A stochastic model for NFL games and point spread assessment

Abstract

Statistical modelling of sports data is indispensable to analyse the sports behaviour and apprehend significant inferences that are helpful to adopt decisive strategies before or during the sports events. This paper introduces a stochastic model as the distribution of difference derived from the Bivariate Affine-Linear Exponential distribution. The distribution of difference is first ever used to model the margin of victory that provides an adequate fitting on the observed data. A simulation study is carried out to observe the stability of the model parameters through their average estimated values, biases, standard errors, root mean square errors and confidence intervals. The performance of the proposed model is examined by applying it on the real data of the National Football League and comparing the results with those of the existing models. Finally, the quantile function of the proposed distribution is used to assess the possible range of point spreads for winning the bet in a particular game.

1. Introduction

Football, being the most popular sport in the USA, generates billions of dollars every year and contributes considerable revenue towards the sports industry as well as the country's economy. Consequently, a massive amount is wagered in betting market where the bettors are curious to make more accurate predictions about the outcomes of the game. According to the American Gaming Association, the Americans gambled $95 billion on the NFL and college football games in 2015 seasons. The National Football League (NFL) is the richest premier football league in the USA having 32 teams divided into two pools: the National Football Conference and the American Football Conference.

While the NFL generates a huge turnover, the betting market associated with the NFL substantially adds to the economy as well. There are many ways to place a bet but point spread is the most popular one for betting in the American football. It helps the wagerer to bet on a team likely to win (favourite), or even on a team likely to lose (underdog) by a certain margin. The point spread is a perceived real number that does not actually predict which team will win or lose but provides a vantage to the bettors to place bets even on weaker teams. For example, if the point spread is 3 for the favourite Washington DC football team versus the underdog New York team, the favourite team should win the match with a margin of more than 3 scores; otherwise, they will lose the bet despite winning the game. If the favourite team wins with a margin of exact 3 scores, the bet is neither won nor lost. This situation is called ‘push’. The basic objective of the point spread betting is to create an equal opportunity to put money on the underdog as well. This is a rational assumption which convinces people to wager on a relatively weaker team. It makes the betting market balanced.

In literature, many statistical models have been proposed using different approaches to predict different aspects of the game i.e. modelling the probability of win, loss or draw, modelling the game score, modelling the difference of scores made by two teams, etc. Warner [29] predicts the margin of victory in the NFL games using the Gaussian process model. Glickman and Stern [7] develop a state-space model for the NFL score assuming that team strength parameters follow the first-order auto regressive process. Matthews [19] studies the effect of data transformation on the paired comparison of Glickman and Stern [7] to reduce the influence of ‘blowouts’ on future predictions. Szalkowski and Nelson [28] investigate the opening and closing lines along with margin of victory and find that the line difference can predict the divisional winner with 75% accuracy. Pelechrinis and Papalexakis [21] provide a descriptive model to find the winning probability of an NFL game and combine it with bootstrap method to make a future matchup prediction scheme. They use Bradley–Terry regression model to study the impact of several factors on winning probability of a game. MacDonald and Dare [17] formulate a generalised model to test the efficiency of the point spread market of football. Baker and McHale [1] exhibit a point process model for forecasting exact end-of-match scores in the NFL by using hazards of scoring and bookmaker point spread to predict exact scores.

The literature review reveals that several aspects of the NFL games have been analysed by using different methodologies and models. These models involve regression, time series and descriptive models in particular but the analysis and prediction of any event are incomplete without using the appropriate probability models. Probability distributions model the random phenomena in real life to foresee the maximum chances of particular outcomes and draw possible inferences about the parameters involved. They observe and estimate the likelihood of the outcomes of a random experiment systematically and provide useful inferences and predictions. The sports activities and events all over the world have become an imperative integrant of everyday life. Thus, the probability models are equally helpful to analyse the sports data and predict different aspects of a game. It is observed that the role of probability distributions in analysing and predicting the global sports events like the American Football is scanty. We find no evidence of using probability distributions to model the margin of victory in the NFL games and assess the point spread so far. Only Stern [26,27] and Carlin [4] use normal distribution to model the margin of victory over point spread and compute the probabilities of winning of the favourite team in the NFL games.

In the present paper, we develop a new stochastic distribution of difference derived from the BALE distribution for modelling the margin of victory in the NFL games. The distribution of difference is first ever used to model the margin of victory that dissociates it from the previous work. Another salient feature of our research is that we provide a point spread assessment scheme relying on the possible inferences obtained from the quantiles of the model. Our approach differs from that of Stern [26,27] in three ways. Stern [26,27] uses the existing normal distribution, models the margin of victory over point spread and computes the probability of winning of the favourite team. On the other hand, we derive a new stochastic distribution of difference, model only the margin of victory and provide the point spread assessment scheme. Indeed, the NFL data can be modelled by using any bivariate distribution but we select the BALE distribution to derive the distribution of difference because the BALE distribution has make out characteristics that it is absolutely continuous; whereas, many well-known distributions, e.g. Marshall and Olkin [18], Gupta and Kundu [8], Sarhan and Balakrishnan [24] etc. have continuous as well as singular parts. Moreover, the BALE distribution has closed form of its distribution function showing that it has vast applications; which is not a case with the distribution proposed by Regoli [23]. An additional merit of the BALE distribution is that it effectively models the data having negative low correlation, see details in Mohsin et al. [20].

The paper is organised as follows: A concise description of the BALE distribution along with the explicit expression for the probability density function and the characteristics of the stochastic distribution of difference are given in Section 2. A simulation study is conducted to observe the consistency of the model parameters in Section 3. For illustration, the application of the proposed model on a real data set of the NFL games and its comparison with other extant distributions are exhibited in Section 4. The computation of quantiles and the assessment of the point spread are presented in Section 5. Finally, conclusion and discussion are stated in Section 6.

2. The methodology

In this section, we first provide the expression and a brief description of the BALE distribution proposed by Mohsin et al. [20] along with some lemmas used. Then, we derive the stochastic distribution of difference and its moments by using the baseline BALE distribution.

The joint pdf and survival function of the BALE distribution are given by

$$f_{\mathrm{XY}}(x,y)=\alpha (\mathrm{\text{β}}+\mathrm{\gamma x})\exp \{-\mathrm{\alpha x}-\mathrm{\text{β}}y-\mathrm{\gamma xy}\},\alpha >0,\mathrm{\text{ β}},\gamma ,x,y\ge 0,\mathrm{\text{ β}}+\gamma >0,$$ (1)

and

$${\overline{F}}_{\mathrm{XY}}(x,y)=\exp \{-\mathrm{\alpha x}\}-\frac{\mathrm{\alpha exp}\{-\mathrm{\text{β}}y\}[1-\exp \{-(\alpha +\mathrm{\gamma y})x\left\}\right]}{(\alpha +\mathrm{\gamma y})},$$ (2)

respectively. The BALE distribution is a quite flexible bivariate distribution obtained by compounding the two exponentially distributed random variables, $X\sim \mathrm{Exp}\left(\alpha \right)$ and $Y|X\sim \mathrm{Exp}(\mathrm{\text{β}}+\mathrm{\gamma X})$. The parameter γ reflects the dependency between the random variables X and Y. For $\gamma =0$, the random variables X and Y are independently and exponentially distributed with the parameters α and β, respectively. The attractive characteristics and vast applications of the BALE distribution propel us to further probe into it and derive a new distribution of the difference by using (1). We use the following lemmas developed by Prudnikov et al. [22] to derive the distribution of difference and its moments.

Lemma 2.1 Prudnikov et al. [22, Vol.1, Equation (2.3.15.7), p.344] —

For $\mathrm{Re}\left(p\right)>0$, $\mathrm{Re}\left(q\right)>0$ and n=0,1,…,

$${\int }_0^{\mathrm{\infty }}{x}^n\exp (-p{x}^2-\mathrm{qx})\mathrm{d}x=\frac{(-1{)}^n}{2}\sqrt{\frac{\pi }{p}}\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{q}^n}[\exp \left(\frac{q^2}{4p}\right)\mathrm{erfc}\left(\frac{q}{2\sqrt{p}}\right)],$$

where $\mathit{erfc}\mathit{\left(}\mathit{x}\mathit{\right)}$ is the complementary error function which is defined as

$$\mathit{erfc}\mathit{\left(}\mathit{x}\mathit{\right)}\mathit{=}\frac{\mathit{2}}{\sqrt{\pi }}{\int }_{\mathit{x}}^{\mathrm{\infty }}\exp (\mathit{-}{\mathit{w}}^{\mathit{2}})\mathrm{d}\mathit{w}\mathit{.}$$

Lemma 2.2 Prudnikov et al. [22, Vol. 1, Equation (2.3.6.9), p. 324] —

If $a,b\in \mathbb{R}$, s>0 and $|\arg c|<\pi$,

$${\int }_0^{\mathrm{\infty }}\frac{x^{a-1}}{(c+x{)}^b}\exp (-\mathrm{sx})\mathrm{d}x=\mathrm{\Gamma }\left(a\right)\left(c{)}^{a-b}\mathrm{\Psi }\right(a,a+1-b;\mathrm{cs}),$$

where $\mathrm{\Psi }(.)$ is Kummer's (confluent hypergeometric) function of second kind which is given by

$$\mathrm{\Psi }(x,y;z)=\frac{1}{\mathrm{\Gamma }\left(x\right)}{\int }_0^{\mathrm{\infty }}\exp (-\mathrm{zt})t^{x-1}(1+t{)}^{y-x-1}\mathrm{d}t.$$

2.1. The proposed model

The following theorems are presented to develop the distribution of difference and derive its moments.

Theorem 2.3

If X and Y are jointly distributed according to (1), the pdf of the difference is given as

$$\begin{array}{rl}& f_D\left(d\right)\\ & =\{\begin{array}{ll}{\displaystyle \frac{\alpha }{4}\exp (-\mathrm{\alpha d})\{2+\sqrt{\frac{\pi }{\gamma }}(\mathrm{\text{β}}+\mathrm{\gamma d}-\alpha )\exp \left(\frac{{\left({\varphi }_1\right(d\left)\right)}^2}{4\gamma }\right)\mathrm{erfc}\left(\frac{{\varphi }_1\left(d\right)}{2\sqrt{\gamma }}\right)\},}& \mathrm{for}d>0,\\ {\displaystyle \frac{\alpha }{2}\exp \left(\mathrm{\text{β}}d\right)+\frac{1}{2}\sqrt{\frac{\pi }{\gamma }}\mathrm{erfc}\left(\frac{{\varphi }_2\left(d\right)}{2\sqrt{\gamma }}\right)\left\{\right(\alpha \mathrm{\text{β}}+\mathrm{\alpha \gamma d})\exp (-\mathrm{\alpha d})}& \\ {\displaystyle \exp \left(\frac{{\left({\varphi }_1\right(d\left)\right)}^2}{4\gamma }\right)-\frac{1}{2}({\alpha }^2+\alpha \mathrm{\text{β}}+\mathrm{\alpha \gamma d})\exp \left(\mathrm{\text{β}}d\right)\exp \left(\frac{{\left({\varphi }_2\right(d\left)\right)}^2}{4\gamma }\right)\},}& \mathrm{for}d<0,\end{array}\end{array}$$ (3)

where ${\varphi }_1\left(d\right)=\alpha +\mathrm{\text{β}}+\mathrm{\gamma d}$ and ${\varphi }_2\left(d\right)=\alpha +\mathrm{\text{β}}-\mathrm{\gamma d}$.

Proof.

From (1), the joint pdf of $(D,W)=(X-Y,Y)$ becomes

$$f(d,w)=\alpha (\mathrm{\text{β}}+\mathrm{\gamma d}+\mathrm{\gamma w})\exp (-\mathrm{\alpha d})\exp (-\gamma w^2-(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d}\left)w\right).$$

The transformation maps x = 0 to w = −d, $x=\mathrm{\infty }$ to $w=\mathrm{\infty }$ and y = 0 to w = 0, $y=\mathrm{\infty }$ to $w=\mathrm{\infty }$. The conditions x>0 and y>0 from the BALE distribution transformed the pdf of D into $-\mathrm{\infty }<d<\mathrm{\infty }$ and $w>-d$ for d>0 and w>0 for d>0. The pdf of D can be written as

$$f_D\left(d\right)=\{\begin{array}{ll}(\alpha \mathrm{\text{β}}+\mathrm{\alpha \gamma d})\exp (-\mathrm{\alpha d})I_0\left(d\right)+\mathrm{\alpha \gamma }\exp (-\mathrm{\alpha d})I_1\left(d\right),& \text{if}d>0,\\ (\alpha \mathrm{\text{β}}+\mathrm{\alpha \gamma d})\exp (-\mathrm{\alpha d})M_0\left(d\right)+\mathrm{\alpha \gamma }\exp (-\mathrm{\alpha d})M_1\left(d\right),& \text{if}d<0,\end{array}$$ (4)

where $I_n\left(d\right)={\int }_0^{\mathrm{\infty }}{w}^n\exp (-\gamma w^2-(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d}\left)w\right)\mathrm{d}w,n=0,1.$ Now, using Lemma 2.1, the integrals $I_n\left(d\right)$ can be calculated as

$$I_n\left(d\right)=\frac{{(-1)}^n}{2}\sqrt{\frac{\pi }{\gamma }}\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{q}^n}{[\exp \left(\frac{q^2}{4\gamma }\right)\mathrm{erfc}\left(\frac{q}{2\sqrt{\gamma }}\right)]}_{q=\alpha +\mathrm{\text{β}}+\mathrm{\gamma d}}$$ (5)

Evaluating the derivative in (5) for n = 0, 1, and simplifying, we get

$$I_0\left(d\right)=\frac{\sqrt{\pi }}{2\sqrt{\gamma }}\exp \left(\frac{{(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d})}^2}{4\gamma }\right)\mathrm{erfc}\left(\frac{(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d})}{2\sqrt{\gamma }}\right)$$ (6)

and

$$I_1\left(d\right)=\frac{1}{2\gamma }-\frac{\sqrt{\pi }(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d})}{4{\gamma }^{3/2}}\exp \left(\frac{{(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d})}^2}{4\gamma }\right)\mathrm{erfc}\left(\frac{(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d})}{2\sqrt{\gamma }}\right).$$ (7)

Similarly

$M_n\left(d\right)={\int }_{-d}^{\mathrm{\infty }}{w}^n\exp (-\gamma w^2-(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d}\left)w\right)\mathrm{d}w,$    $n=0,1,$

follows

$$M_0\left(d\right)=\frac{\sqrt{\pi }}{2\sqrt{\gamma }}\exp \left(\frac{{(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d})}^2}{4\gamma }\right)\mathrm{erfc}\left(\frac{(\alpha +\mathrm{\text{β}}-\mathrm{\gamma d})}{2\sqrt{\gamma }}\right)$$ (8)

and

$$\begin{array}{rl}& M_1\left(d\right)\\ & =\frac{\exp (\mathrm{\alpha d}+\mathrm{\text{β}}d)[2\sqrt{\left(\gamma \right)}-\sqrt{\pi }(\alpha +\mathrm{\text{β}}+\mathrm{\gamma d})\exp \left(\frac{{(\alpha +\mathrm{\text{β}}-\mathrm{\gamma d})}^2}{4\gamma }\right)\mathrm{erfc}\left(\frac{(\alpha +\mathrm{\text{β}}-\mathrm{\gamma d})}{2\sqrt{\gamma }}\right)]}{4{\gamma }^{3/2}}.\end{array}$$ (9)

The result follows by substituting (6), (7),(8) and (9) in (4) and then simplifying the expressions.

The different shapes of the pdf of D for different values of α, β and γ are given in Figure 1. The figure shows that the new distribution of difference is asymmetric and unimodal. It is also observed as α approaches to β, the proposed distribution approaches to symmetry. The new distribution becomes right skewed with lower peak when α decreases, whereas it becomes left skewed with flat peak when β decreases. Also when γ increases, the probability mass shifts to the right and changes its behaviour at the peak in term of its smoothness.

Figure 1.

The probability density function of D for (a) $\alpha =1.0,0.8,0.6,0.4$ and 0.2 with $\mathrm{\text{β}}=1.0$, $\gamma =0.05$, (b) $\mathrm{\text{β}}=0.9,0.7,0.5,0.3$ and 0.1 with $\alpha =1.0$, $\gamma =0.05$.

The distribution of difference is quite useful and it is applied in many fields such as finance, physics, sports, etc. Karlis and Ntzoufras [11] derive the distribution of difference from bivariate Poisson distribution to model the Italian serie A data and water-polo games. Skellam [25] derives the distribution of the difference between two Poisson random variates assuming different means which is the generalisation of Irwin [10]. Cox and Isham [6] give an interesting stochastic connection of the distribution of difference arising in stochastic point process with thinning. For more references, see Kotz et al. [14].

In addition, the moments of $D=X-Y$ when X and Y are distributed according to (1) are presented in the following theorem. These moments are helpful to study the characteristics of the proposed distribution.

Theorem 2.4

If X and Y are jointly distributed according to (1), then it holds:

$$E\left(D^n\right)=\sum _{k=0}^n{(-1)}^k(\genfrac{}{}{0ex}{}{n}{k})\frac{\alpha {\mathrm{\text{β}}}^{2k-n+1}\mathrm{\Gamma }(k+1)\mathrm{\Gamma }(n-k+1)}{{\gamma }^{k+1}}\mathrm{\Psi }(k+1,2k-n+2;\frac{\alpha \mathrm{\text{β}}}{\gamma }).$$ (10)

Proof.

The proof of (10) requires the following result. The product moment of the distribution proposed in (1) is obtained as:

$$\begin{array}{rl}E\left(X^p{Y}^q\right)& =\alpha {\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{x}^p{y}^q(\mathrm{\text{β}}+\mathrm{\gamma x})\exp \{-\mathrm{\alpha x}-\mathrm{\text{β}}y-\mathrm{\gamma xy}\}\mathrm{d}y\mathrm{d}x,p,q=1,2,\dots \\ & \frac{\alpha \mathrm{\Gamma }(q+1)}{{\gamma }^q}{\int }_0^{\mathrm{\infty }}\frac{x^{(p+1)-1}\exp \{-\mathrm{\alpha x}\}}{{(\frac{\mathrm{\text{β}}}{\gamma }+x)}^q}\mathrm{d}x.\end{array}$$

Using Lemma 2.2 and further simplifying we arrive at

$$\begin{array}{rl}E\left(X^p{Y}^q\right)& =\frac{\alpha {\mathrm{\text{β}}}^{p-q+1}}{{\gamma }^{p+1}}\mathrm{\Gamma }(q+1)\mathrm{\Gamma }(p+1)\mathrm{\Psi }(p+1,p-q+2;\frac{\alpha \mathrm{\text{β}}}{\gamma }),\\ & \mathrm{for}p>-1\mathrm{and}q>-1.\end{array}$$ (11)

The result follows by writing $E\left(D^n\right)=E(X-Y{)}^n=\sum _{k=0}^n(-1{)}^{\left(k\right)}(\genfrac{}{}{0ex}{}{n}{k})E\left(X^{n-k}\right)\left(Y^k\right)$ and applying (11) to the expectation in the difference we get the results stated in (10).

For n = 1 and 2, the expressions for $E\left(D\right)$ and $E\left(D^2\right)$ are given as:

$$E\left(D\right)=\frac{\gamma +e^{\frac{\alpha \mathrm{\text{β}}}{\gamma }}({\alpha }^2-2\alpha \mathrm{\text{β}}-2\gamma )E_3\left(\frac{\alpha \mathrm{\text{β}}}{\gamma }\right)}{\mathrm{\alpha \gamma }},$$ (12)

where $E_3$ is exponential integral function which is defined as:

$$\begin{array}{rl}& E_v\left(z\right)=z^{v-1}{\int }_z^{\mathrm{\infty }}{t}^{-v}\exp (-t)d<t;|\mathrm{Arg}\left(z\right)|<\pi .\\ & E\left(D^2\right)=\frac{\mathrm{\text{β}}\left(\begin{array}{c}\alpha \mathrm{\text{β}}{e}^{\frac{\alpha \mathrm{\text{β}}}{\gamma }}(2{\alpha }^3\mathrm{\Gamma }(-3,\frac{\alpha \mathrm{\text{β}}}{\gamma })\\ +(\alpha -\mathrm{\text{β}})(\alpha \mathrm{\text{β}}+2\gamma )\mathrm{\Gamma }(0,\frac{\alpha \mathrm{\text{β}}}{\gamma }))-\gamma (\alpha -\mathrm{\text{β}})(\alpha \mathrm{\text{β}}+\gamma )\end{array}\right)}{{\gamma }^4},\end{array}$$ (13)

where $\mathrm{\Gamma }(-n,z)$ and $\mathrm{\Gamma }(0,z)$ are incomplete gamma functions.

3. Simulation

In this section, simulation is carried out to study the stability of the model parameters. The maximum-likelihood method is applied to estimate the model parameters $\alpha ,\mathrm{\text{β}}$ and γ. We perform numerical method to solve $F\left(x\right)=u$ where $u\sim U(0,1)$ as discussed by Lange [15]. The simulation is run 1000 times for five different combinations of the model parameters to draw the random samples of size n each from the proposed model. The ML estimates are obtained by BFGS method implemented in R package maxLik given by Henningsen and Toomet [9].

Table 1 presents the average estimates (AEs), biases, standard errors (SEs), root mean square errors (RMSEs) and corresponding confidence intervals (CIs) for the samples sizes 50, 200 and 500. It is observed from Table 1 that AEs of the model parameters approach the true values of the parameters as n increases. The biases and SEs for each parameter approach zero as sample size increases. The 95% CIs for all the model parameters tend to contain the estimated values of the parameter as sample size increases. These findings endorse the asymptotic theory (large sample) of the normal distribution showing that the errors of these estimates, as expected, decrease when n increases.

Table 1.

Estimated values, biases, standard errors, root mean square errors and 95% C.Is of the model parameters.

		$\alpha =1.5,\mathrm{\text{β}}=1,\gamma =0.5$			$\alpha =2,\mathrm{\text{β}}=0.5,\gamma =1$			$\alpha =0.7,\mathrm{\text{β}}=0.5,\gamma =0.1$			$\alpha =1,\mathrm{\text{β}}=1,\gamma =1$			$\alpha =2,\mathrm{\text{β}}=1,\gamma =1$
		n			n			n			n			n
		50	200	500	50	200	500	50	200	500	50	200	500	50	200	500
Avg. Estimate	$\hat{\alpha }$	1.5004	1.5003	1.5003	2.0031	2.0026	2.0022	0.7004	0.7003	0.7003	1.0004	1.0003	1.0003	2.0004	2.0003	2.0003
	$\widehat{\mathrm{\text{β}}}$	1.0006	1.0005	1.0005	0.5039	0.5038	0.5045	0.5006	0.5005	0.5004	1.0005	1.0005	1.0005	1.0006	1.0005	1.0004
	$\hat{\gamma }$	0.5008	0.5007	0.5007	1.0048	1.0059	1.0045	0.1007	0.1006	0.1006	1.0007	1.0007	1.0007	1.0006	1.0006	1.0006
Bias	$\hat{\alpha }$	0.0007	0.0007	0.0007	0.0231	0.0005	0.0016	0.0008	0.0008	0.0008	0.0008	0.0008	0.0008	0.0008	0.0008	0.0007
	$\widehat{\mathrm{\text{β}}}$	0.0013	0.0010	0.0010	0.0392	0.0007	0.0038	0.0014	0.0011	0.0010	0.0010	0.0010	0.0012	0.0014	0.0011	0.0010
	$\hat{\gamma }$	0.0019	0.0018	0.0017	0.0426	0.0013	0.0044	0.0015	0.0012	0.0015	0.0018	0.0016	0.0017	0.0015	0.0012	0.0011
std. error	$\hat{\alpha }$	8.4e–05	5.7e–05	0.0001	0.0022	0.0003	0.0004	0.0001	9.3e–05	6.2e–05	0.0001	0.0001	0.0001	0.0001	9.3e–05	6.2e–05
	$\widehat{\mathrm{\text{β}}}$	0.0001	0.0001	0.0001	0.0023	0.0005	0.0011	0.0002	9.7e–05	7.5e–05	9.8e–05	8.6e–05	0.0002	0.0002	9.7e–05	7.5e–05
	$\hat{\gamma }$	0.0002	0.0002	0.0002	0.0034	0.0008	0.0005	0.0002	9.2e–05	0.0002	0.0002	0.0002	0.0002	0.0002	9.2e–05	0.0002
RMSE	$\hat{\alpha }$	0.0037	0.0025	0.0018	0.0158	0.0125	0.0082	0.0051	0.0041	0.0028	0.0051	0.0045	0.0036	0.0051	0.0041	0.0028
	$\widehat{\mathrm{\text{β}}}$	0.0064	0.0052	0.0050	0.0165	0.0213	0.0237	0.0078	0.0043	0.0034	0.0044	0.0038	0.0029	0.0078	0.0043	0.0034
	$\hat{\gamma }$	0.0094	0.0080	0.0073	0.0240	0.0345	0.0119	0.0068	0.0041	0.0068	0.0072	0.0082	0.0082	0.0068	0.0041	0.0068
LCL	$\hat{\alpha }$	1.5002	1.5002	1.5001	1.9987	2.0020	2.0015	0.7002	0.7002	0.7002	1.0002	1.0002	1.0001	2.0002	2.0002	2.0002
	$\widehat{\mathrm{\text{β}}}$	1.0003	1.0003	1.0003	0.4994	0.5029	0.5024	0.5003	0.5003	0.5003	1.0003	1.0003	1.0002	1.0003	1.0003	1.0003
	$\hat{\gamma }$	0.5004	0.5004	0.5003	0.9981	1.0045	1.0034	0.1004	0.1005	0.1004	1.0004	1.0004	1.0003	1.0004	1.0005	1.0004
UCL	$\hat{\alpha }$	1.5005	1.5005	1.5006	2.0075	2.0031	2.0030	0.7006	0.7006	0.7005	1.0006	1.0006	1.0006	2.0006	2.0006	2.0005
	$\widehat{\mathrm{\text{β}}}$	1.0008	1.0007	1.0008	0.5085	0.5048	0.5066	0.5009	0.5007	0.5006	1.0007	1.0007	1.0009	1.0009	1.0007	1.0006
	$\hat{\gamma }$	0.5012	0.5011	0.5011	1.0114	1.0075	1.0055	0.1010	0.1008	0.1010	1.0011	1.0011	1.0011	1.0010	1.0008	1.0010

4. Application

In this section, a real data set of the National Football League (NFL) is analysed to illustrate the application of the proposed model.

4.1. Fitting the model

For modelling the margin of victory, we use the data of 278 NFL games from 3 January 2015 to 7 February 2016 available on the website www.aussportsbetting.com. The data include complete details of the game statistics consisting of scores by the home team and scores by the away team. Though the home team is not necessarily the favourite one always but for the purpose of compatibility, the similar terms are used in the data, i.e. the scores by the home team, and the scores by the away team are named as scores by the favourite team (X), and scores by the underdog team (Y), respectively. The margin of victory (D) is defined as the difference of the scores by the favourite team (X) and the scores by the underdog team (Y), i.e. D= X-Y.

For the fitting of our proposed model, the maximum-likelihood (ML) method is used to estimate the model parameters for the observed data of the difference between the scores by the two teams. If $d_i;i=1,2,\dots ,n$ is a random sample of size n from (3), the ML estimates of α, β and γ are obtained by BFGS method implemented in R package maxLik given by Henningsen and Toomet [9]. The ML estimates of α, β and γ are 0.0998, 0.0721 and 0.0071 along with standard errors 0.0092, 0.0151 and 0.0049 respectively. The ML estimates of α, β and γ for margin of victory look quite stable as their standard errors are smaller than their point estimates. The ML estimate of γ is very small that represents the joint effect of the scores by the favourite team and the scores by the underdog team which is also confirmed by their low and insignificant correlation, i.e. −0.061. The physical interpretation of the parameters is that α and β represent point scoring capabilities of the favourite and the underdog teams, respectively, whereas γ shows the common factors like teams strength, home ground, home crowd, weather conditions, etc. The small value of γ shows that the common factors are least responsible for the margin of victory while α and β significantly affect the margin of victory.

Now (3) is applied to compute the fitted pdf using the ML estimates. This fitted distribution is compared to the histogram of D, difference between the scores by two teams, from the observed data. The fitted pdf of D reasonably follows the general pattern of the histogram. Figure 2(a) shows the histogram of the observed data D along with the fitted pdf of D for each game which depicts that our proposed distribution fits the NFL data adequately. Further to observe the goodness of fit of (3), the observed probabilities are plotted against the predicted probabilities for the proposed model. Figure 2(b) exhibits the probability plot for D, $F_D\left(d_i\right)$ versus $(i-0.375)/(n+0.25)$ as recommended by Blom [2] and Chambers et al. [5] where $d_i$ are the sorted values of D in ascending order. The fit looks reasonably good for the margin of victory since the dots follow the diagonal line closely.

Figure 2.

(a) Fitted values of the pdf of D on the histogram of the difference between the scores by the two teams and (b) Probability plot of D for the difference between the scores by the two teams, for 278 NFL games.

4.2. Comparison with other models

It is important to know that how well the new proposed distribution performs as compared to the other existing distributions. For this comparison, we use asymmetric Laplace, Cauchy, skew t, skew Laplace and odd log-logistic normal distributions because they all belong to asymmetric family and have the same domain for the random variable X as our propose model keeps. The negative log-likelihood, Akaike information criterion (AIC) and Bayesian information criterion (BIC) are used to compare and examine their relative performance using the difference of the scores by the two teams (D) in the NFL data. The model with the highest negative log-likelihood value is taken as the best model. The AIC and BIC express the relative loss of information thus the smaller values of AIC and BIC reflect the better model. These measures can be calculated as: and , where = maximum value of negative log-likelihood, k= number of parameters estimated and n=number of observations. Asymmetric Laplace distribution discussed by Koenker and Machado [13] and Yu and Zhang [30] have three parameters, i.e. location, scale and skewness. The optimize function implemented in R package ald is used for the computation of negative log-likelihood value for the asymmetric Laplace distribution. Similarly, the mledist function implemented in R package fitdistrplus is used to find the value of negative log-likelihood for Cauchy distribution which has two parameters, i.e. location and scale. The skew t distribution has two parameters, whereas skew Laplace distribution has three parameters and to find the values of their negative log-likelihood functions we use R packages skewt, see details King [12], and rmutil, see deatils Lindsey and Swihart [16], respectively. The odd log-logistic normal distribution, given by Braga et al. [3], has three parameters and to find its negative log-likelihood value we use Simulated Annealing (SANN) method implemented in R package maxLik given by Henningsen and Toomet [9]. Table 2 depicts the negative log-likelihood, AIC and BIC values for the difference between the scores by the two teams in the NFL data. The result shows that our proposed model provides the largest value of negative log-likelihood function and smallest values of AIC and BIC as compared to the other models and fits the best on the difference between the scores by the two teams in the NFL data.

Table 2.

Estimated values of negative log-likelihood function, AIC and BIC for different models.

Model	Negative log-likelihood	AIC	BIC
Proposed model	−1136.99	2279.98	2290.86
Asymmetric Laplace	−1927.89	3861.78	3872.66
Cauchy	−1167.28	2338.56	2345.82
Skew-t	−1483.06	2970.12	2977.38
Skew Laplace	−3249.70	6505.40	6516.28
Odd log-logistic normal	−1565.42	3136.84	3147.72

5. Quantiles and point spread assessment

In this section, we provide quantiles of the proposed model by using the arbitrary values of the parameters associated with the pdf of the difference. These quantiles are computed numerically by solving the equation:

$${\int }_{-\mathrm{\infty }}^{z_q}f\left(t\right)\mathrm{d}t=q,\mathrm{where}f\in \left\{f_D\right\}.$$ (14)

The function uniroot in R software is used for the numerical solution of this equation. Table 3 provides tabulated values of $z_q$ using $\alpha =0.5,0.7,1.0,1.5,2.0$, $\mathrm{\text{β}}=0.5,1.0$ and $\gamma =0.1,0.5,1.0$.

Table 3.

The quantiles of the distribution of difference.

q
α	β	γ	0.99	0.95	0.90	0.75	0.50	0.25
0.5	0.5	0.1	8.6212	5.2647	3.7967	1.8213	0.2854	−0.9533
0.7	0.5	0.1	5.9191	3.4993	2.4416	1.0214	−0.0805	−1.3052
1.0	0.5	0.1	3.9053	2.2013	1.4582	0.4638	−0.3821	−1.6305
1.5	0.5	0.1	2.3773	1.2410	0.7472	0.0893	−0.6612	−1.9365
2.0	0.5	0.1	1.6460	0.7970	0.4291	−0.0646	−0.8194	−2.1129
0.5	1.0	0.1	8.7548	5.4580	4.0294	2.1291	0.6799	−0.2014
0.7	1.0	0.1	6.0336	3.7205	2.6972	1.3379	0.3034	−0.4172
1.0	1.0	0.1	4.0909	2.4339	1.7174	0.7668	0.0449	−0.6201
1.5	1.0	0.1	2.5656	1.4620	0.9855	0.3542	−0.1470	−0.8153
2.0	1.0	0.1	1.8231	0.9973	0.6410	0.1693	−0.2584	−0.9304
0.5	0.5	0.5	9.0262	5.7312	4.2888	2.3303	0.7491	−0.3595
0.7	0.5	0.5	6.3419	3.9555	2.9042	1.4657	0.2908	−0.6942
1.0	0.5	0.5	4.3086	2.6064	1.8516	0.8125	−0.0433	−1.0322
1.5	0.5	0.5	2.7129	1.5512	1.0338	0.3195	−0.3437	−1.3865
2.0	0.5	0.5	1.9156	1.0316	0.6377	0.0947	−0.5316	−1.6139
0.5	1.0	0.5	9.0416	5.7607	4.3314	2.4093	0.9033	−0.0255
0.7	1.0	0.5	6.3667	4.0002	2.9658	1.5711	0.4768	−0.2354
1.0	1.0	0.5	4.3467	2.6699	1.9353	0.9439	0.1680	−0.4422
1.5	1.0	0.5	2.7679	1.6351	1.1388	0.4704	−0.0525	−0.6531
2.0	1.0	0.5	1.9810	1.1258	0.7517	0.2493	−0.1732	−0.7847
0.5	0.5	1.0	9.1108	5.8458	4.4234	2.5028	0.9601	−0.0959
0.7	0.5	1.0	6.4461	4.0898	3.0567	1.6482	0.4935	−0.3926
1.0	0.5	1.0	4.4304	2.7538	2.0124	0.9905	0.1340	−0.7044
1.5	0.5	1.0	2.8435	1.6972	1.1854	0.4714	−0.1576	−1.0484
2.0	0.5	1.0	2.0419	1.1645	0.7705	0.2174	−0.3432	−1.2809
0.5	1.0	1.0	9.1155	5.8555	4.4383	2.5342	1.0342	0.0836
0.7	1.0	1.0	6.4543	4.1060	3.0805	1.6949	0.5939	−0.1185
1.0	1.0	1.0	4.4443	2.7796	2.0489	1.0564	0.2635	−0.3181
1.5	1.0	1.0	2.8665	1.7367	1.2385	0.5598	0.0178	−0.5291
2.0	1.0	1.0	2.0724	1.2140	0.8350	0.3189	−0.1047	−0.6658

The physical interpretation of a quantile, which indeed relates the position of the difference of scores by the two teams in a certain game, is point spread. Hence, Table 3 also exhibits point spread assessment scheme. In Table 3, for α=0.5, β=0.5 and γ= 0.1 one can be $95\mathrm{\%}$ confident that the difference between the scores of two teams will not exceed 5.27. It means that favourite team will win by the maximum margin of 5 scores providing an incentive to fix the point spread below 5 to win the bet. The point spread should be fixed below 5 because at 5 or above the chances of push or loosing the bet are certainly greater than those of winning the bet. Although the point spread is an arbitrary number that depends upon the contemporary facts about the particular games and teams, the quantiles help the consultants, bettors, and bookies to assess the possible range of point spreads and finally enable them to make the decisions. The usefulness of these quantiles is not restricted to sports data only but can be extended to many other situations where the behaviour of differences of random variables with linear exponential distribution is of vital importance.

6. Conclusion and discussion

We introduce a new stochastic model to analyse the NFL data and provide a point spread assessment scheme. Since, in an NFL game the correlation between the scores by the favourite and the scores by the underdog is low, i.e. −0.061 therefore we need a subtle distribution like BALE distribution which is quite suitable for modelling the low and moderate negative dependence. The results of the simulation study establish that the model parameters are quite stable and consistent. Also the findings of simulation study show that the estimated values of the model parameters become close to the true values and their standard errors decrease as the sample sizes increase which is in accordance with the asymptotic theory of the normal distribution. In addition, the adequate fitting of the model on the observed data approves its compatibility. The comparison of the proposed model with some existing distributions clearly favours its competency. Therefore, we find that our proposed model is flexible enough to explain all the aspects of the NFL data sufficiently. Moreover, point spreads are estimated by using the quantiles of the proposed model. This estimation of the point spreads not only depends on the values of the parameters but also relates the assumed level of confidence to fix the point spread. As the level of confidence decreases, the point spread also decreases for the favourite team that helps the bookies and the consultants to select the appropriate point spread. The tabulated quantiles are also helpful to draw the important inferences for adopting better strategies to win a bet. The quantiles computed on the basis of previous data of two particular teams help to select point spread for their upcoming football game. In addition, the use of quantiles of the proposed model comes up with a new dimension for the consultants to analyse the NFL data and find the worthwhile clues to fix point spread and adopt appropriate betting strategies.

Sports industry, being one of the largest profit making industries, proves to be an attractive market for bookies, bettors and wagerers. Some authors and analysts state that football betting market is uncertain but majority of the professionals conclude that this market becomes stable if the betting strategy is handled tactfully. The new proposed model and the assessment of point spreads by using quantiles play an important role to handle the betting strategy tactfully. Now bookies, wagerers, bettors and consultants can be more confident while putting money on an NFL game that ensures the stability of the betting market convincing more and more people to wager eagerly.

Disclosure statement

The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.