key: cord-0078481-gg8el8rd
authors: Tanış, Caner
title: Transmuted lower record type inverse rayleigh distribution: estimation, characterizations and applications
date: 2022-05-23
journal: Ricerche mat
DOI: 10.1007/s11587-022-00700-w
sha: 1f607111f6c1050eb9005e7f6254629b9a448abd
doc_id: 78481
cord_uid: gg8el8rd

This study introduces a new lifetime distribution called the transmuted lower record type inverse Rayleigh which extends the inverse Rayleigh distribution and has the potential to model the recovery times of Covid-19 patients.The new distribution is obtained using the distributions of the first two lower record statistics of the inverse Rayleigh distribution. We discuss some statistical inferences and mathematical properties of the suggested distribution. We examine some characteristics of the proposed distribution such as density shape, hazard function,moments, moment generating function, incomplete moments,Rényi entropy, order statistics, stochastic ordering. We consider five estimation methods such as maximum likelihood, least squares, weighted least squares, Anderson-Darling, Cramér-von Mises for the point estimation of the proposed distribution. Then, a comprehensive Monte Carlo simulation study is carried out to assess the risk behavior of the examined estimators. We provide two real data applications to illustrate the fitting ability of the proposed model, and compare its fit with competitor ones. Unlike many previously proposed distributions, the introduced distribution in this paper has modeled the recovery times of Covid-19 patients.

In recent years, many lifetime distributions have been proposed in the literature. The lifetime distributions are useful in various fields such as agriculture, actuarial, biology, engineering, and medical sciences. These distributions have emerged via different methods. One of these methods is the quadratic rank transmutation map (QTRM), B Caner Tanış canertanis@karatekin.edu.tr 1 Department of Statistics, Science Faculty, Çankırı Karatekin University, Çankırı, Turkey C. Tanış   Table 1 Some transmuted distributions in the literature

Author(s) Weibull [2] Lindley [3] Rayleigh [4] Inverse Rayleigh [5] Exponentiated Inverse Rayleigh [6] Generalized Modified Weibull [7] Complementary Exponential Power [8] Exponential Power [9] which is based on order statistics. The QRTM was proposed by [1] to generate a new distribution using the distributions of the first two order statistics. QRTM is summarized as follows: Let X 1:n , X 2:n be the order statistics from a population with cumulative distribution function (cdf) G (.) and probability density function (pdf) g (.) and n denotes sample sizes.

Let us define random variable Y by

where π ∈ (0, 1). The, the cdf of Y is given by F Y (x) = π P (X 1:2 ≤ x) + (1 − π ) P (X 2:2 ≤ x)

Substituting π = 1+λ 2 in (1) the cdf and corresponding pdf are obtained by

and

respectively, where λ ∈ [−1, 1]. The generated distributions by using QRTM are called transmuted distributions. In last decade, many transmuted distributions are suggested in the literature, and some transmuted distributions are given in Table 1 .

In addition to Table 1 , [10] introduced a generalized family of transmuted distribution. [11] proposed a new methodcalled cubic rank map (CRTM) with the motivation of the QRTM. The CRTM is summarized as follows: Let X 1:n , X 2:n and X 3:n be the order statistics from a population with cdf G (x) and pdf g (x).

Let us define a random variable Y by Y d = X 1:3 , with probability π 1 Y d = X 2:3 , with probability π 2 Y d = X 3:3 , with probability π 3 ,

where π 1 + π 2 + π 3 = 1. Thus, the cdf of Y is given by F Y (x) = π 1 P (X 1:3 ≤ x) + π 2 P (X 2:3 ≤ x) + π 3 P (X 3:3 ≤ x)

Substituting 3π 1 = λ 1 , 3π 2 = λ 2 in (4), the cdf and pdf of cubic rank transmuted distribution are given by

and

respectively, where λ 1 ∈ [0, 1] , λ 2 ∈ [−1, 1]. The generated distributions via the CRTM are called cubic rank transmuted distributions. In the literature, some cubic rank transmuted distributions are listed in Table 2 . [17] provided a generalization of cubic rank transmuted distributions. [18] suggested a new method based on the distributions of first two upper records called transmuted upper record type map (TRTM) to produce distributions. They described the TRTM in [18] as follows: Log-logistic [11] Kumaraswamy [12] Inverse Weibull [13] Modified Burr III [14] Modified Burr III Pareto [15] Lindley [16] Let X U (1) and X U (2) be upper records from a population with cdf G (.) and pdf g (.).

Let us describe a random variable Y by

where π 1 + π 2 = 1. In this case, the cdf of Y is given by

where π 2 = p, π 1 = 1 − p, p ∈ (0, 1). Then, the corresponding pdf of Y is

[19] discussed some mathematical properties and estimation methods of a special case based on Weibull distribution of the family of record-based transmuted distributions. Balakrishnan and He [18] also provided the record-based family of distributions which is a mixture based on the distributions of the first two lower record values. This family of distributions is constructed by Let X L (1) and X L(2) be the lower record values from a population with the cdf G (x) .

Let us define a new random variable based on these records:

where U is standard uniform random variable and p ∈ (0, 1). The cdf of Y is obtained by

Balakrishnan and He [18] noticed that the distribution with cdf (9) is called dual record-transmuted distribution. In this paper, we call this family of distributions as the family of transmuted lower record type (TLRT) distributions. The corresponding pdf and hazard function(hf) of TLRT distribution are given by

and

respectively. [20] proposed the first sub-model of the family of TLRT distributions called transmuted lower record type Fréchet distribution. Transmuted lower record type method (TLRTM) defined in (14) and (15) allows to be proposed a new distribution via the mixture of the distributions of the lower record values. For instance, let consider an Olympic athlete who broke more than one record in the Olympics and assume that this athlete achieve first and second records with certain probabilities. The new mixed distribution produced with TLRTM can be associated with this situation in real life. The motivation of this paper is to generate a new special case based on inverse Rayleigh by using TLRTM. We aim to provide a new flexible version of inverse Rayleigh distribution for modelling the data in medical sciences and other fields. Unlike previously proposed distributions, an important advantage of the suggested distribution is that it models the recovery times of COVID-19 patients. The study is organized as follows: In Sect. 2, some distributional properties of the suggested distribution are examined such as density shapes, moments, incomplete moments, moment generating function, Rényi entropy, stochastic ordering, and order statistics. In Sect. 3, we obtain five estimators of the parameters of the introduced distribution such as maximum likelihood estimators (MLEs), least squares estimators (LSEs), weighted least squares estimators (WLSEs), Anderson-Darling estimators (ADEs), and Cramér-von Mises estimators (CvMEs). A comprehensive simulation study is considered to compare the performances of the examined estimators according to mean squared errors (MSEs) and biases in Sect. 4. Section 5 presents two real data applications regarding the recovery times of COVID-19 patients to illustrate the applicability of the introduced distribution, and we show that it has the potential for modelling datasets in medical sciences. Finally, conclusions are given in Sect. 6.

In this section, we introduce a new lifetime distribution called transmuted lower record type inverse Rayleigh (TLRTIR) by using the TLRTM. Let X be a random variable from inverse Rayleigh distribution. The cdf and pdf of X are given as follows:

and

respectively, where, α > 0 and x > 0. Substituting the cdf (12) and pdf (13) into (9) and (10) , then the cdf and pdf of TRTLIR distribution are

and

respectively, where α > 0, p ∈ (0, 1) and x > 0. The distribution with cdf (14) is called "transmuted lower record type inverse Rayleigh (TLRTIR(α, p))"distribution.

In this section, we examine some possible density shapes of TLRTIR (α, p) distribution.

Theorem 2.1 Let X be a random variable from TLRTIR (α, p) distribution. The pdf of TLRTIR (α, p) distribution is unimodal for p > 12 17 .

and

where, 12 17 , and the density of TLRTIR (α, p) distribution has log-concavity property. The proof is completed.

Thus, it can be concluded that the density of the TLRTIR (α, p) distribution is unimodal for p > 12 17 according to Theorem 1. The hf of the TLRTIR(α, p) distribution is given by From Figure 2 , it is seen that the TLRTIR (α, p) distribution has upside bathtub shaped hf for selected parameters. 

The r th raw moment of TLRTIR(α, p) distribution for r ∈ N + is given by

where (·) is a gamma function.

The moment generating function of TLRTIR (α, p) distribution is given by

The incomplete moments of TLRTIR distribution is given by

where (a, x) is incomplete gamma function defined by

The Rényi entropy is proposed by Rényi [21] . The Rényi entropy is a measure of uncertainty, and it is defined by

where ρ > 0 and ρ = 1.

Proof The Rényi entropy of TLRTIR(α, p) distribution can be written using (22) as follows:

Using u = ρα x 2 transformation in (24), the integral is obtained by

Then, using power series expansion of [22] , Eq. (25) is obtained as follows:

By substituting (26) into (24), Thus, the Rényi entropy in (23) and the proof is completed.

Let X 1:n , X 2:n , ..., X n:n be the order statistics from a population with cdf (14) and pdf (15) . The pdf of j th order statistics, X j:n , j = 1, ..., n is given by

C. Tanış

Stochastic and the other ordering are important means for evaluating the comparative properties for a positive continuous random variable [20] . The following theorem shows that the TLRTIR random variables can be ordered with respect to the likelihood ratio.

. If p 1 > p 2 then X is smaller than Y in the likelihood ratio order, i.e., the ratio function of the corresponding pdfs is decreasing in x.

Proof For any x > 0, the ratio of the densities is given by

where,

It is seen that d log(g(x)) dx < 0 for p 1 > p 2 and hence proof is completed. [23] that X is also smaller than Y in the hazard ratio, and stochastic orders under the conditions given in Theorem 2.3.

In order to generate the data from TLRTIR (α, p 1 ) distribution, an acceptance-rejection (AR) sampling method is given in the following algorithm. In this algorithm, the Weibull distribution is chosen as a proposal distribution. The AR algorithm is given as follows: Algorithm 1. A1. Generate data on random variable Y from Weibull distribution with pdf g given as follow:

then set X = Y ("accept"); otherwise go back to A1 ("reject"), where pdf f is given as in (15) and

The output of this algorithm suggests a random data on X from TLRTIR(α, p). It is noticed that the Algorithm 1 is used for all simulations in the paper.

In this section, we examine five estimators for point estimation of the TLRTIR (α, p) distributions such as MLEs, LSEs, WLSEs, ADEs, and CVMEs.

Let X 1 , X 2 , ..., X n be a random sample from TLRTIR(α, p) distribution. The loglikelihood function is given by;

where x = (x 1 , x 2 , ..., x n ). The MLEs,α M L E andp M L E , of α and p are obtained by simultaneously solving the following log-likelihood equations.

The log-likelihood equations (29)-(30) can be solved using numerical methods such as Nelder-Mead,Broyden-Fletcher-Goldfarb-Shanno (BFGS). This algoritm is firstly studied by Fletcher [24] . These methods can be easily employed by optim function in R. The following results regarding to singularity ofα M L E of the parameter α. Theorem 3.1 Let us assume that the parameter p is known. There exists a unique MLE of the parameter α for p ∈ (0, 1) .

Proof We obtained the second derivative of the log-likelihood function (3.1) according to the parameter α as follows:

∂α is a decreasing function. Further, it is clearly seen that lim α→0

the proof is completed.

Let X 1 , X 2 , ..., X n be a random sample from TLRTIR(α, p) distribution, and x 1:n < x 2:n < · · · < x 2:n denote the corresponding observed order statistics. The LSEs , α L SE andp L SE , of α and p are obtained by minimizing

with respect to α and p parameters. The weighted least squares estimators (WLSEs), α W L SE andp W L SE , of α and p are derived by minimizing following equation with respect to α and p parameters.

where cdf F(.) is given as in (14) 3

The ADEs,α ADE andp ADE of α and p are obtained by minimizing following equation with respect to α and p parameters.

The CMEsα C M E andp C M E of α and p are derived by minimizing following equation with respect to α and p parameters.

In this subsection, we perform a Monte Carlo simulation study. In the simulation study, MSEs and biases of five estimators are calculated with 5000 repetitions for the sample sizes such as n=50, 100, 250, 500, 750 and 1000. Five parameter settings are considered as follows:

. In all simulations, the samples are generated from TLRTIR (α, p) distribution by using AR sampling given in Algorithm 1. The biases and MSEs of the estimators are given in Tables 3, 7.  According to Tables 3, 4 , 5, 6 and 7, it is observed that as the sample sizes increase, the biases and MSEs of the all estimators decrease and approach to zero. For parameter p, the performances of MLEs are generally better than the other estimators in terms of MSEs and biases. On the other hand, it is seen that the best estimators is ADE in small sample sizes while MLE is the best estimator in large sample sizes according to MSE and bias for point estimation of parameter α. As a result of the simulation study, we recommend the maximum likelihood method and Anderson-Darling method for point estimation of parameters of TLRTIR (α, p) distribution.

In this section, we perform two real data applications to illustrate the usefulness of TLRTIR (α, p) distribution in modelling real-life data. In real data analysis, two datasets are fitted to TLRTIR(α, p), transmuted inverse Rayleigh (TIR (α, λ) ) [5] , inverse Rayleigh (IR (α)), transmuted Rayleigh (TR (α, λ) ) [4] , transmuted Weibull (TW (α, β, λ) ) [2] , Weibull (α, β), Rayleigh (α), transmuted record type Weibull (TRTW (α, β, p) ) [18] distributions. The pdfs of fitted distributions are listed in Table 8 .

In order to compare the fitted distribution, some selection statistics such as Akaike's information criterion (AIC), Bayesian information criterion (BIC), Anderson-Darling (A*), Cramer von Mises (W*), Kolmogorov-Smirnov (K-S) test statistics and its pvalue are used. Table 8 The list of the lifetime distribution to modelling COVID-19 patient data 

f T I R (x) = 2α x 3 exp − α x 2 1 + λ − 2λ exp − α x 2 , α > 0, λ ∈ [−1, 1] f T R (x) = x α 2 exp − x 2 2α 2 1 − λ + 2λ exp − x 2 2α 2 , α > 0, λ ∈ [−1, 1] f T W (x) = α β x β α−1 exp − x β α 1 − λ + 2λ exp − x β α , α, β > 0, λ ∈ [−1, 1] f W eibull (x) = βαx α−1 exp −βx α , α, β > 0 f Rayleigh (x) = x α 2 exp − x 2 2α 2 , α > 0 f T RT W (x) = βαx α−1 exp −βx α 1 + p βx α − 1 , α, β > 0, p ∈ (0, 1)

The first dataset consists of the time (in days) from the first positive to the first negative COVID-19 PCR test for 50 Israelis (more than 60 years and male).The recovery periods (days) were calculated from an anonymized dataset of recovered COVID-19 patients released to the public by the Israel Ministry of Health on November 25, 2020 [24] . The first dataset is as follows: 16, 16, 16, 14, 36, 9, 10, 11, 8, 9, 12, 10, 22, 5, 11, 17, 20, 12, 29, 12, 15, 25, 25, 24, 18, 13, 44, 14, 20, 19, 11, 10, 18, 21, 31, 9, 29, 12, 10, 10, 13, 12, 19, 33, 37, 16, 63, 9, 28, 16 The MLEs and standard errors (in parenthesis) of the parameters of the fitted distributions are presented in Table 9 and the comparison statistics are given for dataset 1 in Table 10 . Also, Figure 3 provides the fitted cdfs and fitted pdfs for dataset 1. As a result of the analysis of dataset 1, it is observed that the best fitted model is TLRTIR (α, p) according to A*, W*, K-S and its p-value. By assuming the recovery times distribute TLRTIR (α = 249.5468, p = 0.4844) distribution, we estimate the probabilities of recovery times of COVID-19 patients more than 60 years old and male. In this regard, estimated probabilities according to recovery times are given in Table 11 .

From Table 11 , the probability of recovery of a COVID-19 patient (more than 60 years old and male) within the first two weeks after contracting the virus is approximately 45%. The probability of recovery in patients having to the same characteristics within the first three weeks is approximately 72%. It can be concluded that in the first 8 weeks after infection, about 95% of the patients recovered. The expected value of TIRTIR (α = 249.5468, p = 0.4844) distribution is approximately 21. This means the recovery time of the COVID-19 patient (more than 60 years and male) is about 21 days. 

The second dataset is obtained by [25] . For the second dataset, the MLEs and standard errors (in parenthesis) of the parameters of the fitted distributions are given in Table 12 , and the selection criteria statistics are presented in Table 13 . Also, Figure 4 illustrates the fitted cdfs and fitted pdfs for dataset 2. For two datasets, some plots such as Kernel densities, Probability-Probability (P-P) and Quantile-Quantile (Q-Q) plots are respectively given in Figs. 5-7. From Figures 5-7 , we can easily see that there are good fits between TLRTIR (α, p) distribution and datasets 1-2. Also, it is observed that the TLRTIR (α, p) distribution Table 10 and Table 13 .

In this study, we suggest a new lifetime distribution which is useful in modelling medical science data. We obtain five estimators such as MLE, LSE, WLSE, ADE, and CvME of the unknown parameters of the TLRTIR (α, p) distribution. A comprehensive Monte Carlo simulation study is considered to assess the performances of these estimators via biases and MSEs. As a result of the simulation study, we observe that as the sample sizes increase the MSEs and biases of all estimators decrease for all parameter values as expected. We recommend the maximum likelihood and Anderson-Darling method for point estimation of α and p. Two real data applications are performed to show the usefulness of TLRTIR (α, p) distribution. In real data illustration, we provide the data sets regarding the recovery time (in days) of COVID-19 patients. The results of real data applications show that the best fitted model is TLRTIR (α, p) according to K-S, its p-value, A*, and W* for data set 1 while the best fitted model is TLRTIR (α, p) according to all selection criteria for data set 2.

In previous studies, Barman et al. [26] emphasized that the probability of recovery period of a COVID-19 patient within 20 days is about 43%. Sutiningsih et al. [27] reported that the mean recovery time in COVID-19 patients is 20.63 days. Moreover, Voinsky et al. [28] found that the average recovery time in COVID-19 patients is approximately 15 days. Our results support previous studies. We observe that the estimated mean recovery time is found about 21 days for dataset 1. On the other hand, we estimate that the probability of recovery period of a COVID-19 patient within two weeks is about 45%.

In conclusion, we provide a new perspective on the interpretation and statistical evaluation of clinical data on COVID-19 patients. We also have shown that a new lifetime distribution is not only included in statistical theory and that they have the potential to be used in interpreting the COVID-19 data. To our knowledge, this is the first study providing the estimated probabilities of the recovery time in COVID-19 patients using a lifetime distribution. One of the advantages of this study is that the calculated estimates in the real data analysis section are similar to the previous studies. In future times, extensive research should be conducted on the recovery periods of COVID-19 patients.

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The author is thankful to the Israel Ministry of Health for providing public access to anonymized Covid-19 patient records and to Prof. Dr. David Gurwitz for his help in obtaining data of Covid-19 patients. The author sincere thanks to two anonymous reviewers for their helpful comments.