1. Introduction
Because of their flexibility to model several behaviors, the Weibull and the lognormal distributions are two of the most used types of distribution in reliability. However, because the Weibull distribution is based on a non-homogeneous Poisson process, it models additive effect behavior [^{1}]. Similarly, because the lognormal distribution is based on a geometric Brownian motion, then it models multiplicative effect behavior [^{2}]. Therefore, they should not be used interchangeably. Hence, a discrimination process between both distributions is needed. In particular, the negative effect on reliability due to a wrong selection between these distributions is shown by using the stress- strength analysis, where the reliability represents all probabilities that the failure governing strength (S) exceeds the failure governing stress (s) [^{3}]. The stress-strength formulation is given by
In the stress-strength analysis it is assumed that time is not the cause of failure; instead, failure mechanisms are what cause the part to fail [^{4}]. In addition, as can be seen in eq. (1), the estimated reliability depends entirely on the selected stress and strength distributions. Thus, because a wrong selection will overestimate or underestimate reliability, a wrong selection will largely impact the analysis conclusions. To illustrate the impact of a wrong selection on reliability, following data published in Wessels has been used ([^{4}], sec. 7). Table 1 shows the stress data; and Table 2 the strength data.
Finally, the stress-strength analysis for the four possible combinations between the Weibull and lognormal distributions, is presented in Table 3. The estimation of the stress-strength reliability was performed by using the eq. (40-43) given in section 7.
From Table 3, we conclude that because each combination shows a different reliability index, then the accurate discrimination between the Weibull and the lognormal distributions is an issue that must be solved. To this end, researchers have used several selection procedures. Among the oldest ones are the Chi-square, the Anderson-Darling and the Cramer-Von Mises goodness-of-fit tests [^{5}]. On the other hand, the most widely used methods are those based on the maximum likelihood (ML) function as they are those given in [^{6}-^{10}] and recently in [^{11}-^{12}]. In particular, the methods based on probability plot (PP) tests are in [^{13}-^{15}]. Those based on Kolmogorov-Smirnov (KS) test are in [^{16}] and [^{17}], and those based on Bayes analysis are in [^{18}]. The discrimination process between the Weibull and the lognormal distributions depends 1) on the relationship between the Coefficient of Variation (CV) of the observed data and their standard deviation (_{ σx } ), 2) on the mean position of the logarithm of the data (_{ µx } ) and 3) on the dispersion behavior before and after _{ µx } . Unfortunately, since none of the above approaches takes into account the skew behavior of the logarithm of the data, then none of them is effective in discriminating between both distributions.
Based on the fact that the Weibull data logarithm (Gumbel behavior) always presents negatively skewed behavior, the logarithm of lognormal data always presents symmetrical dispersion behavior, the _{ b1ln/b1w } ratio of the estimated lognormal and Weibull coefficients effectively discriminates between the negative and symmetrical dispersion behaviors, a method based on _{ R2 } to effectively discriminate between both distributions is offered by this paper in sec. 4. The reason for the method’s efficiency is that the _{ R2 } index for a fixed sample size (n) depends only on the _{ b1ln/b1w } ratio (see sec. 4.3). That is, because the _{ b1ln/b1w } ratio effectively discriminates between negative and symmetrical dispersion behaviors, the _{ R2 } index effectively discriminates between both distributions also.
This paper is structured as follows. Section 2 shows that the behavior of the logarithm of a Weibull variable is always negatively skewed and that the logarithm of a lognormal variable is always symmetrical. In section 3, based on the data behavior log, the characteristics that completely define whether data follow a Weibull or a lognormal distribution are given. Also, in section 3, the case where the dispersion (Sxx) contribution is not fulfilled is presented also. Section 4 shows the multiple linear regression (MLR) analysis for the Weibull and lognormal distributions. Section 5 presents 1) how via MLR, the _{ b1ln/b1w } ratio efficiently captures the Sxx dispersion behavior, and 2) that because the _{ R2 } index for a fixed n value only depends on the _{ b1ln/b1w } ratio, it captures the Sxx dispersion behavior also. The application of a stress-strength analysis is given in section 6, while Section 7 shows the effect that a wrong selection has over the reliability index. Finally, the conclusions are presented in section 8.
2. Behavior of log-Weibull and log-lognormal variables
Since the discrimination method is based on the logarithm of the Weibull or lognormal observed data and on its dispersion behavior, then in this section, we show that the Weibull data logarithm follows a Gumbel distribution and that it is always negatively skewed. Similarly, we show that the logarithm of the lognormal data follows a Normal distribution and that it is always symmetrical.
2.1. Weibull and Gumbel relationship
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.
2.1.1. Dispersion of the Gumbel distribution
In order to show the dispersion of the log Weibull variable, several Weibull probability density functions (pdf) with fixed scale parameter η=50 and variable shape parameter β are plotted in Fig. 1. Fig. 2, corresponds to the conversion of the Weibull pdf of Fig. 1 on Gumbel pdf.
As can be seen in Fig. 2, the Gumbel distribution is always negatively skewed. Moreover, it is important to highlight that the Gumbel skew is constant at γ_{1}= -1.13955, and as demonstrated by [^{22}], it can be estimated as
On the other hand, as shown in next section, the logarithm of lognormal data follows a Normal distribution.
2.2. Lognormal and normal relationship
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.
2.2.1. Dispersion of the normal distribution
Fig. 3 represents several lognormal pdf for µ_{x}=1 and variable σ_{x}. Plotted Normal pdfs of Fig. 4 correspond to the logarithm of the lognormal pdfs plotted in Fig. 3. By comparing Fig. 3 and Fig. 4, we observe although the lognormal distribution is always positively skewed, its logarithm is always symmetrical.
Therefore, based on the log data behavior, the characteristics that completely define whether data follow a Weibull or lognormal distribution are given in next section.
3. Discrimination properties
This section presents that enough conditions are met in order to show that lognormal data follow a lognormal distribution and that Weibull data follow a Weibull distribution. Additionally, the critical characteristic to discriminate between both distributions when data follow neither a lognormal nor a Weibull distribution is given also.
3.1. Lognormal properties
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:
3.2. Weibull properties
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.
3.3. Weibull or lognormal distribution?
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.
4. Weibull and lognormal linear regression analysis
This section shows that by using MLR, the ratio of the slopes of the lognormal and Weibull distributions (_{ b1ln/b1w } ) is indeed efficient to discriminate between the negative and symmetrical skew behavior. Before showing that, the MLR analysis for the Weibull and lognormal distributions will first be introduced.
4.1. Weibull linear model
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.
4.2. Lognormal linear model
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.
4.3. Multiple linear regression analysis
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.
5. Proposed method
The proposed method is based on the fact that the critical characteristic to discriminate between the Weibull and the lognormal distributions is the Sxx contribution to the log standard deviation 𝜎 𝑥 . Thus, in order to present the steps of the proposed method to discriminate between the Weibull and lognormal distributions, it is necessary first to show that via MLR, the regression coefficients (slopes) b_{ 1ln } /b_{ 1w } ratio completely incorporates the negative skew and the symmetrical behavior of the observed data, and that the multiple linear regression coefficient R^{ 2 } completely depends on the b_{ 1ln } /b_{ 1w } ratio.
5.1. The ratio b _{1ln} /b _{1w} efficiently capture the Sxx behavior
The analysis for the Weibull and lognormal distributions is given below.
5.1.1. Weibull analysis
In order to show that the regression coefficients (slopes) b_{ 1ln } /b_{ 1w } ratio completely incorporates the skew behavior of the Weibull distribution represented by Sxx, it is necessary to first show that based on the b_{ 1ln } /b_{ 1w } ratio given by
For the Weibull distribution, Sxy_{ w } >Sxy_{ ln } . To observe this, it should be remembered that because the Weibull response variable y_{ w } given by y_{ w } =ln[-ln(1-F(t))] is higher weighted in the initial values (lower percentiles), and because for Weibull data, Sxx_{ - } tends to be greater than Sxx_{ +, } then the impact of Sxx_{ -- } over Sxy_{ w } given by Sxy_{ w } =y_{ w } (x-µ) is higher in the initial values. As should be noted, this fact implies that when data follows a Weibull distribution, the difference between Sxy_{ w } and Sxy_{ ln } tends to be higher. Likewise, from eq. (33), this fact implies that for Weibull data the b_{ 1ln } /b_{ 1w } ratio or Sxy_{ ln } /Sxy_{ w } decreases.
5.1.2. Lognormal analysis
In the lognormal case, because the lognormal response variable y_{ ln } , given by y_{ ln } = Φ^{ -1 } (F(t)), is symmetrical around the 50^{th} percentile, then for lognormal data Sxx_{ - } it tends to be Sxx_{ + } (see sec 3.1). As a consequence, the impact of Sxx_{ -- } on Sxy_{ ln } =y_{ ln } (x-µ) is lower than that of the Weibull distribution. This fact implies that for lognormal data, the difference between Sxy_{ w } and Sxy_{ ln } tends to be lower than when data is Weibull. As a result of this lower impact, when data is lognormal in eq. (33), the b_{ 1ln } /b_{ 1w } ratio or its equivalent Sxy_{ ln } /Sxy_{ w } increases.
Thus, because based on the Sxx behavior, for Weibull data the b_{ 1ln } /b_{ 1w } ratio decreases, and for lognormal, data it increases, then we conclude that because Sxy=y(x-µ) clearly captures the behavior of Sxx, then the b_{ 1ln } /b_{ 1w } ratio efficiently captures the behavior of Sxx also.
Now it will be shown that because the R^{ 2 } index depends only on the b_{ 1ln } /b_{ 1w } ratio, then it also captures the behavior of Sxx. Consequently, the R^{ 2 } index can also be used to discriminate between the Weibull and the lognormal distributions.
5.2. The R ^{2} index is completely defined by the b _{1ln} /b _{1w} ratio
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.
5.3. Steps of the proposed method
Because based on the observed data, the R^{2} 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 } .
By using the Weibull (or lognormal) Sxy value from step 1 and the Sxx value from step 2 into eq. (31), estimate the Weibull (or lognormal) slope b_{ 1 } coefficient.
By using the Weibull y vector defined in eq. (21) (or the lognormal y vector defined in eq. (26)), estimate the Weibull (or lognormal) variance of y as Syy=∑(y_{ i } -µ_{ y } )^{ 2 } .
By using the Weibull (or lognormal) slope b_{ 1 } coefficient from step 3, Weibull (or lognormal) Sxy value from step 1 and the Weibull (or lognormal) Syy value from step 4 into eq. (32), estimate the Weibull (or lognormal) coefficient R^{ 2 } .
Compare the Weibull and the lognormal R^{ 2 } indices, select the distribution with higher R^{ 2 } value. If R_{ w } ^{ 2 } >R_{ ln } ^{2} select Weibull distribution; otherwise select lognormal distribution.
6. An application
The efficiency of the R^{ 2 } index to discriminate between the Weibull and the lognormal distribution is shown in a stress-strength analysis by using data in section 1. Table 1 Data corresponds to the stress load in a machine that uses a plunger to press a shaft into a bushing. Table 2 Data corresponds to the strength of the plunger when it is subjected to compression loads [^{26}]. Thus, the selection of the stress distribution by using the proposed method is as follows.
6.1. Stress data analysis
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.
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.
6.2. Strength data analysis
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.
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.
7. Stress-Strength reliability
The stress-strength reliability values of Table 3 were estimated as follow. For the lognormal-lognormal stress-strength, the formulation given in eq. (42) was used
For the lognormal-Weibull stress-strength, the formulation given in eq. (43) was used
For the Weibull-lognormal stress-strength, the formulation given in eq. (44) was used
Finally, for the Weibull-Weibull stress-strength, the formulation given in eq. (45) was used
Where
8. Conclusions
The reliability analysis for the Weibull and the lognormal distributions is performed by using the data logarithm. For the Weibull distribution, the logarithm data is negatively skewed. For the lognormal distribution, the logarithm data is symmetrical. Because for the Weibull distribution, the contribution to the variance before the mean is always greater than the contribution after the mean [Sxx_{ -- } >Sxx_{ + } ], then this behavior is used to discriminate between the Weibull and the lognormal distributions. Since the b_{ 1ln } /b_{ 1w } ratio efficiently represents the contribution behavior, and since the R^{ 2 } index depends only on this ratio, then the R^{ 2 } index is indeed efficient to discriminate between the Weibull and the lognormal distributions. Finally, it is important to highlight that when in the observed data, σ_{ x } =CV, µ_{ x } tends to the 50^{th} percentile and Sxx_{ -- } =Sxx_{ + } , then the lognormal distribution can be directly fitted. And when for the observed data, σ_{ x } ≠CV, µ_{ x } tends to the 36.21th percentile and Sxx_{ -- } >Sxx_{ + } , then the Weibull distribution can be directly fitted.