American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  

http://asrjetsjournal.org/  
 

On the Modification of M-out-of-N Bootstrap Method for 

Heavy-Tailed Distributions 

Hannah F. Opayinkaa*, Adedayo A. Adepoju b 

aFederal College of Education (Special),Nigeria, Phone:+2348034975272 
b Statistics Department, University of Ibadan, Nigeria, Phone:+2348066430258 

afolashadeopayinka@gmail.com  
bpojuday@yahoo.com 

 

Abstract 

This paper is on the modification of 𝑚-out-of-𝑛 bootstrap method for heavy-tailed distributions such as income 

distribution. The objective of this paper is to present a modified 𝑚-out-of-𝑛 bootstrap method (𝑚𝑚𝑜𝑜𝑛) and 

compare its performance with the existing m-out-of-n bootstrap method (𝑚𝑜𝑜𝑛). The nature of the upper tail of 

a distribution is the major reason for the poor performance of classical bootstrap methods even in large samples. 

The ‘𝑚𝑚𝑜𝑜𝑛’ bootstrap method was therefore, proposed as an alternative method to ‘𝑚𝑜𝑜𝑛’ bootstrap method. 

The distribution involved has finite variance. The simulated data sets used was drawn from Singh-Maddala 

distribution. The methodology involved decomposing the empirical distribution and sampling only n⃛ times with 

replacement from a sample size n, such that n⃛  → ∞ as n → ∞, and n⃛/n → 0. The performances are judged 

using standard error; absolute bias; coefficient of variation and root mean square error. The findings showed that 

‘𝑚𝑚𝑜𝑜𝑛’ performed better than 𝑚𝑜𝑜𝑛 in moderate and larger samples and it converged faster. 

Keywords: Bootstrap; Decomposition; Heavy-tailed distributions; Singh-Maddala distribution. 

1. Introduction 

1.1. Background of the Study  

This study is concerned with finding a reliable alternative bootstrap method to heavy-tailed distributions. The 

tail of a distribution, especially the upper tail affects the performance of bootstrap methods. 

 

------------------------------------------------------------------------ 
* Corresponding author.  

E-mail address: folashadeopayinka@gmail.com  

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  1, pp 142-155 

 

Bootstrap is a resampling procedure developed by Efron in 1979. The usual practice when estimating the 

properties of an estimator (such as its variance) is by measuring those properties when sampling from an 

approximating distribution. Empirical distribution of the observed data is a standard choice for an approximating 

distribution. The resampling could be done from an independent and identically distributed (𝑖𝑖𝑑) population. 

Bootstrap is used when parametric assumptions are in doubt, when the formulas for the calculation of standard 

errors are complicated in parametric inference. There is no need to force Gaussian or any other parametric 

distributional assumptions on data. The distribution could be skewed, multimodal, and heavy-tailed; the 

estimator of interest could be complicated [1]. 

The use of bootstrap has been applied to many estimators within cross-section data [2]. In heavy-tailed 

distributions like income distribution, there is frequency of outliers in data sets, which usually cause difficulties 

in the use of asymptotic and bootstrap methods. The nature of the upper-tail of the distribution generally affects 

the performance of the methods [3]. 

The author in [4] proposed the use of the bootstrap for the most commonly applied procedures in inequality, 

mobility and poverty measurements. He suggests that the simplest possible bootstrap procedure should be the 

preferred method in practice because it achieves precision and takes into account the stochastic dependencies in 

the data, without the need of dealing with its covariance structure explicitly. His simulation results suggest that 

the bootstrap performs well in finite samples. He also decomposed income distribution into subgroup such that; 

𝑤𝑖, 𝑥𝑖 are the weights and income sources respectively. His study regarded all subgroups to constitute the 

population and all subgroups are disjoint. 

The authors in [3] studied finite-sample performance of asymptotic and bootstrap inference for both inequality 

and poverty measures. Their simulation results showed that neither asymptotic nor classical bootstrap inference 

for inequality measures perform well, even when the sample size is large enough. They found that the 

performances of both asymptotic and bootstrap are affected by the nature of the upper-tail of the income 

distribution. Authors of some studies [4, 5] involving heavy-tailed distribution recommend the use of bootstrap 

rather than asymptotic methods. 

After the publication of Efron in 1982, research activity on the bootstrap grew so fast with the emergence of 

many theoretical developments on the asymptotic consistency of bootstrap estimate coupled with real-world 

applications. Focus changed in 1990s to finding applications and variants that would perform well in practice. 

Some studies on bootstrap inference for inequality measures were done and the use of bootstrap methods rather 

than asymptotic methods was recommended [1].  

Heavy-tail means that the probability of getting very large values is high. Therefore, heavy-tailed distributions 

typically represent wild as opposed to mild randomness, examples are income distributions, financial returns, 

insurance payouts, reference links on the web etc. A technical difficulty is that, all moments do not exist for 

these distributions. 

Heavy-tailed distributions that are often used include: Burr (Singh-Maddala) distribution, Pareto distribution, 

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  1, pp 142-155 

 

L�́�vy distribution, Weibul distribution, Log-gamma distribution, etc. Singh-Maddala (Burr) distribution is a 

member of a system of continuous distributions introduced by Burr in 1942.The Burr distribution/ Singh-

Maddala distribution is a continuous probability distribution for a non-negative random variable. It is most 

commonly used to model household income [6].  

This study proposes an alternative bootstrap method called ‘𝑚𝑚𝑜𝑜𝑛’ which is expected to perform better than 

𝑚𝑜𝑜𝑛. It has been established that 𝑚𝑜𝑜𝑛 could overcome the inconsistency in the classical method. The study 

would involve estimating the income data by obtaining the chosen estimator in each bootstrap method, 

comparing the statistical inference of 𝑚𝑜𝑜𝑛 and  𝑚𝑚𝑜𝑜𝑛 bootstrap methods. Using simulation would allow one 

to assess the reliability of these methods for empirical work. 

1.2. Limitation of the Study 

The research was carried out using 64-bit Operating system laptop computer. The work would have been faster 

if it had been done on a macro computer. 

2. Methodology 

The research methodology is described as follows. 

2.1. 𝒎-out of–𝒏 (𝒎𝒐𝒐𝒏) bootstrap method  

Authors in [3] regard moon bootstrap method as being useful when the classical bootstrap fails or when it is 

difficult to check its consistency. The author in [7] described 𝒎𝒐𝒐𝒏 as sampling 𝒎 times without replacement 

from a sample size 𝒏, instead of sampling 𝒏 times; such that 𝒎 is much less than 𝒏. Usually the asymptotic 

theory requires 𝒎 → ∞ 𝐚𝐬 𝒏 → ∞, but at a slower rate such that 𝒎/𝒏 → 𝟎. 

In [8] it was called ’the great 𝒎 𝐨𝐮𝐭 𝒏 bootstrap with (𝒎/ 𝒏 →  𝟎 )’ where the bootstrap sample size 𝐦 is 

much smaller than the original size. Mathematically, the requirement is 𝒎 → ∞  and  𝒎/𝒏 → 𝟎,𝒂𝒔 𝒏 → ∞.  In 

theory, the problem is fixed, but in practice, some troubles are involved such as how to choose 𝒎. An obvious 

suggestion is to settle for a fraction of, say 20%. It was pointed out that in good situations, where the regular 

bootstrap performs, such a 𝒎 is not advisable, it could result in loss of efficiency.  

Authors in [9] proposed the 𝐦-out-of-𝐧 bootstrap with or without replacement, where m → ∞  and 𝐦 𝐧⁄   → 0 as 

a way of ensuring consistency when the classical bootstrap is not consistent. 

Authors in [10] explored 𝐦-out-of-𝐧 bootstrapping from the empirical distribution function in nonstandard 

problems and proved the consistency of the method.  

2.2. modified 𝒎-out-of-𝒏 (𝒎𝒎𝒐𝒐𝒏) bootstrap method 

The 𝑚𝑚𝑜𝑜𝑛 bootstrap method is a modification of 𝑚𝑜𝑜𝑛, it involves decomposition of the empirical 

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  1, pp 142-155 

 

distribution (𝐹�𝑛) and sampling only n⃛ times with replacement from a sample size n, such that n⃛  → ∞ as n → ∞, 

and n⃛/n → 0. The method would resample from 𝐻𝑛 distribution which is a decomposed version of 𝐹�𝑛. The 

procedure provides and estimates different measures of statistical precision for an estimator 𝜃�.  The method 

satisfies the conditions in [11].    

2.3. Validity of (𝒎𝒎𝒐𝒐𝒏) bootstrap method 

• Samples must be independently and identically distributed; [1] reported that 𝑖𝑖𝑑 works in large sample. 

• For a bootstrap approach to work well, it was suggested in [12] that the distribution function should have a 

differentiable density. This study makes use of simulated data sets drawn from Singh-Maddala distribution 

which [3] said can quite successfully mimic observed income distributions in various countries. It can be 

shown that the distribution has a differentiable density.  

The cumulative density function (CDF) of the distribution can be written as: 

 𝐹(𝑥)  =  1 −    1

�1+𝑎𝑥𝑏�
𝑐                          {1}   

 Where a = scale parameter; b = shape parameter; c = shape parameter [3].                                               

And the probability density function (pdf) as:  

𝑓(𝑥) = 𝑎𝑏𝑐 𝑥𝑏−1

�1+𝑎𝑥𝑏�
𝑐+1                                       {2}    

• The validity of bootstrap also requires that the estimator (a functional form of the empirical distribution 

function) converges to the true parameter value (the functional form for the true population distribution). 

The commonly used parameters of distribution function can be expressed as functional form of the 

distribution, which includes the mean, the variance etc. Sample estimates such as the sample mean can be 

expressed as the same functional form applied to the empirical distribution. This provides guideline for this 

method in choosing the mean as the functional form of the distribution. A functional form is simply a 

mapping that takes a function F into a real number, examples of such are the mean and variance of a 

distribution [3][13]. 

Assuming mean is used as the functional form in this study, let 𝜇 be the mean for a distribution function 𝐹, then 

𝜇 = ∫ 𝑥𝑑𝐹(𝑥).     It can be shown that the functitonal form of the Singh-Maddala distribution exists:  

𝐸(𝑥) < ∞   ;    𝐸(𝑥) =  ∫ 𝑥 ∙ 𝑝𝑑𝑓 𝑑𝑥     

𝜇 = 𝑐𝑎
1
𝑏Γ�𝑏−1+1�Γ(𝑐−𝑏−1)

Γ(𝑐+1)
  [3] 

• It was shown in [11] that bootstrap principle works for sample mean when finite second moments exist. 

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This provided a stronger justification for this method. The second moment exists for the Singh-Maddala 

distribution considered in this study. 

𝐸(𝑥2) < ∞        ;𝐸(𝑥2) = � 𝑥2𝑓(𝑥)𝑑𝑥 

𝐸(𝑥2) = � 𝑥2
𝑎𝑏𝑐 𝑥𝑏−1

(1 + 𝑎𝑥𝑏)𝑐+1 𝑑𝑥 ;          =
𝑎𝑏𝑐

2 − 𝑐
[1 + 𝑎𝑥𝑏]2−𝑐 

• Bootstrapping actually works (i.e. consistent) if the following holds:  

If 𝑇(𝐹𝑛∗) converges to 𝑇(𝐹) as 𝑛 → ∞, (i.e. the bootstrap estimate is consistent for the population parameter) 

which implies that 

• 𝑇(𝐹𝑛) converges to 𝑇(𝐹) as 𝑛 → ∞, (i.e sample estimate is consistent for the population parameter, when 𝐹𝑛 

converges to 𝐹 uniformly). 

• 𝑇�𝐹�𝑛∗� − 𝑇(𝐹𝑛) → 0   as    𝑛 → ∞, (i.e. the difference between bootstrap estimate and sample estimates tends 

to zero).  

The simulation study can also be used to confirm or deny the usefulness of the bootstrap estimate. The 

performances of the estimates are judged in the empirical work, by obtaining standard error, absolute bias, root 

mean square error and coefficient of variation [13]. 

2.4. Decomposition of Empirical Distribution 

The proposed measurement scenarios are decomposition of the empirical distribution by sub-group (in form of 

strata) which could be income levels, the subgroups are disjoint and all subgroups taken together constitute the 

population.   

Authors in [5] included decomposition by population subgroups due to significant differences in income levels 

among individuals, these differences are caused by some characteristics such as age, race etc. The author in [1] 

recommended stratified sampling as a remedy for inconsistency in bootstrap method. Stratification can be useful 

in reducing the variability of some estimates. 

Stratified sampling has been regarded as a method of variance reduction in computational statistics and that a 

stratified survey could claim to be more representative of the population than a survey of simple random 

sampling or systematic sampling. In stratified sampling, there is assurance that estimates would be made with 

equal precision in different parts of the region, and that comparisons of sub-regions would be made with equal 

statistical power [6].  

The author in [4] decomposed income distribution into subgroups such that 𝑤𝑖, 𝑥𝑖 are the weights and the 

income sources. It is assumed that the subgroups are disjoint and that all subgroups taken together constitute the 

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population (i.e. are mutually exclusive).His statistics of interest are the contribution of income sources to overall 

inequality.  

The observed data is assumed to be of the form 𝑦𝑖 = (𝑟𝑖 ,𝑥𝑖) for 𝑖 =  1 … . . 𝑛, which can be interpreted as 𝑖𝑖𝑑 

sample of size 𝑛 from a joint distribution of 𝐺 of 𝑟 and 𝑥, 𝐺(𝑟𝑖 ,𝑥𝑖), let 𝑟𝑖 denote the decomposition level of 

observational unit 𝑖 and 𝑥𝑖 its income. 

The ideology is to model sampling from a finite population as 𝑖𝑖𝑑 draws from a distribution 𝐺 of decomposition 

levels, 𝑟 and income, 𝑥. The decomposition levels could rank the cases with a particular value of 𝑥. The 

equivalent income associated with an individual rank of 𝑟 =  2 may not necessarily count twice as much as 

incomes associated with an individual rank of 𝑟 =  1, it may count in fraction. 

Let 𝐻𝑛 denotes the distribution function of income that results after the decomposition levels have been taken to 

consideration. 

Let Mean income              =    ∫ 𝑥 𝑑�𝐻𝑛(𝑥)�𝑥         

But  

 𝑑�𝐻𝑛(𝑥)� =     
∫ 𝑟 𝑑�𝐺(𝑟,𝑥)�𝑟

∫ ∫ 𝑟 𝑑�𝐺(𝑟, 𝑥)�𝑟𝑥
  

            

Hence 

� 𝑥𝑑�𝐻𝑛(𝑥)� =
∫ ∫ 𝑟𝑥 𝑑�𝐺(𝑟, 𝑥)�𝑟𝑥

∫ ∫ 𝑟 𝑑�𝐺(𝑟,𝑥)�𝑟𝑥𝑥
 

[4]. 

2.5.  Description of the mmoon Bootstrap Method  

The method 𝑚𝑚𝑜𝑜𝑛 would resample from 𝐻𝑛 distribution which is a decomposed version of 𝐹�𝑛 , the procedure 

provides and estimates different measures of statistical precision for an estimator 𝜃�. Below is the description of 

how the method works.   

Suppose a random sample of size 𝑛 is observed from a completely unspecified probability distribution. 

 𝑋𝑖  = 𝑥𝑖   , 𝑋𝑖 ∼ 𝐹,    𝑖 =  1, … . , 𝑛.    𝑥𝑖  ∼   𝑖𝑖𝑑    

 1. Construct the sample probability distribution 𝐹�, putting mass 1 𝑛�  at each point 𝑥1 , … 𝑥𝑛 . 𝐹� is an empirical 

distribution function (𝐸𝐷𝐹). 

2. Stratify the sample into 𝑟 strata, based on rank of individual 𝑥’𝑠.  

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3. Fix 𝐻𝑛, as described in decomposition of empirical distribution. 

4. With 𝐻𝑛 fixed, draw a random sample of size 𝑛 ; 𝑛 ⃛< n, with replacement from 𝐻𝑛, proportionally from each 

stratum with respect to  𝑛𝑟 𝑛� . This is the bootstrap sample (say 𝑋 ∗ = 𝑥∗).  

5. Approximate the sampling distribution of 𝐻𝑛 by the bootstrapping distribution 𝐻∗ = 𝐻𝑛 �𝑋∗,𝐹�𝑛�. 

6. Repeated realizations of  𝑋∗ are generated, producing 𝑋𝑘∗ (= 𝑥1∗, 𝑥2∗, 𝑥3∗ ,  … . .. 𝑥𝑛∗),  such that 𝑛  <  𝑛;𝑛  →

∞ 𝑎𝑛𝑑   𝑛
𝑛
→ 0 𝑎𝑠 𝑛 → ∞ . Where  𝑋𝑘∗ = 𝑋1∗, 𝑋2∗,   … . .. 𝑋1000∗ . (i.e. 𝑘 independent bootstrap samples, each 

consisting of 𝑛 data drawn with replacement). Evaluating  𝑋∗ will produce 𝜃�∗.    

7. Having chosen a particular 𝜃� (say the mean), obtain empirical bootstrap distribution of 𝜃�∗; (𝜃�1∗, 𝜃�2∗,   … . .. 

𝜃�1000∗ ).  Precision of the estimator could be tested by obtaining:       

(i) Bootstrap estimate of standard errors: After evaluating the corresponding bootstrap replications, estimate the 

standard error of 𝜃� by the empirical standard deviation of the 𝑘 replications, the bootstrap estimate of the 

standard error denoted by 𝑠𝑒� 𝜃�∗ is    

𝑠𝑒� 𝜃�∗ =   �
∑ �𝜃�𝑘∗ −  �̅�∗�2𝐾
𝑘=1

𝐾 − 1
� �

1/2

[1] 

       where:    �̅�∗ =  ∑ 𝜃�𝑘∗𝐾
𝑘=1

𝐾�    

The limit of 𝑠𝑒� 𝜃�∗ as 𝐾 goes to infinity is the ideal bootstrap estimate of 𝑠𝑒 𝜃 :  

lim𝑘→∞ 𝑠𝑒� 𝜃�∗ = 𝑠𝑒 𝜃      

(ii) Bootstrap estimate of Coefficient Variation: The coefficient of variation of a random variable is defined to 

be the ratio of its standard error to the absolute value of its mean. The bootstrap coefficient of variation denoted 

by CV (𝜃�∗) refers to the variation at the resampling (bootstrap) level and at population sampling level.         

 CV (𝜃�∗) =  𝑠𝑒� 𝜃�∗

�̅�∗�       

(iii) Bootstrap estimate of bias; Bias is the difference between the expectation of an estimator 𝜃� and the quantity 

𝜃 being estimated. The bootstrap estimate of bias based on the 𝑘 replications is:  

𝐵𝚤𝑎𝑠�  =  
∑ �𝜃�𝑘∗ −  �̅�∗�𝐾
𝑘=1

𝐾�    

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  1, pp 142-155 

 

where:    �̅�∗ =  ∑ 𝜃�𝑘∗𝐾
𝑘=1

𝐾�  

(iv) The RMSE of an estimator 𝜃� for 𝜃, is  

�𝐸 ��𝜃� − 𝜃�
2� = �𝑠𝑒�𝜃��

2
 +  𝑏𝑖𝑎𝑠�𝜃�,𝜃�

2
 

   = 𝑠𝑒�𝜃��.�1 + �𝑏𝑖𝑎𝑠
𝑠𝑒
�
2
 

when 𝑏𝑖𝑎𝑠 = 0, then RMSE = SE (minimum value) [14]. 

2.6. Simulation Study  

This study makes use of simulated data sets drawn from the Singh-Maddala distribution, which can quite 

successfully mimic observed income distributions in various countries. Two sets of simulation were done for 

large sample and moderate sample such that 𝑛 is 15000 and 500 respectively. The simulation mimic the 

parameter values in [3], such that 𝑎 = 100, 𝑏 = 2.8, 𝑐 = 1.7 (where a, b, and c are defined above). The values of 

𝑚 and 𝑛 are chosen as 20% of original 𝑛 as suggested in [8] and the values increased asymptotically. Authors in 

[15] suggested choosing replication large enough to minimize statistical error, however statistical error is 

unavoidable in most situations. Therefore, the number of bootstrap replications chosen in this study is 𝑘 = 1000.   

3. Results 

The results are presented in tables 1 & 2 and figures 1 to 8. 

 

 

Figure 1: Chart of Standard Error in Large Sample 

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

1

0 5000 10000 15000 20000 25000 30000
n 

Standard error,l 

moon

mmoon

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  1, pp 142-155 

 

 

Table 1: Summary of Bootstrap Estimates in Large Sample 

n 

 

Standard Error Coefficient of variation RMSE Absolute bias 

Moon mmoon moon mmoon moon mmoon 
moon mmoon 

3000 0.86922 0.22806 0.000316 8.29E-05 0.027487 0.002339 9.08E-14 1.70E-13 

4500 0.746153 0.186148 0.000271 6.76E-05 0.023595 0.001909 3.45E-14 8.80E-14 

6000 0.633219 0.156741 0.00023 5.69E-05 0.020024 0.001622 1.37E-13 2.89E-14 

7500 0.594175 0.145941 0.000216 5.30E-05 0.018789 0.001441 8.80E-14 8.05E-14 

9000 0.535212 0.1363 0.000194 4.95E-05 0.016925 0.001303 2.14E-14 5.71E-14 

10500 0.493749 0.126693 0.000179 4.60E-05 0.015614 0.001244 5.60E-15 1.14E-13 

12000 0.452659 0.115947 0.000164 4.21E-05 0.014314 0.001169 7.60E-14 1.22E-13 

13500 0.43135 0.112822 0.000157 4.10E-05 0.01364 0.001063 3.77E-14 1.07E-13 

15000 0.405829 0.105592 0.000147 3.84E-05 0.012833 0.001061 5.22E-14 1.37E-13 

16500 0.408741 0.099758 0.000148 3.62E-05 0.012926 0.000995 4.41E-14 2.26E-14 

18000 0.384636 0.0927 0.00014 3.37E-05 0.012163 0.000961 4.36E-14 2.28E-14 

19500 0.350265 0.09395 0.000127 3.41E-05 0.011076 0.000936 7.00E-14 4.85E-14 

21000 0.650492 0.0887 0.000127 3.22E-05 0.02057 0.000891 2.05E-13 1.39E-13 

22500 0.678618 0.085311 9.02E-05 3.10E-05 0.02146 0.000856 6.29E-14 9.66E-14 

24000 0.711078 0.082506 7.25E-05 3.00E-05 0.022486 0.000823 8.01E-13 4.80E-14 

 

 

 

Figure 2: Chart of Coefficient of Variation in Large Sample 

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0 10000 20000 30000
n 

Coefficient of Variation,l 

moon

mmoon

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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  1, pp 142-155 

 

 

Figure 3: Chart of RMSE in Large Sample 

 

 

Figure 4: Chart of Absolute Bias in Large Sample 

 

 

Figure 5: Chart of Standard Error in Moderate Sample 

0

0.005

0.01

0.015

0.02

0.025

0.03

0 10000 20000 30000
n 

RMSE,l 

moon

mmoon

0.00E+00
1.00E-13
2.00E-13
3.00E-13
4.00E-13
5.00E-13
6.00E-13
7.00E-13
8.00E-13
9.00E-13

0 10000 20000 30000
n 

Absolute Bias,l 

moon

mmoon

0

1

2

3

4

5

0 200 400 600 800 1000
n 

Standard error,m 

moon

mmoon

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Table 2: Summary of Bootstrap Estimates in Moderate Sample 

n 

 

Standard Error Coefficient of variation RMSE Absolute bias 

moon mmoon moon mmoon moon mmoon 
moon mmoon 

100 4.455687 1.16504 0.001661 0.000435 0.140901 0.036842 1.66E-13 2.26E-14 

150 3.635785 0.930918 0.001354 0.000345 0.114974 0.029438 4.80E-14 3.50E-14 

200 3.15536 0.798848 0.001178 0.000299 0.099781 0.025262 8.20E-14 2.87E-14 

250 2.764303 0.716033 0.001031 0.000268 0.087415 0.022643 1.45E-13 1.89E-14 

300 2.40943 0.656948 0.0009 0.000246 0.076193 0.020775 1.10E-13 8.71E-15 

350 2.414765 0.599584 0.0009 0.000224 0.076362 0.018961 2.04E-14 8.46E-15 

400 2.170816 0.572092 0.000809 0.000213 0.068647 0.018091 1.24E-13 2.19E-14 

450 2.193489 0.526211 0.000818 0.000196 0.069364 0.01664 2.65E-14 8.78E-15 

500 1.914705 0.514097 0.000714 0.000191 0.060548 0.016257 6.67E-14 1.48E-13 

550 1.838625 0.474448 0.000685 0.000177 0.058142 0.015003 6.01E-15 5.10E-14 

600 1.783116 0.462876 0.000665 0.000173 0.056387 0.014637 2.16E-14 3.51E-14 

650 1.731724 0.441068 0.000658 0.000164 0.054762 0.013948 7.00E-14 1.36E-13 

700 3.463449 0.420461 0.000127 3.22E-05 0.02057 0.000891 2.05E-13 1.39E-13 

750 3.018738 0.427549 9.02E-05 3.10E-05 0.02146 0.000856 6.29E-14 9.66E-14 

800 2.864389 0.405739 7.25E-05 3.00E-05 0.022486 0.000823 8.01E-13 4.80E-14 

 

Figure 6: Chart of Coefficient of Variation in Moderate Sample 

 

4. Discussion 

Tables 1 and 2 show asymptotic test results for large and moderate samples respectively. Their corresponding 

graphs are presented in Figure 1to 8. Both the tables and the figures show some statistical measures of precision 

on the two bootstrap methods. 

0

0.0005

0.001

0.0015

0.002

0 200 400 600 800 1000
n 

Coefficient of Variation,m 

moon

mmoon

152 
 



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From tables 1 and 2, it is revealed that 𝑚𝑚𝑜𝑜𝑛 bootstrap method performs better than 𝑚𝑜𝑜𝑛 bootstrap method 

in large and moderate samples respectively. The 𝑚𝑚𝑜𝑜𝑛 bootstrap method produces smaller estimates of 

standard error (SE); root mean square error(RMSE); coefficient of variation(CV);absolute bias(ABSB) when 

compared to the  𝑚𝑜𝑜𝑛 bootstrap method. In large sample, when n = 3000: SE = (0.86922;0.22806); CV = 

(0.000316;8.28E-05); RMSE = (0.027487;0.002339); ABSB = (9.08E-14;1.70E-13) for moon and mmoon 

respectively and when n = 1800: SE = (0.384636;0.0927); CV = (0.00014;3.37E-05); RMSE = 

(0.012163;0.000961); ABSB = (4.36E-14;2.28E-14). Also for moderate sample, when n = 100: SE = 

(4.455687;1.16504); CV = (0.001661;0.000435); RMSE = (0.140901;0.036842); ABSB = (1.66E-13;2.26E-14) 

for moon and mmoon respectively and when n = 300: SE = (2.40943;0.656948); CV = (0.009;0.000246); RMSE 

= (0.076193;0.020775); ABSB = (1.10E-13;8.71E-15). 

 

Figure 7: Chart of RMSE in Moderate Sample 

 

Figure 8: Chart of Absolute Bias in Large Sample 

The absolute bias of  𝑚𝑚𝑜𝑜𝑛 bootstrap method in large sample is lesser compared to moderate sample, so also 

the estimate of other measures, these confirmed that, increasing number of samples can reduce effects of 

random sampling errors which can also arise from bootstrap procedure itself., hence 𝑚𝑚𝑜𝑜𝑛 is suggested as a 

preferred method in practice.  

0
0.02
0.04
0.06
0.08
0.1

0.12
0.14
0.16

0 200 400 600 800 1000
n 

RMSE,m 

moon

mmoon

0.00E+00

5.00E-14

1.00E-13

1.50E-13

2.00E-13

0 200 400 600 800 1000
n 

Absolute bias,m 

moon

mmoon

153 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  1, pp 142-155 

 

The charts in figures 1 to 8 show the bootstrap estimate of SE, CV, RMSE and ABSB. The 𝑚𝑚𝑜𝑜𝑛 bootstrap 

method converges at a faster rate and it is asymptotically consistent. The 𝑚𝑜𝑜𝑛 bootstrap method converges at a 

slower rate, until 𝑚 becomes as large as the original 𝑛 (i.e. m = n = 15000. See fig. 1 to 8). But when 𝑚 

becomes larger than 𝑛, 𝑚𝑜𝑜𝑛 diverges, while 𝑚𝑚𝑜𝑜𝑛 still converges even after 𝑛 becomes larger than 𝑛. 

Hence, 𝑚𝑚𝑜𝑜𝑛 is asymptotically consistent. 

5. Conclusion 

The 𝑚𝑚𝑜𝑜𝑛 bootstrap method for heavy-tailed distributions in large and moderate samples is justified through 

its validity and application to empirical work. The bootstrap estimates of standard error and other measures of 

statistical precision (such as absolute bias, coefficient of variation and root mean squared error) confirmed the 

reliability and suitability of the method in practice.  

6. Recommendation 

Bootstrap researchers interested in heavy-tailed distributions should decompose the empirical distribution before 

resampling so as to overcome the difficulties posed by outliers. 

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http://www.jstor.org/stable/27590794
http://arxiv.org/abs/1010.3825
http://thescipubl.com/abstract/10.3844/jmssp.2010.84.88

	= 𝑠𝑒,,𝜃...,1+,,,𝑏𝑖𝑎𝑠-𝑠𝑒..-2..