shape parameter, Table 2 
shows  detailed  results  obtained  for  the  shape 
parameter, classified into three groups of cases: those 
that  have  a  shape  coefficient  less  than  seven,  those 
between seven and ten, and those greater than ten. 
 
Table 2 Observed cases for the Weibull distribution’s shape 
parameter (%) 
10W: ten-weight VIM, >10W: ten-plus-weight VIM 
 
Shape <7  Shape 7-10  Shape >10  Crop 
10W  >10W  10W  >10W  10W  >10W 
Rainfed  72  72  9  9  19  19 
Irrigated  36  36  14  28  50  36 
 
For  the  ten-weight  VIM  (10W),  the  majority  of 
observed cases presented a shape parameter inferior to 
seven for rainfed crops. In the case of irrigated crops, 
half the observed cases presented a shape coefficient 
greater than ten.  
In the case of using ten-plus-weight VIM (>10W), 
the values were identical to the results seen when ten 
weights  were  used,  in  the  case  of  rainfed  crops.  In 
irrigated  crops,  the  cases  in  which  the  shape 
coefficient is less than seven are equal to those which 
have  a  shape  coefficient  greater  than  ten,  with  the 
cases having shape coefficients between seven and ten 
remaining in the minority. 
The shape parameter indicates the level of kurtosis, 
with primarily platykurtic estimations being obtained 
in  rainfed  crops  and  a  less  evident  tendency  in 
irrigated crops, where the functions were flattened in 
14% of cases when the number of weights was other 
than ten. 
The  Wilcoxon  test  (with  a  0.05  level  of 
significance)  for  related  samples  was  used  for  the 
shape and scale parameters, for the ten-weight and ten-
plus-weight  VIM  variants,  indicating  that  the 
similarity hypothesis between the values studied could 
not be rejected in any of the cases, for both rainfed and 
irrigated crops. 
IV. CONCLUSIONS  
When using the VIM for PDF crop yield estimation, 
similar  results  are  obtained  using  ten  weights  or  a 
number of weights freely chosen by the farmer. As a 
consequence,  the  ten-weight  VIM  is  recommended, 
since it is the simplest variant to use.   4 
12
th Congress of the European Association of Agricultural Economists – EAAE 2008 
The  similarity  of  responses  indicates 
methodological  persistence,  one  of  the  logical 
conditions needed to have confidence in the estimates. 
 
REFERENCES 
 
1.  Day RH (1965) Probability Distributions of Field Crop 
Yields. J Farm Econ 47(3):713-741 
2.  Gallagher  P  (1986)  U.S.  Corn  Yield  Capacity  and 
Probability:  Estimation  and  Forecasting  with 
Nonsymmetric  Disturbances.  N  Cent  J  Agr  Econ 
8(1):109-122 
3.  Gallagher  P  (1987)  U.S.  Soybean  Yields:  Estimation 
and  Forecasting  with  Nonsymmetric  Distributions. 
Amer J Agr Econ 69(4):796-803 
4.  Nelson CH, Preckel PV (1989) The Conditional Beta 
Distribution as a Stochastic Production Function. Amer 
J Agr Econ 71(2):370-378 
5.  Nelson  CH  (1990)  The  Influence  of  Distributional 
Assumptions  of  the  Calculation  of  Crop  Insurance 
Premia. N Cent J Agr Econ 12:71-78 
6.  Taylor  CR  (1990)  Two  Practical  Procedures  for 
Estimating Multivariate Nonnormal Probability Density 
Functions. Amer J Agr Econ 72(1):210-218 
7.  Ramírez  OA  (1997)  Estimation  and  Use  of  a 
Multivariate  Parametric  Model  for  Simulating 
Heteroskedastic,  Correlated,  Nonnormal  Random 
Variables: The Case of Corn Belt Corn, Soybean, and 
Wheat Yields. Amer J Agr Econ 79(1):191-205 
8.  Goodwin  BK,  Ker  AP  (1998)  Nonparametric 
Estimation  of  Crop  Yield  Distributions:  Implications 
for Rating Group-Risk Crop Insurance Contracts. Amer 
J Agr Econ 80(1):139-153 
9.  Just RE, Weninger Q (1999) Are Crop Yields Normally 
Distributed?. Amer J Agr Econ 81(2):287-304 
10. Sherrick  BJ,  Zanini  FC,  Schnitkey  GD,  Irwin  SH 
(2004)  Crop  Insurance  Valuation  Under  Alternative 
Yield Distributions. Amer J Agr Econ 86(2):406-419 
11. Pease  JW  (1992)  A  Comparison  of  Subjective  and 
Historical  Crop  Yield  Probability  Distributions. 
Southern Journal of Agricultural Economics 24(2):23-
32 
12. Anderson JR (1997) An ‘ABC’ of risk management in 
agriculture:  overview  of  procedures  and  perspectives. 
In: Risk Management Strategies in Agriculture: State of 
the  Art  and  Future  Perspectives  (Huirne  RBM, 
Hardaker  JB,  Dijkhuizen  AA,  eds).  Wageningen 
Agricultural University, Wageningen, The Netherlands, 
pp. 1-13 
13. Miranda  MJ  (1991)  Area-Yield  Crop  Insurance 
Reconsidered. Amer J Agr Econ 73(2):233-242 
14. Hardaker JB, Huirne RBM, Barry PJ, King RP (1997) 
Risk Management Strategies in Agriculture: the State of 
the Art and Future Perspectives. In: Risk Management 
Strategies in  Agriculture. State of the  Art and Future 
Perspectives  (Huirne  RBM,  Hardaker  JB,  Dijkhuizen 
AA,  eds).  Wageningen  Agricultural  University, 
Wageningen, The Netherlands, pp. ix-xiv 
15. Van  Der  Fels-Klerx  IHJ,  Goossens  LHJ,  Saatkamp 
HW, Horst SHS (2002) Elicitation of quantitative data 
from a heterogeneous expert panel: formal process and 
application in animal health. Risk Analysis 22(1):67-81 
16. Clop-Gallart MM, Juárez-Rubio F (2006) Persistence of 
point crop yield subjective estimates. Spanish Journal 
of Agricultural Research 4(3):202-206 
17. Clop-Gallart MM, Juárez-Rubio F (2007) Elicitation of 
subjective  crop  yield  PDF.  Spanish  Journal  of 
Agricultural Research 5(1):16-24 
18. Francisco EM, Anderson JR (1972) Chance and choice 
West of the Darling. Australian Journal of Agricultural 
Economics 16:82-93 
19. Anderson  JR,  Dillon  JL,  Hardaker  JB  (1977) 
Agricultural  decision  analysis.  The  Iowa  State 
University Press, Ames, Iowa, USA 
20. Bland JM, Altman DG (1995) Comparing methods of 
measurement: why plotting difference against standard 
method is misleading. Lancet 346:1085-1087 
21. Bland JM, Altman DG (1999) Measuring agreement in 
method comparison studies. Stat Methods Med Res 8: 
135-160 
 
 
 
 
Use macro [author address] to enter the address of the 
corresponding author: 
•  Author:  Clop-Gallart M.M. 
•  Institute:  Universitat de Lleida 
•  Street:    Av. Rovira Roure, 191 
•  City:    Lleida (25198), Catalonia 
•  Country:  Spain 
•  Email:  mclop@aegern.udl.cat 
 

A Comparison of Subjective and Historical Crop Yield Probability Distributions.

Agricultural decision analysis.

An ‘ABC’ of risk management in agriculture: overview of procedures and perspectives. In:

Are Crop Yields Normally Distributed?.

Area-Yield Crop Insurance Reconsidered.

Chance and choice West of the Darling.

Comparing methods of measurement: why plotting difference against standard method is misleading.

Crop Insurance Valuation Under Alternative Yield Distributions.

Elicitation of quantitative data from a heterogeneous expert panel: formal process and application in animal health.

Elicitation of subjective crop yield PDF.

Estimation and Use of a Multivariate Parametric Model for Simulating Heteroskedastic, Correlated, Nonnormal Random Variables:

Measuring agreement in method comparison studies. Stat Methods Med Res 8: 135-160 Use macro [author address] to enter the address of the corresponding author:

Nonparametric Estimation of Crop Yield Distributions: Implications for Rating Group-Risk Crop Insurance Contracts.

Persistence of point crop yield subjective estimates.

Probability Distributions of Field Crop Yields.

Risk Management Strategies in Agriculture: the State of the Art and Future Perspectives. In:

The Conditional Beta Distribution as a Stochastic Production Function.

The Influence of Distributional Assumptions of the Calculation of Crop Insurance Premia.

Two Practical Procedures for Estimating Multivariate Nonnormal Probability Density Functions.

U.S. Corn Yield Capacity and Probability: Estimation and Forecasting with Nonsymmetric Disturbances.

U.S. Soybean Yields: Estimation and Forecasting with Nonsymmetric Distributions.

Shape Persistence in Elicited Subjective Crop Yield Probability Density Functions

Shape Persistence in Elicited Subjective Crop Yield Probability Density Functions

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