Gender Wise Distribution of Income Using L-Moments Method

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.


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Published by SCHOLINK INC. 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) 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 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

Materials & Methods
The 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 Electronic copy available at: https://ssrn.com/abstract=3339058 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 The L-moments of a probability distribution are given as And for general (4) The sample L-moments may also be defined as And for the general term Where (5)

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.

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. 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. 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
Electronic copy available at: https://ssrn.com/abstract=3339058  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. 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  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. 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.

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.