A new reciprocal Rayleigh extension: Properties, copulas, different methods of estimation and a modified right-censored test for validation
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Yousof, Haitham M.; Ali, M. Masoom; Goual, Hafida; Ibrahim, Mohamed Article A new reciprocal Rayleigh extension: Properties, copulas, different methods of estimation and a modified rightcensored test for validation Statistics in Transition New Series Provided in Cooperation with: Polish Statistical Association Suggested Citation: Yousof, Haitham M.; Ali, M. Masoom; Goual, Hafida; Ibrahim, Mohamed (2021) : A new reciprocal Rayleigh extension: Properties, copulas, different methods of estimation and a modified right-censored test for validation, Statistics in Transition New Series, ISSN 2450-0291, Exeley, New York, Vol. 22, Iss. 3, pp. 99-121, https://doi.org/10.21307/stattrans-2021-029 This Version is available at: https://hdl.handle.net/10419/266273 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/
STATISTICS IN TRANSITION new series, September 2021 Vol. 22, No. 3, pp. 99–121, DOI 10.21307/stattrans-2021-029 Received – 24.01.2020; accepted – 01.06.2021 A new reciprocal Rayleigh extension: properties, copulas, different methods of estimation and a modified right-censored test for validation Haitham M. Yousof1, M. Masoom Ali2, Hafida Goual3, Mohamed Ibrahim4 ABSTRACT In this article, a new reciprocal Rayleigh extension called the Xgamma reciprocal Rayleigh model is defined and studied. The relevant statistical properties are derived, and the useful results related to the convexity and concavity are addressed. We discussed the estimation of the parameters using different estimation methods such as the maximum likelihood estimation method, the ordinary least squares estimation method, the weighted least squares estimation method, the Cramer-Von-Mises estimation method, and the bootstrapping method. A simulation study was conducted to assess the performances of the proposed estimation methods are investigated through a simulation study. Many bivariate and multivariate type model have also been derived based on Farlie-Gumbel-Morgenstern copula, the Clayton copula, Renyi’s entropy copula and the Ali-Mikhail-Haq copula. A modified Nikulin-Rao-Robson test for right-censored validation is applied to a censored real data set. Key words: Xgamma model, reciprocal Rayleigh model, simulations, bootstrapping, Farlie Gumbel Morgenstern copula, least squares, Cramer-Von-Mises, bootstrapping, Ali-MikhailHaq copula, convexity, concavity. 1. Introduction The probability density function (PDF) and cumulative distribution function (CDF) of the reciprocal Rayleigh (RR) distribution are given, respectively, by 𝑔𝑦2𝜃𝑦𝑒 |∈ℝ, 1 Department of Statistics, Mathematics and Insurance, Faculty of Commerce, Benha University, Benha 13518, Egypt. E-mail: haitha[email protected]. ORCID: https://orcid.org/0000-0003-4589-4944. 2 Department of Mathematical Sciences Ball State University, Muncie, Indiana 47306, USA. E-mail: [email protected]. ORCID: https://orcid.org//0000-0002-0120-9442. 3 Laboratory of Probability and Statistics, University of Badji Mokhtar, Annaba, Algeria. E-mail: [email protected]. ORCID: https://orcid.org/0000-0003-4932-4332. 4 Department of Applied, Mathematical and Actuarial Statistics, Faculty of Commerce, Damietta University, Damietta, Egypt. E-mail: [email protected]. ORCID: https://orcid.org/0000-0003-4893-9669.
100 Haitham M. Yousof et al.: A new reciprocal Rayleigh extension: properties… and 𝐺𝑦𝑒 |∈ℝ, where 𝜃0 refers to the scale parameter. The RR model is a special case from the well-known inverse Weibull distribution. The RR model was originally proposed by Fréchet (1927). It has many applications in accelerated life testing, earthquakes, floods, wind speed, horse racing, rainfall, queues in supermarkets, and sea waves. Gusmao et al. (2011) defined and studied the generalized reciprocal Rayleigh (GRR) distribution. Krishna et al. (2013) proposed some applications of the Marshall-Olkin reciprocal Rayleigh (MORR) distribution. Mahmoud and Mandouh (2013) proposed and studied the transmuted reciprocal Rayleigh (TRR) distribution. Haq et al. (2017) presented a new four-prarameter reciprocal Rayleigh version for modeling extreme values. Korkmaz et al. (2017) studied some theoretical and computational aspects of the odd Lindley reciprocal Rayleigh (OLRR) distribution. Yousof et al. (2018d) defined a new family called the odd reciprocal Rayleigh G (ORR-G) family of distributions. Yousof et al. (2019) defined a new compound version of the reciprocal Rayleigh (OBRR) distribution. Salah et al. (2020) defined and studied a new version of RR model called the odd Burr RR model with different copula, different estimation methods, applications and validation testing. Recently, Cordeiro (2020) proposed and studied the Xgamma-G (Xg-G) family of distribution with CDF and PDF (for 𝜃0) given by 𝐹,𝛏𝑦11𝜃𝜃log1𝐺𝛏𝑦𝜃log1𝐺𝛏𝑦 1𝜃 1𝐺𝛏𝑦|∈ ℝ , (1) and 𝑓 ,𝛏𝑦 𝜃 1𝜃𝑔𝛏𝑦1𝐺𝛏𝑦𝜃12𝜃log1𝐺𝛏𝑦|∈ ℝ , (2) respectively, where 𝑔𝛏𝑦 and 𝐺𝛏𝑦 are the baseline PDF and CDF respectively with a parameter vector 𝛏. To this end, we define the CDF of the Xgamma reciprocal Rayleigh (XgRR) model. Using (1), the CDF of the XgRR can be written as 𝐹,𝑦1 1 1𝜃1𝑒 ⎝ ⎜ ⎛ 1𝜃𝜃log1𝑒 12𝜃log1𝑒 ⎠ ⎟ ⎞ |∈ℝ. (3) The PDF corresponding to (3) reduces to 𝑓 ,𝑦2𝜃𝜃𝑦𝑒 1𝜃1𝑒 𝜃12𝜃log1𝑒 |∈ ℝ . (4)
STATISTICS IN TRANSITION new series, September 2021 101 The XgRR family density in (4) can be expressed as 𝑓 ,𝑦2𝜃𝜃 1𝜃𝑦𝑒 1𝑒 ⎝ ⎜ ⎜ ⎛ 𝜃 21𝜃2𝜃𝑦𝑒 1𝑒 log1𝑒 ; ⎠ ⎟ ⎟ ⎞ |∈ℝ. (5) Consider log1𝑎 𝑎 1 𝑖1 𝑎 𝑎| , (6) and the power series raised to a positive integer 𝑛 (see Gradshteyn and Ryzhik (2002)) 𝑎 𝑢𝑐, 𝑢, (7) where the coefficients 𝑐, (for 𝑗1,2,… ) can be easily determined from the recurrence equation 𝑐,𝑗𝑎𝑚𝑛1𝑗 𝑎𝑐, and 𝑐,𝑎. The coefficient 𝑐, can be calculated from 𝑐,,…,𝑐, and hence from the quantities 𝑎,…,𝑎 . For 1 and 𝑎0, the power series holds 1𝑎 𝑎 𝛤1𝑎 𝑗!𝛤𝑎𝑗1 𝑎 𝑎. (8) Applying (6) to the quantity 𝐴𝑦;𝜃 in the PDF in (5), the PDF can be expressed as 𝑓,𝑦 𝜃 1𝜃2𝜃𝑦𝑒 1𝑒 ⎝ ⎜ ⎜ ⎛ 𝜃 21𝜃2𝜃𝑦𝑒 1𝑒 𝑒 1 𝑖1 𝑒 ;⎠ ⎟ ⎟ ⎞ .
102 Haitham M. Yousof et al.: A new reciprocal Rayleigh extension: properties… Expanding the quantity 𝐵𝑦;𝜃 using (7), the 𝑓,𝑦 can be written as 𝑓,𝑦 𝜃 1𝜃 2𝜃𝑦𝑒 1𝑒 ;, ⎩ ⎪ ⎨ ⎪ ⎧ 𝜃 21𝜃2𝜃𝑦𝑒 𝑐, 𝑒 1𝑒 ;,⎭ ⎪ ⎬ ⎪ ⎫ , where 𝑐, 1/𝑖1. Applying the power series (8) to the quantity 𝐶𝑦;𝜃,𝜃 , we obtain 𝑓 𝑦𝛻𝜋𝑦𝛻, 𝜋𝑦 , (9) where 𝛻1𝜃𝛤𝜃 1𝑗1𝜃𝛤𝜃𝑗, 𝛻,1𝜃𝛤𝜃 𝑐, 21𝜃3𝑖𝑗𝑗!𝛤𝜃𝑗, and 𝜋𝑦 is the RR density with scale parameter 𝜃𝜍 and shape parameter 2. So, the density of 𝑌 is a linear combination of RR densities. The CDF of 𝑌 follows by integrating (8) as 𝐹,𝑦𝛻𝐻𝑦𝛻, 𝐻𝑦 , (10) where 𝐻𝑦 is the RR density with scale parameter 𝜃𝜍 and shape parameter 2. Equations (9) and (10) are the main results of this section. We provide some plots of the PDF and hazard rate function (HRF) of the XgRR model to show its flexibility. Figure 1 displays some plots of the XgRR density for selected parameter values. These plots reveal that the new density can be right skewed with different flexible shapes. The HRF plots of the XgRR distribution can be upside down or increasing. Many other useful real data sets can be found in Aryal and Yousof (2017), Merovci et al. (2017 and 2020), Korkmaz et al. (2017), Hamedani et al. (2017), Brito et al. (2017), Alizadeh et al. (2018), Korkmaz et al. (2018), Yousof et al. (2018a-d), Hamedani et al. (2018), Cordeiro et al. (2018), Hamedani et al. (2019), Ibrahim (2019), Nascimento et al. (2017, 2018 and 2019), Ibrahim et al. (2019), Goual and Yousof (2019), Korkmaz et al. (2019), Alizadeh et al. (2019) and Goual et al. (2020), Ibrahim (2020 a and b) and Yadav et al. (2020). After studying the mathematical properties of the XgRR model, we discussed the estimation of the parameters using different estimation methods such as maximum likelihood estimation method, ordinary least squares estimation method, weighted least-squares estimation method, Cramer-Von-Mises estimation method and
STATISTICS IN TRANSITION new series, September 2021 103 bootstrapping method. Then, the Nikulin-Rao-Robson (N.R.R) statistic and modified N.R.R are discussed. In particular, the modified chi-squared test for composite hypothesis for complete samples was first considered by Nikulin (1973a, 1973b and 1973c), Rao and Robson (1974), among others. On the other hand, several goodnessof-fit tests have been suggested by the statisticians for censored data. 2. Properties 2.1. Moments Let 𝑌 be a rv having density 𝜋𝑦. The 𝑟 ordinary moment of 𝑌, say 𝜇, , follows from (9) as 𝜇, 𝐸𝑌𝛻 𝐸𝑌 𝛻, 𝐸𝑌 . Therefore, 𝜇, 𝜃𝛤1𝑟2𝛻,𝛻, , |, (11) where 𝛻,𝑏 1𝑗 and ∇, ,𝑏, 3𝑖𝑗, and 𝛤1𝜏|∈𝜏!𝜏𝑤 𝑦 𝑒𝑦𝑝𝑦𝑑𝑦. Setting 𝑟1 in (11) gives the mean of 𝑌 𝐸𝑌𝜃𝛤112𝛻 1𝑗 𝛻, 3𝑖𝑗 . 2.2. Incomplete moments The 𝑟 incomplete moment of 𝑌 is defined by 𝑚,𝑡 𝑦 𝑓𝑦𝑑𝑦. We can write from (9) 𝑚,𝑡𝛻 𝑚,,𝑦𝛻, 𝑚,,𝑦 . Therefore, 𝑚,𝑡𝜃⎣ ⎢ ⎢ ⎢ ⎡ 𝛻,𝛾1𝑟2,1𝑗𝜃 𝑡 𝛻, , 𝛾1𝑟2,3𝑖𝑗𝜃 𝑡⎦ ⎥ ⎥ ⎥ ⎤ |, (12)
104 Haitham M. Yousof et al.: A new reciprocal Rayleigh extension: properties… where 𝛾𝜏,𝑞𝑡 𝑒𝑑𝑡𝑞 𝜏𝐹:𝜏;𝜏𝜃1;𝑞1𝑞 𝑗!𝜏𝑗 𝛤𝜏𝛤𝜏,𝑞, 𝛤𝜏,𝑞|𝑡 𝑒𝑦𝑝𝑡𝑑𝑡, and 𝐹:⋅,⋅,⋅ is a confluent hypergeometric function (see Johnson et al. (2005)). Setting 𝑟1 in (12) gives the first incomplete moment 𝑚,𝑡𝜃⎣ ⎢ ⎢ ⎢ ⎡ 𝛻,𝛾12,1𝑗𝜃 𝑡 𝛻, , 𝛾12,3𝑖𝑗𝜃 𝑡⎦ ⎥ ⎥ ⎥ ⎤ Two important applications of the 𝑚,𝑡 are related to the mean deviation about the mean (𝑆) and median and to the Bonferroni and Lorenz curves. The mean deviation about the mean 𝑆,𝐸|𝑌𝐸𝑌|2𝜇, 𝐹𝐸𝑌2𝑚,𝐸𝑌 and about the median 𝑆,𝐸|𝑌𝑀|𝐸𝑌2𝑚,𝑀 where 𝐸𝑌, 𝑀𝑄𝑢𝐹𝑢 is the median of 𝑌, 𝐹𝜇, is easily calculated. 2.3. Moment generating function The moment generating function (MGF) of 𝑌, say 𝑀𝑡𝐸𝑒, is obtained from (9) as 𝑀𝑡𝛻 𝑀,𝑡𝛻, 𝑀,𝑡 . Therefore, 𝑀𝑡∑𝛻𝑡𝜃/𝑟!1𝑗𝛤1 ∑𝛻, 𝑡𝜃/𝑟!3𝑖𝑗𝛤1 , |. 2.4. Convexity and concavity Convex densities play an important role in several areas of mathematics. They are important in studying the “problems of optimization” where they are distinguished by several convenient characteristics. In mathematical analysis, a certain density defined on a certain 𝑛-dimensional interval is called “convex density” if the line between any two points on the graph of the density lies above the graph between the two points. The PDF in (5) is said to be “concave density” if for any 𝑌∼XgRR 𝜃,𝜃 and 𝑌∼ XgRR 𝑐,𝑐 the PDF satisfies 𝑓 𝐛y 𝐛 𝑦𝐛 𝑓 ,𝑦𝐛 𝑓 ,𝑦|𝐛 𝐛 𝐛.
STATISTICS IN TRANSITION new series, September 2021 105 If the function 𝑓𝐛𝑦 𝐛𝑦 is twice differentiable, then if 𝑓//𝐛𝑦 𝐛𝑦 0,∀ 𝑦∈ℝ , 𝑓𝐛𝑦 𝐛𝑦 is “strictly convex density”. If 𝑓//𝐛𝑦 𝐛𝑦 0,∀ 𝑦∈ℝ , then 𝑓𝐛𝑦 𝐛𝑦 is “convex”. The density in (5) is said to be “convex density” if for any 𝑌∼XgRR 𝜃,𝜃 and 𝑌∼XgRR 𝑐,𝑐 the density satisfies 𝑓 𝐛𝑦 𝐛 𝑦𝐛 𝑓 ,𝑦𝐛 𝑓 ,𝑦|𝐛 𝐛 𝐛. If the function 𝑓𝐛𝑦 𝐛𝑦 is twice differentiable, then if 𝑓//𝐛𝑦 𝐛𝑦 0,∀ 𝑦∈ℝ , 𝑓𝐛𝑦 𝐛𝑦 is “strictly convex density”. If 𝑓//𝐛𝑦 𝐛𝑦 0,∀ 𝑦∈ℝ, then 𝑓𝐛𝑦 𝐛𝑦 is “convex”. If 𝑓𝐛𝑦 𝐛𝑦 is “convex” and 𝑐 is a constant, then the function 𝑐𝑓𝐛𝑦 𝐛𝑦 is “convex”. If 𝑓b𝑦 𝐛𝑦 is “convex density”, then 𝑐𝑓𝐛𝑦 𝐛𝑦 is convex for every 𝑐 0. If 𝑓𝐛𝑦 𝐛𝑦 and 𝑔𝐛𝑦 𝐛𝑦 are “convex density”, then 𝑓𝐛𝑦 𝐛𝑦𝑔𝐛𝑦 𝐛𝑦 is also “convex density”. If 𝑓𝐛𝑦 𝐛𝑦 and 𝑔𝐛𝑦 𝐛𝑦 are “convex density”, then 𝑓𝐛𝑦 𝐛𝑦.𝑔𝐛𝑦 𝐛𝑦 is also “convex density”. If the function 𝑓𝐛𝑦 𝐛𝑦 is “convex density”, then the function 𝑓𝐛𝑦 𝐛𝑦 is “convex density”. If 𝑓𝐛𝑦 𝐛𝑦 is “concave density”, then 𝐛 𝐛 is “convex density” if 𝑓𝑦0. If 𝑓𝐛𝑦 𝐛𝑦 is “concave density”, 𝐛 𝐛 is “convex density” if 𝑓𝑦0. If 𝑓𝐛𝑦 𝐛𝑦 is “concave density”, 𝐛 𝐛 is “convex density”. 3. Copulas For modelling the bivariate real data sets, we can consider the bivariate XgRR type generated via the FGM copula, modified FGM copula, Clayton copula and Renyi's entropy copula. Many other types of copula could be considered in separate works. In this Section, we derive some new bivariate type XgRR (BXgRR) model using the theorems of FGM copula, modified FGM copula, Clayton copula and Renyi's entropy. The Multivariate XgRR (MvXgRR) type is also presented. However, future works may be allocated to study these new models. First, we consider the joint CDF (JCDF) of the FGM family, where 𝐶𝑚,𝑤𝑚𝑤1𝜌𝑚∗𝑤∗|∗,∗, where the marginal function 𝑚𝐹𝐹,𝑦, 𝑤𝐹𝐹,𝑦, 𝜌∈1,1 is a dependence parameter and for every 𝑚,𝑤∈0,1, 𝐶𝑚,0𝐶0,𝑤0, which is "grounded minimum", and 𝐶𝑚,1𝑚 and 𝐶1,𝑤𝑤 which is "grounded maximum", 𝐶𝑚,𝑤𝐶𝑚,𝑤𝐶𝑚,𝑤𝐶𝑚,𝑤0. 3.1. Via FGM copula A copula is continuous in 𝑚 and 𝑤 where |𝐶𝑚,𝑤𝐶𝑚,𝑤||𝑚𝑚||𝑤𝑤|,
106 Haitham M. Yousof et al.: A new reciprocal Rayleigh extension: properties… is the stronger Lipschitz condition. For 0𝑚𝑚1 and 0𝑤𝑤1, we have 𝑃𝑟𝑚𝑀𝑚,𝑤𝑊𝑤 𝐶𝑚,𝑤𝐶𝑚,𝑤𝐶𝑚,𝑤𝐶𝑚,𝑤0. Then, setting 𝑚∗𝑆,𝑦1𝐹,𝑦|∗∈, and 𝑤∗𝑆,𝑦1𝐹,𝑦|∗∈,, we can easily obtain the JCDF of the FGM family from 𝐶𝑦,𝑦𝐹,𝑦𝐹,𝑦1𝜌1𝐹,𝑦1𝐹,𝑦, where 𝐹,𝑦1𝒽𝑦 1𝜃1𝜃𝜃log𝒽𝑦 12𝜃log𝒽𝑦, 𝒽𝑦1𝑒 , 𝐹,𝑦1𝒽𝑦 1𝑎1𝑎𝑎log𝒽𝑦 12𝜃log𝒽𝑦, The JPDF can then be derived from 𝑐𝑚,𝑤1𝜌12𝑚12𝑤 or from 𝑓𝑦,𝑦𝐶𝐹,𝐹𝑓𝑓. 3.2. Via modified FGM copula The modified FGM copula is defined as 𝐶𝑚,𝑤𝑚𝑤1𝜌𝑉𝑚𝐴𝑤|∈, or 𝐶𝑚,𝑤𝑚𝑤𝜌𝑉𝑚𝐴𝑤|∈,, where 𝑉𝑚𝑚𝑉𝑚, and 𝐴𝑤𝑤𝐴𝑤, where 𝑉𝑚 and 𝐴𝑤 are being two continuous functions on 0,1 where 𝑉0𝑉1𝐴0𝐴10. Let 𝑐𝑖𝑛𝑓𝜕 𝜕𝑚𝑉𝑚|𝓺𝑚0,𝑐𝑠𝑢𝑝𝜕 𝜕𝑚𝑉𝑚|𝓺𝑚0, 𝑑𝑖𝑛𝑓𝜕 𝜕𝑤𝐴𝑤|𝓺𝑤0 and 𝑑𝑠𝑢𝑝𝜕 𝜕𝑤𝐴𝑤|𝓺𝑤0.
STATISTICS IN TRANSITION new series, September 2021 113 Table 4. AVs and the corresponding MSEs (in parentheses) for n=200 Parameters MLE LS WLS CVM Bootstrap 𝜃=2.0 2.00332 (0.01841) 2.01138 (0.01710) 2.00587 (0.01740) 2.00564 (0.01811) 1.84919 (0.03575) 𝜃=1.5 1.50390 (0.00263) 1.50005 (0.00329) 1.50278 (0.00313) 1.50295 (0.00350) 1.57880 (0.00910) 𝜃=0.9 0.95153 (0.02540) 0.90173 (0.00267) 0.90503 (0.00264) 0.90449 (0.00270) 0.98419 (0.01048) 𝜃=0.3 0.29346 (0.00087) 0.30154 (0.00046) 0.30022 (0.00038) 0.30040 (0.00046) 0.27433 (0.00091) 𝜃=1.2 1.20178 (0.00634) 1.19981 (0.00491) 1.20636 (0.00513) 1.20563 (0.00527) 1.25721 (0.00807) 𝜃=0.6 0.60278 (0.00079) 0.60331 (0.00111) 0.60044 (0.00099) 0.60074 (0.00115) 0.57895 (0.00119) Table 5. AVs and the corresponding MSEs (in parentheses) for n=500. Parameters MLE LS WLS CVM Bootstrap 𝜃=2.0 1.99647 (0.00668) 1.99838 (0.00654) 1.99702 (0.00629) 1.99672 (0.00666) 1.99960 (0.00620) 𝜃=1.5 1.50344 (0.00100) 1.50277 (0.00133) 1.50334 (0.00121) 1.50356 (0.00137) 1.50239 (0.00136) 𝜃=0.9 0.94897 (0.01044) 0.89934 (0.00104) 0.89870 (0.00629) 0.89860 (0.00106) 0.90144 (0.00105) 𝜃=0.3 0.28999 (0.00049) 0.30117 (0.00019) 0.30139 (0.00016) 0.30151 (0.00019) 0.30039 (0.00018) 𝜃=1.2 1.19801 (0.00229) 1.20037 (0.00198) 1.20245 (0.00196) 1.20247 (0.00202) 1.22109 (0.00292) 𝜃=0.6 0.60204 (0.00030) 0.60112 (0.00044) 0.60018 (0.00038) 0.60015 (0.00044) 0.59211 (0.00049) 6. Modified Right-Censored Test for Validation 6.1. The N.R.R statistic test Many goodness-of-fit tests are used to indicate whether or not it is reasonable to assume that a random sample comes from a specific distribution. For this purpose, researchers proposed many different goodness-of-fit tests. For the complete data, Nikulin 1973a,1973b and 1973c and Rao and Robson 1974 separately proposed a statistic known today as the N.R.R statistic. This statistical test is a natural modification of the Pearson statistic. To test the hypothesis 𝐻 we have 𝐻:𝑃𝑇𝑡𝐹𝑡,𝜁|∈, ,,⋯,,
114 Haitham M. Yousof et al.: A new reciprocal Rayleigh extension: properties… where 𝜁 represents the vector of unknown parameters. Nikulin (1973a, 1973b and 1973c) and Rao and Robson 1974 proposed the N.R.R statistic defined as follows: Observations 𝑇,𝑇,⋯,𝑇 are grouped in 𝑟 subintervals and 𝜈𝜈,𝜈,⋯,𝜈 is the vector of frequencies, where 𝜈 is frequency of ith group and ∑𝜈 𝑛. The tests are based on the following Pearson's statistic 𝑌𝜁𝜒𝜁𝑛ℓ𝜁𝚰𝜁𝐉𝜁ℓ𝜁, where 𝜒𝜁𝜈𝑛𝑝𝜁 𝑛𝑝𝜁 ,𝜈𝑛𝑝𝜁 𝑛𝑝𝜁 ,⋯,𝜈𝑛𝑝𝜁 𝑛𝑝𝜁 , and 𝑝𝜁 is the vector of probabilities and 𝜁 is the vector of parameters which can be known (simple hypothesis) or unknown (composite hypothesis). The 𝑌 statistic follows a chi-square distribution with 𝑟1 degrees of freedom (for more details see Nikulin (1973a, 1973b and 1973c)). 6.2. Application to right-censored real data To test the null hypothesis 𝐻, we use the N.R.R statistic. We compute the maximum likelihood estimators 𝜃 0.95473 and 𝜃 1.24885. We then deduce the value of 𝑌11.05847 . The critical value is 𝜒. 6111.0705. Then, the N.R.R 𝑌 statistic value is less than the critical value, we say that taxes revenue data can be fitted by the XgRR model. The modified chi-squared test for composite hypothesis for complete samples was first considered by Nikulin (1973a, 1973b and 1973c), Rao and Robson (1974). Several goodness-of-fit tests have been suggested by the statisticians for censored data. Bagdonavicius and Nikulin 2011𝑎,𝑏 proposed a modification of the N.R.R statistic that takes into account random right censorship and based on the maximum likelihood estimators on the initial data, also follows a limiting Chi-square distribution. In this Section we develop the approach proposed by Bagdonavicius and Nikulin 2011𝑎,𝑏 to confirm the adequacy of XgRR model when the parameters are unknown and data are censored. Let us consider the composite hypothesis 𝐻 : 𝐹𝑡∈𝐹𝐹𝑡,𝜁|∈, ∈⊂, where 𝜁 is an unknown m-dimensional parameter and 𝐹 is a differentiable and completely specified cdf with the support 0,∞. Let us consider a finite time interval, say, 0,𝜏, where 𝜏 is the maximum time of the study, and divide it into 𝑘𝑠 smaller intervals 𝐼𝑎,𝑎 , where 0𝑎𝑎...𝑎𝑎∞.
STATISTICS IN TRANSITION new series, September 2021 115 In this case the estimated 𝑎𝒌 is given by 𝑎𝒌𝛬1 𝑛𝑖1𝐸𝒋 𝛬T,𝜁 ,𝜁,𝑎𝒌t 𝒋1,2,…,𝑘 where 𝜁 is the maximum likelihood estimator of the parameter 𝜁, 𝛬 is the inverse of cumulative hazard function 𝛬 , 𝑇 is the 𝑖 element in the ordered statistics 𝑇,…,𝑇 and 𝐸𝒋𝑛1𝑖𝛬𝑎𝒋,𝜁 𝛬T,𝜁 𝒍 , and 𝑎 are random data functions such as the 𝑘 intervals have equal expected numbers of failures 𝑒 . Usually in real application we fix 𝑘. The test statistic for 𝐻 is given in Goual et al. (2020) and Goual and Yousof (2019). The survival times in days are for the 𝑛51 patients. The data are: 7, 34, 42, 63, 64, 74*, 83, 84, 91, 108, 112, 129, 133, 133, 139, 140, 140, 146, 149, 154, 157, 160, 160, 165, 173, 176, 185*, 218, 225, 241, 248, 273, 277, 279*, 297, 319*, 405, 417, 420, 440, 523*, 523, 583, 594, 1101, 1116*, 1146, 1226*, 1349*, 1412*, 1417. (* censored). We suppose that these data are distributed according to the XgRR distribution, we transform the survival times in months (1 month = 30.438 days), so the maximum likelihood estimates of the parameter vector 𝜁 are 𝜁5.00248,1.378452 We choose 𝑟7 as the number of classes. The elements of the test statistic 𝑌 is presented as follows, we find 𝑌14.000154 and the critical value 𝜒. 7 14.00924. Comparing the critical value and the statistic test 𝑌, we can say that ArmA head and neck cancer data can be adjusted by the XgRR model. 7. Concluding remarks In this article, a new reciprocal Rayleigh extension called the Xgamma reciprocal Rayleigh model is defined and studied. Relevant statistical properties such as raw moments, incomplete moments and moment generating function are derived. After a quick study for their properties, different non-Bayesian estimation methods under uncensored schemes are considered and described such as the maximum likelihood estimation method, ordinary least square estimation method, weighted least square estimation method, Cramér–von-Mises estimation method and Bootstrapping method. The performances of the proposed estimation methods are investigated through a simulation study. Many bivariate and multivariate type models have been also derived based on Farlie Gumbel Morgenstern copula, Clayton copula, Renyi’s entropy copula and Ali–Mikhail–Haq copula. A modified right-censored test for validation is applied to a right-censored real data set.
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