1. Introduction

Graphical presentation of data is a vital tool in sciences. Good graph conveys a great deal of information and can be used to extract new conclusions while bad graph can be misleading and confusing. Given a random sample of univariate data points, a pertinent question is whether this sample comes from some specified distribution F. Decision techniques are based on how close the empirical distribution of the sample and the distribution F are for some sample size n.

A histogram is a graphical representation of the distribution of numerical data and was first introduced by Pearson (1895). The most common form of the histogram is obtained by splitting the range of the data into equal-sized bins. Then for each bin, the numbers of points from the data set that fall into each bin are counted. The classes can either be defined arbitrarily by the user or via some systematic rules. A number of theoretically derived rules have been proposed by Scott (1992). The purpose of a histogram is to graphically summarize the distribution of a univariate data set. The histogram graphically shows the location, spread, skewness and presence of multiple modes in the data.

Quantile-quantile (Q-Q) plot is commonly used device to graphically and informally test the goodness-of-fit of a sample in an exploratory way. It is used to plot the sample quantiles against the theoretical quantiles or other sample quantiles and then a visual check is made to see whether or not the points are close to a straight line; see, Chambers et al (1983), Cleveland (1994) and Cleveland and McGill (1988).The pattern of points in the plot is used to compare the shapes of distributions, providing a graphical view of how properties such as location, scale and skewness. The use of Q–Q plots to compare two samples of data can be viewed as a non parametric approach to comparing their underlying distributions. A Q–Q plot is generally a more powerful approach to do this than the common technique of comparing histogram of the two samples, but requires more skill to interpret; see, Makkonen (2008), Wilk and Gnanadesikan (1968), Wicklin (2011).

The minima-maxima plot (M-M plot) is proposed based on the minimum and maximum of order statistics that can be theoretically computed from a specific distribution and can be estimated from a sample data. The M-M plot depends on giving more weights to the data at the extreme tails than the data at the centre of the distribution; therefore, its philosophy is “most important information for a distribution at the tails”. This plot captures all information not only about the tails of the distribution but also about the whole distribution of the data. The pattern of points in the M-M plot is used to compare the shapes of distributions and providing a graphical view of how properties such as location, scale, skewness and kurtosis. Like Q-Q plot, M-M plot is used to plot the data against theoretical extreme order statistics or sample extreme order statistics and then a visual check is made to see whether or not the points are close to a straight line but the M-M plot has more stability at the tails of the distribution than Q-Q plot. Note that all graphs and programming are done by R-software and the program is given in Appendix A.

The minimum and maximum order statistics and their characteristics to probability distributions are presented in Section 2. The M-M plots are proposed in Section 3 for one variable and two variables. An application to blood plasma levels one hour following chocolate consumption data is studied in Section 4. Section 5 is devoted for conclusion.

2. Minimum and Maximum Order Statistics

Let be a sample from a distribution function F, probability function and quantile function When the are arranged in ascending order of magnitude and then written as

is the order statistic. Since the event occurs if and only if at least of the are less than or equal to is expressible in terms of as the binomial tail probability

The expected value of order statistics is

This can be re-written as

see; David (1981).

Let

Denote the maximum of the first n random variables. Its distribution function is given by

As pointed out by Arnold et al. (2008) and Alpay (2016), clearly knowledge of the distribution of determines completely. This is true since

Moreover, Chan (1967) has shown that if then is uniquely determined by the sequenc

Let

Denote the minimum of the first n random variables. The distribution function is given by

Clearly knowledge of the distribution of determines F completely. This is true since

Also, Chan (1967) has shown that if then is uniquely determined by the sequence

For example,

if and only if is unit exponential

if and only if is triangular and

if and only if is geometric see, for example, Huang (1989).

3. Minima-Maxima Plot

M-M plot is proposed through two functions or curves plotted on the same graph as following.

Single variable

For a given data of size the theoretical minima curve based on the expected value of order statistics is defined as

From Downtown (1966) and Elamir and Seheult (2003) this can be estimated as

The theoretical maxima curve based on the expected value of order statistics is defined as

From Downton (1966) and Elamir and Seheult (2004) this can be estimated as

The M-M plot consists of two curves, the minima curve is plotted as

This curve starts from the average to the minimum value

Also the maxima curve is plotted as

This curve starts from the average to the maximum value

Both curves should tell us the whole picture about the distribution function of random variable for a given data. Also each curve in its own should reflect all the information about the whole distribution for the random variable for a given data.

Figure 1 is created for standard normal distribution at n = 30 based on the quantile fuction using plotting position on the x-axis and M-M values on y-axis using exact extreme order statistics from package EnvStats in R software as

Figure 1. M-M plot for standard normal distribution at n = 30

and

There are a lot of information can be captured from M-M plot about the whole distribution. Some of these features are

1. the joint point between the two curves is the average. Figure 1 shows the average is 0.

2. the gini’s measure of the scale of the distribution can always be obtained from the second value in each curve, . Figure 1 shows that and therefore,

3. if the Gini index of inequality can be estimated as see, Cowell (2011) and Elamir (2013).

4. the skewness can be detected easily. For symmetric distributions the two curves will be at equal distances from the mean line. Figure 1 shows the two curves at the same distances from the mean line. For right skewed distribution the maxima curve will be wider than minima curve from mean line. Also, the minima curve will be wider than maxima curve for the left skewed distribution.

5. the starting and ending of the two curves should give a clear picture about peak and tail of the distribution.

As the order statistics are available for uniform, logistic, Laplace and exponential distributions, Figure 2 shows the behaviour of M-M plot for these distributions at n = 30. The differences among them are clear and some features are

1. the joint point for two curves at averages 0.5, 0,0 and 1.

2. the first three distributions are symmetric about average and fourth one is asymmetric to the right.

3. the sharpness is clear for the second and third distributions while others are more flat. The long tails are clear for third and fourth distributions while short tail is clear for first distribution.

Figure 2. M-M plot for uniform, logistic, Laplace and exponential distributions at n = 30

Also the M-M plot can be done for discrete distributions. From Arnold et al. (2008) the extreme order statistics for binomial distribution can be obtained as

and

From Arnold et al. (2008) the extreme order statistics for Poisson distribution are

and

Figure 3 shows the M-M plot for binomial and Poisson distributions for selected parameters. Some features may be concluded as

1. the discreetness of the distributions is obvious.

2. the joint points or averages are 5, 1, 9.5 and 0.7, respectively.

3. the right skeweness is clear for second and fourth distributions while left skewness for third distribution and symmetric for the first distribution are obvious.

Figure 3. M-M plot for binomial and Poisson distributions for selected parameter. Note that for Poisson

The Pareto distribution is a heavy tailed distribution which is used a lot in economics, has a probability density function that can be written as

Where is the shape parameter which measures the heaviness of the right tail and is a scale parameter; see, Arnold (1983) and Michael (2010). The corresponding cumulative distribution function is

The expected value is

Also from Malik (1966) the order statistics are defined as

For extreme order statistic,

and

Figure 4 shows the M-M plot for Pareto distribution with parameters (10,8), (10,5), (10,2) and (10,1.25). Note the changes for the curves with changes of n terms of skewness and tails especially at the beginning and ending of curves.

Figure 4. M-M plot for Pareto distribution with different values for shape parameter and n = 30

Two variables and M-M line plot

In the sense of Q-Q plot, M-M plot can be used to compare any data with some specified distribution or with any other data. M-M line plot considers the sample as a whole and plots the sample minima and maxima against the theoretical minima and maxima of the specified target distribution F. If a correct target distribution is given, the M-M plot hugs a straight line. Therefore, it can assess if a set of data plausibly came from some theoretical distribution by constructing M-M line plot. If there is another distribution for a random variable Y, therefore, the theoretical versus theoretical is

and theoretical versus estimated is

If this closes to straight line, the data follows the theoretical distribution. Like Q-Q plot, M-M line plot can be plotted without knowing the location and scale parameters where can be computed from standard distributions. Moreover, M-M line plot can be used to check if two data have the same distribution using

The points plotted in a M–M line plot are non-decreasing when viewed from left to right. If the two distributions being compared are identical, the M–M line plot follows the 45° line. If the differences between points e₂₂ and e₁₂ in M–M line plot is more for horizontal variable than vertical variable, the distribution plotted on the horizontal axis is more dispersed than the distribution plotted on the vertical axis and vice versa. If the difference between e₂₂- mean and mean – e₁₂ is nearly zero the variable is symmetric, negative (left skewed) and (positive) right skewed. When M–M line plot is often S shaped this is indicating that one of the distributions has more than one mode and when it is often J shaped this is indicating that one of the distributions has much heavier tails than the other. This can be shown in the following cases.

Case 1: theoretical versus estimated

Figure 5 shows M-M line and Q-Q plots for theoretical and simulated data from Pareto and normal distributions. It is clear that the M-M line plot is more stable than Q-Q plot for heavy tail distributions. For the normal distribution most of the data for M-M plot are concentrated at the extreme tails while few data at the middle. On the other hand most of the data for Q-Q plot are concentrated on the middle and few of them at the extreme tails.

Figure 5. M-M and Q-Q plots for theoretical Pareto (10,3) versus simulated Pareto (10,3) in (a) and (b), and theoretical normal (0,1) versus simulated normal (100,8) in (c) and (d) and n = 50

Case 2: estimated versus estimated

Figure 6 shows two shapes for M-M line when one distribution has more than one mode like Beta distribution (0.5, 0.5), the shape is near from ‘S’ and when one distribution is very heavy, it shows ‘J’ shape.

Figure 6. M-M and Q-Q plots for simulated normal (100,10) versus simulated Beta (0.5,0.5) in (a) and (b), and simulated normal (100,10) versus simulated Pareto (10,2.5) in (c) and (d) and n = 50

Figure 7 shows the M-M line and Q-Q plots for simulated data from lognormal distribution (5, 0.5) and it is clear the difference between two curves at the extreme tails; see, Elamir (2016).

Figure 7. M-M and Q-Q plot for simulated data from lognormal distribution (5,0.5) and n = 500

Also the Min and Max normal plots will complete the picture of QQ-norm plot especially at the extreme tails of the distribution. The Min-norm plot is proposed by plotting the exact minimum order statistics of size from standard normal distribution that can be obtained from package EnvStats in R software versus estimated minimum order statistics from a data as

The pattern of points in the Min-normal and Max-normal plots must show straight line or close to straight line; see for more details, Elamir (2016).

Standard M-M plot

Away from M-M line, M-M plot still has a great value in comparisons between data where it is possible to do more than one M-M plot on the graph paper to compare between actual data and theoretical data but one has to take care of location and scale. In this aspect, standard M-M plot can be proposed as exactly M-M plot except is done for standardised values

where and are the sample mean and standard deviation, respectively. Standard M-M plot is focusing on the shape of the data after excluding the location and scale parameters.

Figure 8 shows M-M and standard M-M plot theoretical data from uniform and normal distributions versus simulated data from same distributions.

Figure 8. M-M and standard M-M plots for simulated data from uniform (75,100), normal (110,5) distributions and n = 25

4. Application

M-M, standard M-M and MM-line plots can be used to compare among several variables. For example, in the analysis of variance the data are assumed to be normally distributed and have equal variances. An article in Nature describes an experiment to investigate the effect of consuming chocolate on cardiovascular health (“Plasma Antioxidants from Chocolate,” Nature, Vol. 424, 2003). The experiment consisted of using three different types of chocolates: 100g of dark chocolate, 100g of dark chocolate with 200mL of full-fat milk, and 200g of milk chocolate. Twelve subjects were used, 7 women and 5 men, with an average age range of years, an average weight of kg, and body-mass index of kg m^-2. On different days a subject consumed one of the chocolate-factor levels and one hour later the total antioxidant capacity of their blood plasma was measured in an assay; see, Montgomery (2013). Data is summarized in Table 1.

Table 1. Blood plasma levels one hour following chocolate consumption*

Figure 9 presents M-M plots for the data from this experiment. The result is an indication that the blood antioxidant capacity one hour after eating the dark chocolate is higher than for the other two treatments. The variability in the sample data from all three treatments seems very similar. Also the shape for three treatments looks similar to normal distribution. Moreover, Table 2 gives some measures can be estimated from the graph.

Figure 9. M-M and standard M-M plots for blood plasma levels one hour following chocolate consumption data

Table 2. Estimated measures from figure 9 for blood plasma levels data

Figure 10 shows M-M line plot using theoretical standard normal distribution versus the three treatments for blood plasma levels one hour following chocolate consumption data with linear regression line added to each line based on

Sample MM values = a+b Theoretical MM values

The same conclusion can be made from this plot as M-M plot.

Figure 10. M-M line plot based on theoretical standard normal distribution versus the three treatments for blood plasma levels one hour following chocolate consumption data

5. Conclusions

Minima-maxima plot depends on extreme order statistics and it is introduced to capture all information not only about the tails of the distribution but also about the whole distribution of the data. The pattern of points in the M-M plot is used to compare the shapes of distributions and providing a graphical view of how properties such as location, scale and skewness. Also, it used to plot the sample data against the theoretical extreme order statistics or sample extreme order statistics and then a visual check is made to see whether or not the points are close to a straight line. The main advantage of M-M plot over Q-Q plot is its stability at the extreme ends of the distribution. Actually, the main idea behind M-M line plot is that the average is a good representative for data at the middle but the extreme tails must be given most attention or weights.

One limitation of M-M plot is when the extreme order statistics are not defined such as the Cauchy distribution where few extreme order statistics are not defined. But the M-M plot may still be plotted by using the available information and ignoring undefined values. Of course, in this case some information will be lost.

Appendix A: R program for M-M plots and Q-Q plot