*2.4. Uniparametric Weibull Distributions*

Consider the uniparametric Weibull distributions family given by the CDF

$$\mathcal{W}\left(\mathbf{x};\mathcal{c}\right) = 1 - \exp\left(-\mathbf{x}^{\mathcal{c}}\right), \quad \mathbf{x} \succ \mathbf{0}, \mathcal{c} \succ \mathbf{0}. \tag{12}$$

 /

The mode is known to be at 0, for *c* ≤ 1 (as a limit, when *c* < 1) and at

$$0 < M\_{\mathcal{E}} = \left(\frac{c-1}{c}\right)^{1/c} < 1\_{\mathcal{E}}$$

for *c* > 1. The expression for *νW* is given by

$$\nu\_W(z;\mathfrak{c}) = \begin{cases} \exp\left[-\left(M\_{\mathfrak{c}} + z\right)^{\mathfrak{c}}\right] + \exp\left[-\left(M\_{\mathfrak{c}} - z\right)^{\mathfrak{c}}\right] - 1, & 0 < z < M\_{\mathfrak{c}} \\\\ \exp\left[-\left(M\_{\mathfrak{c}} + z\right)^{\mathfrak{c}}\right], & z \ge M\_{\mathfrak{c}} \end{cases}$$

On the one hand, when *c* < 1, note that *<sup>ν</sup>W*(*c*) (1) = *e*<sup>−</sup>1, so all these functions intersect at this point. Graphically, it can be seen that there is no ordering by "≥*ν*", and also that *S*<sup>+</sup> (*W* (*c*)) = 1, when *c* < 1. On the other hand, for 1 ≤ *c*1 < *c*2, the following result is obtained.

**Proposition 5.** *Let W* (*<sup>c</sup>*1) *and W* (*<sup>c</sup>*2) *be Weibull distributions with* 1 ≤ *c*1 < *c*2 *and CDF as in (12). Then,*

$$W\left(c\_1\right) \ge\_{\mathbb{V}} W\left(c\_2\right).$$

**Proof.** For 1 ≤ *c*1 < *c*2, the corresponding modes are *M*1 < *M*2. Then, for 0 < *z* < *M*1,

$$\nu\_W\left(z;\mathbf{c}\_1\right) - \nu\_W\left(z;\mathbf{c}\_2\right) = \left\{ \exp\left[-\left(M\_1 + z\right)^{c\_1}\right] - \exp\left[-\left(M\_2 + z\right)^{c\_2}\right] \right\}$$

$$+ \left\{ \exp\left[-\left(M\_1 - z\right)^{c\_1}\right] - \exp\left[-\left(M\_2 - z\right)^{c\_2}\right] \right\} > 0,$$

because each part of the expression inside brackets {·} is positive. If we take *M*1 ≤ *z* < *M*2, then

$$
\nu\_W \left( z; c\_1 \right) - \nu\_W \left( z; c\_2 \right) = \left\{ \exp \left[ - \left( M\_1 + z \right)^{c\_1} \right] - \exp \left[ - \left( M\_2 + z \right)^{c\_2} \right] \right\}
$$

$$
+ \left\{ 1 - \exp \left[ - \left( M\_2 - z \right)^{c\_2} \right] \right\} > 0,
$$

for a similar reason. Finally, if we take *z* > *M*2, then

$$\nu\_W\left(z; \mathbf{c}\_1\right) - \nu\_W\left(z; \mathbf{c}\_2\right) = \exp\left[-\left(M\_1 + z\right)^{\varepsilon\_1}\right] - \exp\left[-\left(M\_2 + z\right)^{\varepsilon\_2}\right] > 0,$$

and the proof is completed.
