Graphical Tools to Assess Goodness-of-Fit in Non-Location-Scale Distributions

Goodness-of-ﬁt (GOF) techniques are used for assessment whether a distribution is suitable to describe a data set or not. These techniques have been studied for distributions belonging to the location-scale family. However, one could be interested in making this assessment for distributions that do not belong to this family. We review the available GOF tests and pro-pose graphical tools based on these tests for censored and uncensored data from non-location-scale distributions. Anderson-Darling, Cramér-von Mises, Kolmogorov-Smirnov, Kuiper, Michael and Watson GOF statistics are considered. We apply the proposed results to real-world data sets to illustrate their potential, with emphasis on some Birnbaum-Saunders distributions.

Several efforts have been made to develop goodness-of-fit (GOF) techniques allowing us to address the problem of fitting the mentioned distributions to different types of data.In general terms, GOF tests permit us to assess whether the distribution under a null hypothesis (H 0 ) is adequate to describe a data set or not.For this hypothesis, there are two options: (i) the distribution can be completely specified (known parameters) or (ii) some (or all) of its parameters are unknown, in which case they need to be estimated with proper methods, such as maximum likelihood (ML).Depending on the distribution under H 0 , ML estimates of its parameters cannot be easily calculated and iterative numerical procedures must be used, but problems of convergence may arise, which have not yet been completely studied; see Castillo & Puig (1997) for more details in this regard.
According to D' Agostino & Stephens (1986, pp. xi-xiii), GOF methods can be of graphical type, of chi-squared type or based on the empirical cumulative distribution function (ECDF), on correlation/regression, on transformations or on moments.Most test statistics used for assessing GOF, such as Anderson-Darling (AD), Cramér-von Mises (CM), Kolmogorov-Smirnov (KS), Kuiper (KU) and Watson (WA), compare the ECDF and the hypothesized theoretical cumulative distribution function (CDF) assumed for the data.KU and WA statistics are modifications of the KS and CM statistics, respectively, in particular for circular data given its invariant property under change of origin in a circle.For more details about AD, CM, KS, KU and WA statistics, see D' Agostino & Stephens (1986, Ch. 4).Some graphical counterparts of the mentioned goodness-of-fit tests are considered by Chen & Balakrishnan (1995), Choulakian & Stephens (2001), Hui, Gel & Gastwirth (2008) and Stehlík, Strelec & Thulin (2014).
A graph allowing us to relate the ECDF with a specified theoretical CDF is the probability versus probability (PP) plot.Analogously, ordered observations corresponding to empirical quantiles can be plotted versus the theoretical quantiles of a specified distribution in a graph known as the QQ plot; see Marden (2004).This author studied such a plot in the case of the normal distribution and provided tools for graphical comparison of two samples, extending also the QQ plot to multivariate data.A disadvantage of the PP plot associated with the KS test is that some points in this graph can be more variable than others.Michael (1983) proposed a modification for the KS test based on the arcsin transformation to stabilize the variance of the points in the PP plot.The graph related to this variance stabilizing transformation is known as the stabilized probability (SP) plot and the statistic associated with the test proposed by Michael (1983) is denoted as MI.This author studied the power of the MI test showing that it proves more powerful than the KS test for certain alternative hypotheses.Stehlík et al. (2014) compared the power and robustness of several classic tests for normality, such as the AD, Jarque-Bera, KU and Shapiro-Wilk, for different distributions in the alternative hypothesis, including the Cauchy, Laplace and Student-t with different degrees of freedom (DF) models.These authors concluded that in general no test can be recommended as the best for all the alternatives considered.
In reliability and survival analyses, it is frequent to find situations where not all individuals or instruments tested complete the event under study which, without the loss of generality, may be referred to as a "failure".Samples involving such situations are named censored data; for literature on this topic see, e.g., Cohen (1991) and Lawless (2003).
When a parametric statistical analysis with censored data needs to validate its distributional assumption, classical GOF test statistics need to be adapted to consider the censorship following two options.The first consists of using GOF tests for uncensored data adapting the type-II right censored data to become an uncensored (complete) data sample, whereas the second option adapts test statistics to type-II right censored data; see, e.g., Malmquist (1950), D'Agostino &Stephens (1986, Chs. 4 and11), Lin, Huang & Balakrishnan (2008) and Barros, Leiva, Ospina & Tsuyuguchi (2014).
The objectives of this article are to review the available GOF tests and propose graphical tools for assessing GOF in NLS distributions based on these tests with uncensored and censored data.These tools can be used for any distribution in the NLS family, as long as their parameters are known or properly estimated.With the provided tools, it is possible to decide what distribution best fits the data.This article is structured as follows.In Section 2, we present some well-known life distributions and the estimation of their parameters with packages of the R software; see www.r-project.org.In Section 3, we introduce GOF tests for NLS distributions with uncensored and censored data.In Section 4, we propose graphical tools based on these GOF tests.In Section 5, we illustrate the graphical tools provided in this paper analyzing real-world data sets from different fields, with emphasis on some Birnbaum-Saunders distributions.We also analyze some empirical robustness aspects with one of these data sets, following the ideas of Hui et al. (2008) and Stehlík et al. (2014).Finally, in Section 6, we sketch some conclusions and mention future research on this topic.

Life Distributions
In this section, we present some well-known life distributions, with α > 0 and β > 0 denoting shape and scale parameters, respectively, and µ denoting the mean of the distribution.

The Gamma Distribution
A RV T following the GA distribution with shape α > 0 and scale β > 0 parameters is denoted by T ∼ GA(α, β).In this case, the CDF of T is where Γ(•) and Γ(•, •) denote the usual and incomplete gamma functions, respectively.Now, the QF given by F −1 (q; α, β), for 0 < q < 1, must be obtained by means of an iterative method.

The Inverse Gaussian Distribution
A RV T following the IG distribution with mean µ > 0 and scale β > 0 parameters is denoted by T ∼ IG(µ, β).In this case, the CDF of T is Again the corresponding QF, expressed as F −1 (q; µ, β) = t(q), must be obtained with a numerical method.

The Weibull Distribution
A RV T following the WE distribution with shape α > 0 and scale β > 0 parameters is denoted by T ∼ WE(α, β).In this case, the CDF of T is

Estimation of Parameters
Using R software (i) exploratory data analysis (EDA) can be conducted for diagnosing statistical features present in the data to be analyzed; and (ii) estimation of the parameters of BS, BS-t, GA, IG, LN, TBS and WE distributions can be carried out by the ML method.Next, we describe the R functions (commands) of gbs, ig and basic packages and briefly illustrate their use.The tbs package is available upon request from the authors.These packages incorporate CDF, probability density function (PDF) and QF of the TBS distribution, as well as a random number generator, the moments and the ML estimation of its parameters for censored and uncensored data.
The R software can be downloaded from CRAN.r-project.org and installed as any other software.It may be used in a simple interactive form with the R commander by installing the Rcmdr package.The gbs, ig and tbs packages must be also installed.Data analyses based on (i) BS and BS-t distributions can be carried with the gbs package, (ii) IG distribution with the ig package and (iii) GA, LN and WE distributions with the basic or fitdistrplus packages.As an example, the gbs package and a data set, x say, must be loaded as

> library(gbs) > data(x)
The data can also be directly imported from text files, from other statistical software or from Excel.Table 1 provides examples of some commands that allow us to work with the BS distribution, whereas similar instructions may be used for the other distributions; for more details on how to use the gbs package, see Barros et al. (2009).
Table 1: Basic functions of the gbs package.

EDA descriptiveSummary(x)
A descriptive summary of data x is obtained.histogram histgbs(x, boxPlot=T, pdfLine=T) Histogram, boxplot and fitted PDF for x are given.

GOF Tests with Censored and Uncensored Data
In this section, we define the notations and provide some transformations useful for obtaining the graphical tools proposed in the paper.In addition, we establish the hypotheses of interest and the corresponding test statistics to assess GOF for NLS distributions with censored and uncensored data.

Un/Censored Data, Transformations and Order Statistics
Let T 1 , . . ., T n be a sample of size n extracted from a RV T following a distribution with CDF F (•). Also, let T 1:n ≤ • • • ≤ T n:n be the order statistics (OSs) of T 1 , . . ., T n , with t 1:n ≤ • • • ≤ t n:n and t 1 , . . ., t n being their corresponding observations (data).The ECDF is defined by (1) where As it is well-known, the RV U given by the transformation follows a uniform distribution in [0,1], denoted by U(0,1), for any continuous F (•), which is known as probability integral transformation (PIT).
Another transformation associated with the U(0, 1) distribution was proposed by Michael (1983), who noted that, if U ∼ U(0, 1), then the RV given by the transformation which is known as stabilized probability transformation, follows a distribution with PDF f S (s) = 2 π sin(πs), 0 < s < 1. (4) The OSs S 1:n ≤ • • • ≤ S n:n associated with a sample of size n from the distribution of the RV S given in (3) have a constant asymptotic variance, because as n goes to ∞ and j/n to q, Var[n S j:n ] goes to 1/π 2 , which is independent of q, for j = 1, . . ., n; see Michael (1983).Consider a sample with censoring proportion p, which conducts to r failure (uncensored) data and n − r censored data.Note that n and p are controlled by the researcher.Assume a type-II right censorship, so that, in this case, r is fixed and n − r observations are greater than the censoring point t r:n .Then, the smallest r OSs of the type-II censored sample of size n and n − r observations are greater than u r:n = F (t r:n ).Let U j:n = F (T j:n ) be the jth OS of a sample of size n extracted from a RV U ∼ U(0, 1) and u j:n = F (t j:n ) be its observed value, for j = 1, . . ., n.Consider the transformation where B r,n−r+1 (x) = I x (r, n − r + 1) is the Beta(r, n − r + 1) CDF, with I x (•, •) being the incomplete beta ratio function.Hence, the OSs U 1:n , . . ., U r:n obtained from the transformation given in ( 5) are distributed as the OSs from a complete sample of size r from U ∼ U(0, 1); see details in Theorem 1 of Lin et al. (2008), Theorem 1 of Michael & Schucany (1979) and Theorem 8 of Fischer & Kamps (2011).

Hypotheses
Consider the hypotheses: H 0 : "The data come from a RV T with CDF F (•)" versus H 1 : "The data do not come from this RV".The hypothesized distribution with CDF F (•) is indexed by a parameter vector θ that can contain location (µ), scale (β), shape (α) parameters, or any other parameter not necessarily of location and scale, that is, T belongs to the NLS family.If the CDF is completely specified in H 0 , that is, θ is assumed to be known, the data must be transformed for testing uniformity.On the contrary, the parameters must be consistently estimated and the data transformed for testing normality of the distribution under H 0 .

Test Statistics for GOF with Uncensored Data
In order to test H 0 established in Subsection 3.2, where F (•) is completely specified, and then to assess GOF of a distribution to a censored or uncensored data set, we consider test statistics based on the ECDF F n (•) defined in (1).The most common statistics constructed with the ECDF use vertical differences between F n (t) and F (t) by means of the supremum and quadratic classes.Statistics that consider the mentioned classes are AD, CM, KS, KU and WA given by Revista Colombiana de Estadística 37 (2014) 341-365 By considering (3) and ( 8), we can define the MI statistic as Now, by considering (2), AD, CM, KS, KU, MI and WA statistics defined in ( 6)-( 11) can be implemented in practice by the formulas where w j:n and U j:n are as given in (1) and ( 5), respectively.Further details on the expressions provided in ( 6)-( 17) can be found in D 'Agostino & Stephens (1986, Ch. 4) and Michael (1983).Quantiles of the distribution of the statistics AD, CM, KS, KU, MI and WA must be obtained under H 0 .However, if the distribution under this hypothesis is not completely specified, its parameters must be properly estimated and the AD, CM, KS, KU, MI and WA statistics must be modified for the distribution under H 0 .These modified statistics are denoted by AD , CM , KS , KU , MI and WA , and their calculated values by ad , cm , ks , ku , mi and wa , respectively.In this case, new quantiles of the distribution of AD , CM , KS , KU , MI and WA must be computed under H 0 .

Test Statistics for GOF with Censored Data
To test H 0 where F (•) is completely specified and then to assess GOF in practice with r uncensored data and n − r type-II right censored data, we use the results presented in D' Agostino & Stephens (1986, Ch. 4) and adapt the statistics given in ( 12)-( 17 KS r,n = max max 1≤j≤r KU r,n = max 1≤j≤r MI r,n = max max The quantiles of the distribution of the AD r,n , CM r,n , KS r,n , KU r,n , MI r,n and WA r,n statistics given in ( 18)-( 23) must be obtained under H 0 .However, if the distribution under H 0 is not completely specified, its parameters must be properly estimated, taking into account the censorship, and the statistics must be modified for each case under H 0 .We denote these statistics by AD r,n , CM r,n , KS r,n , KU r,n , MI r,n and WA r,n , and their calculated values by ad r,n , cm r,n , ks r,n , ku r,n , mi r,n and wa r,n , respectively.Also, new quantiles of the distribution of AD  2010) and Castro-Kuriss (2011).In the next subsections, we mention that, for NLS distributions, analogous results for assessing GOF with both censored and uncensored data can be considered.

GOF Tests for NLS Distributions with Uncensored Data
If the hypotheses of interest H 0 is F (t) = Φ([t − µ]/β) with unknown parameters, we can consider the procedure detailed in Algorithm 1.
We consider a procedure that can be applied to NLS distributions based on the work proposed by Chen & Balakrishnan (1995), which provides an approximate GOF method.This method first transforms the data to normality and then applies Algorithm 1, generalizing it.Testing normality in H 0 allows us to compute the critical values of the corresponding test statistics, independently of the parameter Algorithm 1 GOF test for normality with uncensored data 1: Collect data t 1 , . . ., tn and order them as t 1:n , . . ., tn:n; 2: Estimate µ and β of Φ([t − µ]/β) by µ and β, respectively, with t 1 , . . ., tn; 3: Obtain u j:n = Φ( z j ), with z j = [t j:n − µ]/ β, for j = 1, . . ., n; 4: Evaluate AD , CM , KS , KU , MI and WA statistics at u j:n ; 5: Compute the p-values of the AD , CM , KS , KU , MI and WA statistics; 6: Reject H 0 : F (t) = Φ([t − µ]/β) for a specified significance level based on the obtained pvalues.
estimators, if they are consistent and the sample size is large enough.To test the hypotheses of interest defined in Subsection 3.2, for α > 0 and β > 0 unknown, we consider a generalization of Algorithm 1 based on Chen & Balakrishnan (1995)'s method, which is detailed in Algorithm 2. Following Chen & Balakrishnan (1995); in general, we recommend a sample size n > 20, so that the approximations work well.This is also valid for the algorithms presented in the next sections.

GOF Tests for NLS Distributions with Censored Data
As mentioned, GOF tests for NLS distributions with uncensored data can be considered for censored data adapting them or the GOF statistics.
To test the hypotheses of interest defined in Subsection 3.2, for α > 0 and β > 0 both of them unknown and type-II right censored data, we first transform censored data into uncensored data by using (5).Algorithm 3 details the corresponding GOF procedure.
Second, as mentioned, another way to perform a GOF test for NLS distributions with censored data can be obtained adapting the GOF statistics, as detailed in Algorithm 4.
Algorithm 4 GOF test 2 for NLS distributions with censored data 1: Repeat Steps 1-5 of Algorithm 2; 2: Evaluate AD r,n , CM r,n , KS r,n , KU r,n , MI r,n and WA r,n statistics at u j:n ; 3: Determine the p-values of AD r,n , CM r,n , KS r,n , KU r,n , MI r,n and WA r,n statistics; 4: Reject the corresponding H 0 for a specified significance level depending on the obtained p-values.

Graphical Tools
In this section, based on Algorithm 4, we provide acceptance regions for the KS and MI statistics that allow graphical tools to be obtained for assessing GOF in NLS distributions.

PP and SP Plots
PP and QQ plots are well known, however this is not the case of the SP plot.Note that, if the distribution under H 0 is U(0,1), then the corresponding QQ plot is essentially the same as the PP plot; see Castro-Kuriss et al. (2009).Michael (1983) used the arcsin transformation to stabilize the variance of points on probability graphs associated with the KS test to propose the SP plot; see PDF given in ( 4) and comments relating thereto in Subsection 3.1.Formulas to construct PP and SP plots are provided in Table 2.In this table, w j:n is as given in (1) and u j:n as given above (5).

Acceptance Regions for Probability Plots
Acceptance regions for PP and SP plots can be constructed by means of KS and MI statistics.Thus, we can display acceptance bands to assess whether the data can come from the distribution under H 0 with these two statistics; see Castro-Kuriss et al. (2009) and Castro-Kuriss et al. (2010).Formulas to construct 100 % acceptance regions on PP and SP plots with right type-II censored data, based on KS r,n and MI r,n , are displayed in Table 3.In this table, w and x are continuous versions of w j:n and x j:n given in Table 2 to construct the acceptance bands.If all of the r data points lie within the constructed acceptance bands, then H 0 cannot be rejected at the 1 − level.Also, if a noticeable curvature is detected, we can question such a hypothesis.Table 3 may be adapted to the uncensored case with r = n and the quantiles must be replaced by the quantiles of the distribution of the corresponding statistics without censorship.
To test H 0 defined in Subsection 3.2, for some α > 0 and β > 0 and type-II right censored data, we propose a graphical tool which procedure is detailed in Algorithm 5 based on Algorithm 4 and Tables 2 and 3, which is valid for censored or uncensored data.We consider the general case for unknown parameters of the distribution under H 0 , although it may also be used when the parameters are known.
Algorithm 5 Acceptance regions to test GOF for NLS distributions with censored data 1: Repeat Step 1 of Algorithm 4; 2: Draw the PP plot with points w j:n versus u j:n , for j = 1, . . ., r and r = 1, . . ., n; 3: Draw the SP plot with points x j:n = [2/π] arcsin( √ w j:n ) versus s j:n = [2/π] arcsin( u j:n ); 4: Construct acceptance bands according to Table 3 specifying a 1 − level; 5: Decide if H 0 must be rejected for the corresponding significance level; 6: Corroborate decision in Step 5 with the p-values after evaluating KS r,n and MI r,n statistics at u j:n .

Examples
In this section, we consider several real-world data sets and NLS distributions under H 0 and decide whether such data may reasonably come from the hypothesized distribution.The results are also displayed by means of the probability plots with the acceptance bands proposed in Section 4.

Example 1: Uncensored Sea Data
These data are drawn from sea surface temperatures (in o K), generated by an advanced very high resolution radiometer.We name data "sea".The sample size is n = 88 and the truncation point is κ = 278.187o K; see DePriest (1983).An EDA for sea data is provided in Table 4, including the coefficients of variation (CV), skewness (CS) and kurtosis (CK) and the standard deviation (SD).Figure 1 displays their histogram and boxplot.This EDA indicates that the TBS distribution can be suitable for describing such data, as a competitor of the TN distribution.Some atypical data are detected in the boxplot, but their study is not considered here since it is beyond the scope of this work.We consider the TBS and TN models under H 0 with parameters estimated.We use the tbs and truncnorm R packages, respectively, for making this estimation with the ML method.The corresponding estimates and the p-values from the GOF tests are displayed in Table 5.According to these p-values, both distributions perform a good fitting to the data.We show the PP and SP plots in Figure 2 for the TBS distribution defined in Section 2. From this figure, note that the points are well aligned, as expected, due to the high p-values obtained for the TBS distribution, and that all the points fall inside the 95% acceptance bands, which confirms the good fitting of the TBS distribution to sea data.

Example 2: Uncensored Forestry Data
Data correspond to the diameter at breast height (DBH, in cm) of loblolly pine trees from a plantation in the Western Gulf Coast.We name data "forestry".The sample size is n = 75 and the left-truncation point is κ = 6 cm; see Leiva, Ponce, Marchant & Bustos (2012).Table 6 provides an EDA of forestry data, which indicates once again that the TBS distribution can be a good model for the data.
From the histograms displayed in Figure 3, note that the fit of the TBS distribution is apparently better than for the TN distribution.The corresponding estimates and the p-values from the GOF tests are displayed in Table 7.According to these p-values, both distributions perform a reasonable fit to the data, but clearly the TBS distribution has a better performance.Figure 4 shows the PP and SP plots for the TBS distribution.From this figure, note that once again the points are well aligned, as expected, due to the p-values obtained for the TBS distribution, and all the points fall within distribution to forestry data.

Example 3: Uncensored Survival Data
These data are drawn from survival times (in days) of pigs injected with a dose of tubercle bacilli, under a regimen corresponding to 4.0 × 10 6 bacillary units per 0.5 ml (log(4.0× 10 6 ) = 6.6).We name data "survival".The sample size is n = 72 guinea pigs infected with tubercle bacilli in regimen 6.6; see Azevedo et al. (2012).Table 8 and Figure 5 provide an EDA of survival data.From this EDA, we detect a distribution skewed to the right with the presence of some outliers.We are considering an outlier in the sense defined by Hubert & Vandervieren (2008).We propose the BS and BS-t distributions defined in Section 2 for analyzing survival data.ML estimates of the BS and BS-t parameters, which are obtained using the gbs package, and the p-values from the GOF tests are displayed in Table 9. Clearly the BS distribution does not properly fit these data, whereas the BS-t distribution provides a better fit.Figures 6 and 7 show the PP and SP plots for the BS and the BS-t distributions, respectively.From Figure 6, note that the points are not well aligned, particularly in the center, and that one observation (number 15) is outside the KS band.In Figure 7, the points are better aligned, as expected, due to the p-values obtained for the BS-t distribution, and all the points fall within the 95% acceptance bands, indicating the good fitting of the BS-t distribution to survival data.6: PP (left) and SP (right) plots with 95% acceptance bands for the BS distribution with survival data.

Example 4: Uncensored Survival Data with Outliers
Next, we conduct a simple empirical robustness study.First, we add a large value (outlier) to the data and call them "survival1", so that we now have a sample  of size n = 73.This new observation is greater than all the observed values (t 73:73 = 580).Second, we add another large value (t 74:74 = 750) to the data and call them as "survival2', so that we now have a sample of size n = 74.Then, we add one more outlier, corresponding to the value t 75:75 = 1000, and call these data as "survival3'.Figure 8 displays usual and adjusted boxplots and stripcharts for survival data, as well as for the data with one, two and three outliers, that is, for survival1, survival2 and survival3.The adjusted boxplot is often used for skewed data because it includes a robust measure of skewness; see Hubert & Vandervieren (2008) and Hubert & Vanderveeken (2008).The stripchart is a scatterplot in one dimension, where all the observations are plotted.From these graphs we can visualize the effect of the outliers added to the data.In the adjusted boxplot for survival1 (see Figure 8-center), it is possible to note that the first added value is an outlier, but when a second atypical value is added for survival2, only this second value is detected as outlier, the first is no longer an outlier for this data set.With the third value being part of the sample, which is much greater than the others, the presence of two outliers is detected.ML estimates and the p-values of the GOF tests for survival1, survival2 and survival3 are provided in Table 10.Note that minor changes in the estimated parameters and in the bounds for the p-values of the tests are detected.For survival3 data, the differences are more noticeable.We conclude that the GOF tests are relatively robust to outliers when the BS-t distribution is considered under H 0 , in particular the MI test, nonetheless a more extensive study on this issue should be carried out.
Figure 9 shows the PP and SP plots based on the BS-t distribution for the survival data with three outliers added.From this figure, note that the points are still well aligned, particularly in the center, but now observation number 15 is not near the bands and there is one observation (number 42) outside the KS band.If we compare Figures 7 and 9, there are no apparently visual differences with minor different alignments in the points, specifically because rejection is due to points in the center and not in the tails of the sample where the outliers are located.Rojas, Galea & Sanhueza (2014).We consider a censored fatigue data sample, such as in Barros et al. (2014), so that we have r = 80 failures and n − r = 21 data censored.From the histogram displayed in Figure 10, note that the distribution of fatigue data is clearly skewed to the right and that the BS distribution shows a good fitting to these data, which must be corroborated.

Conclusions and Future Research
We reviewed goodness-of-fit tests for life distributions not belonging to the location-scale family with uncensored and type-II right censored data.We considered the most used tests to assess goodness-of-fit based on the empirical distribution function.These tests are Anderson-Darling, Cramér-von Mises, Kolmogorov-Smirnov, Kuiper, Michael and Watson, which we have used for assessing if a non-location-scale distribution is suitable to fit a data set.We adopted the two options available in the literature on the topic to construct goodness-of-fit tests with censored data, that is, transforming the censored sample into an uncensored one thereafter applying the usual tests, or adapting the goodness-of-fit test to censored data.We proposed graphical tools related to goodness-of-fit tests based on Kolmogorov-Smirnov and Michael statistics, useful for diverse practitioners who in the past were limited to fit distributions in the location-scale family.Now, they may take into account any distribution with parameters properly estimated and sample size large enough.We illustrated the use of the results proposed in the paper, with emphasis in some Birnbaum-Saunders distributions, by means of several censored and uncensored real-world data sets, which analyses have shown their potential and how the goodness-of-fit tests and the associated graphs can be used in practice.Also, we analyzed some empirical robustness aspects through an example.We would like to mention some future research on this topic.We specifically propose: • to study the sensitivity of the tests to outliers with different distributions under the null hypothesis.
• to analyze the robustness of the tests for different alternative hypothesis.
• to establish for distributions not considered by Chen & Balakrishnan (1995) what sample size and censoring proportion can be recommended, for approximate tests to work well.
• to study the power of the Michael test and compare it to other considered tests.
• to extend the graphical tools to multivariate data.

Figure 1 :
Figure 1: Histogram, boxplot and estimated PDF from TN (left) and TBS (right) distributions for sea data.

Figure 2 :
Figure 2: PP (left) and SP (right) plots with 95% acceptance bands for the TBS distribution with sea data.

Figure 3 :Figure 4 :
Figure 3: Histogram, boxplot and estimated PDF from TN (left) and TBS (right) distributions for forestry data.

Figure 5 :
Figure 5: Histogram, boxplot and estimated PDF from BS (left) and BS-t (right) distributions for survival data.

Figure 7 :
Figure 7: PP (left) and SP (right) plots with 95% acceptance bands for the BS-t distribution with survival data.

Figure 10 :
Figure 10: Histogram, boxplot and estimated BS PDF for fatigue data.

Figure 11 :
Figure 11: PP (left) and SP (right) plots with 95% acceptance bands for the BS distribution with fatigue data.

Table 2 :
formulas for the indicated probability plot.

Table 3 :
100 % acceptance regions for the indicated plot and statistic with 100 th quantiles ks r,n, and mi r,n, .

Table 4 :
Descriptive statistics for sea data.

Table 5 :
Estimated parameters and values of the statistics for the indicated distribution under H0 with sea data.where ks and mi are the estimated values of the KS and MI statistics.

Table 6 :
Descriptive statistics for forestry data.

Table 7 :
Estimated parameters and statistic values for the indicated distribution under H0 with forestry data.

Table 8 :
Descriptive statistics for survival data.

Table 9 :
Estimated parameters and statistic values for the indicated distribution under H0 with survival data.