









































Communication, Society and Media 
ISSN 2576-5388 (Print) ISSN 2576-5396 (Online) 

Vol. 2, No. 1, 2019 
www.scholink.org/ojs/index.php/csm 

29 
 

Original Paper 

Gender Wise Distribution of Income Using L-Moments Method 
Muhammad Alam1, Saeed Ullah Jan2* & Alam Zeb1 

1 Faculty of Basic & Applied Sciences, Department of Statistics, International Islamic University, 

Islamabad, Pakistan 
2 Department of Computer Science & IT, University of Malakand, Chakdara, Pakistan 
* Saeed Ullah Jan, Department of Computer Science & IT, University of Malakand, Chakdara, Pakistan 

 

Received: January 31, 2018    Accepted: February 9, 2019    Online Published: February 16, 2019 

doi:10.22158/csm.v2n1p29                     URL: http://dx.doi.org/10.22158/csm.v2n1p29 

 

Abstract 

The main purpose of this work is to explore the income distribution of both male and female in 

Pakistan over the period of 2010-2011. For this purpose, the lognormal distribution with known 

parameters is used as a model and its unknown parameters are estimated by three methods that are 

likelihood, moments and L-moments. The results show that citizens of Pakistan are not equal in income 

and the probability plot suggested that the income of the male is greater than that of a female in 

Pakistan. Moreover, for small sample size, the best method of parameters estimation is the L-moments, 

while, for large sample size the best method is a maximum likelihood. Findings of the study suggest 

that suitable policy is required to maintain equality in income distribution in the country. It will 

consequently reduce the gap among rich and poor and will certainly improve social welfare. 

Keywords 

distribution, lognormal, likelihood, l-moments, ratio, population 

 

1. Introduction 

The importance of income throughout the world is undeniable. It is a backbone for a country, especially 

in Pakistan is important as breathing air, without air there is no hope for life. Without having a 

reasonable source of earning there cannot be even a single ray of hope for survival. The lognormal 

distribution (Note 1) is used in several fields of life for estimation of parameters, e.g., they are used in 

Statistics, Geology, Medical Science, Environmental Science, Technology, Ecology, Social science, 

and income. Incomes within countries generally adopt a skewed distribution with a long heavy tail. 

Lognormal distribution has been found best fit on the average for income distribution. This can also 

prove through the goodness of fit test such as Chi-square test, Anderson Darling test (Albreht & 

Klazinga, 2009). 



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The lognormal distribution is a continuous probability distribution. The probability density function of 

the three parametric lognormal distributions is given as: 

            (1) 

Where - 0 

Whereas parameters, 

 Scale, 

 Shape and  

 Location or Threshold 

The L-moments (Note 2) can be used for characterization and explanation of theoretical probability 

distributions, narration, explanation of experimental data samples, assessment of different parameters, 

assessing of probability distributions and hypothesis testing. The major benefit of L-moments over 

traditional moment is that the L-moments illustrate a broad range of distributions, suffered less from 

the effect of sampling variability and are more powerful than when there are outliers in the data by 

(Bílková, 2008). It is used for extreme events such as rainfall, flood, earth quick, droughts, heat waves, 

snowfall, famine, and income distribution. The L-moments technique gives the correct result in the 

estimation of parameters as compared to the maximum likelihood, method of moments (Bílková, 

2011). 

Let suppose  is a vector of observations in a sample which can be described by the model , 

where  is the parameter value. The given model , will give us the probabilities of the various 

values of . When we define the maximum likelihood then we have to define the likelihood function 

first (Bílková, 2012). The likelihood function for a given value of  is  as a function of . The 

MLE is a reasonable method of estimation because it locates that value of  for which the observed 

data are most probable. The MLE enjoys some good properties in large sample size such as normality, 

efficiency, consistency, and un-biasedness. Suppose a random sample  are drawn 

from a distribution with probability density function ,  where  is parameter space 

(Bílková, 2012). The likelihood function is:  

                               (2) 

Let suppose we have random samples ,  from a distribution having probability density 

function is , and ),  which are considered as unknown vector parameters. 

We then define the  population moment at the origin “0” as, for continuous distribution: 

                       (3) 



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The advantages of the moment’s method are that it is an old and simple technique used for the 

estimation of parameters and a very simple method for finding the estimators of parameters. Its 

estimators are consistent and best method in case of point estimation parameters. It is an easy method 

to find an estimator and also in the case when the other methods fail to find. It is used in case of large 

sample size. Its first moment equal to mean, second to variance, third to skewness and forth one is 

equal to kurtosis (Arltová, 2013).  

The drawbacks of the moment’s method are that this method is not available for every distribution. 

Moments method estimators are not necessary to be sufficient. 

The Hosking et al. (2005) is used for the data of Household Income and Expenditure Survey (HIES) 

and Pakistan Integrated Household Survey (PIHS) w.e.f 1963 to 2002 for both rural, urban citizen and 

was decided that the main goal of economic growth is to improve the living standard of ordinary men, 

but only economic growth is not sufficient for the standard of living of a common men, the 

improvement in the income distribution is also playing a vital role in the betterment of a layman life.  

But unluckily the ratio of poverty becomes an increase in the 1990s and the economist focuses to find 

the ratio of the poor population instead of improving the measurement of the income distribution. 

In Pakistan, the income inequalities have also been estimated for both methods, i.e., Pakistan Integrated 

Household Survey (PIHS) and Household Income and Expenditure Survey (HIES). 

The wage data of the Czech Republic from 2004-2005 apply different methods of parameters for the 

aforesaid estimation such as quantile method, moment method, and maximum likelihood method. By 

applying lognormal distribution one can easily decide the results of the lognormal distribution (Kemal, 

2009). 

Rehman et al. (2008) analyzed that the distribution of income, growth and its development in Pakistan 

is because of income inequality and create financial hurdles to all them respectively. In the early 

periods, several attempts were made to make a relation between income inequality and economic 

growth, but there is no proper mechanism was developed. Rehman et al. (2008) concluded that there is 

a reverse correlation between income and economic growth.  

The data on household income per capita in the Czech Republic from 1992-2008 decided that the 

method of L-moments for estimation of parameters gives us the correct result as compared to other 

methodologies like maximum likelihood, moment subject to the condition that the data should be 

separate, for grouped data, these methods will give similar results of Arltová (2013). 

The Census data record from Czech Statistical Office, income data from statistical surveys and wage 

data taken from Czech’s official website shows that the L-moments technique gives the correct result in 

the estimation of parameters as compared to the maximum likelihood, method of moments and quantile 

method for individual data subject to the condition that data should not be in group because the in 

grouped data all the four methods give a similar result like that of Bílková (2008). 



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Also, the income data of the Czech Republic discussed by Steinführer et al. (2010) suggested that 

method of L-moments for estimation of parameters provides precise consequence as compared to the 

other methods of estimation, such as the method of moment, a method of maximum likelihood in case 

of ungrouped data; for grouped data, these methods give the same results. The household per capita of 

the Czech Republic proposed by Kemal (2008) that for income and wage distribution the L-moments is 

the best method for estimation of parameters as compared with the method of moment, quantile method, 

and method of maximum likelihood. 

Furthermore, the data of the Czech Republic’s household income from 1992-2008 has also been used 

for the lognormal distribution. Also, apply four methods of parameters estimation, which includes 

L-moments, quantile, maximum likelihood, and moment method. They decided that the method of 

L-moments gives more accurate results than other methods by Albreht (2008). 

There are two types of data; the first one is the Czech Republic per capita household income in years 

1992, 1996 and 2002, and the second is the statistical survey micro-census of years 2005, 2006, 2007 

and 2008. They used the three parameters of the lognormal distribution. And conclude that the 

accuracy of the method of L-moments is better than the other methods of estimation, that are maximum 

likelihood and the quantile method, a method of the moment for individual data. But for the grouped 

data the method of L-moments, a method of maximum likelihood, a method of moments, and the 

quantile method gives similar consequences by Langhamrová and Bílková (2008). 

 
2. Materials & Methods 

The Income data are taken from Pakistan Social and Living Standard Measurement Survey (PSLSMS) 

(Note 3) of 2010-2011 of the Pakistan Bureau of Statistics (PBS) Islamabad. The calculation of a 

national income is important. From the national income, we can find the performance of a country 

during that year. From the national income, we can also find out whether the financial system is 

increasing or decreasing. The national income depends upon the wages, profits, rents, interest, and 

business. Due to these factors, we can improve our national income and from national income, we 

improve the per capita income and also the standard of living. The income distribution also used to find 

the proportion of low, middle and high incomes workers in a country. 

The main goal of economic growth is to improve the standard of life, but only economic growth is not 

sufficient for the standard of living of peoples; the improvement in the income distribution is also 

playing an important role in the betterment of the human life. But unluckily the ratio of poverty 

becomes an increase in the 1990s and the economist focuses to find the ratio of the poor population 

instead of improving the measurement of the income distribution. Income inequality is a main financial 

problem to all the word. In the early periods, several attempts have been made to make out a relation 

between the income inequality and the economic growth, but there is no conclusion find between the 



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income inequality and the economic growth but conclude that there is a reverse correlation between 

income and economic growth. 

The conventional moments are used to estimate the parameters of a distribution but the conventional 

moments for small sample size are not always convenient, moreover, we can also specify a distribution 

by its L-moments even if its conventional moments do not exist. The sample convenient moments are 

used to summarize the mean, variance, skewness and kurtosis, similarly, the sample L-moments are 

used to summarize the location, scale, skewness, and kurtosis for a specified distribution. 

Let  be a random variable with cumulative distribution function and quantile function  

and let  be the order statistics of a random sample of size “ ” taken from the 

distribution of . The L-moments of  can be defined 

     

    

   

  

The general term can be written as 

 
Now the expectation of an order statistic may be defined as  

         (2) 

The unique specification of L-moments is that it specifying the distribution if some of its conventional 

moments do not exist. 

The L-moments ratios (Note 4) can be defined as: 

,    

The L-CV is                        

The probability weighted moments is defined as  

                   (3) 

The L-moments of a probability distribution are given as 

     

  

   

  

 



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And for general 

                   (4) 

The sample L-moments may also be defined as 

      

     

        

   

And for the general term 

  Where 

                       (5) 

 

3. Results and Discussion 

In Table 1,  is the mean of the data, the value of  is 0.4047 that indicates that there is inequality in 

the male income data.  is the L-skewness and its value is 0.4399 which is positive so the male 

income data distribution will be positively skewed, it indicates that most of the peoples income will be 

on the left side of the mean and the extreme income means the high income will be on the right side of 

the mean, , ,  are the three quartiles of the male income data as shown in the table given below. 

 

Table 1. L-Moments Ratios and Quartiles for Male Income in Pakistan 

       

131900 0.4047008 0.4399626 0.3190783 62400 96000 151200 

79352.53 0.5651828 0.5235219 0.3122305 24000 36000 96000 

 

Table 2. Parameters Estimation by Different Methods of Three Parameters Lognormal 

Distribution for Male Income in Pakistan 

 

 

 

 

 
 
 
 

 
 

Methods of estimation    
L-moments 11.14 0.94 24366.17 

Moments 11 1 31449.8 

Maximum likelihood 11.5 0.7 -2001 



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Figure 1. Probability Distribution Plot of Male Income in Pakistan 

 

From the above figure, we conclude that 25% of male income is less than 62000, 50% of peoples have 

income below than 96000. And 75% of the total population income is less 150000, and the last 25% of 

the population that have income above than 150000. 

 

Table 3. L-Moments Ratios and Quartiles for Female Income in Pakistan 

       

79352.53 0.5651828 0.5235219 0.3122305 24000 36000 96000 

131900 0.4047008 0.4399626 0.3190783 62400 96000 151200 

 

In Table 2, the average income of female is less than the male income; the value of  is 0.5652 that 

indicates that there is inequality in the female income data. As we compare the variation of male 

income with female income then there is a large variation in female income as compared with male 

income, the L-CV of male income is less than the female, and also the skewness of female income is 

greater than male income,  is the L-skewness and its value is 0.5235 which is positive so the female 

income data distribution will be positively skewed. It indicates that most of the female income will be 

on the left side of the mean and the extreme income means the high income will be on the right side of 

the mean, , ,  are the three quartiles of the female income data. 

 

 



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Table 4. Parameters Estimation by Different Methods of Three Parameters Lognormal 

Distribution for Female Income in Pakistan 

Methods of estimation    
L-moments 10.58 1.15 2663.74 

Moments 11.4 0.8 -38872.5 

Maximum likelihood 10.7 1.1 605 

 

From the below female income plot shown in Figure 2, we see that the first 25% of the population has 

income below than 24000 and 50% of the population income less than 36000. And 75% of the total 

female population income is less than 96000 and the last 25% of the female population that have 

income greater than 96000. As we see that the male income is greater than female income in Pakistan. 

 
 
 
 
 
 
 

  
 
 
 

 
 
 
 

 
Figure 2. Probability Distribution Plot of Female Income in Pakistan 

 

 

 

 

 

 



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Table 5. Absolute Bias and RMSE for Different Methods of Estimation of Parameters for Sample 

Size n=20 

Methods of 

estimation 
    RMSE 

L-moments  11.59 0.76 -8390 35892 162128 

Moments 12.40 0.40 -127937 151951 231421 

Maximum 

likelihood  

11.64 0.72 -11172 69622 211036 

 
When n=20, small sample size the best method of estimation is L-moments because its absolute Bias 

and RMSE are less than the Moments method and Maximum likelihood method. 

In last we compare the parameters of lognormal distribution obtained by L-moments method, Moments 

method, and Maximum likelihood method, with respect to absolute Bias and RMSE. We take different 

sample sizes from the data and also take the random numbers of the same size that we selected from the 

real income data and then find the absolute Bias and RMSE, so the best method will be that whose 

absolute Bias and RMSE are small. So for a small sample size the best the L-moments method, and for 

large sample size the best method is the Maximum Likelihood. 

 

Table 6. Absolute Bias and RMSE for Different Methods of Estimation of Parameters for Sample 

Size n=30 

Methods of 

estimation 
    RMSE 

L-moments 11.2

5 

1.1

0 

15673 39066 233841 

Moments 12.0

6 

0.7

3 

-68074 74522 293810 

Maximum 

likelihood 

11.2

2 

1.1

8 

17360 83800 479023 

 

When n=30 then, in this case, the L-moments method for estimation of parameters is best because its 

absolute Bias and RMSE are less than the other methods. 

Similarly, for n=50 the best method of parameters estimation is the maximum likelihood because its 

absolute Bias and RMSE are less than the L-moments method and moments method as shown in the 

below table. 

 



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Table 7. Absolute Bias and RMSE for Different Methods of Estimation of Parameters for Sample 

Size n=50 

Methods of 

estimation 
    RMSE 

L-moments  11.5

9 

0.71 -8310 44740 182374 

Moments 12.3

4 

0.39 -116996 129186 192291 

Maximum 

likelihood 

11.4

8 

0.78 980 12203 152556 

 

Table 8. Absolute Bias and RMSE for Different Methods of Estimation of Parameters for Sample 

Size n=100 

Methods of estimation     RMSE 

L-moments  11.51 0.70 380 13826 145428 

Moments 11.18 0.88 23264 27413 157953 

Maximum likelihood 11.62 0.63 -8860 3176 119216 

 

But when n=100 means when sample size increases the best method of estimation of parameters is 

maximum likelihood method because its absolute Bias and RMSE are less than other methods that are 

L-moments and moments method. 

 

Table 9. Absolute Bias and RMSE for Different Methods of Estimation of Parameters for Sample 

Size n=250 

Methods of 

estimation 
    RMSE 

L-moments  11.11 0.91 20654 17299 233256 

Moments 11.10 0.98 14758 3962 199615 

Maximum 

likelihood  

11.52 0.65 -6640 2095 158376 

 

And when n=250 or above than 250 means by increases sample size the best method of estimation of 

parameters is maximum likelihood method than L-moments method and moments method because its 

absolute Bias and RMSE are less. 

 



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4. Conclusion 

Incomes within countries generally adopt a skewed distribution with a long heavy tail; lognormal 

distribution has been found best fit on the average for income distribution in Pakistan. We estimate the 

parameters of lognormal distribution by different methods of estimation that are the L-moments 

method, moments method and maximum likelihood method. We find the parameters of these three 

methods for all male and female in the whole Pakistan. We also find the probability distribution plot for 

male and female income data. The citizens of Pakistan are not equal in income, a few groups of peoples 

are very rich and some are very poor. Finally, we compare the parameters of lognormal distribution 

obtained by L-moments method, moments method, and maximum likelihood method, with respect to 

absolute Bias and RMSE for different sample size. So the best method of parameters estimation will be 

that whose absolute Bias and RMSE are less. So in the case of small sample size, the best method of 

parameters estimation is the L-moments, and for large sample size the maximum likelihood method. 

The moments method is also used in case of large sample size but its estimators are not efficient and 

not necessary to be sufficient. 

 

References 

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Hosking, J. R. M., & Wallis, J. R. (2005). Regional frequency analysis: An approach based on 

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Rehman, H. U., Khan, S., & Ahmed, I. (2008). Income distribution, growth, and financial development: 

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Notes 

Note 1. The Lognormal distribution is used to claim that the distribution is positively skewed or not. 

The distribution of income is also positively skewed. So we use the lognormal distribution for income 

data. 

Note 2. L-moments are the outlook of statistics of the order of certain linear combinations and can be 

defined for every random variable whose mean exists. And can be used for characterization and 

explanation of theoretical probability distributions 

Note 3. The data are taken from Pakistan Social and Living Standard Measurement Survey (PSLSMS) 

of 2010-2011. 

Note 4. The L-moment ratio can be obtained by higher order L-moment divided by a measure of 

dispersion that as t_r=l_r/l_2 . The t_3 used for skewness and t_4 used for kurtosis, l_1 denoted for the 

mean of the data, t is for L-CV which is similar to the coefficient of variation and the value of t will 

0≤t<1. 


