1 Introduction

The Weibull distribution appears in many fields of science such as reliability (Almeida 1999; Fok et al. 2001; Queeshi and Sheikh 1997), survival analysis (Carroll 2003), hydrology (Heo et al. 2001), social sciences (Roed and Zhang 2002), wind energy industry (Kang et al. 2018), actuarial science (Bolancé and Guillen 2021), financial mathematics (Chen and Gerlach 2013; Gebizlioglu et al. 2011). In the last one, it may be used to describe the distribution of the return rates of certain investments.

This paper is concerned with estimation of the parameters and quantiles of the Weibull distribution on the basis of complete data. Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed (i.i.d.) random variables from the two-parameter Weibull distribution with the probability density function defined by

$$\begin{aligned} f(x) = \frac{\beta }{\sigma }\left( \frac{x}{\sigma }\right) ^{\beta -1} \exp \Big [-\left( \frac{x}{\sigma }\right) ^{\beta }\Big ], \quad x\ge 0, \beta>0, \sigma >0. \end{aligned}$$

We propose three estimators of the shape parameter \(\beta \) of the Weibull distribution defined above, when the scale parameter \(\sigma \) is unknown. One of them is based on a modification of the equation from which the maximum likelihood estimator of the shape parameter is determined. The other two estimators are obtained on the basis of the non-parametric estimators of the Gini coefficient. The estimators proposed lead to new estimators of the scale parameter and new plug-in estimators of quantiles. The main purpose of this paper is to compare the performance of these estimators of the Weibull distribution parameters and plug-in estimators of Weibull distribution quantiles with those recommended in the literature. We use the bias and the mean squared error as criteria for comparison.

The paper is organized as follows. In Sect. 2 we present a review of known estimators of the Weibull distribution parameters and papers in which their accuracy were compared. In Sect. 3 we propose new estimators of the shape parameters of the Weibull distribution which lead to new estimators of the scale parameter and plug-in estimators of quantiles. Some comments on plug-in estimators of quantiles of the Weibull distribution are provided in Sect. 4. The estimators of the parameters and quantiles of the Weibull distribution are compared in the simulation study, and results of this study are presented in Sect. 5. Section 6 includes a real data analysis. Some concluding remarks are presented in Sect. 7.

2 A review of estimators in two-parameter Weibull distribution model

In the literature, over a dozen methods have been applied to estimation of the Weibull distribution parameters. These methods include: the method of moments (MM) (Cran 1988), the modified MM (Cohen and Whitten 1982), the method of probability weighted moments (Greenwood et al. 1979; Singh et al. 1990), the maximum likelihood estimation (ML) (Cohen 1965; Balakrishnan and Kateri 2008), the weighted maximum likelihood estimation (Jacquelin 1993), a few modifications of the ML estimation (Cohen and Whitten 1982), specially the Tiku’s modified ML (TMML) estimator (Gebizlioglu et al. 2011), method based on L-moments (LM) (Teimouri et al. 2013), the logarithmic moments (Johnson et al. 1994), the method of entropy (Singh 1987), the least squares (LS) (Bain and Antle 1967; Swain et al. 1988), the weighted least squares (WLS) (Hung 2001; Lu et al. 2004; Swain et al. 1988; Van Zyl and Schall 2012; Zhang et al. 2008; Kantar 2015), the generalized least squares (Engeman and Keefe 1982; Kantar 2015), the percentiles method (Hassanein 1971; Seki and Yokoyama 1993), a method based on U-statistics (U) (Sadani et al. 2019), the median rank regression method (Genschel and Meeker 2010; George 2014), the method based on second-kind statistics (log-cumulant estimator) (Sun and Han 2010), the generalized spacing method (GS) (Ghosh and Jammalamadaka 2001; Gebizlioglu et al. 2011). In parentheses, we gave abbreviations of estimators, which we will consider in the simulation study.

Closed form expressions of estimators are only given for L-moments estimators, estimators of logarithmic moment method, percentile estimators, Tiku’s modified ML estimators, and the estimators based on U-statistics.

Comparisons of various estimators of the Weibull distribution parameters have been performed in the following papers (in chronological order): Gross and Lurie (1977), Gibbons and Vance (1981), Al-Baidhani and Sinclair (1987), Singh et al. (1990), Wu et al. (2006), Genschel and Meeker (2010), Gebizlioglu et al. (2011), Teimouri et al. (2013), Akram and Hayat (2014), George (2014), Pobocikova and Sedliackova (2014), Kantar (2015), and Sadani et al. (2019). However, the conclusions of these comparisons differ. For example, Al-Baidhani and Sinclair (1987) compared the generalised least squares, maximum likelihood, two estimators proposed in Bain and Antle (1967), and two mixed methods of estimating the parameters of the two-parameter Weibull distribution. The comparison was made using the observed relative efficiency of parameter estimates to summarize the results of 1000 simulated samples of sizes 10 and 25. The results were that: generalised least squares is the best method of estimating the shape parameter, the best method of estimating the scale parameter depends on the size of the shape parameter. Gebizlioglu et al. (2011) compared nine estimators and recommended ML estimator and TMML as the best for estimating both parameters. Teimouri et al. (2013) proposed LM and compared it with four other methods. They concluded that LM and ML estimators had the best performance. Pobocikova and Sedliackova (2014) generally recommended ML estimators, but for small samples (\(n=10\) or \(n=20\)) suggested using WLS estimators. Wu et al. (2006) compared ML estimators with methods based on linear regression, considering different approximations to obtain them. They noticed that ML and MM estimators have tendency to overestimate rather than underestimate the parameters. Furthermore they showed that certain approximations are better than others in methods based on linear regression.

To the best of our knowledge, comparisons of various estimators of the Weibull distribution quantiles of order p have been performed only in the following papers: (Gibbons and Vance 1981) (\(n=10, n=25\) and \(p=0.1, p=0.9\)), Al-Baidhani and Sinclair (1987) (\(n=10,\) \(n=25\)), Singh et al. (1990) (\(n\in \{10,50,100,1000\},\) \(p\in \{0.9,0.99,0.999\}\)). Gibbons and Vance (1981) concluded that for estimating quantile of order .9, the best are the ML estimator and the best linear invariant estimator, followed by the best linear unbiased estimator. The results of their simulations suggest that estimator proposed by Bain and Antle (1967) and Gumbel (1958) are the best for estimating the quantile of order. 1. Al-Baidhani and Sinclair (1987) showed that ML estimator outperforms the least squares estimator and the generalised least squares estimator of quantiles of order 0.95 and 0.99.

3 New estimators of the shape parameter

3.1 Modified maximum likelihood estimators

Let \(x_{1},\ldots ,x_{n}\) be the realizations of i.i.d. random variables \(X_{1},\ldots ,X_{n}\) from the Weibull distribution with the scale parameter \(\sigma \) and the shape parameter \(\beta ,\) and denote by L the likelihood function. The likelihood equations are

$$\begin{aligned} \frac{\partial \ln L}{\partial \sigma }= & {} -\frac{n\beta }{\sigma }+\frac{\beta }{\sigma ^{\beta +1}}\sum _{i=1}^{n}x_{i}^{\beta }=0, \end{aligned}$$
(1)
$$\begin{aligned} \frac{\partial \ln L}{\partial \beta }= & {} \frac{n}{\beta }-n\ln \sigma +\sum _{i=1}^{n}\ln x_{i} +\frac{\ln \sigma }{\sigma ^{\beta }}\sum _{i=1}^{n}x_{i}^{\beta }-\frac{1}{\sigma ^{\beta }}\sum _{i=1}^{n}x_{i}^{\beta }\ln x_{i}=0. \end{aligned}$$
(2)

From Eq. (1) we have a closed-form expression for the ML estimator of \(\sigma ,\) namely

$$\begin{aligned} {{\hat{\sigma }}}_{ML}=\Big (\frac{1}{n}\sum _{i=1}^{n}x_{i}^{{{\hat{\beta }}}_{ML}}\Big )^{1/{{\hat{\beta }}}_{ML}}, \end{aligned}$$
(3)

where the ML estimator \({{\hat{\beta }}}_{ML}\) of the parameter \(\beta \) is a solution to the equation

$$\begin{aligned} \sum _{i=1}^{n}x_{i}^{\beta } +\frac{1}{n}\Big [\sum _{i=1}^{n}\ln (x_{i}^{\beta })\Big ]\sum _{i=1}^{n}x_{i}^{\beta } -\sum _{i=1}^{n}x_{i}^{\beta }\ln (x_{i}^{\beta }) = 0. \end{aligned}$$
(4)

Lemat 1

The equation (4) is not an unbiased estimating equation, i.e.,

$$\begin{aligned} E_{\sigma ,\beta }\Big \{\sum _{i=1}^{n}X_{i}^{\beta } +\frac{1}{n}\Big [\sum _{i=1}^{n}\ln (X_{i}^{\beta })\Big ]\sum _{i=1}^{n}X_{i}^{\beta } -\sum _{i=1}^{n}X_{i}^{\beta }\ln (X_{i}^{\beta })\Big \}\ne 0. \end{aligned}$$
(5)

Proof

Let us denote \(Y_{i}=X_{i}^{\beta }.\) Then the left side of (5) is equal to

$$\begin{aligned} E_{\sigma ,\beta }\Big [\sum _{i=1}^{n}Y_{i} +\frac{1}{n}\Big (\sum _{i=1}^{n}\ln Y_{i}\Big )\sum _{i=1}^{n}Y_{i} -\sum _{i=1}^{n}Y_{i}\ln Y_{i}\Big ]. \end{aligned}$$
(6)

Using the fact that if X has the Weibull distribution with the shape parameter \(\beta \) and the scale parameter \(\sigma \) then \(X^{\beta }\) has the exponential distribution with the scale parameter \(\sigma ^{\beta },\) we have that

$$\begin{aligned} E_{\sigma ,\beta }(Y_{i})=\sigma ^{\beta }, \end{aligned}$$
(7)
$$\begin{aligned} E_{\sigma ,\beta }(\ln Y_{i})=\psi (1)-\ln (\sigma ^{\beta }) \end{aligned}$$
(8)

and

$$\begin{aligned} E_{\sigma ,\beta }(Y_{i}\ln Y_{i})=\sigma ^{\beta }[\psi (1)+1-\ln (\sigma ^{\beta })], \end{aligned}$$
(9)

\(i=1,\ldots ,n,\) where \(\psi \) is the digamma function. Thus, expression (6) is equal to

$$\begin{aligned} \sum _{i=1}^{n}E_{\sigma ,\beta }(Y_{i}) +\frac{1}{n}\sum _{i=1}^{n}E_{\sigma ,\beta }(Y_{i}\ln Y_{i})+\frac{1}{n}\sum _{i\ne j}E_{\sigma ,\beta }(Y_{i}\ln Y_{j}) -\sum _{i=1}^{n}E_{\sigma ,\beta }(Y_{i}\ln Y_{i})=\sigma ^{\beta }. \end{aligned}$$

\(\square \)

From the proof of Lemma 1, it can be easily seen that the following equation

$$\begin{aligned} \frac{n-1}{n}\displaystyle \sum _{i=1}^n x_i^{\beta } + \frac{1}{n}\Big [\displaystyle \sum _{i=1}^n\ln (x_i^{\beta })\Big ] \displaystyle \sum _{i=1}^n x_i^{\beta }- \displaystyle \sum _{i=1}^n x_i^{\beta }\ln (x_i^{\beta }) = 0 \end{aligned}$$
(10)

is an unbiased estimating equation for the parameter \(\beta .\)

Theorem 1

The solution to equation (10) exists and is unique.

Proof

The existence and uniqueness of the solution to equation (10) can be proved analogously to the existence and uniqueness of the ML estimator given in Balakrishnan and Kateri (2008). \(\square \)

We take the unique solution to equation (10) as the modified maximum likelihood (MML) estimator \({{\hat{\beta }}}_{MML}\) of the shape parameter \(\beta .\) As the MML estimate of the parameter \(\sigma \) we take

$$\begin{aligned} {{\hat{\sigma }}}_{MML}=\Big (\frac{1}{n}\sum _{i=1}^{n}x_{i}^{{{\hat{\beta }}}_{MML}}\Big )^{1/{{\hat{\beta }}}_{MML}}. \end{aligned}$$
(11)

Theorem 2

With probability 1, \({{\hat{\beta }}}_{MML}<{{\hat{\beta }}}_{ML},\) and \({{\hat{\sigma }}}_{MML}<{{\hat{\sigma }}}_{ML}.\)

Proof

Equation (4) can be expressed alternatively as

$$\begin{aligned} 1=\frac{\sum _{i=1}^{n}x_{i}^{\beta }\ln (x_{i}^{\beta })}{\sum _{i=1}^{n}x_{i}^{\beta }} -\frac{1}{n}\sum _{i=1}^{n}\ln (x_{i}^{\beta })=:H_\textbf{x}(\beta ), \end{aligned}$$
(12)

and equation (10) as

$$\begin{aligned} \frac{n-1}{n}=H_\textbf{x}(\beta ). \end{aligned}$$
(13)

It can be shown that for each \(\textbf{x}=(x_{1},\ldots ,x_{n}),\) the function \(H_\textbf{x}(\beta )\) of the argument \(\beta \) increases from 0 to infinity. Hence \({{\hat{\beta }}}_{MML}<{{\hat{\beta }}}_{ML}\) with probability 1. The second part of the theorem follows from the form of \({{\hat{\sigma }}}_{ML},\) \({{\hat{\sigma }}}_{MML},\) the relation between \({{\hat{\beta }}}_{ML}\) and \({{\hat{\beta }}}_{MML},\) and the power mean inequality. \(\square \)

Remark 1

It can be easily shown that \({{\hat{\beta }}}_{MML}(X_{1},\ldots ,X_{n})={{\hat{\beta }}}_{MML}(cX_{1},\ldots ,cX_{n}),\) and \({{\hat{\sigma }}}_{MML}(cX_{1},\ldots ,cX_{n})=c{{\hat{\sigma }}}_{MML}(X_{1},\ldots ,X_{n}),\) for an arbitrary \(c>0.\)

Remark 2

In Cohen and Whitten (1982) five modifications of the ML estimator of the three-dimensional Weibull distribution parameter are considered. They were obtained by replacing one of the likelihood equations (the one derived by differentiating the likelihood function with respect to the location parameter) with alternate functional relationships. Two of these modifications, under the assumption that the location parameter equals zero, are considered in Gebizlioglu et al. (2011) in the case of the two-parameter Weibull distribution. In Gebizlioglu et al. (2011) the TMML estimator is also considered which is based on linearizing the likelihood equations around the first two terms of the Taylor series. The proposed MML estimator is constructed by a slight modification of the equation used to determine the ML estimate of the shape parameter and differs from the modifications of the ML estimator considered in the literature.

3.2 Estimators based on the Gini index

Both equations (4) and (10) can only be solved numerically. Motivated by the need to determine an appropriate starting point for finding solutions to these equations, we look for a simple estimate of the shape parameter \(\beta .\) We took into account the fact that the Lorenz curve (see for example Sarabia 2008), and consequently the Gini index G, does not depend on the scale parameter (here \(\sigma \)). For the Weibull distribution \(G=1-2^{-1/\beta }\), so

$$\begin{aligned} \beta =[-\log _2(1-G)]^{-1}. \end{aligned}$$
(14)

One of the nonparametric estimators of G is

$$\begin{aligned} {\widehat{G}}=\frac{1}{n^2{\bar{X}}}\displaystyle \sum _{i=1}^n(2i-n-1)X_{i:n}, \end{aligned}$$

where \(X_{i:n}\) is the i-th order statistic of the sample \((X_{1},\ldots ,X_{n}).\) The estimator \({\widehat{G}}\) is biased and in Davidson (2007) an approximated expression for bias of \({\widehat{G}}\) is given. Based on the result in Davidson (2007), in Mirzaei et al. (2017) the following bias-corrected estimator of G is proposed

$$\begin{aligned} {\widetilde{G}}=\frac{n}{n-1}{\widehat{G}}. \end{aligned}$$

The estimator \({\widetilde{G}}\) is not unbiased, but its bias is of order \(\frac{1}{n}\). We consider the following two simple estimators of the parameter \(\beta \)

$$\begin{aligned} \hat{\beta }_{{\widehat{G}}}=[-\log _2(1-{\widehat{G}})]^{-1} \qquad \text {and}\qquad \hat{\beta }_{{\widetilde{G}}}=[-\log _2(1-{\widetilde{G}})]^{-1}. \end{aligned}$$

As the estimates of the parameter \(\sigma \) we take

$$\begin{aligned} \hat{\sigma }_{{\widehat{G}}}=\left( \frac{1}{n}\displaystyle \sum _{i=1}^n x_i^{\hat{\beta }_{{\widehat{G}}}}\right) ^{1/\hat{\beta }_{{\widehat{G}}}} \qquad \text {and}\qquad \hat{\sigma }_{{\widetilde{G}}}=\left( \frac{1}{n}\displaystyle \sum _{i=1}^n x_i^{\hat{\beta }_{{\widetilde{G}}}}\right) ^{1/\hat{\beta }_{{\widetilde{G}}}}. \end{aligned}$$

It can be easily shown that \(\hat{\beta }_{{\widehat{G}}}>\hat{\beta }_{{\widetilde{G}}}\) with probability 1, and based on the power mean inequality, it implies that \(\hat{\sigma }_{{\widehat{G}}}>\hat{\sigma }_{{\widetilde{G}}}\) with probability 1.

Fact 1

The estimator \(\hat{\beta }_{{\widetilde{G}}}\) has the same form as the L-moment estimator \(\hat{\beta }_{LM}\) proposed by Teimouri et al. (2013).

Proof

The estimator \(\hat{\beta }_{LM}\) of the parameter \(\beta ,\) proposed by Teimouri et al. (2013), is of the form

$$\begin{aligned} \hat{\beta }_{LM}&=-\frac{\log 2}{\log (1-m^0_2/m^0_1)}, \end{aligned}$$

where

$$\begin{aligned} m^0_1 = {\bar{X}} \qquad \text {and} \qquad m^0_2 = \frac{2}{n(n-1)}\sum _{i=1}^n (i-1)X_{i:n} - {\bar{X}}. \end{aligned}$$
(15)

Hence,

$$\begin{aligned} \hat{\beta }_{LM}&=-\frac{1}{\log _2(1-m^0_2/m^0_1)}= \left[ -\log _2\left( 1-\frac{1}{n(n-1){\bar{X}}}\sum _{i=1}^n(2i-2)X_{i:n}-{\bar{X}} \right) \right] ^{-1}\\&=\left[ -\log _2\left( 1- \frac{1}{n(n-1){\bar{X}}}\sum _{i=1}^n(2i-2)X_{i:n}- \frac{1}{n(n-1){\bar{X}}}\sum _{i=1}^n(n-1)X_{i:n} \right) \right] ^{-1}\\&=\left[ -\log _2\left( 1- \frac{1}{n(n-1{\bar{X}}}\sum _{i=1}^n(2i-n-1)X_{i:n} \right) \right] ^{-1} = [-\log _2(1-{\tilde{G}})]^{-1}=\hat{\beta }_{{\widetilde{G}}}. \end{aligned}$$

\(\square \)

Remark 3

Although \(\hat{\beta }_{LM}=\hat{\beta }_{{\widetilde{G}}},\) the estimator of the parameter \(\sigma \) proposed by Teimouri et al. (2013) is not of the same form as \({{\hat{\sigma }}}_{{\widetilde{G}}}.\)

4 Estimators of quantiles of the Weibull distribution

Another issue we investigated is the estimation of the extreme quantiles of the Weibull distribution. We decided to investigate the extreme quantiles, as they are widely used for durability tests (Gibbons and Vance 1981), in hydrology (Smith 1987), and in financial mathematics, specifically for estimating risk measures (Gebizlioglu et al. 2011; Alemany et al. 2013).

The quantile of order p of the Weibull distribution is given by

$$\begin{aligned} Q(p)=\sigma [-\ln (1-p)]^{1/\beta }, \end{aligned}$$

where \(p\in (0,1).\) The plug-in estimators \({{{\widehat{Q}}}}\) of the quantiles can be obtained substituting the unknown parameters by their estimators. For example, the MML estimator of the quantile of order p is

$$\begin{aligned} {{\hat{Q}}}_{MML}(p)={{\hat{\sigma }}}_{MML}[-\ln (1-p)]^{1/{{\hat{\beta }}}_{MML}} =\Big [-\frac{1}{n}\ln (1-p)\sum _{i=1}^{n}x_{i}^{{{\hat{\beta }}}_{MML}}\Big ]^{1/{{\hat{\beta }}}_{MML}}. \end{aligned}$$

In the simulation study, presented in Sect. 5, we compare various plug-in estimators of quantiles obtained by using estimators of parameters which are recommended in the literature and those proposed in Sect. 3.

5 Simulation study

To investigate the performance of the new proposed estimators of the Weibull distribution parameters and quantiles relative to the estimators recommended in the literature we conducted a simulation study. The measures that we use to compare the estimators are the mean squared error (MSE) and bias. We performed the simulations using the R programming language.

5.1 Simulation design

For 26 values of shape parameter varying between 0.5 and 3 and 20 sample sizes varying between 10 and 200, we simulated 2000 samples from Weibull distribution. Without loss of generality we set scale parameter equal to 1. For each sample we calculated the values of the following parameter estimators recommended in the literature: MM, ML, WLS, LM TMML, GS, LS, U and the new estimators proposed: MML, G1, G2. Then we used the estimates of the parameters to compute the plug-in estimates of quantiles of order .9, .95, .99 and .999.

Most of the estimators considered require numeric computation. To execute them we used functions uniroot from library rootSolve and optim from library stats in R package. Some of the estimators considered, namely MM, ML, GS, LM and U estimators, are implemented in the ForestFit (Teimouri et al. 2020) library in R. Determining the values of some estimators considered in the literature may be problematic due to the lack of roots or having many roots, resulting in nonconvergence or convergence to incorrect values. This issue was discussed in Kantar and Şenoğlu (2008) (in the case of estimating the location and scale parameters with known shape parameter of the Weibull distribution) and Gebizlioglu et al. (2011). For example the equation used to determine the MML2 estimator of the shape parameter (proposed in Cohen and Whitten (1982), considered in Gebizlioglu et al. (2011) but not considered in this paper) often has no solution when the sample size and shape parameter are small. The estimators given by explicit formulas (i.e. G1, G2, TMML) and those whose existence and uniqueness has been theoretically proven (e.g. ML and MML) do not show such problems. During the simulation study, no problems with computing the values of other compared estimators (MM, WLS, LM, GS, LS, U) have been encountered.

5.2 Comparison of the estimators of the parameters

In Fig. 1 we present the empirical MSE’s ratios of selected estimators of \(\beta \) with respect to the ML estimator and empirical biases of the estimators of \(\beta \) for different values of \(\beta \) and \(\sigma =1.\) Lines denoted by G1 and G2 correspond to estimators based on \({\widehat{G}}\) and \({\widetilde{G}},\) respectively. The values on the x axis has been customised to present explicitly the results of several best estimators. Consequently, when a certain estimator is significantly worse than others only a fragment of its plot is included in the chart (e.g. the plot corresponding to the LS estimator in the top right chart in Fig. 1) or it is not included at all (e.g. the plot corresponding to the LS estimator in the top right chart in Fig. 2).

Fig. 1
figure 1

Empirical MSE’s ratios of selected estimators of \(\beta \) to ML estimator and empirical biases of the estimators of \(\beta \) for different values of \(\beta \) and \(\sigma =1\), for \(n=10\) (on the left), \(n=50\) (in the middle), \(n=200\) (on the right)

Full size image

Figure 2 contains a comparison of the scale parameter estimators analogous to the one presented in Fig. 1.

Fig. 2
figure 2

Empirical MSE’s ratios of selected estimators of \(\sigma \) to ML estimator and empirical biases of the estimators of \(\sigma \) for different values of \(\beta \) and \(\sigma =1\), for \(n=10\) (on the left), \(n=50\) (in the middle), \(n=200\) (on the right)

Full size image

Figures 3 and 4 include heatmap plots related to the empirical MSE’s of the shape and scale parameter estimators, respectively. The closer the color is to blue, the better the MML estimator is compared to the selected estimator. The closer the color is to red, the worse the MML estimator is compared to the selected estimator. If the color is white, the compared estimators have comparable empirical MSE’s.

Fig. 3
figure 3

Comparison of empirical MSE’s of MML estimator and selected estimators of the shape parameter when \(\sigma =1\)

Full size image
Fig. 4
figure 4

Comparison of empirical MSE’s of MML estimator and selected estimators of the scale parameter when \(\sigma =1\)

Full size image

Our simulations concerning the parameter \(\beta \) show that:

  • the MML estimator has the smallest empirical MSE’s for samples of sizes \(n\ge 50\) (see Fig. 3);

  • the MML estimator has the smaller empirical MSE’s than ML estimator for all cases considered; for \(n = 10\) the ratio of these errors is approximately 3/4 and increases to 1 as n increases;

  • for all considered cases, the empirical bias of the MML estimator is approximately half that of the ML estimator;

  • the G2, LM, TMML, GS estimators have smaller empirical MSE’s than MML estimator when \(n=10\) (see Fig. 3);

  • the LS estimator has the smallest empirical bias, close to zero (see Fig. 1);

  • most of the estimators tend to overestimate the shape parameter, while GS estimator underestimates it (see Fig. 1).

5.3 Comparison of the estimators of extreme quantiles

Figure 5 contains a comparison of MSE’s and biases of selected estimators of quantile \(q=Q(p)\) of order \(p=.99.\) The values displayed in the plots are the ratios of MSE’s (on top) (or biases — bottom) of quantile estimator considered and ML estimator of q. This means, that if the curve corresponding to the estimator, say \({{\hat{q}}},\) is below 1,  the estimator \({{\hat{q}}}\) is better (in terms of MSE) than ML estimator of q.

Fig. 5
figure 5

Empirical MSE’s ratios and empirical biases ratios of quantiles of order .99 estimators to ML estimator for different values of \(\beta \) and \(\sigma =1,\) for \(n=10\) (on the left), \(n=50\) (in the middle), \(n=200\) (on the right)

Full size image
Fig. 6
figure 6

Comparison of empirical MSE’s of the MML estimator and selected estimators of the quantile of order .99 when \(\sigma =1\)

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The results are similar for all four investigated orders (.9, .95, .99 and .999) of quantiles, hence we only present results for one order \(p=.99\) of quantile. The conclusions are as follows:

  • when \(n\ge 50\) and \(\beta \ge 1.5\) ML, MML and TMML estimators have almost identical values of MSE’s (see Figs. 5, 6);

  • when \(\beta >0.8\) MML estimator has much smaller, and TMML estimator has much larger bias than ML estimator (see Fig. 5);

  • for small sample sizes (\(n=10\) and \(n=20\)) the G1 estimator has the smallest MSE (see Figs. 5, 6);

  • the LM, G2, U estimators and for larger sample sizes the WLSE have much smaller bias than ML, MML and TMML estimators, although they have significantly larger MSE’s (see Fig. 5);

  • for all cases considered, the MML estimator has slightly larger MSE’s than ML estimator;

  • for \(\beta >0.75\) the MML estimator has much smaller bias (\(50\%\)) than ML estimator;

  • the G2 estimator has the smallest bias when \(\beta >1.5.\)

5.4 Comparison of the CPU times of the estimators

We consider the central processing unit (CPU) times to compare the computational complexity of the considered estimators. Some estimators, namely G1, G2, LM, and TMML, are given by explicit formulas, while others require iterative methods to find the root of the corresponding equation. For most estimators (except MM, G1, G2, LM, and TMML), the time required increases significantly with sample size. Table 1 shows the comparison of average CPU times (in miliseconds) when shape parameter is equal to 2. Similar results have been achieved with several different values of this parameter. CPU times of the estimators given by explicit formulas (i.e. G1, G2, TMML) and the MM estimator are much shorter than those of other estimators, especially when the sample size is large. The shape and scale estimators based on U-statistic, which outperform several other estimators in the sense of bias, when the sample size is large is the slowest one in the experiment and it might be extremely slow (see (Sadani et al. 2019)). In case when two estimators have similar efficiency, the one which requires less time to be computed may be chosen.

Table 1 The average CPU times (in miliseconds) per sample for selected sample sizes and estimators when \(\beta =2\)
Full size table

6 Real data analysis

The data, given in Table 2, has been taken from Murthy et al. (2004) (data set 4.1). It consists of 20 observations of the time till failure. The data is complete, which means that it has not been censored.

Table 2 Real data consisting of 20 observations of the time till failure
Full size table

We applied the Anderson-Darling test (implemented in goftest package) to verify the null hypothesis that the observed times to failure are realizations of the random variables from the Weibull distribution. The obtained p value in this test is equal to 0.8257, so we do not reject it on the significance level 0.05.

Table 3 includes the estimates of the shape and scale parameters and quantiles.

Table 3 Estimates of the parameters and extreme quantiles
Full size table

The estimated values of the shape parameter \(\beta \) are between 1.42 and 1.73. According to the results of our simulations, the MM and G1 estimators of quantiles are recommended for such combination of the sample size \(n=20\) and the shape parameter \(\beta .\) The MM estimator though gives significantly greater values of quantiles, so we recommend using the G1 estimator of quantiles.

7 Concluding remarks

We proposed three Weibull distribution shape parameter estimators, which lead to three scale parameter estimators and three new quantile estimators of this distribution. One of the proposed estimators (the MML) is a modification of the ML estimator. Based on the simulations performed, we can conclude that it has both a smaller mean squared error and a smaller bias in relation to the ML estimator. However, the plug-in estimators of the Weibull distribution quantiles based on the proposed modification of the ML estimator of the shape parameter turned out to be worse (due to the mean squared error) compared to the plug-in estimators based on ML estimators of the parameters. We also proposed two estimators of the shape parameter which have closed form expressions. The main aim of this proposal was to obtain a good starting point for the numerical computation of the ML and MML estimators. They are based on the nonparametric estimators of the Gini coefficient, thus we did not expect that they would have smaller errors than known estimators, in particular the ML estimator. It turned out, however, that one of the proposed estimators leads to good extreme quantile estimates (better than the ML) in the case of small sample sizes.

In general, the plug-in estimator of extreme quantiles based on the ML estimators of the parameters is the best or as good as other estimators (due to the mean squared error), except the case of small samples (up to 30), where we recommend G1 and MM estimators. For parameters estimation we recommend using MML estimator.

The approach presented in this paper can also be applied to censored data. For example, in case of Type-I censoring, Type-II censoring or progressive Type-II censoring, the ML estimator of the scale parameter \(\sigma \) can be expressed as a function of the ML estimator of the parameter \(\beta \) (see e.g. Cohen (1965), Balakrishnan and Kateri (2008)), which is the solution to an equation (depending on the censoring type) which can only be solved numerically. However, analogously to the case of complete data, relationship (14) between this parameter and the Gini index can be used. As the estimator of the Gini index we can take one of the non-parametric estimators of this index with right censored data considered, for example, in Gigliarano and Muliere (2013), Lv et al. (2017), Hong et al. (2018), and recently (Kattumannil et al. 2021). To the best of our knowledge, this approach has not been used to estimate the Weibull distribution parameters based on censored data. Applying the approach analogous to that presented in Sect. 3.1, especially in the case of Type-I censored data, seems to be a more difficult task. This is because it results in having a sum of a random number of functions of order statistics in an equation for the ML estimator of the parameter \(\beta \). Comparison of estimators of the parameters and quantiles of the Weibull distribution based on censored data, obtained using the methods sketched above with the estimators known from the literature, e.g. Gibbons and Vance (1981), Zhang et al. (2008), Balakrishnan and Kateri (2008), Genschel and Meeker (2010), Guure and Ibrahim (2013), Yu and Peng (2013), Jia (2020), Jiang (2022), is an interesting task which we intend to undertake.