Estimation of three-parameter Fréchet distribution for the number of days from drug administration to remission in small sample sizes

Abstract

In medical research, it is common to estimate parameters for each group and then evaluate the estimated parameters for each group without comparing the groups. However, researchers frequently want to determine whether the two distributions using the estimated parameters differ significantly between the two groups. For the Weibull distribution, the two-sample Kolmogorov-Smirnov test (two-sided) was used to examine whether the two distributions were significantly different between the two groups. Based on this, we developed a method to compare the two groups using a three-parameter Fréchet distribution. The number of days from drug administration to remission frequently followed a Fréchet distribution. It is appropriate to use a three-parameter Fréchet distribution with a location parameter because patients typically go into remission after several days of drug administration. We propose a minimum variance linear estimator with a hyperparameter (MVLE-H) method for estimating a three-parameter Fréchet distribution based on the MVLE-H method for estimating a three-parameter Weibull distribution. We verified the effectiveness of the MVLE-H method and the two-sample Kolmogorov-Smirnov test (two-sided) on the three-parameter Fréchet distribution using Monte Carlo simulations and numerical examples.

Keywords

Parameter estimation remission small sample sizes three-parameter Fréchet distribution two-sample Kolmogorov-Smirnov test

1. Introduction

The Fréchet distribution (Fréchet, 1927) is known to fit data such as life tests (Malathi & Muthulakshmi, 2017), survival time (Abbas et al., 2019; Dey et al., 2019), and the number of days from drug administration to remission (Mathew, 2020). Many researchers have studied parameter estimation methods for two-parameter Fréchet distributions without a location parameter (Abd-Elfattah & Omima, 2009; Nasir & Aslam, 2015; Ramos et al., 2020). This study focused on the number of days from drug administration to remission. Because patients typically enter remission several days after drug administration, the location parameter is $>0$ , implying that a three-parameter Fréchet distribution is appropriate. Medical data may be limited by small sample sizes, such as in early-phase clinical trials. A three-parameter Fréchet distribution may not be estimated using the well-known maximum likelihood estimation (MLE) in small sample sizes. Therefore, estimation methods for the three-parameter Fréchet distribution have been studied by various scholars. Wu et al. (2020) estimated a three-parameter Fréchet distribution using the probability weighted moment (PWM) proposed by Greenwood et al. (1979). They validated their parameter estimation method using two numerical examples: lamp lifetime and time to ball-bearing failure. Although data of both the examples had only positive, location parameters were estimated as negative. Abbas et al. (2019) proposed four Bayesian estimators for a three-parameter Fréchet distribution with a non-informative or gamma prior under linear exponential and general entropy (GE) loss functions. They compared four Bayesian estimators, and the best estimator was the GE loss function with a non-informative prior (BGENP). They validated their parameter estimation method using a numerical example of a group of patients with from head and neck cancer’s survival time. Although data of the example had only positive, whereas the location parameter was estimated to have a negative value. If a location parameter is negative, there is a probability of obtained negative value.

Existing estimation methods have the disadvantage of estimating the location parameter as negative even though the data did not assume negative. In a three-parameter Weibull distribution, the shape parameter estimated using the minimum-variance linear estimator with a hyperparameter (MVLE-H) method was observed to be close to the population shape parameter (Ogura et al., 2020; Sugiyama & Ogura, 2022). The MVLE-H method was extended to estimate a three-parameter Fréchet distribution. Because the MVLE-H method is constructed with a hyperparameter, we used Monte Carlo simulations (MCSs) to determine the optimal hyperparameter. Additionally, we derived an estimation method for the scale and location parameters using the estimated shape parameter estimated.

In a Weibull distribution, Ogura & Shiraishi (2023) showed the method to examine whether two Weibull distributions estimated from two groups are different using the two-sample Kolmogorov-Smirnov (K-S) test (two-tailed) (Kolmogorov, 1933; Smirnov, 1939). Based on their method, we derive the two-sample K-S test (two-sided) to examine whether the two three-parameter Fréchet distributions estimated from the two groups differ. Statistical hypothesis testing is commonly performed using data obtained from patients; however, this study compared the two estimated distributions. Before using numerical examples, we validated the test using MCSs in various settings.

In Section 2, we propose the MVLE-H method for estimating a three-parameter Fréchet distribution and use MCSs to determine the optimal hyperparameter. In Section 3, we review the three existing estimation methods (MLE, PWM, and BGENP) and compare them with the MVLE-H method. In Section 4, we derive a method to examine whether the two Fréchet distributions estimated from the two groups differ using the two-sample K-S test (two-sided). In Section 5, the validity of the MVLE-H method and the two-sample K-S test (two-sided) is verified through a comparison of the three existing estimation methods using MCSs. In Section 6, we present numerical examples to demonstrate an attempt to estimate a three-parameter Fréchet distribution and perform a two-sample K-S test (two-sided). Finally, the conclusions of this study are presented in Section 7.

2. MVLE-H method

2.1 Shape parameter

The MVLE-H method for the three-parameter Fréchet distribution was developed based on a procedure for deriving the MVLE-H method for the three-parameter Weibull distribution (Ogura et al., 2020). A three-parameter Fréchet distribution consists of the shape ( $m>0$ ), scale ( $\eta>0$ ), and location ( $\gamma<x$ ) parameters, where $x$ is a variable in the probability density function of the three-parameter Fréchet distribution, expressed as follows:

$\displaystyle f(x;m,\eta,\gamma)=\frac{m}{\eta}\left(\frac{x-\gamma}{\eta}% \right)^{-1-m}\exp\left[-\left(\frac{x-\gamma}{\eta}\right)^{-m}\right].$ (1)

The cumulative distribution function is expressed as follows:

$\displaystyle F(x;m,\eta,\gamma)=\exp\left[-\left(\frac{x-\gamma}{\eta}\right)% ^{-m}\right].$ (2)

Let random samples from $f(x;m,\eta,\gamma)$ be denoted as $X_{1},\ldots,X_{n}$ , where $n\geqslant 2$ is sample size. The random samples are sorted as $\gamma<X_{(1)}\leqslant\cdots\leqslant X_{(n)}$ , where $X_{(i)}$ denotes the $i$ -th order statistic ( $i=1,\ldots,n$ ). Using the inverse transform method, order statistic of random sample and uniform random sample are expressed as follows:

$\displaystyle\exp\left[-\left(\frac{X_{(i)}-\gamma}{\eta}\right)^{-m}\right]=U% _{(i)},∼{}i=1,\ldots,n,$ (3)

where $U_{(i)}$ denotes the $i$ -th order statistic of uniform random sample. By performing the logarithmic transformation twice on both sides, Eq. (3) is transformed as follows:

$\displaystyle m\log\left(\frac{X_{(i)}-\gamma}{\eta}\right)=-\log[-\log(U_{(i)% })],∼{}i=1,\ldots,n.$ (4)

To express this without using $\eta$ , Eq. (4) is divided into $i=1$ and $i=2,\ldots,n$ . Subtracting Eq. (4) with $i=1$ from Eq. (4) with $i=2,\ldots,n$ provides the following equation:

$\displaystyle m\log\left(\frac{X_{(i)}-\gamma}{X_{(1)}-\gamma}\right)=-\log% \left[\frac{-\log(U_{(i)})}{-\log(U_{(1)})}\right],∼{}i=2,\ldots,n.$ (5)

Subsequently, $X_{(1)}$ is treated as a constant. The expected value is calculated as follows:

$\displaystyle\text{E}\left[\log\left(\frac{X_{(i)}-\gamma}{X_{(1)}-\gamma}% \right)\right]=-\frac{1}{m}\text{E}\left\{\log\left[\frac{-\log(U_{(i)})}{-% \log(U_{(1)})}\right]\right\}=-\frac{k(i,n)-k(1,n)}{m},$ (6)

where

$\displaystyle k(i,n)=\int_{0}^{1}\log\left[-\log\left(u\right)\right]\frac{n!}% {(i-1)!(n-i)!}u^{i-1}(1-u)^{n-i}\textit{du}.$ (7)

From the first and third terms in Eq. (6), $1/m$ can be expressed as follows:

$\displaystyle T_{(i)}=\frac{1}{m}=-\log\left(\frac{X_{(i)}-\gamma}{X_{(1)}-% \gamma}\right)\bigg{/}[k(i,n)-k(1,n)],∼{}i=2,\ldots,n,$ (8)

where $T_{(i)}$ is the unbiased estimator of $1/m$ for the $i$ -th order statistic. Because $T_{(i)}$ is expressed with $\gamma$ included, $m$ cannot be estimated from $T_{(i)}$ in Eq. (8). Based on the solution to the similar problem in the MVLE-H method for a three-parameter Weibull distribution, the following formula is introduced:

$\displaystyle\frac{1}{X_{(1)}-\gamma}=c\left(\frac{1}{U}-1\right),$ (9)

where $U$ and $c>0$ represent the standard uniform distribution and hyperparameter, respectively. Substituting (9) into Eq. (8), we obtain the following:

$\displaystyle T_{(i)}=-\frac{\log\left[c\left(1-U\right)\left(X_{(i)}-X_{(1)}% \right)+U\right]-\log(U)}{k(i,n)-k(1,n)}.$ (10)

Therefore, $T_{(i)}$ can be expressed without $\eta$ and $\gamma$ . This implies that it is possible to estimate $m$ first. The optimal $c$ is discussed in Section 2.4. Based on the minimum-variance linear unbiased estimator, $m$ is estimated as follows:

$\displaystyle\hat{m}={\bm{1}}^{\prime}{\bm{S}}^{-1}{\bm{1}}\bigg{/}{\bm{1}}^{% \prime}{\bm{S}}\left(\begin{array}[]{c}\int_{0}^{1}T_{(2)}dU\\ \vdots\\ \int_{0}^{1}T_{(n)}dU\\ \end{array}\right),$ (11)

where $\text{\boldmath$1$}^{\prime}=(1∼{}\cdots∼{}1)$ is ( $n-1$ )-dimensional vector,

$\displaystyle{\bm{S}}=\left(S_{(ij)}\right)=\text{cov}\left(T_{(i)},T_{(j)}% \right)=\frac{1}{k(i,n)-k(1,n)}\frac{1}{k(j,n)-k(1,n)}[\text{cov}\left(\log(X_% {(i)}-\gamma),\log(X_{(j)}-\gamma)\right)-\text{cov}\left(\log(X_{(i)}-\gamma)% ,\log(X_{(1)}-\gamma)\right)-\text{cov}\left(\log(X_{(j)}-\gamma),\log(X_{(1)}% -\gamma)\right)+\text{var}\left(\log(X_{(1)}-\gamma)\right)],∼{}∼{}i,j=2,% \ldots,n,\text{cov}\left(\log(X_{(i)}-\gamma),\log(X_{(j)}-\gamma)\right)=% \text{E}\left\{\log(X_{(i)}-\gamma)-\text{E}\left[\log(X_{(i)}-\gamma)\right]% \right\}\left\{\log(X_{(j)}-\gamma)-\text{E}\left[\log(X_{(j)}-\gamma)\right]% \right\}=\int_{u}^{1}\int_{0}^{1}\left\{\frac{-\log\left[-\log(u)\right]}{m}-% \left[-\frac{k(i,n)}{m}\right]\right\}\left\{\frac{-\log\left[-\log(v)\right]}% {m}-\left[-\frac{k(j,n)}{m}\right]\right\}\times\frac{n!u^{i-1}(v-u)^{j-i-1}(1% -v)^{n-j}}{(i-1)!(j-i-1)!(n-j)!}\textit{dudv},\text{var}\left(\log(X_{(i)}-% \gamma)\right)=\text{E}\left[\log(X_{(i)}-\gamma)\right]^{2}$ (13) $\displaystyle=\int_{0}^{1}\left\{\frac{-\log\left[-\log(u)\right]}{m}-\left[-% \frac{k(i,n)}{m}\right]\right\}^{2}\frac{n!u^{i-1}(1-u)^{n-i}}{(i-1)!(n-i)!}% \textit{du}.$ (14)

Although Eqs (13) and (14) may appear to be dependent on $m$ , it is canceled by substituting into Eq. (11).

2.2 Location parameter

After the $\hat{m}$ in Eq. (11) is estimated, $\gamma$ is estimated. The estimated $\hat{m}$ is treated as known. Substituting the $m=\hat{m}$ into Eq. (8), we obtain the following:

$\displaystyle\frac{1}{\hat{m}}=-\log\left(\frac{X_{(i)}-\gamma}{X_{(1)}-\gamma% }\right)\bigg{/}[k(i,n)-k(1,n)].$ (15)

The location parameter is estimated as follows:

$\displaystyle\hat{\gamma}=\text{E}\left[\hat{\gamma}_{(i)}\right]=\text{E}% \left[\frac{X_{(i)}-X_{(1)}\exp\left[-\frac{k(i,n)-k(1,n)}{\hat{m}}\right]}{1-% \exp\left[-\frac{k(i,n)-k(1,n)}{\hat{m}}\right]}\right].$ (16)

2.3 Scale parameter

After the $\hat{m}$ and $\hat{\gamma}$ are estimated, $\eta$ is estimated. The estimated $\hat{m}$ and $\hat{\gamma}$ are treated as known. The scale parameter is estimated using the MLE in Section 3.1. Substituting the $m=\hat{m}$ and $\gamma=\hat{\gamma}$ into Eq. (23), we obtain the following:

$\displaystyle L_{SS}=\left[\frac{n}{\hat{m}}+n\log\eta-\sum_{i=1}^{n}\log(X_{i% }-\hat{\gamma})+\sum_{i=1}^{n}\log\frac{X_{i}-\hat{\gamma}}{\eta}\left(\frac{X% _{i}-\hat{\gamma}}{\eta}\right)^{-\hat{m}}\right]^{2}+\Bigg{[}\frac{n\hat{m}}{% \eta}-\frac{\hat{m}}{\eta}\sum_{i=1}^{n}\left(\frac{X_{i}-\hat{\gamma}}{\eta}% \right)^{-\hat{m}}\Bigg{]}^{2}+\left[(\hat{m}+1)\sum_{i=1}^{n}\frac{1}{X_{i}-% \hat{\gamma}}-\frac{\hat{m}}{\eta}\sum_{i=1}^{n}\left(\frac{X_{i}-\hat{\gamma}% }{\eta}\right)^{-\hat{m}-1}\right]^{2}.$ (17)

The scale parameter estimated using the MVLE-H method is defined as $\hat{\eta}$ , which minimizes $L_{SS}$ in Eq. (17).

2.4 Optimal

c

in the MVLE-H method

The optimal $c$ in Eq. (10) was determined using MCSs. Based on the MCS settings of Abbas et al. (2019), the population three parameters were set to 16 patterns with combinations of $(\eta,\gamma)=$ (1.0,4.0), (1.5,3.0), (2.0,3.0), and (2.0,4.0) and $m=$ 0.5, 1.0, 1.5, and 2.0. As the target of this study was small sample sizes, the sample size was set to $n=$ 5, 10, 15, and 20. The replication size was 10,000. We used the software R version 4.1.3 (R Core Team, 2022). This MCS settings were also used for MCS settings in Sections 3.4, 5.1, and 5.2.

The $c$ value, which minimized the bias $(=\text{estimation}-\text{population})$ and root-mean-square error ( $\text{RMSE}=\sqrt{\text{variance}+\text{bias}^{2}}$ ), was considered optimal. It could be observed that the optimal $c$ varied greatly with regard to $n$ . It also varied with regard to $m$ , but was less affected by $m$ than $n$ . When data were obtained from patients, $n$ was known and $m$ was unknown. Therefore, it is practically reasonable to change the optimum $c$ based on $n$ . From the MCS results on the bias and RMSE, $c=$ 1.8, 2.2, 3.0, and 4.5 were recommended for $n=$ 5, 10, 15 and 20, respectively.

Sugiyama & Ogura (2022) used the MVLE-H method of the Weibull distribution to search for the optimal $c$ in five increments of $n$ , and complemented $c$ for the remaining sample sizes with linear regression. We discovered that the optimum $c$ can be determined using a quadratic equation of the $n$ instead of a linear regression line in the MVLE-H method of the Fréchet distribution. A quadratic equation was used to supplement the $c$ for the remaining sample sizes as follows: $\hat{c}=2.025-0.097n+0.011n^{2}(n=5-20)$ .

3. Existing estimation methods

The three existing estimation methods (MLE, PWM, and BGENP) were used for comparison with the MVLE-H method.

3.1 MLE

The likelihood function in Eq. (1) is expressed as follows:

$\displaystyle L=\prod_{i=1}^{n}\left\{\frac{m}{\eta}\left(\frac{X_{i}-\gamma}{% \eta}\right)^{-1-m}\exp\left[-\left(\frac{X_{i}-\gamma}{\eta}\right)^{-m}% \right]\right\}.$ (18)

To facilitate maximization, the likelihood function is performed using a logarithmic transformation, and is expressed as follows:

$\displaystyle l=\log L=n\log m+\textit{nm}\log\eta-(1+m)\sum_{i=1}^{n}\log(X_{% i}-\gamma)-\sum_{i=1}^{n}\left(\frac{X_{i}-\gamma}{\eta}\right)^{-m}.$ (19)

We take the partial differentiation of $l$ in Eq. (19) with respect to $m$ , $\eta$ , and $\gamma$ , and additionally set them equal to 0 as follows:

$\displaystyle l_{m}=\frac{n}{m}+n\log\eta-\sum_{i=1}^{n}\log(X_{i}-\gamma)+% \sum_{i=1}^{n}\log\frac{X_{i}-\gamma}{\eta}\left(\frac{X_{i}-\gamma}{\eta}% \right)^{-m}=0,$ (20) $\displaystyle l_{\eta}=\frac{\textit{nm}}{\eta}-\frac{m}{\eta}\sum_{i=1}^{n}% \left(\frac{X_{i}-\gamma}{\eta}\right)^{-m}=0,$ (21) $\displaystyle l_{\gamma}=(m+1)\sum_{i=1}^{n}\frac{1}{X_{i}-\gamma}-\frac{m}{% \eta}\sum_{i=1}^{n}\left(\frac{X_{i}-\gamma}{\eta}\right)^{-m-1}=0.$ (22)

Although MLE is obtained by solving three simultaneous equations in Eqs (20)–(22), it may not be possible to solve them when $n$ is small. To avoid an inability to estimate the three parameters, we define the sum of squares $L_{SS}$ of Eqs (20)–(22) as follows:

$\displaystyle L_{SS}=\left[\frac{n}{m}+n\log\eta-\sum_{i=1}^{n}\log(X_{i}-% \gamma)+\sum_{i=1}^{n}\log\frac{X_{i}-\gamma}{\eta}\left(\frac{X_{i}-\gamma}{% \eta}\right)^{-m}\right]^{2}+\Bigg{[}\frac{\textit{nm}}{\eta}-\frac{m}{\eta}% \sum_{i=1}^{n}\left(\frac{X_{i}-\gamma}{\eta}\right)^{-m}\Bigg{]}^{2}+\left[(m% +1)\sum_{i=1}^{n}\frac{1}{X_{i}-\gamma}-\frac{m}{\eta}\sum_{i=1}^{n}\left(% \frac{X_{i}-\gamma}{\eta}\right)^{-m-1}\right]^{2}.$ (23)

This study treats $\hat{m}_{\text{MLE}}$ , $\hat{\eta}_{\text{MLE}}$ , and $\hat{\gamma}_{\text{MLE}}$ , which minimizes $L_{SS}$ , as the three parameters estimated using the MLE.

3.2 PWM method

Wu et al. (2020) estimated the three-parameter Fréchet distribution using the PWM method proposed by Greenwood et al. (1979). The PWM has the form as follows:

$\displaystyle M_{\textit{prs}}=\int_{0}^{1}x^{p}G^{r}(1-G)^{s}\textit{dG},$ (24)

where $p$ , $r$ , and $s$ are real numbers. The inverse function in Eq. (2) is expressed as $x=\gamma+\eta(-\log G)^{-1/m}$ , where $G=F^{-1}(x;m,\eta,\gamma)$ . When $p=1$ and $s=0$ , $M_{\textit{prs}}$ is expressed as follows:

$\displaystyle M_{1r0}=\frac{\gamma+\eta(1+r)^{1/m}\Gamma(1-1/m)}{1+r}.$ (25)

Substituting $r=0$ , 1, and 2 into Eq. (25), the three equations can be expressed as follows:

$\displaystyle M_{100}=\gamma+\eta\Gamma(1-1/m),$ (26) $\displaystyle M_{110}=\frac{\gamma+2^{1/m}\eta\Gamma(1-1/m)}{2},$ (27) $\displaystyle M_{120}=\frac{\gamma+3^{1/m}\eta\Gamma(1-1/m)}{3}.$ (28)

Landwehr et al. (1979) showed the unbiased estimator of $M_{1r0}$ as follows:

$\displaystyle\hat{M}_{1r0}=\frac{1}{n}\sum_{i=1}^{n}X_{(i)}{j-1\choose r}\bigg% {/}{n-1\choose r}.$ (29)

Thereafter, $\hat{M}_{1r0}$ is substituted into the left side of Eqs (26)–(28) and the simultaneous equations are solved. Then, $\hat{\gamma}_{\text{PWM}}$ cannot be expressed simply, and is obtained by solving the following equation:

$\displaystyle\frac{1}{\log 2}\log\left(\frac{2\hat{M}_{110}-\hat{\gamma}_{% \text{PWM}}}{\hat{M}_{100}-\hat{\gamma}_{\text{PWM}}}\right)=\frac{1}{\log 3}% \log\left(\frac{3\hat{M}_{120}-\hat{\gamma}_{\text{PWM}}}{\hat{M}_{100}-\hat{% \gamma}_{\text{PWM}}}\right).$ (30)

By contrast, $\hat{m}_{\text{PWM}}$ and $\hat{\eta}_{\text{PWM}}$ are simply expressed as follows:

$\displaystyle\hat{m}_{\text{PWM}}=\frac{\log 2}{\log(2\hat{M}_{110}-\hat{% \gamma})-\log(\hat{M}_{100}-\hat{\gamma})},$ (31) $\displaystyle\hat{\eta}_{\text{PWM}}=\frac{\hat{M}_{100}-\hat{\gamma}}{\Gamma(% 1-1/\hat{m})}.$ (32)

3.3 BGENP method

According to Abbas et al. (2019), the three parameters estimated using the BGENP method are defined as follows:

$\displaystyle\hat{m}_{\text{BGENP}}=\bigg{\{}\hat{m}_{\text{MLE}}^{-k_{2}}% \bigg{[}1-\frac{k_{2}}{\hat{m}_{\text{MLE}}}\bigg{(}-\frac{1}{\hat{m}_{\text{% MLE}}}\sigma_{mm}-\frac{1}{\hat{\eta}_{\text{MLE}}}\sigma_{m\eta}-\frac{1}{% \hat{\gamma}_{\text{MLE}}}\sigma_{m\gamma}-\frac{k_{2}+1}{2\hat{m}_{\text{MLE}% }}\sigma_{mm}\bigg{)}+\frac{1}{2}(\Lambda_{1}\sigma_{mm}+\Lambda_{2}\sigma_{% \eta m}+\Lambda_{3}\sigma_{\gamma m})\bigg{]}\bigg{\}}^{-(1/k_{2})},$ (33) $\displaystyle\hat{\eta}_{\text{BGENP}}=\bigg{\{}\hat{\eta}_{\text{MLE}}^{-k_{2% }}\bigg{[}1-\frac{k_{2}}{\hat{\eta}_{\text{MLE}}}\bigg{(}-\frac{1}{\hat{m}_{% \text{MLE}}}\sigma_{\eta m}-\frac{1}{\hat{\eta}_{\text{MLE}}}\sigma_{\eta\eta}% -\frac{1}{\hat{\gamma}_{\text{MLE}}}\sigma_{\eta\gamma}-\frac{k_{2}+1}{2\hat{% \eta}_{\text{MLE}}}\sigma_{\eta\eta}\bigg{)}+\frac{1}{2}(\Lambda_{1}\sigma_{m% \eta}+\Lambda_{2}\sigma_{\eta\eta}+\Lambda_{3}\sigma_{\gamma\eta})\bigg{]}% \bigg{\}}^{-(1/k_{2})},$ (34) $\displaystyle\hat{\gamma}_{\text{BGENP}}=\bigg{\{}\hat{\gamma}_{\text{MLE}}^{-% k_{2}}\bigg{[}1-\frac{k_{2}}{\hat{\gamma}_{\text{MLE}}}\bigg{(}-\frac{1}{\hat{% m}_{\text{MLE}}}\sigma_{\gamma m}-\frac{1}{\hat{\eta}_{\text{MLE}}}\sigma_{% \gamma\eta}-\frac{1}{\hat{\gamma}_{\text{MLE}}}\sigma_{\gamma\gamma}-\frac{k_{% 2}+1}{2\hat{\gamma}_{\text{MLE}}}\sigma_{\gamma\gamma}\bigg{)}+\frac{1}{2}(% \Lambda_{1}\sigma_{m\gamma}+\Lambda_{2}\sigma_{\eta\gamma}+\Lambda_{3}\sigma_{% \gamma\gamma})\bigg{]}\bigg{\}}^{-(1/k_{2})},$ (35)

where $\hat{m}_{\text{MLE}}$ , $\hat{\eta}_{\text{MLE}}$ , and $\hat{\gamma}_{\text{MLE}}$ are estimated using the MLE, and the explanations of $\sigma_{mm}$ , $\sigma_{m\eta}$ , $\sigma_{m\gamma}$ , $\sigma_{\eta m}$ , $\sigma_{\eta\eta}$ , $\sigma_{\eta\gamma}$ , $\sigma_{\gamma m}$ , $\sigma_{\gamma\eta}$ , $\sigma_{\gamma\gamma}$ , $\Lambda_{1}$ , $\Lambda_{2}$ , and $\Lambda_{3}$ can be found in Appendix of Abbas et al. (2019). However, there was no explanation for $k_{2}$ in their paper. Using MCSs, we search for the optimal $k_{2}$ for each $n$ .

3.4 Optimal

k_{2}

in BGENP method

The optimal $k_{2}$ may vary depending on the sample size and population three parameters. Thus, the MCSs were used to search for the optimal $k_{2}$ value in various settings. The MCS settings were as shown in Section 2.4. We searched for the optimal $k_{2}$ in the range of 0.1 to 1.0 in 0.1 increments for each pattern and sample size. The three parameters estimated using the MLE were used to estimate the three parameters in the BGENP method. If the MLE-estimated three parameters are far from the population three parameters, then the BGENP-estimated three parameters would also be far from the population three parameters. Therefore, $\hat{m}_{\text{MLE}}$ , $\hat{\eta}_{\text{MLE}}$ , and $\hat{\gamma}_{\text{MLE}}$ in Eqs (33)–(35) used population three parameters in the search for the optimum $k_{2}$ .

When $k_{2}=$ 0.3, 0.4, 0.6, 0.7, 0.8, and 0.9, the three parameters could not often be estimated in small sample sizes because there was the $-k_{2}$ th root on the right-hand side in Eqs (33)–(35). When $k_{2}=$ 0.2 and 1.0, the shape and scale parameters were often estimated to be negative values. Therefore, the candidate for the optimal $k_{2}$ was 0.1 or 0.5. The RMSE of the three parameters estimated using the BGENP method with $k_{2}=0.5$ was smaller than that with $k_{2}=0.1$ . We used the BGENP method with $k_{2}=0.5$ for the small sample sizes.

4. Derivation of two-sample Kolmogorov-Smirnov test

A three-parameter Fréchet distribution was estimated for each group for the number of days from drug administration to remission. Subsequently, the two-sample K-S test (two-sided) was used to test whether the two distributions estimated from the two groups were different. In the first group, the cumulative distribution function of three-parameter Fréchet distribution estimated from random samples ( $X_{1}^{(1)},\ldots,X_{n_{1}}^{(1)}$ ) is expressed as $F(x;\hat{m}_{1},\hat{\eta}_{1},\hat{\gamma}_{1})$ , where $n_{1}$ denotes the sample size. Similarly, in the second group, the cumulative distribution function of three-parameter Fréchet distribution estimated from random samples ( $X_{1}^{(2)},\ldots,X_{n_{2}}^{(2)}$ ) is expressed as $F(x;\hat{m}_{2},\hat{\eta}_{2},\hat{\gamma}_{2})$ , where $n_{2}$ denotes the sample size. Corder & Foreman (2014, p.81) presented the calculation procedure from the test statistic to $P$ -value. We applied their procedure to this study. The absolute value divergence $D$ between two groups is expressed as follows:

$\displaystyle D(x)=|F(x;\hat{m}_{1},\hat{\eta}_{1},\hat{\gamma}_{1})-F(x;\hat{% m}_{2},\hat{\eta}_{2},\hat{\gamma}_{2})|.$ (36)

Using the largest divergence $D_{\max}=\max_{0<x<\infty}D(x)$ , the Kolmogorov-Smirnov test statistic $Z$ is defined as follows:

$\displaystyle Z=D_{\max}\sqrt{\frac{n_{1}n_{2}}{n_{1}+n_{2}}}.$ (37)

The $P$ -value (two-sided) is calculated as follows:

$\displaystyle P=\left\{\begin{array}[]{ll}>0.999&(0.00\leqslant Z<0.27)\\ 1-\frac{2.506628}{Z}\left(Q_{1}+Q_{1}^{9}+Q_{1}^{25}\right)&(0.27\leqslant Z<1% .00)\\ 2\left(Q_{2}-Q_{2}^{4}+Q_{2}^{9}-Q_{2}^{16}\right)&(1.00\leqslant Z<3.10)\\ <0.001&(3.10\leqslant Z)\end{array}\right.,$ (38)

where $Q_{1}=\exp\left(-1.233701Z^{-2}\right)$ and $Q_{2}=\exp\left(-2Z^{2}\right)$ . When the significance level was set at 0.05, the null hypothesis was rejected at $Z>1.3581$ .

5. Monte Carlo simulations

5.1 Parameter estimation

We compared the three-parameter Fréchet distributions estimated using four methods (MVLE-H, MLE, PWM, and BGENP). The MCS settings were as shown in Section 2.4 except for $m$ , which was set to $m=0.5$ , 1.0, and 2.0. The sample size was set to $n=5$ , 10, and 20. Tables 1–3 (Appendix) list the bias and RMSE for the three parameters for each pattern. Because the MLE relied on the initial value of the three parameters, we set the initial value to the population three parameters for the optimal estimation results. This was to improve the accuracy of the estimation results using the MLE. Because the BGENP method used the estimation results using the MLE, this led to the improvement of the accuracy of the estimation results using the BGENP method. In the MLE, PWM, and BGENP methods, the estimated three parameters were often far from the population three parameters. The bias and RMSE increased significantly when the parameter was estimated far away from the population parameter even once. To prevent this, we set the lower and upper limits of parameter estimation. When the estimated shape, scale, and location parameters were $<0.001$ , they were treated as estimated to be 0.001. When the estimated shape, scale, and location parameters were $>7$ , they were treated as estimated to be 7. For the MVLE-H method, no lower and upper limits on parameter estimation were set. The effectiveness of the MVLE-H method was clear if the RMSEs of the MVLE-H method were smaller than those of the existing methods under these conditions. MCSs were conducted using the following steps:

1.
Generate random samples $(x_{1},\ldots,x_{n})$ from $f(x;m,\eta,\gamma)$ .
2.
Estimate the three parameters using each method.
3.
Independently, repeat Steps 1–2 10,000 times.
4.
Calculate the bias and RMSE of the three parameters.

The RMSEs of the shape parameter estimated using the MVLE-H method were the least among the four methods for all settings. The RMSEs of the scale parameter estimated using the MVLE-H method were the least among the four methods when $m\geqslant 2.0$ . The RMSEs of the location parameter estimated using the MVLE-H method were the least among the four methods when $m\geqslant 1.0$ . The accuracy of estimated three parameters using the MVLE-H method was the highest, although conditions favored existing methods.
5.2 Two-sample Kolmogorov-Smirnov test

The validity of the two-sample K-S test (two-sided) was assessed using the MCSs. The MCS settings were as shown in Section 2.4. As the MLE, PWM, and BGENP methods are often unestimable, the MCSs of the two-sample K-S test (two-sided) were performed using the MVLE-H method. MCSs were performed based on the following steps:

1.
Generate two random samples $(x_{1},\ldots,x_{n_{1}})$ and $(y_{1},\ldots,x_{n_{2}})$ from $f(x;m,\eta,\gamma)$ .
2.
Estimate the three parameters from $(x_{1},\ldots,x_{n_{1}})$ and $(y_{1},\ldots,x_{n_{2}})$ , respectively.
3.
Calculate the K-S test statistic $Z$ in Eq. (37) using the three parameters estimated in Step 2.
4.
Determine whether $Z>1.3581$ is satisfied and count the number of times it is satisfied.
5.
Independently, repeat Steps 1–4 10,000 times.
6.
The count in Step 4 is divided by 10,000 and compared with a significance level of 0.05.

When the probability obtained in Step 6 was $\leqslant 0.05$ , the two-sample K-S test (two-sided) was considered to strictly adhere to the significance level. Table 4 (Appendix) lists the probability of satisfying $Z>1.3581$ . Strict adherence to the significance level 0.05 of the two-sample K-S test (two-tailed) was confirmed in all settings.
6. Numerical examples

We demonstrate the estimation of the three-parameter Fréchet distribution using three numerical examples. Because the number of days from drug administration to remission was not presented in published papers, it was read from the Kaplan-Meier curve. Therefore, the data we used may have been off by a few days from the actual data. From the read data, it is probable that the judgement of remission was made daily in Examples 1 and 2, and the once a week in Example 3. As remission was determined during regular hospital visits, data from once a week might be closer to the clinical setting. We used the software R to estimate the three parameters using the four methods. Because the number of days from drug administration to remission was positive, the lower limits of the shape, scale, and location parameters were set to 0.001 in the MLE, PWM, and BGENP methods.

6.1 Example 1

The first was the number of days to achieve remission after combination therapy with rituximab, low-dose cyclophosphamide, and prednisone for idiopathic membranous nephropathy (Cortazar et al., 2017). Regarding the position of rituximab for membranous nephritis, there has been a report of moderate to severe membranous nephritis with steroid resistance; however, evidence for single-agent and concomitant use of steroids and endoxan has not yet been established. Although rituximab has been used as a treatment for primary membranous nephropathy, rituximab monotherapy achieved complete remission in some patients but not in many others (Fervenza et al., 2010; Ruggenenti et al., 2012). Cortazar et al. (2017) examined the efficacy of combination therapy with rituximab, low-dose cyclophosphamide, and prednisone for idiopathic membranous nephropathy. The patients were divided into two groups (initial therapy and second-line therapy (Cattran et al., 2012)) and Kaplan-Meier curve was drawn. Reading the number of days from the Kaplan-Meier curve, the initial therapy group for 7 patients was 12, 13, 20, 31, 52, 58, and 198 (day), and the second-line therapy group for 8 patients was 11, 51, 66, 71, 128, 281, 303, and 312 (day).

6.2 Example 2

The second was the number of days to achieve remission with rituximab, low-dose cyclophosphamide, and prednisone for primary membranous nephropathy (Zonozi et al., 2021). The patients were divided into three groups (phospholipase A2 receptor (PLA2R), non-PLA2R-associated, and PLA2R-association unknown) and Kaplan-Meier curve was drawn. Reading the number of days from the Kaplan-Meier curve, the non-PLA2R-associated group for 7 patients was 16, 28, 36, 50, 159, 180, and 1080 (day), and the PLA2R-association unknown for 9 patients was 33, 56, 65, 70, 232, 276, 307, 335, and 393 (day). Although the PLA2R-associated group included 44 patients, they were excluded as the target of this study was $n\leqslant 20$ .

6.3 Example 3

The third was the number of days to remission with oral and intravenous cyclophosphamide (CPA) in children with steroid-resistant nephrotic syndrome (Hidayati et al., 2011). CPA is an alkylating agent widely used in steroid-dependent nephrotic syndrome, either orally or intravenously (Bircan & Kara, 2003). Intravenous CPA is an effective form of therapy with significantly fewer side effects than oral CPA (Sümegi et al., 2008). Hidayati et al. (2011) compared the number of days to remission between intravenous and oral CPA. The patients were divided into two groups (oral CPA and intravenous CPA) and Kaplan-Meier curve was drawn. Reading the number of days from the Kaplan-Meier curve, the oral CPA group for 14 patients was 14, 35, 49, 56, 70, 84, 98, 105, 112, 119, 126, 140, 154, and 217 (day), and the intravenous CPA group for 15 patients was 28, 28, 84, 112, 119, 119, 133, 133, 147, 147, 154, 203, 203, 238, and 252 (day).

6.4 Results

The estimated three-parameter Fréchet distributions using Examples 1–3 are listed in Table 5 (Appendix). Although the results of the three parameters estimated using the MLE depended on the initial values, the results were almost the same even when the initial values were changed.

If the lower limit of the location parameter was not set, negative values were estimated where the location parameter written 0.001 in Table 5. If the location parameter estimated using the MLE was negative, the location parameter of the BGENP method could not be estimated. This is because the $-0.5$ power of a negative value appeared in Eq. (35). The MLE, PWM, and BGENP methods did not work in many cases. Because the location parameter estimated using the MVLE-H method had positive in Examples 1–3, the MVLE-H method worked under all cases. We compared the methods using numerical Examples 1–3 in which the location parameters were estimated $\geqslant 0.002$ . In the initial therapy group in Example 1 and the non-PLA2R-associated group in Example 2, only one patient had a longer day than the others. The scale parameters estimated using the MVLE-H method were greater than those estimated using the BGENP method and MLE. This implies that the probability of the three-parameter Fréchet distribution estimated using the MVLE-H method declined slowly to accommodate the fact that the data contained 198 day for Example 1 and 1080 day for Example 2. By contrast, the probabilities of the three-parameter Fréchet distribution estimated using the BGENP method and MLE were very low at 198 day in Example 1 and 1080 day in Example 2.

The two-sample K-S test (two-sided) was used to examine whether there was a significant difference between the two Fréchet distributions estimated from the two groups in Examples 1–3. None of the examples rejected the null hypothesis at the significance level of 0.05.

7. Conclusions

We proposed the MVLE-H method to estimate the three-parameter Fréchet distribution in small sample sizes. The optimal hyperparameters in the MVLE-H method were searched for each sample size using the MCSs. In addition, we used the two-sample K-S test (two-sided) to examine whether the two Fréchet distributions estimated from the two groups were different. Using the MCSs, it was confirmed that the test strictly adhered to the significance level. The MCS results showed that the RMSEs of the three parameters estimated using the MVLE-H method were the smallest among the four methods under numerous settings. In numerical examples, it is often difficult to estimate the three parameters using the MLE, PWM, and BGENP methods. This is because the location parameters estimated using these three methods are often negative if a lower limit of the location parameter is not set. When the location parameter has negative, data with a negative value may be obtained. However, a negative value can not be obtained for the number of days from drug administration to remission. By contrast, the location parameter estimated using the MVLE-H method was positive, and the scale and location parameters estimated using the MVLE-H method were reasonable. Therefore, the MVLE-H method is recommended for small sample sizes.

Footnotes

Acknowledgments

The authors are thankful learned reviewers for their valuable comments towards the improvements of the manuscript.

Appendix

Table 1

Comparison of bias and RMSE of the three parameters estimated using the four methods ( $n=5$ )

				MVLE-H		MLE		PWM		BGENP
Parameter	$m$	$\eta$	$\gamma$	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
Shape	0.5	1.0	4.0	0.206	0.407	0.026	1.195	1.290	1.878	$-$ 0.446	0.594
		1.5	3.0	0.130	0.323	$-$ 0.045	0.944	1.075	1.469	$-$ 0.453	0.591
		2.0	3.0	0.079	0.262	$-$ 0.090	0.846	0.990	1.306	$-$ 0.447	0.624
		2.0	4.0	0.075	0.258	$-$ 0.045	0.950	1.061	1.443	$-$ 0.448	0.605
	1.0	1.0	4.0	0.057	0.387	0.521	2.260	2.126	3.022	$-$ 0.958	1.004
		1.5	3.0	$-$ 0.092	0.311	0.293	1.851	1.522	2.255	$-$ 0.962	1.000
		2.0	3.0	$-$ 0.175	0.312	0.206	1.689	1.359	2.020	$-$ 0.963	1.002
		2.0	4.0	$-$ 0.175	0.312	0.287	1.850	1.517	2.256	$-$ 0.961	1.005
	2.0	1.0	4.0	$-$ 0.453	0.656	0.952	3.069	3.221	3.908	$-$ 1.969	1.983
		1.5	3.0	$-$ 0.711	0.799	0.711	2.817	2.260	3.184	$-$ 1.969	1.985
		2.0	3.0	$-$ 0.848	0.903	0.627	2.683	1.978	2.921	$-$ 1.974	1.982
		2.0	4.0	$-$ 0.846	0.901	0.764	2.850	2.285	3.211	$-$ 1.970	1.985
Scale	0.5	1.0	4.0	1.898	2.885	$-$ 0.416	1.626	3.009	3.917	$-$ 0.956	1.049
		1.5	3.0	1.648	2.826	$-$ 0.778	1.912	2.775	3.667	$-$ 1.466	1.516
		2.0	3.0	1.210	2.644	$-$ 1.165	2.242	2.636	3.537	$-$ 1.967	2.004
		2.0	4.0	1.207	2.646	$-$ 1.020	2.351	2.935	3.779	$-$ 1.968	2.000
	1.0	1.0	4.0	0.829	1.632	0.180	1.943	2.251	3.108	$-$ 0.897	1.055
		1.5	3.0	0.913	1.907	$-$ 0.139	2.027	1.825	2.705	$-$ 1.459	1.493
		2.0	3.0	0.889	2.084	$-$ 0.353	2.354	1.863	2.840	$-$ 1.978	1.990
		2.0	4.0	0.882	2.072	$-$ 0.258	2.508	2.211	3.234	$-$ 1.978	1.985
	2.0	1.0	4.0	$-$ 0.103	0.502	0.250	1.578	2.611	3.148	$-$ 0.482	1.399
		1.5	3.0	$-$ 0.253	0.817	0.120	1.841	1.635	2.362	$-$ 1.293	1.497
		2.0	3.0	$-$ 0.399	1.124	0.024	2.174	1.554	2.456	$-$ 1.889	1.956
		2.0	4.0	$-$ 0.406	1.120	0.157	2.405	2.135	3.085	$-$ 1.883	1.948
Location	0.5	1.0	4.0	$-$ 0.265	0.860	0.150	1.108	$-$ 2.388	2.950	$-$ 0.126	1.441
		1.5	3.0	$-$ 0.088	0.965	0.343	1.237	$-$ 1.985	2.402	0.045	1.347
		2.0	3.0	0.024	1.073	0.509	1.363	$-$ 2.061	2.461	0.117	1.458
		2.0	4.0	$-$ 0.006	1.039	0.428	1.384	$-$ 2.682	3.196	$-$ 0.167	1.820
	1.0	1.0	4.0	0.084	0.410	$-$ 0.256	1.561	$-$ 2.074	2.778	$-$ 0.663	2.059
		1.5	3.0	0.308	0.646	$-$ 0.019	1.530	$-$ 1.583	2.194	$-$ 0.407	1.838
		2.0	3.0	0.546	0.908	0.137	1.710	$-$ 1.641	2.283	$-$ 0.369	2.028
		2.0	4.0	0.545	0.881	0.044	1.905	$-$ 2.111	2.917	$-$ 0.756	2.524
	2.0	1.0	4.0	0.315	0.405	$-$ 0.270	1.408	$-$ 2.677	3.217	$-$ 0.626	2.044
		1.5	3.0	0.627	0.724	$-$ 0.162	1.607	$-$ 1.631	2.280	$-$ 0.551	1.979
		2.0	3.0	0.953	1.057	$-$ 0.083	1.875	$-$ 1.540	2.316	$-$ 0.634	2.231
		2.0	4.0	0.950	1.056	$-$ 0.220	2.125	$-$ 2.163	3.034	$-$ 1.090	2.805

Table 2

Comparison of bias and RMSE of the three parameters estimated using the four methods ( $n=10$ )

				MVLE-H		MLE		PWM		BGENP
Parameter	$m$	$\eta$	$\gamma$	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
Shape	0.5	1.0	4.0	0.188	0.283	$-$ 0.058	0.455	0.840	1.016	1.476	3.089
		1.5	3.0	0.126	0.217	$-$ 0.065	0.400	0.771	0.872	1.461	3.080
		2.0	3.0	0.084	0.177	$-$ 0.082	0.363	0.743	0.819	1.477	3.109
		2.0	4.0	0.086	0.180	$-$ 0.070	0.400	0.771	0.864	1.423	3.043
	1.0	1.0	4.0	0.057	0.245	0.314	1.412	1.304	1.990	0.197	2.116
		1.5	3.0	$-$ 0.070	0.207	0.206	1.080	1.027	1.483	0.202	2.135
		2.0	3.0	$-$ 0.144	0.222	0.154	0.936	0.918	1.286	0.229	2.165
		2.0	4.0	$-$ 0.147	0.224	0.210	1.109	1.027	1.492	0.222	2.156
	2.0	1.0	4.0	$-$ 0.468	0.541	0.953	2.549	2.537	3.387	$-$ 1.696	1.989
		1.5	3.0	$-$ 0.691	0.724	0.759	2.299	1.711	2.651	$-$ 1.686	1.996
		2.0	3.0	$-$ 0.816	0.838	0.664	2.105	1.501	2.384	$-$ 1.687	2.003
		2.0	4.0	$-$ 0.818	0.839	0.776	2.307	1.692	2.635	$-$ 1.686	1.997
Scale	0.5	1.0	4.0	1.699	2.410	$-$ 0.276	1.119	3.669	4.330	$-$ 0.771	1.257
		1.5	3.0	1.641	2.476	$-$ 0.460	1.460	3.620	4.158	$-$ 1.339	1.562
		2.0	3.0	1.410	2.415	$-$ 0.685	1.791	3.619	4.062	$-$ 1.831	2.024
		2.0	4.0	1.421	2.411	$-$ 0.663	1.804	3.775	4.220	$-$ 1.838	2.021
	1.0	1.0	4.0	0.463	0.961	0.156	1.285	1.793	2.580	$-$ 0.602	1.248
		1.5	3.0	0.589	1.333	0.068	1.419	1.865	2.575	$-$ 1.365	1.471
		2.0	3.0	0.640	1.574	$-$ 0.020	1.667	2.016	2.754	$-$ 1.921	1.963
		2.0	4.0	0.647	1.571	0.076	1.848	2.311	3.129	$-$ 1.917	1.961
	2.0	1.0	4.0	$-$ 0.144	0.349	0.344	1.308	1.961	2.639	1.169	2.649
		1.5	3.0	$-$ 0.330	0.601	0.330	1.573	1.297	2.041	$-$ 0.687	1.583
		2.0	3.0	$-$ 0.465	0.882	0.321	1.824	1.320	2.163	$-$ 1.610	1.855
		2.0	4.0	$-$ 0.474	0.870	0.456	2.104	1.689	2.684	$-$ 1.585	1.827
Location	0.5	1.0	4.0	$-$ 0.422	0.825	0.063	0.488	$-$ 2.757	3.119	$-$ 0.038	0.881
		1.5	3.0	$-$ 0.333	0.719	0.127	0.556	$-$ 2.407	2.596	$-$ 0.041	0.932
		2.0	3.0	$-$ 0.265	0.729	0.197	0.636	$-$ 2.545	2.694	$-$ 0.056	1.082
		2.0	4.0	$-$ 0.294	0.830	0.176	0.685	$-$ 3.229	3.477	$-$ 0.210	1.401
	1.0	1.0	4.0	$-$ 0.024	0.268	$-$ 0.158	1.054	$-$ 1.600	2.221	$-$ 0.178	1.355
		1.5	3.0	0.126	0.360	$-$ 0.071	1.054	$-$ 1.582	2.045	$-$ 0.278	1.445
		2.0	3.0	0.276	0.514	$-$ 0.009	1.187	$-$ 1.720	2.176	$-$ 0.469	1.712
		2.0	4.0	0.275	0.511	$-$ 0.096	1.391	$-$ 2.100	2.714	$-$ 0.807	2.212
	2.0	1.0	4.0	0.234	0.303	$-$ 0.339	1.210	$-$ 1.986	2.678	0.162	1.223
		1.5	3.0	0.492	0.561	$-$ 0.326	1.437	$-$ 1.275	1.950	$-$ 0.105	1.592
		2.0	3.0	0.748	0.824	$-$ 0.314	1.643	$-$ 1.283	2.022	$-$ 0.427	1.943
		2.0	4.0	0.751	0.827	$-$ 0.449	1.923	$-$ 1.664	2.571	$-$ 0.754	2.412

Table 3

Comparison of bias and RMSE of the three parameters estimated using the four methods ( $n=20$ )

				MVLE-H		MLE		PWM		BGENP
Parameter	$m$	$\eta$	$\gamma$	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
Shape	0.5	1.0	4.0	0.112	0.160	$-$ 0.014	0.161	0.678	0.718	2.878	3.835
		1.5	3.0	0.067	0.121	$-$ 0.017	0.155	0.642	0.668	2.842	3.820
		2.0	3.0	0.042	0.102	$-$ 0.013	0.156	0.634	0.656	2.868	3.824
		2.0	4.0	0.040	0.101	$-$ 0.016	0.158	0.645	0.672	2.822	3.791
	1.0	1.0	4.0	$-$ 0.069	0.151	0.091	0.633	0.813	1.203	2.603	3.812
		1.5	3.0	$-$ 0.159	0.195	0.076	0.547	0.712	0.972	2.586	3.796
		2.0	3.0	$-$ 0.211	0.235	0.075	0.510	0.668	0.880	2.566	3.794
		2.0	4.0	$-$ 0.212	0.236	0.071	0.533	0.709	0.963	2.585	3.793
	2.0	1.0	4.0	$-$ 0.699	0.714	0.565	1.832	1.765	2.701	$-$ 1.362	1.877
		1.5	3.0	$-$ 0.850	0.859	0.528	1.712	1.231	2.093	$-$ 1.400	1.846
		2.0	3.0	$-$ 0.942	0.948	0.440	1.551	1.074	1.853	$-$ 1.378	1.849
		2.0	4.0	$-$ 0.940	0.947	0.518	1.698	1.218	2.068	$-$ 1.391	1.854
Scale	0.5	1.0	4.0	1.126	1.721	$-$ 0.041	0.633	4.391	4.794	$-$ 0.819	1.085
		1.5	3.0	1.292	2.012	$-$ 0.076	0.913	4.383	4.651	$-$ 1.448	1.458
		2.0	3.0	1.304	2.139	$-$ 0.118	1.182	4.267	4.459	$-$ 1.969	1.971
		2.0	4.0	1.349	2.201	$-$ 0.090	1.205	4.406	4.584	$-$ 1.966	1.973
	1.0	1.0	4.0	0.164	0.551	0.021	0.667	1.337	1.940	0.459	2.023
		1.5	3.0	0.248	0.851	0.014	0.842	1.674	2.238	$-$ 1.073	1.325
		2.0	3.0	0.371	1.182	0.013	1.081	1.963	2.564	$-$ 1.791	1.822
		2.0	4.0	0.379	1.178	0.011	1.096	2.163	2.853	$-$ 1.791	1.819
	2.0	1.0	4.0	$-$ 0.249	0.348	0.212	0.926	1.285	2.030	5.148	5.427
		1.5	3.0	$-$ 0.420	0.572	0.278	1.217	0.977	1.678	2.239	3.416
		2.0	3.0	$-$ 0.561	0.808	0.280	1.431	1.072	1.854	$-$ 0.272	1.892
		2.0	4.0	$-$ 0.553	0.804	0.363	1.608	1.304	2.230	$-$ 0.093	1.914
Location	0.5	1.0	4.0	$-$ 0.245	0.642	0.043	0.151	$-$ 3.180	3.391	0.099	0.205
		1.5	3.0	$-$ 0.208	0.568	0.069	0.202	$-$ 2.729	2.801	0.138	0.285
		2.0	3.0	$-$ 0.153	0.515	0.094	0.266	$-$ 2.809	2.862	0.154	0.431
		2.0	4.0	$-$ 0.192	0.669	0.089	0.264	$-$ 3.627	3.725	0.130	0.527
	1.0	1.0	4.0	0.051	0.189	$-$ 0.019	0.507	$-$ 1.171	1.642	0.271	0.481
		1.5	3.0	0.160	0.269	$-$ 0.007	0.592	$-$ 1.426	1.805	0.349	0.665
		2.0	3.0	0.275	0.389	$-$ 0.001	0.737	$-$ 1.650	2.027	0.317	0.946
		2.0	4.0	0.274	0.386	$-$ 0.002	0.769	$-$ 1.904	2.414	0.278	1.069
	2.0	1.0	4.0	0.291	0.324	$-$ 0.208	0.863	$-$ 1.285	2.039	0.542	0.584
		1.5	3.0	0.513	0.551	$-$ 0.269	1.128	$-$ 0.951	1.594	0.549	1.132
		2.0	3.0	0.747	0.791	$-$ 0.268	1.310	$-$ 1.034	1.734	0.551	1.472
		2.0	4.0	0.741	0.785	$-$ 0.352	1.492	$-$ 1.267	2.119	0.597	1.543

Table 4

Probability of rejecting the null hypothesis at significance level 0.05 when the null hypothesis is true using the two-sample K-S test (two-sided)

$m$	$\eta$	$\gamma$	$n_{1}=n_{2}=5$	$n_{1}=n_{2}=10$	$n_{1}=n_{2}=15$	$n_{1}=n_{2}=20$
0.5	1.0	4.0	0.0049	0.0085	0.0315	0.0407
	1.5	3.0	0.0025	0.0021	0.0195	0.0372
	2.0	3.0	0.0020	0.0013	0.0133	0.0377
	2.0	4.0	0.0018	0.0015	0.0270	0.0368
1.0	1.0	4.0	0.0047	0.0072	0.0089	0.0078
	1.5	3.0	0.0031	0.0069	0.0099	0.0093
	2.0	3.0	0.0022	0.0043	0.0065	0.0084
	2.0	4.0	0.0024	0.0034	0.0056	0.0071
1.5	1.0	4.0	0.0040	0.0055	0.0045	0.0037
	1.5	3.0	0.0022	0.0041	0.0057	0.0057
	2.0	3.0	0.0033	0.0033	0.0058	0.0082
	2.0	4.0	0.0040	0.0042	0.0062	0.0081
2.0	1.0	4.0	0.0025	0.0033	0.0034	0.0027
	1.5	3.0	0.0032	0.0036	0.0033	0.0039
	2.0	3.0	0.0028	0.0038	0.0060	0.0053
	2.0	4.0	0.0025	0.0039	0.0052	0.0060

Table 5

Three-parameter Fréchet distribution estimated using the four methods and the $P$ -value of the two-sample K-S test (two-sided) in Examples 1–3

	Shape	Scale	Location	Shape	Scale	Location	$P$ -value
Example 1	Initial therapy			Second-line therapy
MVLE-H	0.471	15.480	11.147	0.407	272.756	5.442	0.494
MLE	0.220	0.572	11.999	0.860	55.805	0.001
PWM	1.494	20.309	0.001	1.783	75.761	0.001
BGENP	0.059	0.148	11.143	0.001	0.001	0.001
Example 2	Non-PLA2R-associated			Unknown
MVLE-H	0.386	85.674	14.170	0.402	226.784	28.756	$>0.999$
MLE	0.635	22.470	12.659	1.192	89.246	0.003
PWM	1.204	40.369	0.001	1.978	109.534	0.001
BGENP	0.348	0.001	0.015	0.001	0.001	0.001
Example 3	Oral CPA			Intravenous CPA
MVLE-H	0.534	191.901	7.215	0.429	216.672	23.321	$>0.999$
MLE	1.232	56.640	0.001	1.304	84.073	0.001
PWM	2.553	66.980	0.001	2.905	101.886	0.001
BGENP	0.001	0.001	0.001	0.001	0.001	0.001

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