Goodness–of–fit tests based on the min–characteristic function

Computational Statistics and Data Analysis 197 (2024) 107988 Available online 16 May 2024 0167-9473/© 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Goodness–of–fit tests based on the min–characteristic function S.G. Meintanisa,b,∗, B. Miloševi´ cc, M.D. Jiménez–Gamero d aDepartment of Economics, National and Kapodistrian University of Athens, Athens, Greece bPure and Applied Analytics, North–West University, Potchefstroom, South Africa cFaculty of Mathematics, University of Belgrade, Belgrade, Serbia dDepartment of Statistics and Operations Research, University of Seville, Seville, Spain A R T I C L E I N F O A B S T R A C T Keywords: Min–characteristic function Extreme–value distributions Goodness–of–fit test Invariant tests Tests of fit for classes of distributions that include the Weibull, the Pareto and the Fréchet families are proposed. The new tests employ the novel tool of the min–characteristic function and are based on an 𝐿2–type weighted distance between this function and its empirical counterpart applied on suitably standardized data. If data–standardization is performed using the MLE of the distributional parameters then the method reduces to testing for the standard member of the family, with parameter values known and set equal to one. Asymptotic properties of the tests are investigated. A Monte Carlo study is presented that includes the new procedure as well as competitors for the purpose of specification testing with three extreme value distributions. The new tests are also applied on a few real–data sets. 1. Introduction The min–characteristic function (minCF) of a univariate random variable 𝑋>0is defined as Ψ𝑋(𝑡) =𝖤(min{1, 𝑡𝑋}), 𝑡 >0. The minCF is a novel tool introduced recently by Falk and Stupfler (2021), who show uniqueness, boundedness, and other interesting properties of Ψ𝑋(⋅). The same authors introduce and prove uniform strong consistency and weak convergence over compact intervals of the associated empirical minCF, Ψ𝑛(𝑡)= 1 𝑛 𝑛 ∑ 𝑗=1 min{1,𝑡𝑋𝑗},(1) where (𝑋𝑗, 𝑗=1, ..., 𝑛)denote independent copies of 𝑋. Note in this connection that for certain random variables, the minCF is often easier to compute than the classical characteristic function. This feature combined with the uniqueness of Ψ𝑋(⋅)for positively supported distributions, motivates goodness–of–fit (GOF) tests based on some distance between the population minCF and the empirical minCF, Ψ𝑛(⋅). However, as with tests based on the characteristic function, the drawback with many such tests is with composite null hypotheses involving parameters that actually occur in the population minCF of the distribution under test. These parameters are considered unknown and must be estimated from the data (𝑋𝑗, 𝑗=1, ..., 𝑛)and it is often the case that the finite–sample as well as the asymptotic distribution of the resulting tests depends on the unknown true values of these parameters. If so, then GOF tests must be performed by bootstrap resampling * Corresponding author at: Department of Economics, National and Kapodistrian University of Athens, Athens, Greece. E-mail address: [email protected] (S.G. Meintanis). https://doi.org/10.1016/j.csda.2024.107988 Received 17 October 2023; Received in revised form 10 May 2024; Accepted 12 May 2024 Computational Statistics and Data Analysis 197 (2024) 107988 2 S.G. Meintanis, B. Miloševi´c and M.D. Jiménez–Gamero methods that incur a certain computational cost on the implementation of the test procedure (see e.g. Jiménez-Gamero et al. (2009); Jiménez-Gamero and Kim (2015)). In this paper we suggest minCF–based GOF tests for a certain classes of composite null hypotheses that do not have the aforementioned computational drawback. The paper is organized as follows. In Section 2we introduce the null hypothesis to be tested and the corresponding tests statistics. Then in Section 3we prove that the resulting tests are parameter–free if the distributional parameters are replaced by maximum likelihood estimators (MLE), while in Section 4computational formulae are provided for the test statistic with specific instances of distributions under test. In Section 5we present weak convergence of the test statistic under the null hypothesis and its behavior under alternatives. Section 6presents the results of a Monte Carlo study on the power of the new tests, as well as comparisons with other GOF tests. Real–data examples are included in Section 7, and the paper concludes in Section 8with discussion. All proofs are deferred to the Appendix. 2. Goodness–of–fit tests Consider the null hypothesis that the law of 𝑋belongs to , for some 𝜗 ∈Θ, where ={𝐹𝜗; 𝜗 ∈Θ} is the particular class of distribution functions indexed by a parameter 𝜗assumed to take values in Θ, the parameter space. The standard approach to GOF testing is by means of a distance measure between the population distribution function (DF) and its empirical counterpart, the empirical DF; a classic reference for empirical DF–based tests is D’Agostino and Stephens (1986). Another approach which has become popular recently is the one based on the empirical characteristic function, while in the context of positive random variables the empirical Laplace transform has also attracted attention. The interested reader is referred to the papers of Ndwandwe et al. (2023b), Ebner et al. (2022), Cupari´ c et al. (2022), Bothma et al. (2021), Lee et al. (2019), Meintanis and Ushakov (2016), Miloševi´ c and Obradovi´ c (2016), Meintanis et al. (2015), Meintanis and Iliopoulos (2003), among others. Meintanis (2016)gives a review of such GOF tests, while some earlier contributions for tests based on the characteristic function may be found in (Ushakov, 1999, §3.9-3.10). We will introduce our test in 𝐿2(𝑤), the (separable) Hilbert space setting of measurable functions 𝑓∈, 𝑓∶ℝ+→ℝ, which are squared integrable with respect to a weight function 𝑤(⋅). In this setting the inner product of 𝑓, 𝑔∈is defined as <𝑓,𝑔> 𝑤=⎛⎜⎜⎝ ∞ ∫ 0 𝑓(𝑡)𝑔(𝑡)𝑤(𝑡)d𝑡⎞⎟⎟⎠ 1∕2 , with corresponding norm ‖𝑓‖𝑤=(<𝑓,𝑓>)1∕2 =⎛⎜⎜⎝ ∞ ∫ 0 𝑓2(𝑡)𝑤(𝑡)d𝑡⎞⎟⎟⎠ 1∕2 . In complete analogy with the aforementioned tests based on the empirical DF (or the empirical characteristic function or the empirical Laplace transform), a reasonable test procedure would be to reject the null hypothesis for large values of the test statistic 𝑇𝑛,𝑤 =‖√𝑛(Ψ𝑛−Ψ  𝜗𝑛)‖2 𝑤,(2) where Ψ𝜗(⋅)is the minCF corresponding to 𝐹𝜗, Ψ𝑛(⋅)is the empirical minCF defined in (1), and  𝜗𝑛∶=  𝜗𝑛(𝑋1, ..., 𝑋𝑛)denotes some consistent estimator of 𝜗. Here however we will consider families of distributions with parameters 𝜗 =(𝑐, 𝜑), and such that the corresponding DF may be written as 𝐹𝑐,𝜑(𝑥)=𝐹0((𝑥 𝑐)𝜑),(3) for some 𝑐, 𝜑 >0, and some fixed DF 𝐹0(⋅). Clearly the corresponding family, say 0, is completely determined by 𝐹0, and then the null hypothesis may be restated as Υ0∶The law of 𝑋∈0,forsome𝑐,𝜑 > 0.(4) Specific instances of 0to be considered in this work are the Weibull, the Pareto type I, and the Fréchet distributions, all belonging to the class of extreme–value distributions; see Kotz and Nadarajah (2000)for an account of extreme–value distributions. 3. Data transformations based on the MLE In this section we investigate certain equivariance properties of the MLE under the class 0with DF as in (3). These properties will have a direct impact on the proposed test, and specifically it will be shown that the test statistic will be rendered free of parameter values if the empirical minCF is applied on suitably standardized data. To this end, let (𝑐𝑛, 𝜑𝑛)be the MLE of (𝑐, 𝜑), and consider the standardized observations Computational Statistics and Data Analysis 197 (2024) 107988 3 S.G. Meintanis, B. Miloševi´c and M.D. Jiménez–Gamero  𝑌𝑗=(𝑋𝑗 𝑐𝑛)𝜑𝑛 ,𝑗=1,...,𝑛. (5) In the next result we investigate the aforementioned equivariance properties of the MLE; for a different proof we refer to Matsui and Meintanis (2023). A general discussion of equivariant estimators is provided by (Borovkov, 1998, §29), and (Romano and Lehmann, 2005, Chapter 6). Proposition 3.1. Let (𝑋𝑗, 𝑗=1, ..., 𝑛)be independent copies of 𝑋>0, and assume that the law of 𝑋∈0, for some 𝑐, 𝜑 >0. Assume further that the density of 𝑋exists, and that the MLEs 𝑐𝑛(𝑋1, ..., 𝑋𝑛)and 𝜑𝑛(𝑋1, ..., 𝑋𝑛)of the parameters 𝑐and 𝜑also exist, and that they are unique. Then the MLE corresponding to ( 𝑌𝑗, 𝑗=1, ..., 𝑛)satisfy 𝑐𝑛( 𝑌1, ...,  𝑌𝑛) =1and 𝜑𝑛( 𝑌1, ...,  𝑌𝑛) =1. An important consequence of Proposition 3.1 is that any GOF procedure for families satisfying (3)that depends on (𝑋𝑗, 𝑗=1, ..., 𝑛) only via the observations ( 𝑌𝑗, 𝑗=1, ..., 𝑛)defined by (5) satisfies 𝑇𝑛(𝑎𝑋1∕𝑏 1,...,𝑎𝑋1∕𝑏 𝑛)=𝑇𝑛(𝑋1,...,𝑋𝑛),(6) for each 𝑎, 𝑏 >0, and consequently and without loss of generality, we may perform the test by assuming that 𝑐and 𝜑are fixed at 𝑐=𝜑 =1. In this connection we suggest to apply the test defined by (2)by replacing the empirical minCF by 𝜓𝑛(𝑡)=1 𝑛 𝑛 ∑ 𝑗=1 min{1,𝑡 𝑌𝑗},(7) and the estimated population minCF Ψ 𝜗𝑛(⋅) ∶= Ψ𝑐𝑛,𝜑𝑛(⋅)figuring in 𝑇𝑛,𝑤 by 𝜓0(𝑡) ∶= Ψ1,1(𝑡). As a consequence, the finite–sample as well as the asymptotic distribution of the resulting test statistic, say 𝑛,𝑤, will be independent of the true but unknown values of (𝑐, 𝜑). On a practical level, (6)implies that a potentially much simpler test may be invoked for testing families satisfying (3), such as in the case of the Weibull distribution whereby any test for exponentiality applied on ( 𝑌𝑗, 𝑗=1, ..., 𝑛)may be used. Of course the idea of data–transformation in the context of GOF testing is not new. By way of example, for Pareto, generalized Pareto, and Weibull, distributions, log–transformations may be applied that have a similar effect, i.e. they reduce to exponentiality tests on the log–transformed data or to tests for the extreme–value distribution; the reader is referred to Ndwandwe et al. (2023a), Chu et al. (2019), Meintanis and Bassiakos (2007), and Cabaña and Quiroz (2005)for such tests. 4. Calculation of the test statistic In this section we particularize the families to be tested as per the null hypothesis in (4)and the functional form in 𝐹0(𝑥)figuring in (3). In this connection note that in the context of testing the null hypothesis Υ0and in view of the results of the previous section, we may perform the GOF test based on the test statistic 𝑛,𝑤 ∶= ‖√𝑛(𝜓𝑛−𝜓0)‖2 𝑤=𝑛(‖𝜓𝑛‖2 𝑤+‖𝜓0‖2 𝑤−2<𝜓 𝑛,𝜓0>𝑤),(8) for any given family 0. In fact if we use 𝑤(𝑡) =e −𝛾𝑡, 𝛾>0, as weight function, the resulting test statistic, say 𝑛,𝛾 , may be written as 𝑛,𝛾 =1 𝑛 𝑛 ∑ 𝑗,𝑘=1 Λ𝛾( 𝑌𝑗, 𝑌𝑘)+𝑛𝛾−2 𝑛 ∑ 𝑗=1 𝜆𝛾( 𝑌𝑗),(9) where Λ𝛾(𝑧1,𝑧2)= ∞ ∫ 0 min{1,𝑡𝑧1}min{1,𝑡𝑧2}e−𝛾𝑡d𝑡, 𝛾= ∞ ∫ 0 𝜓2 0(𝑡)e−𝛾𝑡d𝑡, and 𝜆𝛾(𝑧)= ∞ ∫ 0 min{1,𝑡𝑧}𝜓0(𝑡)e−𝛾𝑡d𝑡. The families to be considered are: •the Weibull with 𝐹0(𝑥) =1 −e −𝑥, 𝑥 >0, and 𝜓0(𝑡) =𝑡(1 −e −1∕t ), •the Pareto with 𝐹0(𝑥) =1 −𝑥−1, 𝑥 >1, and 𝜓0(𝑡) =𝑡(1 −log𝑡), 𝑡 ≤1, and 𝜓0(𝑡) =1, 𝑡 >1, •the Fréchet with 𝐹0(𝑥) =e −𝑥−1 , 𝑥 >0, and 𝜓0(𝑡) =1 −e −𝑡+𝑡Γ[𝑡], where Γ[𝑧] =∫∞ 𝑧𝑢−1e−𝑢d𝑢. Computational Statistics and Data Analysis 197 (2024) 107988 4 S.G. Meintanis, B. Miloševi´c and M.D. Jiménez–Gamero Clearly, the first integral Λ𝛾(𝑧1, 𝑧2)does not depend on the family 0being tested and depends solely on the data ( 𝑌𝑗, 𝑗=1, ..., 𝑛) and the value of the weight parameter 𝛾. This integral is straightforward to compute. Specifically after some straightforward algebra we obtain Λ𝛾(𝑧1,𝑧2)= 𝑧1 𝑧2 2𝑧2 2−e −𝛾∕𝑧2(2𝑧2 2+2𝛾𝑧2+𝛾2) 𝛾3+𝑧1 𝑧2 e−𝛾∕𝑧2(𝛾+𝑧2) 𝛾2−𝑧1 e−𝛾∕𝑧1 𝛾2,𝑧 1≤𝑧2. For 𝑧1>𝑧 2we simply replace 𝑧1(resp. 𝑧2) with 𝑧2(resp. 𝑧1). On the other hand, the second and third integrals depend on the family under test, with 𝛾been independent of the observations whereas the third integral 𝜆𝛾(⋅), is the only one that involves both the family 0as well as the observations. Specifically for the Weibull distribution it follows after some long but otherwise routine calculations that the test statistic may be written as W 𝑛,𝛾 =1 𝑛 𝑛 ∑ 𝑗,𝑘=1 Λ𝛾( 𝑌𝑗, 𝑌𝑘)+𝑛W 𝛾−2 𝑛 ∑ 𝑗=1 𝜆W 𝛾( 𝑌𝑗),(10) where W 𝛾=2 𝛾3+4√2𝐾3[√8𝛾]−4𝐾3[√4𝛾] 𝛾3∕2 , 𝜆W 𝛾(𝑧)= 2𝑧−e −𝛾∕𝑧(𝛾+2𝑧) 𝛾3−(𝑧𝑀W 𝛾,1(𝑧)+𝑀W 𝛾,2(𝑧)), 𝑀W 𝛾,1(𝑧)= 1∕𝑧 ∫ 0 𝑡2e−(𝑡−1+𝛾𝑡)d𝑡, and 𝑀W 𝛾,2(𝑧)= ∞ ∫ 1∕𝑧 𝑡e−(𝑡−1+𝛾𝑡)d𝑡, where 𝐾𝜈[𝑧] = (1∕2)(𝑧∕2)𝜈∫∞ 0e−𝑡−(𝑧2∕4𝑡)𝑡−(𝜈+1)d𝑡, stands for the modified Bessel function of the second kind of order 𝜈. By analogous calculations the test statistic for the Pareto distribution may be written as P 𝑛,𝛾 =1 𝑛 𝑛 ∑ 𝑗,𝑘=1 Λ𝛾( 𝑌𝑗, 𝑌𝑘)+𝑛P 𝛾−2 𝑛 ∑ 𝑗=1 𝜆P 𝛾( 𝑌𝑗),(11) where P 𝛾=e−𝛾 𝛾+e−𝛾(4 − 𝛾2)+4(log𝛾+Γ[𝛾]+𝛾E−1) 𝛾3+𝐼P 𝛾, 𝜆P 𝛾(𝑧)=𝑧(2−e −𝛾(2 + 2𝛾+𝛾2) 𝛾3−3−2𝛾E−e −𝛾(3 + 𝛾)−2Γ[𝛾]−2log𝛾 𝛾3)+𝑧 𝛾2(e−𝛾(1 + 𝛾)−e −𝛾∕𝑧),𝑧≤1, 𝜆P 𝛾(𝑧)=2𝑧 𝛾3−e−𝛾∕𝑧(2𝑧2+2𝛾𝑧+𝛾2) 𝑧𝛾3−𝑧𝑀P 𝛾,1(𝑧)+ e−𝛾∕𝑧(𝛾+𝑧) 𝛾2𝑧−e−𝛾 𝛾2−𝑀P 𝛾,2(𝑧),𝑧>1, 𝐼P 𝛾= 1 ∫ 0 𝑡2log2𝑡e−𝛾𝑡d𝑡, 𝑀P 𝛾,1(𝑧)= 1∕𝑧 ∫ 0 𝑡2log𝑡e−𝛾𝑡d𝑡, and 𝑀P 𝛾,2(𝑧)= 1 ∫ 1∕𝑧 𝑡log 𝑡e−𝛾𝑡d𝑡, where 𝛾E(≈ 0.57721) denotes the Euler constant. Computational Statistics and Data Analysis 197 (2024) 107988 5 S.G. Meintanis, B. Miloševi´c and M.D. Jiménez–Gamero Finally, for the Fréchet distribution we have F 𝑛,𝛾 =1 𝑛 𝑛 ∑ 𝑗,𝑘=1 Λ𝛾( 𝑌𝑗, 𝑌𝑘)+𝑛F 𝛾−2 𝑛 ∑ 𝑗=1 𝜆F 𝛾( 𝑌𝑗),(12) where F 𝛾=1 2+𝛾+𝛾+ 2(log(1 + 𝛾)+(1+𝛾)−1 −1) 𝛾2−2(1 + 𝛾+ log(2 + 𝛾)+(2+𝛾)−1 −1) (1 + 𝛾)2+𝐼F 𝛾, 𝜆F 𝛾(𝑧)= 𝑧(1 − e−𝛾∕𝑧) 𝛾2−𝑧(1 − e−(1+𝛾)∕𝑧) (1 + 𝛾)2+𝑧𝑀F 𝛾,1(𝑧)+𝑀F 𝛾,2(𝑧), 𝐼F 𝛾= ∞ ∫ 0 𝑡2Γ2[𝑡]e −𝛾𝑡d𝑡, 𝑀F 𝛾,1(𝑧)= 1∕𝑧 ∫ 0 𝑡2Γ[𝑡]e−𝛾𝑡d𝑡, and 𝑀F 𝛾,2(𝑧)= ∞ ∫ 1∕𝑧 𝑡Γ[𝑡]e −𝛾𝑡d𝑡. In the next section we investigate the asymptotic behavior of the test 𝑛,𝑤 defined by (8), both under the null hypothesis Υ0as well as under alternatives. Remark 4.1. More general distributions with extra parameters may be included in our framework of testing GOF. For instance the Burr type XII distribution belongs to the class of distributions defined by (3), with DF 𝐹0(𝑥) =1 −(1 +𝑥)−𝜉, 𝜉>0, and thus also satisfies Proposition 3.1. Specifically if 𝑋follows a Burr type XII distribution with parameters (𝑐, 𝜑, 𝜉)then 𝑎𝑋1∕𝑏follows a Burr type XII distribution with parameters (𝑎𝑐1∕𝑏, 𝑏𝜑, 𝜉). Other classes include members of the exponentiated class of distributions, like the exponentiated Weibull distribution with 𝐹0(𝑥) =(1 −e −𝑥)𝜉, 𝜉>0(see Mudholkar et al. (1995)), and the generalized gamma distribution with density 𝑓𝑐,𝜑,𝜉(𝑥) =(𝜑𝑥𝜑𝜉−1)∕(𝑐𝜑𝜉Γ(𝜉))e−(𝑥∕𝑐)𝜑, 𝜉>0. 5. Limit null distribution and consistency of the test Here and in what follows, the notation  ⟶and  =mean convergence and equality in distribution of random elements and random variables, respectively, P ⟶means convergence in probability, a.s. ⟶means almost sure convergence, o𝖯(1) stands for convergence in probability to 0, and O𝖯(1) denotes a bounded in probability random variable or process. All limits are understood as 𝑛 →∞, where 𝑛stands for the sample size. While for computational convenience we have suggested to actually perform the test based on the weight function 𝑤(𝑡) =e −𝛾𝑡, 𝛾> 0, nevertheless the limit behavior of the statistic will be studied for arbitrary weight functions satisfying the following assumption. Assumption 5.1. 𝑤 ∶[0, ∞) ↦(0, ∞), is continuous on (0, ∞) and satisfies ∫∞ 0𝑤(𝑡) d𝑡 <∞. Recall also that the test figuring in (8) implicitly involves the MLE via the transformed observations ( 𝑌𝑗, 𝑗=1, ..., 𝑛)defined in (5). In this connection we assume that 𝑐𝑛and 𝜑𝑛admit the asymptotic representations in the next Assumption. Assumption 5.2. √𝑛(𝑐𝑛−𝑐0)= 1 √𝑛 𝑛 ∑ 𝑗=1 C(𝑋𝑗)+o P(1),(13) and √𝑛(𝜑𝑛−𝜑0)= 1 √𝑛 𝑛 ∑ 𝑗=1 Φ(𝑋𝑗)+o P(1),(14) where 𝖤0(C(𝑋1)) =0 =𝖤0(Φ(𝑋1)), 𝖤0(C2(𝑋1)) <∞, 𝖤0(Φ2(𝑋1)) <∞, and 𝖤0(⋅)indicating expectation taken under 𝐹𝑐0,𝜑0, with (𝑐0, 𝜑0)being the true parameter values. Computational Statistics and Data Analysis 197 (2024) 107988 6 S.G. Meintanis, B. Miloševi´c and M.D. Jiménez–Gamero Remark 5.3. If the family of distributions satisfy certain regularity conditions (see, for example Theorem 5.1 in Lehmann and Casella (1998)), then the MLEs of the parameters 𝑐and 𝜑satisfy Assumption 5.2 with (C(𝑥) Φ(𝑥))=𝐼−1(𝑐0,𝜑0)(𝑙𝑐(𝑥;𝑐0,𝜑0) 𝑙𝜑(𝑥;𝑐0,𝜑0)), where 𝐼(𝑐, 𝜑)is the information matrix and 𝑙𝑐(𝑥; 𝑐, 𝜑) =𝜕 𝜕𝑐 log 𝑓(𝑥; 𝑐, 𝜑), 𝑙𝜀(𝑥; 𝑐, 𝜑) =𝜕 𝜕𝜑 log𝑓(𝑥; 𝑐, 𝜑). This is the case for the families Weibull and Fréchet. Moreover, the likelihood equations associated with these two families have a unique solution (see Ramos et al. (2020)for the Fréchet family and Balakrishnan and Kateri (2008)for the Weibull family). The Pareto family does not satisfy the above cited regularity conditions. Nevertheless, the MLE of 𝑐and 𝜑still fulfills Assumption 5.2. Moreover, in this family, the MLE of the parameter 𝑐is 𝑋(1) =min{𝑋1, … , 𝑋𝑛}, and it may easily be checked that √𝑛(𝑋(1) −𝑐)=o P(1), and thus (13) trivially holds with C(𝑥) =0, ∀𝑥. As already emphasized and due to test invariance with respect to (𝑐, 𝜑), the study of the asymptotic null distribution of the test statistic 𝑛,𝑤 provided by Theorem 5.5 below, is carried out by setting (𝑐0, 𝜑0) =(1, 1). It will be also assumed that 𝐹0has a density (with respect to the Lebesgue measure) as expressed in the next assumption. Assumption 5.4. Let 𝑎 =inf{𝑥 >0 ∶𝐹0(𝑥) >0}, and assume that 𝐹0has a density 𝑓0such that 𝑓0(𝑎) = lim 𝑥→𝑎+𝑓0(𝑥)exists and is continuous on [𝑎, 𝑏], ∀𝑏 >𝑎. Moreover suppose that ∫log𝜅(𝑢)𝑢±𝜌𝑓2 0(𝑢)d𝑢<∞, for 𝜅=0and 𝜅=2, and for some 0 <𝜌 <1. Theorem 5.5. Let (𝑋𝑗, 𝑗=1, ..., 𝑛)be independent copies of 𝑋>0, and assume that the law of 𝑋∈0, for some 𝑐, 𝜑 >0. Suppose that Assumptions 5.1, 5.2 and 5.4 hold, that 𝖤0(log4(𝑋)) <∞and that there exists 0 <𝜀 <1such that ∞ ∫ 1 𝑡8𝜀∕(1−𝜀)𝑤(𝑡)d𝑡<∞.(15) Then, 𝑛,𝑤 −𝑛,𝑤,0=o P(1),(16) and 𝑛,𝑤  ⟶‖‖2 𝑤,(17) where 𝑛,𝑤,0=‖√𝑛𝑉𝑛‖2 𝑤,𝑉 𝑛(𝑡)=1 𝑛 𝑛 ∑ 𝑗=1 𝑔(𝑋𝑗;𝑡),(18) 𝑔(𝑥;𝑡)=min(1,𝑡𝑥)−𝜓0(𝑡)−𝜇(𝑡)C(𝑥), 𝜇(𝑡)=−𝑡 1∕𝑡 ∫ 0 𝑢𝑓0(𝑢)d𝑢, and is a centered Gaussian random element of having covariance kernel 𝑐(𝑡,𝑠)=𝖤0(𝑔(𝑋;𝑡)𝑔(𝑋;𝑠)).(19) Remark 5.6. We have that ‖‖2 𝑤  =∑𝑗≥1𝜆𝑗𝜒2 1𝑗, where 𝜒2 11, 𝜒2 12, …are independent chi-squared random variables each with one degree of freedom and {𝜆𝑗}𝑗≥1are the eigenvalues of the operator 𝐴defined on 𝐿2(𝑤)through 𝐴𝜙(𝑥)= ∞ ∫ 0 ℎ∗(𝑥,𝑦)𝜙(𝑦)𝑓0(𝑦)d𝑦, with Computational Statistics and Data Analysis 197 (2024) 107988 7 S.G. Meintanis, B. Miloševi´c and M.D. Jiménez–Gamero ℎ∗(𝑥,𝑦)= ∞ ∫ 0 𝑔(𝑥;𝑡)𝑔(𝑦;𝑡)𝑤(𝑡)d𝑡. Explicit expressions of ℎ∗(𝑥, 𝑦)can be calculated for the Weibull, Pareto and Fréchet families of distributions. The easiest case is obtained for the Pareto family because, as observed in Remark 5.3, in this case C(𝑥) =0, ∀𝑥, and thus ℎ∗(𝑥,𝑦)P=Λ P 𝛾(𝑥,𝑦)−𝜆P 𝛾(𝑥)−𝜆P 𝛾(𝑦)+P 𝛾, where ΛP 𝛾, 𝜆P 𝛾and P 𝛾are as defined in Section 4. We did not obtain the eigenvalues of the operator associated to the considered families. Nevertheless, they can be approximated by using a result in Dehling and Mikosch (1994)(see Remark (c) after Corollary 3.3), which states that if (𝑋𝑗, 𝑗=1, ..., 𝑛)are independent copies of 𝑋>0, and the law of 𝑋∈0, then the eigenvalues of the random matrix 𝑀=(𝑛−1ℎ∗(𝑋𝑗,𝑋𝑘))1≤𝑗,𝑘≤𝑛converge a.s. to those of the operator 𝐴; a related work is Ngatchou-Wandji (2009). Nevertheless approximating these eigenvalues is a non–trivial numerical exercise that, in the context of goodness–of–fit testing, has been carried out in only a few selective cases. We refer to Cupari´ c et al. (2022), Ebner and Henze (2023)and Meintanis et al. (2023). Remark 5.7. Notice that the weight function 𝑤(𝑡) =e −𝛾𝑡 adopted in Section 4satisfies Assumption 5.1 as well as (15), ∀ 0 <𝜀 <1, for each 𝛾>0. Remark 5.8. In Section 4we considered three special instances of distributions 𝐹0. It can be easily checked that each of these distributions satisfies Assumption 5.4, ∀ 𝜌 ∈(0, 1), and that 𝖤0(log4(𝑋)) <∞. We now consider the asymptotic behavior of 𝑛,𝑤 under fixed alternatives to the null hypothesis Υ0. The following result implies the consistency of the test that rejects Υ0for large values of 𝑛,𝑤. Theorem 5.9. Let (𝑋𝑗, 𝑗=1, ..., 𝑛)be independent copies of 𝑋>0. Assume that (𝑐𝑛, 𝜑𝑛) P ⟶(𝑐𝑋, 𝜑𝑋), for some 𝑐𝑋∈(0, ∞), 𝜑𝑋∈(0, ∞), and that 𝖤(| log(𝑋)|) <∞. Then, 𝑛,𝑤 𝑛 P ⟶‖𝜓𝑋−𝜓0‖2 𝑤∶= Δ𝑤,(20) where 𝜓𝑋(𝑡) =𝖤(min{1, 𝑡(𝑋∕𝑐𝑋)𝜑𝑋}). Remark 5.10. If in the assumptions of Theorem 5.9, convergence in probability is replaced by almost sure convergence, i.e. if (𝑐𝑛, 𝜑𝑛) a.s. ⟶(𝑐𝑋, 𝜑𝑋), then we may write a.s. ⟶in equation (20). Under the standing assumptions, Δ𝑤is positive unless 𝜓𝑋(𝑡)is equal to 𝜓0(𝑡)identically in 𝑡, i.e. if the minCF of (𝑋∕𝑐𝑋)𝜑𝑋is equal to 𝜓0(⋅). In turn by the uniqueness of the minCF this implies that (𝑋∕𝑐𝑋)𝜑𝑋∈0, which, since the transformation 𝑋↦(𝑋∕𝑐𝑋)𝜑𝑋 is one–to–one, entails 𝑋∼𝐹𝑐,𝜑 with (𝑐, 𝜑) =(𝑐𝑋, 𝜑𝑋). Thus Δ𝑤>0unless Υ0is true, and consequently the test that rejects the null hypothesis Υ0for large values of 𝑛,𝑤 is consistent against each fixed alternative distribution satisfying the assumptions stated in Theorem 5.9. Finally, the next result gives the asymptotic behavior of the test statistic under contiguous alternatives that approach Υ0at the rate 𝑛−1∕2. To simplify notation, we identify the distribution function of a random variable with the associated probability measure. It will be assumed that the data come from a variable having distribution function 𝐹𝑛that meets the following: 𝐹𝑛is a measure dominated by some fixed 𝐹0∈0, with Radon-Nikodym derivative d𝐹𝑛∕d𝐹0=1 +𝑛−1∕2𝑎, for some measurable bounded function 𝑎 ∶[0, ∞) ↦ℝsatisfying ∞ ∫ 0 𝑎(𝑢)𝑓0(𝑢)d𝑢 <∞.(21) The boundedness of 𝑔implies 𝑑𝐹𝑛∕𝑑𝐹0≥0for large enough 𝑛. Theorem 5.11. Let (𝑋𝑛,𝑗 , 𝑗=1, ..., 𝑛)be independent copies of 𝑋𝑛>0with distribution function 𝐹𝑛satisfying (21). Suppose that Assumptions 5.1, 5.2 and 5.4 hold, that 𝖤0(log4(𝑋)) <∞and (15)also hold. Then, 𝑛,𝑤  ⟶‖+𝑑‖2 𝑤, where is the Gaussian random element figuring in Theorem 5.5, and 𝑑(𝑡) =𝖤0(𝑔(𝑋; 𝑡)𝑎(𝑋)). As a consequence of Theorem 5.5 and Theorem 5.11, the test that rejects the null hypothesis Υ0for large values of 𝑛,𝑤 is able to detect contiguous alternatives satisfying (21). Computational Statistics and Data Analysis 197 (2024) 107988 8 S.G. Meintanis, B. Miloševi´c and M.D. Jiménez–Gamero Table 1 Distributions included in the power study. Distribution Notation Density function Weibull 𝑊(𝜑,𝑐)𝜑𝑥𝜑−1 𝑐𝜑𝑒−( 𝑥 𝑐)𝜀,𝑥>0; 𝜑>0,𝑐>0; Pareto 𝑃(𝜑,𝑐)𝜑𝑐𝜑 𝑥𝜑+1 𝑥>𝑐;𝜑>0,𝑐>0; Gamma Γ(𝜑,𝑐)𝑥𝜑−1 𝑐𝜑Γ(𝜑)𝑒−𝑥 𝑐,𝑥>0; 𝜑>0,𝑐>0; Lognormal 𝐿𝑁(𝜇,𝜎)1 𝑥𝜎√2𝜋𝑒−(ln(𝑥)−𝜇)2 2𝜎2,𝑥>0; 𝜇∈𝐑,𝜎>0; Halfnormal 𝐻𝑁(𝜃)√2 𝜃√𝜋𝑒−𝑥2 2𝜃2,𝑥>0; 𝜃>0; Linear failure rate 𝐿𝐹𝑅(𝜃)(1+𝜃𝑥)𝑒−𝑥−1 2𝜃𝑥2,𝑥>0; 𝜃>0; Chen 𝐶𝐻(𝜃)2𝜃𝑥𝜃−1𝑒𝑥𝜃−2(1−𝑒𝑥𝜃),𝑥>0; 𝜃>0; Fréchet 𝐹(𝜑,𝑐)𝜑 𝑐(𝑥 𝑐)−1−𝜑𝑒−( 𝑥 𝑐)−𝜑,𝑥>0; 𝜑>0,𝑐>0; 6. Monte Carlo study In this section we study the finite–sample performance of the new GOF tests, simply denoted it by 𝛾, in comparison with other tests. In order to perform comparisons, all tests are applied on standardized data (5)with sample size 𝑛 =20and 𝑛 =50. In this connection and since the null distributions of tests statistics do not depend on unknown parameters, the tests’ powers are obtained using a Monte Carlo procedure with 𝑁= 10, 000 replicates, with nominal level of significance 𝛼=0.05. The study includes the following distributions: Weibull, Pareto, Gamma, Lognormal, Halfnormal, Linear failure rate, Fréchet, as well as Chen distributions, for some specific choices of distributional parameters. These particular distributions occur in analogous Monte Carlo studies; see e.g. Cupari´ c et al. (2020), Allison et al. (2022), and Ndwandwe et al. (2023b). Their density functions are given in Table 1. For the Weibull distribution we include the Anderson–Darling test (often the best test amongst the classical tests based on the empirical DF), and the test of Öztürk and Korukogu (1988), which are labeled as AD and OK, respectively. We also include a recent test based on a Stein-type characterization. This test is denoted by 𝑆𝑎, and we adopt the value 𝑎 =1for the value of tuning parameter following the authors’ recommendation; see Ebner et al. (2022). We have also included an empirical–Laplace–transform based test that utilizes transformation of a Weibull variate to extreme value distribution of type I, labeled as  𝐿𝑇 2(see Krit (2014)). Another test which is related to the test based on the minCF utilizes the max–characteristic function defined as Ξ𝑋(𝑡) =𝖤(max{1, 𝑡𝑋}); see Falk and Stupfler (2017). The maxCF exists provided that 𝖤(𝑋) <∞, and then the corresponding test, denoted by FS𝛾, has a direct connection to a test based on the minCF, since Ξ𝑋(𝑡) =1 +𝑡𝖤(𝑋) −Ψ 𝑋(𝑡). Here we consider the maxCF test FS𝛾constructed analogously to our test with weight function 𝑤(𝑡) =𝑒−𝛾𝑡, with 𝛾=1. We highlight that tests based on the maxCF are also considered for the first time in the literature. The figures in Table 2show percentage of rejection of the null hypothesis of a Weibull distribution for each test considered rounded to the nearest integer. Overall, the best performing test appears to be the OK test. For sample size 𝑛 =50our test with 𝛾=5 is a good contender of OK, and preferable to the next best test, i.e. the AD test. For 𝑛 =20, our test 5is less of a contender, but still ranks close to the AD test which is second best. In this connection we wish to point out that since the best performing 𝛾–test across 𝛾is a formidable opponent of the OK test, data–driven selections of the weight parameter 𝛾such as the ones suggested by Allison and Santana (2015)and Tenreiro (2019), seem to hold further promise for the new test. However more research is needed in this direction both in the actual implementation of such a test as well as for its asymptotic. For the Pareto type I distribution we compare our test with the 𝑀𝑎–test that is based on the difference between the empirical characteristic function of transformed data and that of the standard uniform distribution (see Meintanis et al. (2014)and Ndwandwe et al. (2023b)for details), and the test based on the Mellin transform proposed by Meintanis (2009), denoted by 𝐺𝑎, which is amongst the best performers in the Monte Carlo study of Ndwandwe et al. (2023a). We also consider a recent characteristic-function based test proposed in Ndwandwe et al. (2023b), denoted by 𝑆(1) 3,2. Additionally, we include the likelihood ratio–based test 𝑍𝐶proposed by Zhang (2002)and the test 𝐼2which is one of the best performers among the recent characterization–based tests utilizing the V-empirical distribution approach (see Allison et al. (2022)). The figures in Table 3show clearly that the new tests based on 𝛾are the most powerful or second best nearly uniformly over alternatives and for all values of 𝛾with both sample sizes, followed by the 𝑀1–based test. Finally, for the Fréchet distribution the proposed GOF test is compared against the likelihood ratio–based test 𝑍𝐴of Zhang (2002) and the AD test. The figures shown in Table 4show a similar picture to that of Table 2, i.e., the new tests rank second best after the 𝑍𝐴test, being mostly more powerful than the AD–test, but not by a wide margin. 7. Real–data applications The Weibull distribution is often used in the context of reliability analysis. In this connection, Wu et al. (2021)consider the Weibull distribution for strength–modeling of two glass fibers, of length 1.5 cm and 15 cm, respectively. The datasets are originally taken from Smith and Naylor (1987). The main goal of their study was to consider different methods for parameter estimation, while the fit of the Weibull distribution has been justified using the classical Kolmogorov-Smirnov test. However, it is not clear how this test was used, i.e. it is not known whether the fact that it is not distribution free in the presence of estimation of unknown parameters was taken into account. Here we apply the tests utilized in our power study to check goodness–of–fit to the Weibull distribution. Computational Statistics and Data Analysis 197 (2024) 107988 9 S.G. Meintanis, B. Miloševi´c and M.D. Jiménez–Gamero Table 2 GOF tests for the Weibull distribution: Empirical percentage of rejection. 𝑛=20 𝑛=50 Dist 0.515AD OK 𝑆1 𝐿𝑇 2FS10.515AD OK 𝑆1 𝐿𝑇 2FS1 𝑊(1,0.5) 5555555 45655555 5 𝑊(0.5,1) 5556555 55555555 5 Γ(0.8,1) 5556565 55666565 6 Γ(2,1) 6676757 589129117118 Γ(3,1) 8 7 9 7 9 5 9 7 13 13 18 13 13 10 13 12 𝐿𝑁(1) 23 20 26 23 30 17 27 20 56 53 63 56 70 47 62 52 𝐿𝑁(2.5) 24 21 26 22 30 18 27 19 57 54 65 56 70 47 61 53 𝐻𝑁(1) 7 8 8 9 6 108 7 11 131613 13 1313 12 𝐿𝐹𝑅(0.2) 55 47 43 55 56 39 43 47 95 92 89 97 96 94 86 92 𝐿𝐹𝑅(0.5) 49 41 39 48 52 32 38 41 92 88 85 94 93 85 81 87 𝐿𝐹𝑅(0.8) 45 37 37 43 47 31 35 38 89 85 82 93 92 82 77 85 𝐿𝐹𝑅(1) 44 36 36 43 46 29 35 36 89 83 81 91 91 80 77 84 𝐶𝐻(0.8) 7 7 8 8 5 9 8 7 11 12 16 12 13 11 12 12 𝐶𝐻(1) 7 8 8 8 6 9 8 7 11 12 15 12 12 11 12 12 𝐶𝐻(1.5) 8 8 8 8 6 9 7 7 11 12 16 12 12 11 21 12 Table 3 GOF tests for the Pareto distribution: Empirical percentage of rejection. 𝑛=20 𝑛=50 Dist 0.515AD 𝑆(1) 3,2𝑀1𝐺2𝑍𝐶𝐼20.515AD 𝑆(1) 3,2𝑀1𝐺2𝑍𝐶𝐼2 𝑃(1,1) 555555554555555555 𝑃(2,1) 555555554555555555 𝑊(0.5,1) + 1 29 27 19 55 25 34 34 51 39 59 57 43 91 72 66 89 89 79 𝑊(0.8,1) + 1 6 5 7 5 13 3 2 6 4 16 17 24 13 25 12 7 13 6 𝑊(1.2,1) + 1 38 36 38 13 49 27 17 5 22 94 94 95 84 93 91 88 61 80 𝑊(1.5,1) + 1 62 61 63 39 73 53 38 12 42 100 100 100 98 99 99 98 93 97 Γ(0.8,1) + 1 7 7 8 4 15 4 2 6 4 22 23 29 15 29 16 10 12 8 Γ(1,1) + 1 18 18 19 5 29 12 7 4 9 66 66 70 44 67 59 51 22 40 Γ(1.2,1) + 1 34 32 34 24 46 24 15 4 18 91 91 93 78 90 88 83 53 76 𝐻𝑁(1) + 1 47 51 55 20 64 37 25 9 29 98 98 99 91 97 95 92 75 87 𝐿𝑁(1) + 1 15 14 13 4 24 11 7 2 9 63 64 56 51 66 65 61 23 55 𝐿𝑁(1.2) + 1 7 7 6 2 13 5 3 2 5 32 31 28 21 35 32 29 7 24 𝐿𝑁(1.5) + 1 444273234667496434 𝐿𝑁(2.5) + 1 24 26 18 42 19 31 31 34 39 55 52 38 79 58 64 65 66 72 𝐿𝐹𝑅(0.2) + 1 12 12 14 4 22 8 4 3 5 50 48 53 30 52 4 35 13 27 𝐿𝐹𝑅(0.5) + 1 18 18 19 6 29 13 6 4 5 65 65 69 45 67 58 49 20 41 𝐿𝐹𝑅(0.8) + 1 22 22 23 6 33 15 8 4 8 76 75 79 54 75 68 59 27 50 𝐿𝐹𝑅(1) + 1 24 23 25 7 36 17 4 9 11 79 80 81 59 80 62 64 32 54 𝐶𝐻(0.8) + 1 6 6 8 5 14 4 2 8 3 19 19 26 13 28 13 5 13 10 𝐶𝐻(1) + 1 22 22 25 7 35 16 4 5 7 77 78 82 57 78 67 59 33 48 𝐶𝐻(1.5) + 1 73 74 76 41 82 64 47 19 50 100 100 100 100 100 99 99 98 99 The results are presented in Table 5. The p–values reported suggest that for Dataset 2 the Weibull distribution may be a reasonable model supported by all tests considered. On the other hand, the fit of the Weibull distribution to Dataset 1 is questionable, with the exception of the OK test,  𝐿𝑇 2test and marginally so by the new test for 𝛾=5. Next we turn to the fit to the Fréchet distribution. In this connection, Ramos et al. (2020) present an overview of the properties of this distribution and, as an illustration, they considered five real data sets related to minimum monthly flows of water (m3/s) in the Piracicaba River, located in São Paulo state, Brazil. They concluded, using several information criteria, that in all cases the Fréchet distribution is the best of the models considered. The p–values reported in Table 6seem to corroborate the conclusions of Ramos et al. (2020)by means of the formal GOF tests applied herein. As an extremal event with a potentially good Pareto fit, we apply our tests to the monetary expenses incurred as a result of wind related catastrophes in 40 separate instances during 1977 (after the de-grouping algorithm used in Allison et al. (2022)). The results presented in Table 7indicate that the Pareto distribution may be one of the possible choices for modeling these data. Another case of extremity is top income earnings, and as such, a typical example is the earnings of top athletes. In this connection, we tested the earnings of the top 20 tennis players as of January 29, 2024, sourced from https://www.spotrac.com/atp/. The results presented in Table 8confirm the suitability of the Pareto distribution for this dataset. Computational Statistics and Data Analysis 197 (2024) 107988 16 S.G. Meintanis, B. Miloševi´c and M.D. Jiménez–Gamero Krit, M., 2014. Goodness-of-fit tests for the Weibull distribution based on the Laplace transform. J. Soc. Fr. Stat. 155 (3), 135–151. 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