A Class of Bell-Shaped Distributions with Bounded Supports

Abstract

In this article we introduce a new family of parametrized density functions that are bell-shaped and supported by a finite interval (0, 1). The corresponding random variable X assumes the structure where (W, V) is a bivariate Dirichlet random vector with parameters and is any symmetric random variable on (0, 1). We provide the formulation of our bell-shaped density function in terms of Euler’s and Appell’s integral representations of hypergeometric functions and derive closed form expressions for the density for specific parameter values. We provide closed form characterization for the moments of X. Graphical illustrations are provided for different choice of . We provide a practical simulation procedure for generating values for X. The distribution’s symmetric shape along with its bounded support, make it an attractive alternative to the normal distribution when modelling real-life phenomena observed over a finite interval. We demonstrate the distribution’s suitability for such situations by using anthropometric data. The class of bell-shaped densities constructed in this work define a rich class of kernel functions that can be used for estimation purposes in many statistical methods, such as regression, time series and density estimation.

Introduction

In this paper we present a family of parameterized symmetric bell-shaped probability density functions defined on bounded intervals.

The continuous random variables for this family of densities assume the following structure. Let (W, V) be a bivariate Dirichlet random vector with parameters on (0, 1) ; . Let be a symmetric continuous random variable on (0, 1) independent of (W, V) . We define

.

As a random function, X is linear in with random intercept W and slope V. The resulting density functions assume a smooth bell-shaped density function on (0, 1). Surprisingly the irregularities of the shape of the density of becomes smooth through the transformation , providing a fairly simple and board method to construct Bell-Shaped densities on finite intervals.

In this article, We consider the cases where assumes a beta, uniform, and triangular distribution. This makes this family of distributions particularly suitable for modeling real-life phenomena that are observed over finite ranges, avoiding the problems of over-representation or misrepresentation inherent when using distributions with infinite support to model such data.

Symmetric data on a finite interval are usually modelled with probability distributions that have infinite support. In most cases, little accuracy is lost, however, when the random variable represents cost, consumer based indicators, or rates the left side of the distribution tail can be grossly misrepresented. Unbounded support can lead to over representation of the tails of the distribution attributing non-zero probabilities to impossible outcomes and obscuring the probability of extreme or rare events that are possible. Inferences drawn from this approach lead to misleading results or results that lie outside of the variables possible support. For bounded data, the log-normal and truncated normal distribution are often times used as alternatives, although they provide better representation of the data characteristics, these alternatives do not necessarily provide better solutions. Common pitfalls include limitations on the value of the scale and shape parameters and overly robust or overly sound models [8]. The parameters in the formulation of our bell-shaped density functions provide the possibility for the density function to have light or thick tails near the interval boundaries. This is an important feature in probabilistic modelling.

The normal distribution is commonly used as the prior distribution for the unknown parameter in inference problems using Bayesian analysis. This is somewhat not realistic as the normal distribution is supported by the entire linear line, while there are typically restrictions on the possible values assumed by the population parameters. For instance, suppose the parameter of interests is , the AR(1) parameter in an autoregressive model of order one, then . In this situation is not feasible to assume that the normal distribution can be used as a prior distribution for . Some researchers inevitably reduce the variance of the normal distribution to fit it into the desired interval where the unknown parameter takes value [10].

The bell-shaped densities that are established in this article are suitable for overcoming this issue. Our bell-shaped density functions can be adjusted to have support on a given finite interval and vanishing on the boundaries of that interval. This class of bell-shaped distribution is applied in many areas of research a review of which can be found in Bahatti and Do [2] while dealing with copula bivariate distributions (see also the work of Nguyen and Bhatti [1]).

This paper is organized as follows. Section 2 provides the formulation of the probability distribution and its properties. An explicit form for the probability density function and its properties are established in Sect. 3. We explore the implications of using different distributions for the centering parameter in Sect. 4. Section 5 discusses how this distribution can be applied in modelling applications and demonstrates its use on real data. Finally, concluding remarks are given in Sect. 6.

Family of Bell-Shaped Density Functions

Suppose that (W, V) is a bivariate random vector that follows a Dirichlet distribution of order with parameters , respectively. The joint probability density function of (W, V) iswhere andA random factor component is introduced and functions as a centering parameter. It is assumed that follows a symmetric probability distribution function defined over an interval (0, 1) independent of (W, V). This pivotal assumption will hold throughout the course of this work and is a key factor in the derivation of the results presented. The random variable X is then defined as a linear transformation of (W, V) and , specifically;

2.1

We refer to X as a standard bell-shaped density (BSD) random variable with paramteres , and centering random parameter , abbreviated bywhere the interval (0, 1) defines the support of X. Conclusively, X has a bell-shaped symmetric distribution for any class of symmetric distributions assumed for defined over (0, 1). The formulation of X in this manner intrinsically subsumes a reasonably large class of bell-shaped distributions defined on a finite interval (0, 1) and symmetric about 1/2. The standard BSD random variable is transformed byto take values on a given interval (a, b). When the support is defined on (a, b), we refer to X as a BSD random variable with parameters, a, b, , and centering random parameter , denoted byThe expression of X given in (2.1) is particularly fortuitous; establishing symmetry, enabling straightforward derivation of the density function and also providing a formulation for random variate propagation.

Properties of the BSD Random Variable

This section provides an investigation of the BSD distribution and its properties. It is shown how the BSD random variable as a construct of bivariate Dirichlet random variables and a symmetric centering variable ensures the symmetry of the probability density function. The conditional distribution of X for any given is first derived from which the probability density function of the BSD random variable is extracted along with its moments. This is followed by an examination of the distribution’s properties and their implications.

Establishing Symmetry

Theorem 3.1

The distribution of the BSD random variable defined on (0, 1) is symmetric.

Proof

Recall that , where (W, V) is a bivariate Dirichlet random vector such that (W, V) possesses joint density functionwhere andIt is assumed that follows a symmetric distribution independent of (W, V). To prove the symmetry of the BSD distribution, it suffices to show that X and possess the same distribution. Note that (W, V) and possess the same distribution and since is assumed to be symmetric, it follows that and also follow the same distribution. The following result is obtained:  

When defined over (0, 1), the distribution of X is symmetric around 1/2.

The Probability Density Function

Expressions for the conditional density functions of X given the centering variable , and subsequently the marginal density function of X are given below. As will be become apparent in the next theorem, the density functions are expressed in terms of special hypergeometric functions. The notations ₂F₁ and are used to denote Euler’s and Appell’s integral representations of the hypergeometric function respectively:

3.1

3.2

where , [9].

Theorem 3.2

If a random variable X is of the form and construct given in Eq. (2.1), then the conditional density of X, , defined on (0, 1) can be expressed as:

3.3

where

3.4

3.5

and the functions and are defined as

3.6

3.7

The form of the function ₂F₁ used in Eqs. (3.4) and (3.5) is given in (3.1).

Proof

To begin, the joint conditional density of (W, X) given is specified as,where and , andIt follows that for a given value of , , is by definition

3.8

Let and let . Expressing in terms of y gives:which reduces to the following formSubstituting with Euler’s integral function given in Eq. (3.1), the above integral becomesproviding the equivalent of .

The same intuition is used to derive the expression given in . Namely, for a given value of , ,

3.9

By defining

3.10

the following expressions can be derivedSubstituting the above terms in (3.9), becomes

3.11

where

3.12

With some algebraic manipulation, (3.11) reduces to

3.13

where

3.14

Substituting with the hypergeometric function given in (3.1), (3.11) becomes;

3.15

which is equal to .

Corollary 1

If and , then the conditional density function , is in fact the density function of the generalized two-sided power distribution given in Soltani and Homei [11].

Theorem 3.3

Let . Assume that is a continuous and symmetric random variable defined over (0, 1). Then the probability density function of X is given bywhere g(x) is of the form

3.16

3.17

and is as defined in (3.6).

Proof

Expressing the marginal density in terms of the conditional density function obtained in Theorem 3.2 gives:Assume that and , the marginal density can then be expressed as . It must be shown that .Suppose that , then becomeswhich is equal togiving the expression of . And hence, .

Corollary 2

.

Remark 1

Note that which in turn gives .

Remark 2

When defined over the interval [a, b], .

Remark 3

The BSD is a unimodal distribution that is symmetric around its mode and bell-shaped with one inflection point at the mode for , , and two inflection points for , , giving that if then as well. This can be can be verified numerically by plotting the first and second derivatives for f(x).

Note that the density function is expressed in terms of an improper integral and as such does not have a closed form whereby explicit formulations for estimation methods and Monte Carlo simulation can be employed. Contriving X from expression (2.1) alleviates these concerns.

The Moments

The moments of the BSD random variable are derived in terms of the distribution of , where it is assumed that is a continuous random variable with a symmetric distribution.

Theorem 3.4

Let . Then

1. ;

2. ;

3. 3.18

   where

   3.19

Proof

To prove Theorem 3.4, we refer to the formation of X as a weighted expression of a bivariate Dirichlet distribution given in Eq. (2.1) and exploit the characterizations and assumptions therein. Primarily, the moment about the origin of the joint distribution of (W, V) isand it follows that the marginal expectations of W and V are and . Since the expectation of is equal to 1/2, it follows that the expectation of X is simply the evaluation of the expression :which reduces to 1/2, completing the proof of part (1) of Theorem 3.4.

The moment of X given in statement (3.18) of Theorem 3.4 is derived by evaluating the expression, using the binomial theorem:Evaluating gives the expression of given in (3.19). Evaluating for provides the expression in statement 2 of Theorem 3.4. Since is a continuous random variable that follows a symmetric distribution function, it follows that . The second moment,is expressed in terms of the second moment of , .

The BSD Distribution for Certain Distributions

Once the distribution of is specified, a complete expression for the BSD distribution can be obtained and the moments evaluated. There are numerous continuous distributions that can act as functional representations of the centering parameter complying with the required assumptions of symmetry and finite support defined over (0, 1). The complexity of the density and the number of parameters it adds to the resulting distribution need to be considered when selecting . The parameters in the formulation of our bell-shaped density function facilitate the capacity for the density function to have light or thick tails near the interval boundaries. This feature is also important in distribution fitting. This feature of our bell-shaped density functions is illustrated in the figures presented in this section.

With this in mind, two specific distributions for are examined to complete the expression of the BSD distribution:

• the beta distribution, ,

• and the symmetric triangular distribution, Tri(1/2).

Both distributions are defined over (0, 1).

BSD with a Beta Distributed Centering Parameter

With reference to the expression given in (3.3) which states that the marginal density of X is:The following theorem gives the form for g(x) when the centering parameter follows a beta distribution, .

Theorem 4.1

Let where . Then, the function g(x) given in Theorem 3.3 becomes:where

4.1

4.2

4.3

4.4

4.5

and is defined in (3.6).

Proof

Note that g(x) is given bywhereFurthermore, for , it follows that . Hence, for , isand g(x) becomes,

4.6

whereFurther evaluation of Eq. (4.6) gives the following form for g(x):whereUsing the representation of the First Hypergeometric Function of Appell given in (3.2), g(x) can be expressed as

4.7

where

4.8

and , , and are as defined in (4.1), (3.6), (4.4), (4.2) and (4.3), respectively, giving the desired expression of g(x) completing the proof.

It is evident from Eq. (4.7) that the number of distribution parameters changes. When the random centering parameter follows a beta distribution with shape parameters, and , the random variable X follows the BSD distribution with parameters .

Corollary 3

Attributing a value of 1 to the shape parameter, , reduces to a uniformly distributed random variable producingwherereducing the parameters of the corresponding BSD distribution to two.

Figure 1 illustrates the variations in the form of the BSD density function with a distributed centering parameter for selected values of the parameters , and . The blue line in the plots represents the special case when and . The density function clearly maintains a bell-shaped form with the tails terminating at the (0, 1), however, the graphs demonstrate that a decrease in the value of increases the distribution of the probability in them. In contrast, increases in the value of and lend to an increase in the distribution’s peakedness, giving it a sharper shape. Of particular interest is the rightmost plot in the top row of Fig. 1. We see that the bell-shaped density with parameters exhibits light (or thin) tails at the (0, 1) boundaries, this is in direct contrast to the thick (or fat) tails of the bell-shaped density with parameters .

[See PDF for image]

Fig. 1

The probability density function of the BSD distribution with for values of , , and specified

With the distribution of specified, Theorem 3.4 can be used to evaluate the moments of X. The mean of the BSD distribution is fixed at 1/2, however, the variance of X remains a function of the parameters and in addition to the parameter(s) introduced through the distribution of . When , the variance of X is,The parameter dynamics and their impact on the value of the variance is examined in Fig. 2. The blue line in the plots represents the special case when and . It can be seen how changes in each parameter value affects the variance. The plots show an increase in the variance as the value of increases and the value of decreases. Increases in the value of also produce sharp decreases in the distribution’s variance.

[See PDF for image]

Fig. 2

The behaviour of the variance of the BSD distribution with for values of , , and indicated above

Closed Form Expressions for the Density Functions

We present proper formulas for the BSD densities when has a uniform (0, 1) distribution, and . The table below provides the corresponding BSD density functions, , when .

n	f(x)
1
2
3
4
5
6
7
8
9
10

The plots in Fig. 3 depict the BSD denisty functions, , when listed in the table.

[See PDF for image]

Fig. 3

Bell Shaped-Densities, . The legend on the right of the graph gives the value of n

The BSD Distribution with Triangularly Distributed Centering Parameter

Suppose that the centering parameter follows a symmetric triangular distribution defined on (0, 1), Tri(1/2), with a density function given as follows:

4.9

Recall from Theorem 3.3 that the density function of isTheorem 4.2 gives the density function of the BSD distribution under the assumption that follows a triangular distribution.

Theorem 4.2

Let have the symmetric triangular distribution given in (4.9) defined over (0, 1). Then, the function g(x) in Theorem 3.3 assumes the following formulation.

4.10

where

4.11

4.12

and the expressions for , are given in (3.6) and (4.4), respectively.

Proof

The density is derived by substituting for the density of in the expression of g(x) given in (3.17) in Theorem 3.3.

One obvious advantage to using a symmetric triangular distributed centering parameter is that the resulting distribution maintains the same number of parameters. Changes in the BSD density’s peakedness in response to different parameter values can be observed in Fig. 4. Plots of the variance and how it changes with respect to different parameter values are given in Fig. 5.

[See PDF for image]

Fig. 4

The probability density function of the BSD distribution with for values of and specified

[See PDF for image]

Fig. 5

The behaviour of the variance of the BSD distribution with for values of and indicated above

Estimation, Distribution Fitting and Modeling

The density of the BSD distribution does not posses a closed form. Nevertheless, the BSD distribution may still be used effectively to model real data and draw inference. The simulation of random variates, estimation of the distribution parameters and distribution fitting are elucidated in this section.

The formation of as a linear construct defined in (2.1) facilitates the generation of BSD random variates. This is a simple three step process. For a random sample of size n: Step 1

Generate n random bivariate observations for (W, V), ;

Step 2

Generate n random observations from the distribution of , ;

Step 3

formulate as .

The parameter estimation required to fit the BSD distribution to real data can be performed using the search via simulation (SVS) method outlined in Aboukhamseen and Soltani [3]. The SVS is a simulation based technique that circumvents analytical intractability to find suitable estimates for data that follow a given distribution.

To demonstrate the performance of the BSD distribution we apply the SVS estimation technique to fit two real data sets that are assumed to be normally distributed. One of the data sets is samples of anthropometric children’s measurements of weight. The second data set consists of residuals produced from a fitted autoregressive model of order one, AR(1).

Anthropometric Data

Analysis of weight and height of children of all ages are essential to the assessment of child health and nutritional status. Measurements of the weight and height of children are collected through routinely conducted national and international surveys such as the National Health and Nutrition Examination Surveys (NHANES) conducted by the Center for Disease Control, the National Center for Health Statistics, Student Health Service (SHS) of Hong Kong, and the World Health Organization (WHO) among many others. The data collected are used in the development of growth charts. These measurements are also analyzed longitudinally to determine changes and trends in infant survival and children’s growth rates (see for example [7]).

Historically, weight and height models are assumed to be normally distributed [5]. The residuals of the models commonly used to generate growth charts are also assumed to be normally distributed [5]. The World Health Organization’s methodology for computing percentiles and z-scores for age-specific weight, height and body mass index indicators restricts the derivation of percentiles to only those with a corresponding z-scores within the interval of , thus limiting the Box-Cox normal distribution and essentially imposing a truncation on the distribution WHO [12]. "This approach avoids making assumptions about the distribution of data beyond the limits of the observed values" (WHO Multicentre Growth Reference Study Group, 2006) Group [6]. A.Slavskii et al. [4] also discuss the limits of the normal distribution in the modelling of human height.

The weight of an infant at birth, in particular, is an important indicator of the infant’s chances of survival. The BSD distribution is used to fit data on the weight of full-term male newborns (live births delivered at a 40 week gestation period) taken from the National Center for Health Statistics 1968 Vital Statistics Natality Birth Data survey. Table 1 provides a summary of the complete record of all males born at 40 weeks gestation included in the data file and those of a random sample of size 1000 used in the modelling.

Table 1. Summary statistics weight of males at birth in grams

	Population	Sample
Size	145, 382	1000
Skewness	0.071	0.005
Minimum	225	1701
Maximum	7031	5406
Mean	3444.8	3426.83
Std. dev	483.05	458.24

The scaled data is fitted to a BSD distribution of the form , where follows a triangular distribution. The SVS method estimates the parameters values and with selection criteria . Figure 6 depicts a histogram of the scaled sample data along with the fitted BSD distribution seen in green. It is clear that the BSD distribution depicts the distribution of the data well and provides low probability in the tails.

[See PDF for image]

Fig. 6

Histogram of weight of newborn males; fitted BSD and normal densities

The distribution of modeling errors

Linear models often assume normal distributed errors. Nematollahi et al. [10] fitted data on 60 values of quarterly change in business inventories using a first order autoregressive model. The model residuals are assumed to be normal. The BSD distribution is fitted to the scaled standardized residuals using a uniform centering parameter. The resulting fitted distribution is imposed on the histogram of the residuals depicted in Fig. 7 using estimated parameter values and . Compared to the normal distribution, the BSD provides a better fit to the location and spread.

[See PDF for image]

Fig. 7

Histogram of residuals from an AR(1) model fitted using the BSD and normal densities

In both data examples the BSD distribution fits the distribution of the data well. The primary benefit, however, is that the distribution will not attribute probability beyond the feasible scope of the measured phenomenon.

Concluding Remarks

In this article we presented and developed a new class of parametric bell-shaped density functions on finite intervals that vanish on and beyond the interval endpoints. Such a density is needed to explain data that are symmetric in shape but are bounded. This offers an alternative to the Normal distribution as an approximate distribution to fit bounded data. The density function and its basic statistics are provided. The characterization of the Bell-Shaped density random variable given in (2.1) enables the inclusion of a random centering parameter. The specification of the centering parameter’s distribution opens up a broad class of distributions. We examine the cases where the centering parameter takes the form of a triangular, beta and uniform random variable. Closed form expressions for special cases of the BSD density functions when is uniform (0, 1) are also presented.

The characterization enables easy facilitation of sample simulation and parameter estimation. When applied to real data, we show that the Bell-Shaped density distribution fits symmetric data, often assumed to be normal, quite well.

The implications and applications of this density are far reaching and much needed. They can be applied in Bayesian analysis as symmetric bell-shaped prior distributions on finite intervals. This density can be used to define a parameterized Kernel function on which can be applied in regression, time series and density estimation. The parametric family of bell-shaped distributions established in this article define, indeed, a rich family of kernel functions that can be applied in non-parametric statistics and in time series spectral analysis. The parameters of our kernel function make it flexible to obtain the desired Mean Square Error or Mean Integrated Square Error.

Acknowledgements

The authors would like to thank the Editor and Reviewers for their insights that have helped to improve the quality of the work presented.

Author Contributions

A.R.S. conceptualized the work presented in this manuscript. A.R.S. and S.A. conducted the formal analysis, methodology development and review of both the writing and content. S.A. conducted the computational analysis, applications and interpretations, drafted the manuscript and prepared all figures and tables. Abrar Al-Mukhazeem performed the mathematical derivations along with A.R.S and S.A. and participated in the analysis.

Funding

The authors declare that no funding or any form of financial support was received for this work.

Data Availability

No datasets were generated or analysed during the current study.

Declarations

Competing Interests

All authors declare that they have no competing interests (financial or otherwise) regarding the creation of this manuscript and the work presented therein.All authors declare that they have no competing interests (financial or otherwise) regarding the creation of this manuscript and the work presented therein.

Ethics Approval and Consent to Participate

Not applicable

Consent for Publication

Not applicable

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