The Weibull distribution is given by

In eq. (2), t >0 and β and η are the Weibull shape and scale parameters respectively. On the other hand, the Gumbel distribution is given by

In eq. (3)-∞< x <∞ with x=ln(t) and ^_μG is the location parameter and ^_σG is the scale parameter [19]. Thus, based on eq. (2) and eq. (3), the relation between both distributions is as follows.

Theorem: If a random variable t follows a Weibull distribution [t~W(β, η)], then its logarithm x=ln(t) follows a Gumbel distribution [ ^{_x~G(μG,σG)} ] [20].

Proof: Let F(ln(t)) = P(ln(t) ≤ ln(T)) be the cumulative function of x = ln(t), with T representing the failure time value. Thus, in terms of x, F(ln(t)) = Pr[ln(t) ≤ x]; F(x) = Pr[t ≤ exp(x)]. Then by substituting t = exp(x), F(x) is

Finally, based on the relations between the Weibull and Gumbel parameters given by [20].

and by taking ^{_{W=((x-µG)/σG)}} , eq. (4) is given by F(x) = 1-exp{-exp{(x-ln(η))·β}} which in terms of W is

from eq. (7), the reliability function is

and the density function is given by

clearly, eq. (9) in terms of W is

Since eq. (10) is as in eq. (3), we conclude that the logarithm of Weibull data follows a Gumbel distribution. On the other hand, by using the moment method [21] (sec. 1.3.6.6.16), the parameters of eq. (10) are given by:

where ^_µY and ^_σY are the mean and the standard deviation of the log data.

As it is well known, the lognormal data logarithm follows a Normal distribution [19]. If ^{_{Y~N(µ, σ2)}} , then ^_X=eY (non-negative) has a lognormal distribution. Thus, because the logarithm of X yields a Normal variable (Y=ln(X)) then the lognormal distribution is given by

In eq. (13) ^_μx and ^_σx are the log mean and log standard deviation. Similarly, the Normal distribution is given by

Note that, although the Normal distribution is the most widely used distribution in statistics, it is rarely used as lifetime distribution. However, in reliability the Normal distribution is used as a model for ln(t), when t has a lognormal distribution.

In order to select the lognormal distribution as the best model to represent the data, the following characteristics have to be met. First, the coefficient of variation has to be equal to the log-standard deviation ^_σx ( ^_σx=CV ). Thus, because based on the mean and on the standard deviation of the observed data defined as

the CV index is given by

Then from eq. (17) clearly ^{_{σx ≈ CV}} . Second, the log mean ^_µx should be located at the 50^th percentile. The reason is that the lognormal data logarithm follows a Normal distribution (see sec. 2.2). Third, since the total sum square (Sxx) is cumulated by the contribution before ( ^_Sxx-- ) and after ( ^_Sxx+ ) the mean ^_µx is as follow

Then, due to the symmetrical behavior of the lognormal data logarithm, then in the lognormal case, the contribution before and after the mean must be equal; it is to say for the lognormal case ^{_{Sxx--= Sxx+}} .

Thus, because when ^{_{σx ≈ CV}} , ^_µx is located in the 50^th percentile and ^{_{Sxx--= Sxx+}} , we should directly fit the lognormal model. Similarly, the characteristics to be met for the Weibull distribution are as follow:

In the Weibull case, because the Weibull data logarithm follows a Gumbel distribution, and because the Gumbel distribution is always negatively skewed (See sec 2.1.1), then the following characteristics have to be met. First, the coefficient of variation should be different from the standard deviation of the data logarithm ( ^{_{σx ≠ CV}} ). Second, the log mean ^_µx should be located around the 36.21^th percentile. Third, the contribution to Sxx before ^_µx is always greater than the contribution after ^_µx ; in other words, due to the negative skewness of the Gumbel distribution, in the Weibull case ^_Sxx-->Sxx+ . Thus, because ^{_{σx ≠ CV, µx}} is located around the 36.21^th percentile and ^_Sxx-->Sxx+ , then we should directly fit the Weibull distribution. Nonetheless, the next section will describe what happens when the above statements do not hold at all.

The discrimination process, when data neither completely follow a Weibull distribution nor completely follow a lognormal distribution, is based on the following facts. 1) For a Weibull shape parameter β≥2.5, the Weibull pdf is similar to the lognormal pdf [23]. 2) For β≥2.5, the log-standard deviation ^_σx tends to be the CV ( ^{_{σx ≈ CV}} ), and ^_µx tends to be located near the 50^th percentile. 3) For Weibull data, regardless of the β value, the contribution before and after the mean tends to be different ( ^_Sxx-->Sxx+ ). Now for the Normal distribution we always expect that ^_Sxx--=Sxx+ and for the Gumbel distribution we always expect that ^_Sxx-->Sxx+ ; thus, because from eq. (18), ^_Sxx-- captures the skewness of the Gumbel distribution, then based on the MLR analysis, in the proposed method the product of the y vector with the ^_Sxx-- and ^_Sxx+ contribution is used as the critical variable to discriminate between the Weibull and the lognormal distributions. In order to show that, the linear regression analysis on which the proposed method is based must first be introduced.

The Weibull and lognormal distributions can be analyzed as a regression model of the form

The linear form of the Weibull distribution is based on the cumulative density function, given by

Thus, by applying double logarithm, its linear form is

where ^_F(ti) is estimated by the median rank approach [24] given by

From eq. (21), the shape parameter β is directly given by the slope ^_b1 , and the scale parameter η is given by

Additionally, it is necessary to note that in eq. (21)y=ln(-ln(1-F(t))) represents the behavior of the Gumbel distribution (negative skew), and that once the Weibull parameters β and η are known, the expected data can be estimated as

Clearly, from eq. (24), the ^_ln(ti) value depends only on y. And since from the double logarithm the y values before F(t)=1-e^-1=0.6321 are always negatively skewed, then in order for that data follows a Weibull distribution, its logarithm has to be negatively skewed as well. This fact implies that in the Weibull case, ^_Sxx-->Sxx+ is always true. On the other hand, the analysis for the lognormal distribution is as follows.

Since for the lognormal distribution the cumulative density function is given by

Then the lognormal linear relationship is given by

where ^_µx is given by ^_µx=-b0/b1 , and ^_σx is given by ^_σx=1/b1 and ^_F(ti) is estimated as in eq. (22). On the other hand, ^_µx and ^_σx can respectively be estimated directly from the data as

From eq. (26) ^{_y=Φ-1(F(t))} represents the behavior of the Normal distribution (symmetrical behavior). Thus, once the lognormal parameters ^_µx and ^_σx are known, the expected data can be estimated from eq.(26) as follows:

On the other hand, since ^_ln(ti) in eq. (29) follows a normal distribution, then its behavior is always symmetrical, and as a consequence of the lognormal case, the contribution to the Sxx variable is equivalent before and after μ x . In other words, in the lognormal case, Sxx _-- =Sxx ₊ . Now that it has been seen that for the Weibull distribution Sxx _-- >Sxx ₊ , and that for the lognormal distribution Sxx _-- =Sxx ₊ , let us describe the linear regression analysis to show that the ratio of the Weibull and lognormal regression coefficients efficiently represents the Sxx _-- and Sxx ₊ behavior.

In order to discriminate between the Weibull and lognormal distributions, first, the Weibull parameters of eq. (21) and the lognormal parameters of eq. (26) have to be estimated by using linear regression analysis as follows

The related multiple determination coefficient (R ² ), is

Thus, since from eq. (30) and eq. (31), we observe that the estimated coefficients are based on the key variable Sxx, then we conclude that the regression coefficients b ₀ and b ₁ represent the Sxx behavior also. Based on these parameters, the proposed method is outlined in the next section.

The analysis for the Weibull and lognormal distributions is given below.

In order to show that the R ² index is completely defined for the b _1ln /b _1w ratio, it will be first be noted that based on eq. (32), the relationship between b _1w parameter and the Weibull R _w index and the relationship between the b _1ln parameter and the lognormal R _ln index can be formulated by the following relation

Secondly, in doing this it should be observed that by taking away Sxy=b ₁ Sxx from eq. (31), and by replacing it in eq. (34), b ₁ is directly related with σ _x , σ _y and R ² , as follows

where

Thus, from eq. (35), the Weibull b _1w and R _w values are related with the lognormal b _1ln and R _ln values as follows

Next, it will be shown that because the R _ln /R _w ratio depends only on the b _1ln /b _1w ratio, then the R ² index can be used to efficiently discriminate between the Weibull and the lognormal distributions. Having done this, it should also be noted from eq. (38) and eq. (39) that σ _x is the standard deviation of the data logarithm, and that it is the same for both distributions. This fact (σ _x = σ _x ) implies from eq. (38) and eq. (39) that

Therefore, based in eq. (40), the relationship between the R _ln and the R _w indices is given by

And because the σ _{y w} /σ _{y ln} ratio is constant in the analysis, we conclude that the R _ln /R _w ratio depends only on the b _1ln /b _1w ratio. Consequently, the R ² index is efficient to discriminate between the Weibull and the lognormal distributions. Additionally, it is important to highlight that the σ _yw /σ _yln ratio in eq. (41) is constant also, and that this is so because σ _y defined in eq. (36) depends only on the sample size n. Thus, once n is known (or selected, see [25] eq. (13)), σ _y is constant.

Because based on the observed data, the R² index efficiently represents the Sxx behavior, then based on the observed data, the steps of the proposed method to discriminate between the Weibull and the lognormal distributions are as follows.

By using the Weibull y vector defined in eq. (21) (or the lognormal y vector defined in eq. (26)) and the observed data logarithm (ln(t)=x), the Weibull (or lognormal) correlation is estimated as Sxy=∑y_i(x_i-µ_x).

From the logarithm of the observed data, estimate the variance of x as Sxx=∑(x _i -µ _x ) ² .

From the stress observed data shown in Table 4, we note that because 1) the σ _x ≈CV (σ _x =CV=0.0055), 2), µ _x is located near the 50^th percentile, and 3) Sxx _-- = 53% ≈ Sxx ₊ = 47%. Then, from section 3.1, it is reasonable to expect that the lognormal distribution represents the data.

The above statement is verified by applying the proposed method to the Table 1 data. The required values Sxy, Sxx and Syy to apply the method are estimated by applying the MLR analysis to the stress data. The values for the Weibull and the lognormal distributions are given in Table 4.

Table 4

Source: The authors

Thus, by using data of Table 4, the lognormal analysis is as follows. From step 1, Sxy _ln =7.1714 (column 9). From step 2, Sxx=1.3412 (column 10). From step 3, and eq. (31), b _1ln =5.3471. From step 4, Syy _ln =39.4812 (column 13). Therefore, from step 5, and eq. (32), R _ln ² =0.9712.

Similarly, by applying the proposed method to the Weibull distribution, we have from step 1, Sxy _w =8.8996 (column 8). From step 2, Sxx, as in the lognormal case, is also Sxx=1.3412 (column 10). From step 3, and eq. (31), b _1w =6.6357. From step 4, Syy _w =61.9775 (column 12). Therefore, from step 5, and eq. (32), R _w ² =0.9528.

Finally, as expected, by comparing the Weibull and lognormal R ² indices, in step 6, we have that R _ln ² =0.9712>R _w ² =0.9528. Thus, we conclude that the failure governing the stress distribution is the lognormal distribution. On the other hand, the selection of the strength distribution by using the proposed method is as follows.

The strength data is given in Table 5. From this data, we note that while 1) the σ _x ≈CV (σ _x =CV=0.0077) and 2) the µ _x is located near the 50^th percentile, 3) the Sxx _-- contribution is greater than the Sxx ₊ contribution Sxx _-- =61%> Sxx ₊ =39%. Thus, because from section 3.3 the characteristics of the lognormal distribution are not completely met, then we conclude that data can be better represented by the Weibull distribution. However, the estimation of the R ² index is necessary. When doing this, the values of Sxy, Sxx and Syy are estimated by using an MLR analysis of the strength data. The MLR analyses for the Weibull and the lognormal distributions are summarized in Table 5.

Table 5

Source: The authors

By using Table 5 data and by applying the proposed method for the lognormal distribution, we have that, from step 1, Sxy _ln =1.9402 (column 9). From step 2, Sxx=0.3298 (column 10). From step 3, and eq. (31), b _1ln =5.8820. And from step 4, Syy _ln =12.2451 (column 13). Therefore, from step 5, and eq. (32) R ² =0.9319.

Similarly, by applying the proposed method for the Weibull distribution, we have that, from step 1, Sxy _w =2.4320 (column 8). From step 2, Sxx as in the lognormal case, is Sxx=0.3298 (column 10). From step 3, and eq. (31), b _1w =7.3730. And from step 4, Syy _w =18.5330 (column 12). Therefore, from step 5, and eq. (32), R _w ² =0.9675.

Finally, by comparing the R ² indices as in step 6, we have that R _w ² =0.9675>R _ln ² =0.9319. Thus, the failure governing the strength distribution is the Weibull distribution.

As a summary, because the stress data follows a lognormal distribution and the strength data follows a Weibull distribution, then for the stress-strength analysis the lognormal-Weibull combination has to be used. Therefore, from Table 3, the corresponding lognormal-Weibull reliability is R(t)=0.9860. Finally, the effect that a wrong selection of the distribution has over the estimated reliability is given in Table 3. Although the reliability values given in Table 3 were estimated by using the Weibull++ software, the next section provides the formulas to estimate such values.