# Ristic-Balakrishnan extended exponential distribution/Distribuicao Ristic- Balakrishnan exponencial estendida.

IntroductionIt is hardly necessary to emphasize that a probabilistic model is commonly employed to attack practical situations in which a deterministic model is not feasible. This definition, albeit implicitly, had already been part of common sense since the Renaissance era, in which the notion of probability was unconsciously employed to propose solutions in games of chance (Bernstein, 1996). In fact, this intrinsic sense of probability lies at the heart of scientific methodology. Here it is worth quoting the classic book 'The logic of scientific discovery' by Sir Karl Popper:

The most important application of the theory of probability is to what we may call chance-like or random events, or occurrences. These seem to be characterized by a peculiar kind of incalculability which makes one disposed to believe after many unsuccessful attempts that all known rational methods of prediction must fail in their case. We have, as it were, the feeling that not a scientist but only a prophet could predict them. And yet, it is just this incalculability that makes us conclude that the calculus of probability can be applied to these events (Popper, 1959, p. 167).

Taking a leap forward in time, we see that probabilistic models still arouse the fascination of applied scholars and researchers. This interest materializes in the great amount of works that are dedicated to the proposal of new distributions. In particular, those dealing with distribution generators. Our research presented below is related to the generalization of probabilistic models through generators of distributions. In the generator approach, we refer to the following papers: Marshall and Olkin (1997) for the 'Marshall-Olkin' class; Eugene, Lee, and Famoye (2002) for the 'beta' class; Zografos and Balakrishnan (2009) for the 'Gamma' class; Cordeiro and Castro (2011) for the 'Kumaraswamy' class and Cordeiro, Ortega, and Cunha (2013) defined the 'exponentiated generalized' class of distributions.

Recently, Gomez, Bolfarine, and Gomez (2014) introduced a new extended exponential (EE for short) distribution. For x > 0, its cumulative density function (cdf) and probability density function (pdf) are given by Equation 1 and 2:

G (x; [alpha], [beta]) = [alpha] + [beta]-([alpha] + [beta] + [alpha][beta]x)[e.sup.-[alpha]x]/[alpha]+[beta] (1)

g (x; [alpha], [beta]) = [[alpha].sup.2] (1+[beta]x)[e.sup.[alpha]x]/[alpha]+[beta] (2)

where:

[alpha] > 0 and [beta] [greater than or equal to] 0. Several mathematical properties of the EE distribution, including expectation, variance, moment generating function (mgf), asymmetry and kurtosis coefficients, among others, were studied by Gomez et al. (2014). In particular, they proved that the density of the EE model is a mixture of the exponential and gamma densities.

We believe that the addition of parameters to the EE model may generate new distributions with great adjustment capability and, for this reason, we propose a generalization of it. On the other hand, Ristic and Balakrishnan (2012) defined the 'Ristic-Balakrishnan' -G (RB-G for short) family for x [member of] R and a > 0 having, respectively, pdf and cdf given by Equation 3 and 4:

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

where: g(x, = dG(x, [xi]) with [xi] a parametric vector, [mathematical expression not reproducible] is the gamma function and [mathematical expression not reproducible] denotes the lower incomplete gamma function. The main motivation for this family is that, for a = n [member of] N, Equation 3 is the pdf of the nth lower record value of a sequence independent and identically distributed variables from a population with density g(x [xi]).

In this paper we propose a new lifetime model called 'Ristic-Balakrishnan extended exponential' (RBEE) distribution by taking Equation 1 in 4. As we will see later, the proposed model is quite flexible and its failure rate function can accommodate both inverted bathtub and bathtub shapes, which are important for reliability, life time, biological and medical sciences, among others. In addition, the new density may be expressed as a mixture of 'Erlang' densities. Thus, many properties can be derived using this simple representation. As will also be clear later, many important distributions are obtained as a special case of our model. Finally, we prove the new model is very superior in terms of adjustment to real data, when compared to the base model and other important models well established in the literature.

Material and methods

The RBEE distribution

Let X be a random variable with support on the positive real line having the RBEE distribution, say X ~ RBEE(a, [alpha], [beta]). The cdf of X is defined by inserting Equation 1 in Equation 4, according Equation 5:

[mathematical expression not reproducible] (5)

where:

[mathematical expression not reproducible]. The density of X, for x > 0, can be reduced to Equation 6:

[mathematical expression not reproducible]. (6)

We write F(x) = F(x; a, [alpha], [beta]) in order to eliminate the dependence on the model parameters. Clearly, the EE model is a special case of Equation 5 when a = 1. The exponential and Lindley distributions arise as special cases when [beta] = o and [beta] = o, respectively, in addiction to a = 1. If [beta] = o, [beta] = 1 and a [not equal to] 1, we obtain the RB-exponential and RB-Lindley respectively.

Some plots of the pdf Equation 6 are displayed in Figure 1. These plots reveal that the RBEE pdf is quite flexible and can take various forms reinforcing the importance of the proposed model.

The survival function is Equation 7:

S(x) = [gamma](a, [rho](x)]/[GAMMA](a). (7)

The hazard rate function (hrf) and reversed hazard rate function (rhrf) of X are given by Equation 8 and 9:

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

respectively. Some plots of the hrf Equation 8 are displayed in Figure 2. Non-monotone forms such as bathtub and inverted bathtub are particularly important because of its great practical applicability. For example, the time of human life is one phenomena that the bathtub form is applicable.

Asymptotic and shapes of the RBEE

For a detailed mathematical approach for the RBEE model, we investigate the shapes of its pdf and hrf using their first and second derivatives.

The first derivatives of log {f(x)} and log {h(x)} for the RBEE model are given by Equation 10 and 11:

[mathematical expression not reproducible] (10)

[mathematical expression not reproducible] (11)

with [phi](x) = [alpha] + [beta]- ([alpha] + [beta] + [alpha][beta]x)[e.sup.- [alpha]x].

Hence, the critical values off(x) and h(x) are the roots of the Equation 12 and 13:

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

respectively. The values [x.sub.0] and [x'.sub.0] which solves the Equations 12 and 13 above can be a maximum, minimum or inflection point. To check this, we evaluate the signs of the second derivatives of log {f(x)} and log {h(x)}, respectively, at x = [x.sub.0] and x = [x'.sub.0]. We have Equation 14 and 15:

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

It is common to obtain numerical solutions with high accuracy through optimization routines in most mathematical and statistical platforms.

Quantile function

For many practical purposes, it is important to make explicit the quantile function (qf) of X. The RBEE qf, say q(u) can be obtained by inverting Equation 5 (for 0 < u < 1) as Equation 16:

[mathematical expression not reproducible] (16)

where:

z = [Q.sup.-1] is the inverse function of Q(a,z) = 1 [gamma](a,z)/[GAMMA](a) and W(*) denotes the Lambert [GAMMA](a)

W-function. In a recent paper, Nadarajah, Bakouch, and Tahmasbi (2011) used the Lambert W-function to derive the qf of the exponentiated Lindley distribution. For any complex t, the Lambert W-function is defined as the inverse of the function g(t) = te'. For more details, see http://mathworld. wolfram.com/LambertW-Function.html. An implementation in R software is available through the 'LambertW package. See http://cran.rproject.org/web/packages/LambertW/LambertW.pdf. In the 'Mathematica platform', the 'LambertW is available through the function 'ProducLog[z]', which gives the principal solution for W in z = [we.sup.w]. By using the Lagrange inversion theorem, we can write an expansion for the qf of X as follows Equation 17:

[mathematical expression not reproducible]. (17)

Note that the above equation can be easily implemented in computational platforms that have numerical elementary routines.

The applications of qf are diverse and include: calculation of the moments, estimation of parameters, simulations, calculation of asymmetry and kurtosis measurements, among others. For illustration, we use the qf of X to determine the Bowley skewness (Kenney & Keeping, 1962) (B) and Moors kurtosis (Moors, 1988) (M). The Bowley skewness is based on quartiles B = [Q(3/4)-2Q(1/2) + Q(1/4)]/[Q(3/4)-Q(1/4)], whereas the Moors kurtosis is based on octiles M = [Q(7/8)-Q(5/8)-Q(3/8) + Q(1/8)]/[Q(6/8)Q(2/8]. These two measures are less sensitive to outliers and they exist even for distributions without moments.

In Figure 3 and 4, we present 3D plots of B and M measures for several parameters values. These plots are obtained using the 'Wolfram Mathematica' software. Based on these plots, it is possible to conclude that changes in the additional parameter a have a considerable impact on the skewness and kurtosis of the RBEE distribution, thus showing its greater flexibility.

Properties

A useful representation

We provide useful linear representations for Equation 5 and 6 based on the exponentiated class of distributions. Mathematical properties of the exponentiated distributions have been published by many authors in the 90s and more recently. See, for example, Gupta and Kundu (1999) for exponentiated exponential, Nadarajah et al. (2011) for exponentiated Lindley, Sarhan and Kundu (2009) for exponentiated linear failure rate and, more recently, Lemonte (2013) for the exponentiated Nadarajah-Haghighi distributions. For an arbitrary baseline cdf G(x) a random variable [Y.sub.a] has the exp-G class with power parameter a > 0 say [Y.sub.a] ~ exp-G(a), if its cdf and pdf are given by [H.sub.a](x) = G[(x).sup.a] and ha (x) = ag(x)G[(x).sup.a-1] respectively. For a comprehensive discussion about the exponentiated class, see a recent paper by Tahir and Nadarajah (2015).

By using results presented in Cordeiro and Bourguignon (2016) we can be expressed the pdf f(x) as Equation 18:

F(x) = [[infinity].summation over (j=0)] [h.sub.j+1] (x) (18)

With

[mathematical expression not reproducible] (19)

where:

quantities d(a-1) (for I [greater than or equal to] 0) determined by [d.sub.0](c)=c/2, [d.sub.1](c)=c(3c+5)/24, [d.sub.2](c)=c(c2+5c + 6)/48, [d.sub.3](c)=c(15[c.sup.2]+150[c.sup.2]+485c+502)/5760, etc.

Note that, by integrating Equation 18, we can express F(x) as Equation 20:

F(x) = [[infinity].summation over (j=0)] [w.sub.j][H.sub.j+1] (x) (20)

where:

[H.sub.j+1](x) denotes the exp-EE cumulative distribution with power parameter j+1. Here, [h.sub.j+1](x) is the exp-EE density function with power parameter j+1, and is given by (for j[greater than or equal to] 0) [mathematical expression not reproducible] By interchanging [mathematical expression not reproducible] in the last equation, where Equation 21:

[mathematical expression not reproducible], (21)

and, after a simple algebraic manipulation, we obtain Equation 22 and 23:

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible] (23)

where:

[pi](x; i + 1, (k+1)[alpha]) denotes the pdf of the Erlang distribution with parameters i+1 (for i[greater than or equal to] 0) and (k + 1)a. If Z is an Erlang random variable with parameters s(=1, 2, 3, ...) and [lambda]>0, its pdf is given [mathematical expression not reproducible]. Changing [mathematical expression not reproducible], the density function reduces to Equation 24:

[mathematical expression not reproducible] (24)

where [mathematical expression not reproducible]. Equation 24 is the main result of this section.

Moments, incomplete moments and generating function

Then, the nth moment of X and its incomplete moments, respectively, are given by Equation 25 and 26:

[mathematical expression not reproducible] (25)

[mathematical expression not reproducible] (26)

where [GAMMA](a,z) = [[integral].sup.[infinity].sub.z] [t.sup.a-1][e.sup.-t]dt denotes the upper incomplete gamma function.

The moment generating function (mgf) of X can be determined from Equation 24, according Equation 27:

[mathematical expression not reproducible]. (27)

Then, for all t < (k + 1)[alpha], we have Equation 28:

[mathematical expression not reproducible]. (28)

Order statistics

By using results presented in Cordeiro and Bourguignon (2016) the density function [f.sub.i:n] (x) of the ith order statistic, say [X.sub.i:n], for i = 1, ..., n, from a random sample [X.sub.1], ..., [X.sub.n] having the RBEE distribution, can be expressed as Equation 29:

[mathematical expression not reproducible] (29)

with

[mathematical expression not reproducible] (30)

where [w.sub.r] is defined by Equation 19, and

[mathematical expression not reproducible].

Reliability

In reliability the stress-strength model describes the life of a component which has a random strength [X.sub.1] that is subjected to a random stress [X.sub.2]. The component fails at the instant that the stress applied to it exceeds the strength, and the component will function satisfactorily whenever [X.sub.1] > [X.sub.2]. Hence, R = P([X.sub.2] < [X.sub.1]) is a measure of component reliability. When [X.sub.1] and [X.sub.2] have independent RBEE([a.sub.1], [alpha], [beta]) and RBEE([a.sub.2], [alpha], [beta]) models the reliability is defined by R = [[integral].sup.[infinity].sub.0] [f.sub.1](x)[F.sub.2](x)dx. The pdf of [X.sub.1] and cdf of [X.sub.2] are expressed from Equations 18 and 19 as Equation 31:

[mathematical expression not reproducible] (31)

where (for 5 = j, k; p = i, n and q = 1, 2)

[mathematical expression not reproducible] (32)

Thus, we have Equation 33:

[mathematical expression not reproducible] (33)

where [mathematical expression not reproducible].

Hence, after a simple algebraic manipulation, the reliability of the RBEE distribution is given by Equation 34:

[mathematical expression not reproducible] (34)

Entropy

The Renyi entropy is defined (for [delta] > 0 and [delta] [not equal to] 0), according Equation 35:

[mathematical expression not reproducible]. (35)

Let [bar.G](x) = 1-G(x) be the baseline survival function. Following similar idea given in Nadarajah, Cordeiro, and Ortega (2015) (Section 10), we have Equation 36:

[mathematical expression not reproducible] (36)

where:

The constants [p.sub.j,k] can be calculated recursively by Equation 37:

[mathematical expression not reproducible] (37)

for k = 1, 2, ..., [p.sub.j,0] = 1 and [c.sub.k]=[(-1.sup.)k+1][(k+1).sup.-1]. By using Equation 36 and generalized binomial expansion we obtain the Renyi entropy for the family as Equation 38 and 39:

[mathematical expression not reproducible] (38)

where,

[I.sub.i] = [[integral].sup.[infinity].sub.0] G'(x)[g.dup.[delta]]dx (39)

comes from the baseline distribution. Based on the cdf Equation 1 and pdf Equation 2, we can express Equation 39 as Equation 40:

[mathematical expression not reproducible] (40)

where E[n,z] = [[integral].sup.[infinity].sub.1] [t.sup.-n] [e.sup.-zt] is 'exponential integral function'.

Estimation and inference

The maximum likelihood method is the one that stands out most among the estimation methods admitting good asymptotic properties. The maximum likelihood estimators (MLEs) can be used when constructing confidence intervals and regions and also in test statistics. Let [x.sub.1], ..., [x.sub.n] be a random sample of size n from the RBEE (a, [alpha], [beta]) model. The log-likelihood function for the vector of parameters [THETA] = [(a, [alpha], [beta]).sup.T] can be expressed from Equation 41:

[mathematical expression not reproducible] (41)

where:

[mathematical expression not reproducible].

The components of the score vector U([THETA]) are given by Equation 42, 43 and 44:

[U.sub.a] ([THETA]) = -n[psi](a) + [N.summation over (i=1)] [phi]([x.sub.i]) (42)

[mathematical expression not reproducible] (43)

[mathematical expression not reproducible] (44)

where

[psi](*) is the digamma function and [mathematical expression not reproducible].

The information matrix is given by J/([THETA]) = {-[U.sub.rs]} and its elements [U.sub.rs]([THETA]) = [[partial derivative].sup.2]l([THETA])/[partial derivative]r[partial derivative]s for r, s [member of] {a, [alpha], [beta]} can be obtained from the authors upon request.

Results and discussion

Application

We consider an uncensored data set from Murty, Xie and Jiang (2004) (page 180) used in the industry, representing the failure time (in weeks) of 50 components put into use at time. The data are: 0.013, 0.065, 0.111, 0.111, 0.163, 0.309, 0.426, 0.535, 0.684, 0.747, 0.997, 1.284, 1.304, 1.647, 1.829, 2.336, 2.838, 3.269, 3.977, 3.981, 4.520, 4.789, 4.849, 5.202, 5.291, 5.349, 5.911, 6.018, 6.427, 6.456, 6.572, 7.023, 7.087, 7.291, 7.787, 8.596, 9.388, 10.261, 10.713, 11.658, 13.006, 13.388, 13.842, 17.152, 17.283, 19.418, 23.471, 24.777, 32.795, 48.105. Table 1 provides some descriptive statistics.

For Murthy et al. (2004)'s data, we compared the RBEE model with the EE Gomez et al. (2014) and Lindley sub-models and other commonly used models in survival analysis, namely the log-logistic, Frechet and Birnbaum-Saunders (BS) distributions. The densities of these models are given, for example, in the Wolfram alpha website (https://www.wolframalpha.com).

Table 2 gives the MLEs of the fitted models to the current data with their corresponding standard errors, in addition to the AIC, BIC and CAIC statistics. Table 3 lists the values of the A* and W* statistics. In general, it is considered that lower values of these criteria fit better the data.

Additionally, we took into consideration the Anderson-Darling (A*) and Cramer-von Mises (W*) statistics (Chen & Balakrishnan, 1995). Chen and Balakrishnan (1995) proposed a general approximate goodness-of-fit test for the hypothesis [H.sub.0]: [X.sub.1], ..., [X.sub.n] with [X.sub.i] following F(x; [theta]), where the form of F is known but the p-vector [theta] is unknown. To obtain the statistics A* and W*, we can proceed as follows: (1) compute [r.sub.i] = F([x.sub.i], [theta]), where [x.sub.i]'s are in ascending order, and then [[gamma].sub.i] = [[phi].sup.1]([r.sub.i]), where [phi]() is the standard normal cumulative distribution; (2) compute [u.sub.i] = [phi] ([y.sub.i]-[bar.y]/s) where [mathematical expression not reproducible]. Table 3 lists the values of the A* and W* statistics. In general, it is considered that lower values of these criteria fit the data better.

Table 3 presents the mean, variance, asymmetry and kurtosis for the RBEE, EE and Lindley adjusted models. As we can see, the empirical and estimated means and variances do not differ considerably. This shows that the models are adequate to explain this data.

The figures in Table 2 and 4 reveals that the R BEE model has the lowest AIC, BIC, CAIC, A* and W* values among all fitted models. Thus, the proposed RBEE distribution is the best model to explain these data. Finally, Figure 5 displays the histogram of the data and the estimated pdf and cdf of the R BEE model. These plots reveal that the proposed model is quite suitable for these data.

Conclusion

In this article, we introduce and study a new model of lifetime, called the 'Ristic-Balakrishnan extended exponential' distribution. The proposed model has three parameters and generalizes important distributions. We provide a comprehensive study of the mathematical and statistical properties of the new model. In addition, the practical utility of the new model was empirically demonstrated. We hope that the RBEE model can be useful for applied statisticians and other researches who refer to a model with few parameters but flexible to accommodate supported data in real positives.

Doi: 10.4025/actascitechnol.v40i1.34963

Acknowledgements

We thank two anonymous referees and the associated editor for their valuable suggestions, which certainly contributed to the improvement of this paper. Additionally, Thiago A. N. de Andrade is grateful the financial support from CAPES (Brazil).

References

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Cordeiro, G. M., & Castro, M. (2011). A new family of generalized distribution. Journal of Statistical Computation and Simulation, 81(7), 883-893.

Cordeiro, G. M., Ortega, E. M. M., & Cunha, D. C. (2013). The exponentiated generalized class of distributions. Journal of Data Science, 11(1), 1-27.

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Gomez, Y. M., Bolfarine, H., & Gomez, H. W. (2014). A new extension of the exponential distribution. Revista Colombiana de Estadistica, 37(1), 25-34.

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Kenney, J. F., & Keeping, E. S. (1962). Mathematics of statistics (3rd ed.). Trenton, NJ: Chapman & Hall.

Lemonte, A. J. (2013). A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function. Computational Statistics and Data Analysis, 62, 149-170. doi 10.1016/j.csda.2013.01.011

Marshall, A. N., & Olkin, I. (1997). A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families. Biometrica, 84(3), 641-652.

Moors, J. J. (1988). A quantile alternative for kurtosis. Journal of the Royal Statistical Society. Series D, 37(1), 25-32.

Murthy, D., Xie, M., & Jiang, R. (2004). Weibull models. Wiley series in probability and statistics. Trenton, NJ: John Wiley and Sons.

Nadarajah, S., Bakouch, H. S., & Tahmasbi, R. (2011). A generalized Lindley distribution. Sankhya B, 73(2), 331-359.

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. M. (2015). The Zografos-Balakrishnan-G family of distributions: Mathematical properties and applications. Communications in Statistics-Theory and Methods, 44(1), 186-215. doi 10.1080/03610926.2012.740127

Popper, K. (1959). The logic of scientific discovery. Viena: Taylor & Francis e-Library.

Ristic, M. M., & Balakrishnan, N. (2012). The gamma exponentiated exponential distribution. Journal of Statistical Computation and Simulation, 82(8), 1191-1206. doi 10.1080/00949655.2011.574633

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Received on January 25, 2017.

Accepted on September 19, 2017.

Frank Gomes-Silva (1) *, Thiago Alexandro Nascimento de Andrade (2) and Marcelo Bourguignon (3)

(1) Departamento de Informatica e Estatistica, Universidade Federal Rural de Pernambuco, Rua Manuel de Medeiros, s/n., 52171-900, Recife, Pernambuco, Brazil. (2) Departamento de Estatistica, Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil. (3) Departamento de Estatistica, Universidade Federal do Rio Grande do Norte, Natal, Rio Grande do Norte, Brazil. * Author for correspondence. E-mail: franksinatrags@gmail.com

Caption: Figure 1. Plots of the R BEE density function for some parameter values.

Caption: Figure 2. Plots of the R BEEhazard function for some parameter values.

Caption: Figure 3. Plots of the Bowley skewness for the RBEE distribution for some parameter values.

Caption: Figure 4. Plots of the Moors kurtosis for the R BEE distribution for some parameter values.

Caption: Figure 5. Estimated pdf of the RBEE model; Estimated cdf of the RBEE model.

Table 1. Descriptive statistics for number of successive failure times of 50 components. Statistics n Mean Median Variance Minimum Maximum 50 7.821 5.320 84.76 0.013 48.100 Table 2. MLEs (and the corresponding standard errors in parentheses), AIC, BIC and CAIC statistics for number of successive failures for the air conditioning system. Distributions [??] [??] [??] RBEE 0.02792 0.4257 4.5200 (0.02486) (0.6573) (2.2804) EE 0.1279 2.338E-7 1 (--) (0.0352) (0.0302) Lindley 0.2317 1 1 (--) (0.590) (--) [??] [??] Log-logistic 4.0938 1.0834 (0.9218) (0.1304) [??] [??] Frechet 1.2802 0.4791 (0.4028) (0.04541) [??] [??] BS 2.7621 1.2576 (0.2973) (0.2721) Distributions AIC BIC CAIC RBEE 306.0 311.7 306.5 EE 309.7 313.5 309.9 Lindley 324.6 326.5 324.6 Log-logistic 316.0 319.8 316.3 Frechet 341.3 345.1 341.5 BS 327.4 331.2 327.7 Table 3. Mean, Variance, Skewness and Kurtosis for the three main distributions. Distributions Mean Variance Skewness kurtosis R BEE 7.82154 82.7757 -0.23734 1.18290 EE(a = 1) 7.81862 61.1309 -- -- Lindley 7.81997 36.5953 0.17305 1.13473 Table 4. Goodness-of-fit tests. Models Statistics W* A* RBEE 0.0539 0.2709 EE 0.0658 0.3295 Lindley 0.0657 0.3284 Log-logistic 0.2572 1.3816 Frechet 0.6097 3.3138 BS 0.2794 1.5364

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Title Annotation: | STATISTICS |
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Author: | Gomes-Silva, Frank; de Andrade, Thiago Alexandro Nascimento; Bourguignon, Marcelo |

Publication: | Acta Scientiarum. Technology (UEM) |

Date: | Jan 1, 2018 |

Words: | 4258 |

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