INTRODUCTION
Risk is a concept that denotes a potential negative impact to some characteristic
of value that may arise from a future event, or we can say that risks are events
or conditions that may occur and whose occurrence, if it does take place, has
a harmful or negative effect. Exposure to the consequences of uncertainty constitutes
a risk. In everyday usage, risk is often used synonymously with the probability
of a known loss. Risk communication and risk perception are essential factors
foe all human decision making (Cooper et al., 2005).
A systematic process of risk management is divided into risk identification,
risk analysis and risk response (Li and Liao, 2007; Duijne
et al., 2008). Risk identification requires recognizing and documenting
the associated risk. Risk analysis examines each identified risk issue, refines
the description of the risk and assesses the associated impact. Finally, risk
response identifies, evaluates, selects and implements strategies in order to
reduce the likelihood of occurrence or impact of risk events.
Risk analysis has several objectives (Cooper et al.,
2005):
• 
It gives an overview of the general level and pattern
of risk facing the project 
• 
It focuses management attention on the highrisk items in the list 
• 
It helps to decide where action is needed immediately and where
action plans should be developed for future activities; and it facilitates
the allocation of resources to support management’s action decisions 
Depending on the available data, risk analysis can be performed qualitatively
or quantitatively or semi quantitatively (Chapman and Ward,
2004; Groen et al., 2006). Chun
and Ahn (1992) and Smith (1999) are trying to propose
risk analysis techniques in various environments. Risks are prioritized according
to their potential implications for meeting the stakeholders’ objectives. The
typical approach to prioritizing risks is to use a lookup table or a probability
and impact matrix. The better results emerge in the cost and time planning fields,
in which the causal distribution of random events is analyzed to improve predictions.
Although numerous techniques are at present available to practitioners for risk
assessment (Dikmen et al., 2008) sophisticated
simulation techniques or statistical techniques can be with difficulty adapted
to technical risk multidimensionality.
Risk management has been developed mostly on the basis of cost and time risk,
while technical risk analysis has not yet aroused wide interest on nonquality
risk. Risk management is becoming an important management method in the planning
of a reliable, suitable, adequate and subsequently more efficient real system,
as it plays a key role in the quality management field toward a suitable, adequate
and subsequently more efficient quality system for building in conformity to
specifications (Kerzner, 2006). Quality planning, environmental
control and safety planning require holistic approaches in process representation
and a basic qualitative risk assessment. The Failure Mode and Effects Analysis
(FMEA) (Hu et al., 2008), plays an effective role
for a qualitative failure process analysis and provides a systematic, indexed
order of technical risks.
Resampling techniques have been conventionally used as a means of tackling
problems which are too complicated to be solved analytically. Over the past
30 years, the theoretical foundations for this technique have been expanded
and substantiated (Efron and Tibshirani, 1993). These techniques
are particularly suitable for hypothesis testing and for determining the accuracy
of nonparametric or complex statistics for which closedform formulae, if they
exist, depend on extensive assumptions.
On the other hand, risk data analysis often encounters situations in
which:
• 
It cannot be answered in a parametric framework for
which closedform formulae for accuracy exist 
• 
It may need to be examined by standard, existing tools, but the
results exhibit a bias that influences inference 
• 
It can only be assessed by specially tailored algorithms or procedures
that, in turn, require objective validation. As the occurrence and
impact of risks are random; therefore, statistical approaches are
required for analyzing risks effectively 
For these reasons, as well as due to the availability of fast computers, in
this paper bootstrap resampling approach is proposed to use for analyzing risks.
This approach is flexible, easy to implement, applicable in nonparametric settings
and requires a minimal set of assumptions (Tak, 2004).
In this study we hope to contribute to this area by providing a comprehensive
framework for the application of bootstrap technique to data obtained from experts’
judgments. Reduction of SD for risks is shown in this study significantly by
using nonparametric bootstrap technique. Moreover, the nonparametric bootstrap
has been applied to estimate confidence intervals for the Risk Factors (RFs)
in risk management process.
The normal (Gaussian) distribution is characterized by two parameters; the
mean and SD. Statistical techniques that assume the Gaussian distribution of
data are called parametric. Nonparametric or distributionfree statistical techniques
are used to analyze data that do not assume a particular family of probability
distributions. It is in this latter category of data that bootstrap techniques
are valuable (Efron and Tibshirani, 1993; Henderson,
2005).
The bootstrap resampling technique developed by Efron (1979)
has been used widely in statistical problems. It can be used where standard
techniques cannot be applied, for instance in situations in which few data are
available, so that approximate large sample techniques are not applicable. The
bootstrap has subsequently been used to solve many other problems that would
be too complicated for traditional statistical analysis (AitSahalia
and Duarte, 2003; Stark and Abeles, 2005). In simple
words, the bootstrap does with the computer what the experimenter would do in
practice, if it was possible, he or she would repeat the experiment (Modarres
et al., 2006; Walters and Campbell, 2005).
The main advantage of using the bootstrap resampling technique is that good
estimates can be obtained, regardless of the complexity of the data processing.
In this study, we show that the bootstrap resampling technique is well suited
for estimating and decreasing SD for risk data.
PROPOSED APPROACH
Risk data sizes are always too small and also there are no parametric distributions
on which significance can be estimated for risks data; therefore, nonparametric
bootstrap technique is extremely valuable in situations where data sizes are
too small. Moreover, the bootstrap is a powerful tool for assessing the accuracy
of a parameter estimator in situations where conventional techniques are not
valid (Armitage et al., 2002; Heiermann
et al., 2005; William and Joseph, 2005).
Having considered all above mentioned reasons, here one practical approach
is proposed to use in risk management process in three steps. In first
step, principle of nonparametric bootstrap is described in order to resample
risks data from original observed risks data. In second step, the bootstrap
principle for estimating the SD of RFs is demonstrated in order to compare
bootstrap resampled risk data with original observed risks data and finally
in third step, the bootstrap principle for calculating a confidence interval
for the mean of RFs is presented for better decision making in risk management
process.
The nonparametric bootstrap principle (Step 1): Based on the
first step of proposed approach, the bootstrap technique is a tool for
uncertainty analysis based on resampling of experimentally observed data.
Application of the bootstrap is justified by the socalled plugin principle,
which means to take statistical properties of experimental results (=
sample) as representative for the parent population. The main advantage
of the bootstrap is that it is completely automatic. It is described best
by setting two Worlds, a Real World where the data is obtained and a Bootstrap
World where statistical inference is performed, as shown in Fig.
1.
The nonparametric bootstrap principle is as follows:
• 
Conduct the experiment to obtain the random sample
and calculate the estimate
from sample x 
• 
Construct the empirical distribution, ,
which puts equal mass, 1/n, at each observation, X_{1} = x_{1},
X_{2} = x_{2}, ….,X_{n} = x_{n} 
• 
From the selected ,
draw a sample, ,
called the bootstrap resample 
• 
Approximate the distribution of
by the distribution of
derived from x*. 
The bootstrap principle for estimating the SD of RF (Step 2):
Based on the second step of proposed approach, the bootstrap principle
for estimating the SD of RFs is as follows:
• 
Experiment. Conduct the experiment and collect the random
data into the sample x = {X_{1}, X_{2},…,X_{n}} 

Fig. 1: 
Schematic diagram of the bootstrap technique according
to Efron and Tibshirani (1993) 
• 
Resampling. Draw a sample of size n, with replacement,
from x 
• 
Calculation of the bootstrap estimate. Evaluate the bootstrap estimate
^{*}
from x* calculated in the same manner as
but with the resample x* replacing x 
• 
Repetition. Repeat steps 1 and 2 many times to obtain the total
B bootstrap estimates
Typical value for B are between 25 to 200 
• 
SD estimation of .
Estimate the SD,
of ,
by the sample SD of the B bootstrap estimates: 
The bootstrap principle for calculating a confidence interval for
the mean of RF (step 3): In accordance with third step of proposed
approach, the bootstrap principle for calculating a confidence interval
for the mean of RFs is as follows:
• 
Experiment. Conduct the experiment. Suppose present
sample is x = {X_{1}, X_{2}, …, X_{n}} with
,
the mean of all values in x 
• 
Resampling. The bootstrap principle 
• 
Calculation of the bootstrap estimate. Calculate the mean of all
values in x* 
• 
Repetition. Repeat steps 2 and 3 a large number of times to obtain
a total of n bootstrap estimates 
• 
Approximation of the distribution of .
Sort the bootstrap estimates in to increasing order to obtain ,
where
is the kth smallest of 
• 
Confidence interval. The desired (1α) 100% bootstrap confidence
interval is ,
where q_{1} = [Nα/2] and q_{2} = Nq_{1}+1 
Finally, for better understanding, Fig. 2 shows the
proposed approach. As it is evident, the approach for risk analysis consists
of two main steps; risk observation data or original samples and nonparametric
bootstrap. It is highly appropriated to mention that this approach is
iterative, this means that we have to resample original samples with different
B until SD for risks will be decreased, typical value for B are between
25 to 200. For instance B can be selected from B = {m_{1} = 25,
m_{2} = 35, …., m_{i} = 200} or any other similar set.

Fig. 2: 
Proposed nonparametric statistical approach for risk
evaluation (Iterative Process) 
APPLICATION EXAMPLE
Risk management process can be applied in various fields such as financial,
insurance, project, operational, business, market, Health, safety, environment
and so forth. Here, proposed approach based on nonparametric bootstrap is applied
in project field. A project, as defined in the field of project management,
consists of a temporary endeavor undertaken to create a unique product, service
or result (Cooper et al., 2005). Project management
tries to gain control over project’s variables such as risk; therefore, risk
analysis is essential for all projects.
Apply the proposed approach in project risk analysis: The risk management
process aims to analyze risks in order to enable them to be understood clearly
and managed effectively (Han et al., 2008).
Table 1: 
Risk observed data 

There
are many commonly used techniques for risk analysis (Cho et al., 2002; Majdara and Nematollahi, 2008; Duijne et al., 2008), these techniques generate
list of risks that often do not directly assist the manager in knowing where
to focus risk management attention. Quantitative assessment can help to prioritize
identified risks by estimating their probability and impacts, exposing the most
significant risks. In this section, an application example which can analyze
project risks in nonparametric environment is introduced. Here, we show how the proposed approach can be used in risk analysis
according to lack of risk sample data and periodic features of the projects.
Hence, the comparison of the mean and the SD between the original sample
distribution and the bootstrap resampled distribution can produce a better
result.
In risk analysis two indexes, i.e., probability and impact, are considered.
The probability of a risk is a number between 01 but the impact of a
risk is qualitative. Though, it must be changed to quantitative number,
just like probability, a number between 01.
The RF_{ij} for ith risk in jth observation is calculated as the follow
(Wang and Elhag, 2006, 2007):
RF_{ij} = P_{ij}xI_{ij} 
(2) 
Five different risks have been assumed for which we contemplate five
probabilities and five impacts each that form our sample. It means that
according to (2) we have P_{ij} which is the probability of the
ith risk in jth observation and I_{ij} which is the impact of
the ith risk in jth observation. The assumed data is presented in Table
1.
A sampling distribution is based on many random samples from the population.
In place of many samples, from the population, create many resamples by
repeatedly sampling with replacement from this one random sample. Each
resample is the same size as the original random sample. Sampling with
replacement means that after we randomly draw an observation from the
original sample, we put it back before drawing the next observation. Think
of drawing a number from a hat, then putting it back before drawing again.
As a result, any number can be drawn more than once, or not at all. If
we sampled without replacement, we would get the same set of numbers we
started with, though in a different order. In practice, we draw hundreds
or thousands of resamples, not just five.
The sampling distribution of a statistic collects the values of the statistic
from many samples. The bootstrap distribution of a statistic collects
its values from many resamples. The bootstrap distribution gives information
about the sampling distribution.
The true value of the population characteristic is denoted by RF. A set
of n values are randomly sampled from the population. The sample estimate
is based on the 5 values (P_{1}, P_{2}, …, P_{5})
and (I_{1}, I_{2}, …, I_{5}). Sampling 5 values
with replacement from the set (P_{1}, P_{2}, …, P_{5})
and (I_{1}, I_{2}, …, I_{5}) provides a bootstrap
sample
and .
Observe that not all values may appear in the bootstrap sample. The bootstrap
sample estimate RF* is based on the 5 bootstrap values
and .
The sampling of (P_{1}, P_{2}, …, P_{5}) and
(I_{1}, I_{2}, …, I_{5}) with replacement is
repeated many times (say B times), each time producing a bootstrap estimate
RF*.
Call the means of these resamples
in order to distinguish them from the mean
of the original sample. Find the mean and SD of the ’s
in the usual way. To make clear that these are the mean and SD of the
means of the B resamples rather than the mean
and SD of the original sample, we use a distinct notation:
The Bias can also be calculated for all the resamples population which
is the difference between the mean of the resample mean and the original
sample. This delineates that the resampled mean is not far from the original
sample and it will not deviate from the original sample. The data for
this resamples are available in Table 35,
respectively for resample size 50, 100 and, 200.
Due to the fact that a sample consists of few observed samples, which
is the nature of the projects, we use bootstrap resampling technique to
ameliorate the accuracy of the calculation of the mean, SD and confidence
interval for the RF of the risks which may occur in a project.
RESULTS AND DISCUSSION
To do the resampling replications, we used resampling Stat Addin of
Excel. We compare the original sample and the bootstrap resample of 50,
100 and 200 population of the data provided by the Excel Addin to see
what differences it makes.
Table 2: 
Statistical data of the original sample 

Table 3: 
Statistical data of the 50 resample 

Table 4: 
Statistical data of the 100 resample 

Table 5: 
Statistical data of the 200 resample 

Table 6: 
Confidence intervals for the three resamples 

In Table 2, the statistical
data of the original sample is presented.
After 50, 100 and 200 resampling replications, we obtain the mean for
P, I and RF and then the SD for them. Moreover, we calculate RF_{Bizs}
to show the mean provided by the resampling is not far from the original
sample mean. The data are reported in Table 3, 4
and 5 as follows.
The confidence interval of the resamples with 50, 100 and 200 replications
are calculated with α = 5%. The q_{1} and q_{2} are
presented for each of the risks in Table 6.
We analyze risks using proposed approach bases on nonparametric bootstrap
technique in project. The inference of results is applicably feasible,
appealing and interesting in risk management.

Fig. 3: 
SD Comparison between Original Sample and BResample
(B = 50, 100 and 200) for Project RFs 
We calculate SD for RFs
of each risk for original sample and Bresample (B = 50, 100, 200), as
shown in Fig. 3. As it is clear the SDs are reduced
remarkably and it shows the efficiency of nonparametric bootstrap technique
in risk analysis. The results show that the proposed approach is reasonable
for estimating the SD.
SD Reduction Rate: Comparison between the SD of the original sample
and the three resampled SD with 50, 100 and 200 replications show that
SD for each risk has been reduced remarkably through nonparametric bootstrap
technique, for instance the SD of risk 1 of the original sample is 0.066
where the SD of the same risk with 200 resample is 0.030, can depict that
the bootstrap technique is making a better result in accuracy of the RF
for each risk. And then, SD reduction rate is calculated as follows:
where, SD_{Red} (%) denotes the rate of SD reduction through
nonparametric bootstrap technique, SD_{Org} represents SD for
original risk factor data sample and SD_{B} indicates SD for B
size bootstrap. Rate of SD reduction is presented in Table
7 for each risk.
For instance, the comparison between original risk factor data sample
and Bresample (B = 50) as SD reduction point of view is shown in Fig.
4.
Moreover, the span of the confidence interval of the risks is calculated,
although the confidence interval between the resamples with 50, 100 and,
200 replications are not different by far. For the risks with smaller
SD, the confidence interval is smaller too. So the results are more precise
for the resampled data.
Table 7: 
Rate of SD reduction for each risk 


Fig. 4: 
SD Comparison between Original Sample and BResample
(B = 50) for Project RFs 
We have shown that resamplingbased procedures can be easily applied
to lots of different types of problems yielding meaningful results, results
that often cannot be obtained using conventional approaches. Routines
for implementing the procedures described in this paper were calculated
in Stat Addin of Excel.
It is firmly recommended to take advantage of this proposed approach
in large projects; because of the fact that mega projects have following
characteristics:
• 
They are unique 
• 
They are contemporary 
• 
They have elaborative progress 
• 
Investment and financing are main issues 
• 
They are being managed in risky environment 
• 
Projects’ data and samples are too small 
• 
Distribution of projects’ data and samples are not always definite 
Having considered all different aspects involved in projects’ characteristics,
nonparametric statistical approach particularly bootstrap is very useful
for risk analysis in each project, because it provides accurate calculation.
CONCLUSION AND FURTHER RESEARCH
Nonparametric statistical approach was presented to use in risk management
process, proposed approach had two specific sections; first risk observation
data or original risk data was evaluated, then in second step nonparametric
bootstrap was applied for original risk data. On the other hand, section
two had three main steps including; nonparametric bootstrap technique,
SD calculation for each risk and calculation of confidence interval for
each risk. We found that bootstrap has greater accuracy for estimating
SD of RFs and greater accuracy in terms of level of significance than
analyzing original risks data. SDs for RFs were remarkably reduced when
nonparametric bootstrap was applied. In application example section,
SD reduction rate was calculated and acceptable results were conducted,
for instance the reduction rate for risk 5 in B = 50 resampling process
was about 60%. Moreover, RF_{Bias} was calculated to show the
mean provided by the resampling is not far from the original sample mean.
The bootstrap is extremely an attractive tool because it requires very
little assumptions in the way of modeling, assumptions, or analysis and
it can be applied in an automatic way. Further, bootstrap technique is
extremely valuable in situations where data sizes are too small, which
is often the real case in risk analysis applications.
In the future work, we may work on the topic that considers the nonparametric
regression model for project RFs and we may compare different nonparametric
resampling techniques for choosing the best way for analyzing RFs.
ACKNOWLEDGMENT
The authors thank Mr. M. Heydar for his helpful comments and suggestions,
which helped to improve the work.