The Tan-G Family of Distributions with Illustration to Data in the Health Sciences

Introduction

The modification of trigonometric functions to give new statistical distributions is becoming a popular technique in the literature, and for examples see [1, 2, 3, 4, 5, 6]. In Kumar D, et al. [1], they transformed the sine function into a new statistical distribution called SS transformation with the following CDF

$$G(x) = \sin\left(\frac{\pi}{2} F(x)\right)$$

The new distribution is shown to be a better fit to data in the health sciences as compared to some well-known distributions available in the statistical literature through Akaike information criteria (AIC), Bayesian information criteria (BIC), log-likelihood and the associated Kolmogorov-Smirnov (KS) test values. The sine function, as is, violates the properties of a distribution function, hence the argument of the sine function is modified accordingly taking into account any baseline CDF. The crucial properties can be easily checked, in particular it is enough to check that $G(-\infty) = 0$ and $G(\infty) = 1$. Since the baseline CDF is a distribution function, we know $F(-\infty) = 0$ and $F(\infty) = 1$, and since $\sin(0) = 0$ and $\sin\left(\frac{\pi}{2}\right) = 1$ we can see $G(x)$ is a proper distribution function. By this observation, the present paper proposes a new statistical distribution by modifying the tangent function.

The rest of this paper is organized as follows. In Section 2, we introduce the new family, and illustrate a member of this family in Section 3. In Section 4, we compare the new distribution with a modification of Ampadu-G [7] (the modification appearing in the parameter space). The last section is devoted to the conclusions.

The New Family

The CDF of Tan-G is given by

$$F(x; \xi) = \tan\left(\frac{\pi}{4} G(x; \xi)\right)$$

where $x$ R, and the baseline distribution has CDF $G(x; \xi)$ and PDF $g(x; \xi)$. The parameter space of $\xi$ depends on the chosen baseline distribution. By differentiating the CDF, we get the PDF of Tan-G as follows

$$f(x; \xi) = \frac{1}{4} g(x; \xi) \pi \sec^2\left(\frac{\pi G(x; \xi)}{4}\right)$$

Practical Illustration

We assume the baseline distribution is Weibull with CDF

x f g e G x f g − =− where x, f, g > 0, then the Tan-G family defined in the previous section implies the following Definition 5.1. The CDF of Tan-Weibull is given by ( ) ( ) 1 ; , x f $$ ) = \tan \left(\frac {1}{4} \pi (1 - e ^ {- (1)} \right. $$ ( ) 1 ( ) tan( (1 )) 4 ; , e f g g π F x where x, f, g>0 Obviously, the PDF of Tan-Weibull can be obtained by differentiating the CDF Remark 5.2. We write J ∼ TW (f, g), if J is a Tan-Weibull random variable (Figure 1).

Click to enlarge

Comparison with Modified Ampadu-G

At first we recall the following Definition 6.1. Let λ > 0, ξ > 0 be a parameter vector all of whose entries are positive [7], and x ∈ R. A random variable X will be said to follow the Ampadu-G family of distributions if the CDF is given by ( )2 ; 1 ; , λ ξ − − G x e F x ( ) $$ ; \lambda , \xi) = \frac {1 - e}{1 - e ^ {- \lambda}} $$ − −

1 e and the PDF is given by

2 ; 2 ; ( ; ; ,

λ ξ λ ξ ξ λξ λ

$$ ) = \frac {2 \lambda g (x ; \xi) G (x ; \xi) e ^ {- \lambda}}{1 - e ^ {- \lambda}} $$ ) g x g x G x e f x ( ) ( )( )

1 e where the baseline distribution has CDF G(x; ξ) and PDF g(x; ξ) Remark 6.2. When λ ∈(-∞, 0) $$ \mathrm {E} = \mathrm {E} _ {1} + \mathrm {E} _ {2} + \dots + \mathrm {E} _ {n} $$ (0, ∞ ) and the parameter space of ξ depends on the baseline distribution with CDF G and PDF g, we call the distribution in the previous definition the Modified Ampadu-G Family of distributions In the rest of this section we compare the Modified Ampadu-Weibull and the Tan Weibull family of distributions in fitting the breast cancer patients data, Section 5.2 [8]. However, we have the following Proposition 6.3. The CDF of Modified Ampadu-Weibull is given by x f g e e F x f g $$ ) = \frac {1 - e ^ {- \lambda \left(1 - e ^ {- \lambda}\right)}}{1 - e ^ {- \lambda}} $$

where x, f, g>0 and λ ∈(-∞, 0) ( ) 2 (1 ) 1 ( ; , , ) λ λ λ

1 e $$ \mathrm {E} = \mathrm {E} _ {1} + \mathrm {E} _ {2} + \dots + \mathrm {E} _ {n} $$ (0, ∞ ) Remark 6.4. We write Q~MAW (f, g, λ), if Q is a Modified Ampadu-Weibull random variable (Figure 2).

Click to enlarge

In order to compare the two distribution models, we used the following criteria: -2(Log- likelihood) and AIC (Akaike information criterion), AICC (corrected Akaike information criterion), and BIC (Bayesian information criterion) for the data set. The better distribution corresponds to the smaller-2(Log-likelihood) AIC, AICC, and BIC values:

AIC = 2k − 2l

$$ C = A I C + \frac {2 k (k + 1)}{n - k - 1} $$

BIC = klog(n) − 2l

where k is the number of parameters in the statistical

model, n is the sample size, and l is the maximized value

of the log-likelihood function under the considered model.

2 ( 1) 1 k k AICC AIC n k From Table 2 above, it is clear that the TW distribution has the smallest values across all criteria considered, hence we see the newly discovered Tan-Weibull distribution is a better fit than the Modified Ampadu- Weibull distribution to the breast cancer patients data.

Concluding Remarks

In the present paper we introduced a so-called Tan-G family of distributions, and have shown fit to real-life data. The future interesting problem is to investigate some properties and application of this new class of statistical distributions.