A note on finding peakedness in bivariate normal distribution using Mathematica

Peakedness measures the concentration around the central value. A classical standard measure of peakedness is kurtosis which is the degree of peakedness of a probability distribution. In view of inconsistency of kurtosis in measuring of the peakedness of a distribution, Horn (1983) proposed a measure of peakedness for symmetrically unimodal distributions. The objective of this paper is two-fold. First, Horn’s method has been extended for bivariate normal distribution. Secondly, to show that computer algebra system Mathematica can be extremely useful tool for all sorts of computation related to bivariate normal distribution. Mathematica programs are also provided.


Introduction
Well-known characteristics of any probability distribution are location, dispersion, skewness and peakedness.Peakedness is a statistical measure and its notion is easy to understand; however, is not uniquely defined in the literature.The term 'peakedness' is usually employed synonymously with 'concentration' or inversely with 'dispersion' or 'scatter' (Wang and Serfling, 2005) but may be somewhat confusing because in essence it is a measure of the fatness of the tails of the density function.
A classical standard measure of peakedness of a probability distribution is kurtosis which is the degree of 'flatness' or 'peakedness' of a univariate probability distribution (Sahai and Khurshid, 2002).There is much confusion about how kurtosis is related to the shape of distributions.Many researchers have asserted that kurtosis is a good measure of the peakedness of distributions, which is not strictly true.One can generate examples of distributions that are flatter or more peaked, but have the same kurtosis.There seems to be disagreement about the meaning and interpretation of kurtosis which has been described as 'vague concept' (Mosteller and Tukey, 1977).Kaplansky (1945) voiced concern about the way kurtosis is typically interpreted and with examples, demonstrates that the peakedness of different probability distributions does not go with the common interpretations of kurtosis which is not as straight forward to interpret as is commonly thought.Balanda and MacGillivray (1988) wrote "it is best to define kurtosis vaguely as the location-and scale-free movement of probability mass from the shoulders of a distribution into its centre and tails."See also Balanda and MacGillivany (1990), Darlington (1970), De Carlo (1997), Groeneveld and Meeden (1984), Hogg (1974), Oja (1981), Ruppert (1987) for various criticisms on the inconsistency of kurtosis in the meaning the peakedness of a distribution.
Several alternative measures of peakedness have been proposed in the literature, for example Horn (1983), Ruppert (1987) and recently by Wu (2002) and Schmid and Trede (2003).In view of inconsistency of kurtosis in measuring of the peakedness of a distribution, Horn (1983) proposed a measure of peakedness for symmetric unimodal distributions defined by the ratio for a given probability distribution ) (x f , p and p A as shown in Figure 1. Kurtosis is usually of interest only when dealing with approximately symmetric distributions.Skewed distributions are always leptokurtic (Hopkins and Weeks, 1990).

Figure 1: Symmetrical curves
In bivariate distributions, the property of peakedness is important and has two impending problems i.e. the marginal distributions corresponding to a bivariate distribution would have peakedness (i) separately and (ii) jointly with relationship parameter, usually given by the linear correlation coefficient.Kurtosis is one such property which has not been studied for a bivariate model according to the theory developed by Horn (1983).Quraishi and Haq (1999) and Hussain et al. (2000) modified the Horn's measure for bivariate discrete probability distributions and bivariate normal distribution respectively.In this article Horn's measure of peakedness is modified for the bivariate normal distribution.It is also shown that computer algebra system Mathematica can easily be used in computation for peakedness and plotting its contour (Wolfram, 1991).

Bivariate Normal Distribution
Bivariate normal distribution provides a useful visual model for bivariate relationships just as the univariate normal distribution provides a useful probability model for a single variable.A pair of random variables 1 X and 2 X have a bivariate normal distribution if their joint density is given by X respectively gets smaller and smaller and as a consequence the peak of the probability model, which is located at ( 1 μ and 2 μ ) gets higher and higher.ρ is a correlation coefficient between 1 X and 2 X .An overview about bivariate normal distribution and its properties and applications is provided by Kotz et al. (2000).One may also refer to Rose and Smith (1996) with Mathematica implementation.A brief review of Mathematica and commands used in this article are provided in Appendix 1.

It has five parameters namely
By substituting  From the above graphs it is evident that for ρ = 0.75, distribution is highly peaked and as values of ρ decrease, it becomes flattened.

Standard bivariate normal distribution contour
The bivariate surface is perfectly understood only when the counter are drawn which are obtained by cutting bivariate normal surface with planes that are parallel to the xy-plane, and intersect at the given Z-values are heights of the parallel planes, at which these intersect the surface, as shown in Figure 4.For the Mathematica statement for contours see Appendix 3. -

Peakedness with Mathematica
The mode in a standardized bivariate normal distribution occurs at (0, 0); and a major portion of probability is concentrated, on the volume defined by cubic: where The probability that is concentrated on volume of standardized random variable 1 x and 2 x is given by To evaluate Equation ( 4), it can be expressed as a mixed difference equation which is a function of ρ , where For some selected intervals for calculated joint probabilities are displayed in Table 1.(For Mathematica code see Appendix 3).
Table 1: Joint probabilities for ρ = 0.00, 0.25, 0.50, 0.75 , another quantity, T V , can be found, which defines the volume that stands on the same area as given above, but whose vertical height is equal to which is also the function of .where 0 V is the probability/volume captured in given interval by bivariate normal distribution, and T V the total cubic volume over the given interval .

Computation for Peakedness
Some values of the parameters have been selected for the standardized bivariate normal distribution for which peakedness measure, ) (ρ M , has been found, as shown in the Table 2 using Mathematica (For Mathematica code see Appendix 3).It is important to note the relationship between the correlation coefficient ρ and the shape of the joint normal density.In order to get some idea as to how the From the graphs in it is obvious that the more squashed the density (and the ellipses) the higher the correlation.

Conclusion
Peakedness in probability distributions has been discussed extensively by using the kurtosis.There are various definitions of kurtosis in literature.However, Pearsons's measure and Fisher's measure are dependent on the values of the mathematical expectations or moments (i.e.

ρ 1 X and 2
Of these, 1 μ and 2 μ are location parameters at which the maximum ordinate of the probability distribution is located.Changes in the values of 1 μ and 2 μ do not change the peakedness of the probability distribution.However, 1 σ and 2 σ are those parameters which effect the variability of the bivariate model of probability and certainly change the peakedness.For smaller values of 1 σ and 2 σ , the variability in the random variables

Figure 4 :
Figure 4: Standard bivariate normal contours plots for ρ=0.0, 0.25, 0.5 & 0.75 It is obvious from Figure 4 that, for , 0 = ρ the surface is symmetrical and the contours are circular; but for , 0 ≠ ρ the contours are ellipses which are more concentrated. ρ