WLinfer: Statistical Inference for Weighted Lindley Distribution
Yu-Hyeong Jang
Korea UniversitySungBum Kim
Konkuk UniversityHyun-Ju Jung
Konkuk UniversityHyoung-Moon Kim (Corresponding author)
Konkuk University2023-01-17
Abstract
New distributions are still being suggested for better fitting of a distribution to data, as it is one of the most fundamental problems in terms of the parametric approach. One of such is weighted Lindley (WL) distribution (Ghitany et al. 2011). Even though WL distribution has become increasingly popular as a possible alternative to traditional distributions such as gamma and log normal distributions, fitting it to data has rarely been addressed in existing R packages. This is the reason we present the WLinfer package that implements overall statistical inference for WL distribution. In particular, WLinfer enables one to conduct the goodness of fit test, point estimation, bias correction, interval estimation, and the likelihood ratio test simply with theWL function which is at the core of this package. To assist
users who are unfamiliar with WL distribution, we present a brief review
followed by an illustrative example with R codes.
Introduction
Weighted Lindley (WL) distribution has recently received considerable attention since it provides a more flexible fit to data from various fields than traditional widely-used distributions such as exponential, log normal, and gamma distributions (Ghitany et al. 2011; Mazucheli, Louzada, and Ghitany 2013). The probability density function (pdf) of WL distribution is given by \[f(x) = \frac{\lambda^{\phi+1}}{(\lambda+\phi)\Gamma(\phi)}{x}^{\phi-1}(1+{x})\exp(-\lambda {x}), x>0,\lambda>0,\phi>0,\] which can be interpreted as a mixture of two gamma distributions \[\begin{aligned} &f(x) = \frac{\lambda}{\lambda+\phi}f_{1}(x)+\frac{\phi}{\lambda+\phi}f_{2}(x),\quad x>0,\lambda>0,\phi>0, \end{aligned}\] where \[\begin{aligned} &f_{i}(x) = \frac{\lambda^{\phi+i-1}}{\Gamma(\phi+i-1)}x^{\phi+i-2} \exp (-\lambda x),\quad x>0,\lambda>0,\phi>0,\quad i=1,2. \end{aligned}\] Due to its nature as a gamma mixture, however, statistical inference for WL distribution, such as parameter estimation, bias correction, interval estimation, and statistical test, is overall more cumbersome and tedious than those of the aforementioned distributions.
Despite this difficulty, there is no existing R package that implements this comprehensive process for WL distribution. To the best of our knowledge, mle.tools (Mazucheli, Menezes, and Nadarajah 2017) is the only R package that enables one to obtain maximum likelihood (ML) estimates for WL distribution with asymptotic variance and bias correction. However, they are just limited to ML estimates. An R package fitdistrplus (Delignette-Muller and Dutang 2015) which fits certain univariate distributions to data sets is not applicable to WL distribution, while LindleyR package (Mazucheli, Fernandes, and Oliveira 2016) is exclusively for the use of pdf, cumulative distribution function (cdf), quantile, and random number generation, not statistical inference itself.
Based on this motivation, we present an R package WLinfer by providing various estimation methods in addition to maximum likelihood estimator (MLE), such as method of moment estimator (MME), modified method of moment estimator (MME\(_m\)), and closed form MLE-like estimator (MLE\(_c\)). For bias correction, WLinfer encompasses Firth’s method and the bootstrap method which were not considered in mle.tools, as well as Cox and Snell’s method. Furthermore, WLinfer provides the goodness of fit and likelihood ratio tests.
The remainder of this paper is organized as follows. First, we
briefly review the theoretical results for each inferential step ranging
from the goodness of fit test to the likelihood ratio test, while
introducing relevant arguments of the WL function. We then
provide an example to illustrate the whole output of WL
function and how to use WL function which implements the
whole statistical inference at once. Finally, we conclude with
summarizing remarks.
The WLinfer package is available from the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/package=WLinfer. R code for the examples demonstrated herein has been provided as supplementary material. The supplementary code has been tested with WLinfer version 1.0.0, and results presented herein have been produced with this version.
Goodness of fit test
The first step in fitting a distribution to data is to check whether
the data can be assumed as generated from the distribution. For this
purpose, many goodness of fit tests have been developed. Among them,
WLinfer includes three popular tests: the Kolmogorov-Smirnov
test, the Anderson-Darling test, and the Cramér-von Mises test.
Uncertainty that could occur from using estimates instead of true values
are not considered here. The argument dist\(\_\)test of WL
function is specified by one of either "ks",
"cvm", "ad", or "all". The
default is dist\(\_\)test="ks" while
dist\(\_\)test=all" returns the
results of all three tests.
Point estimation
WLinfer package considers four estimators: MLE, MLE\(_{c}\), MME and MME\(_{m}\). To the best of our knowledge, these are the only estimators of which asymptotic distribution has been analytically proven. In this study, we only review the estimation formulas. Please refer to (Kim and Jang 2020; Ghitany, Song, and Wang 2017; Mazucheli, Louzada, and Ghitany 2013) for more specific theoretical results.
- MLE of WL distribution is obtained by \[\widehat{\lambda}_{MLE}=\frac{-\widehat{\phi}_{MLE}(\overline{X}-1)+\sqrt{\left[\widehat{\phi}_{MLE}(\overline{X}-1)\right]^{2}+4
\widehat{\phi}_{MLE}\left(\widehat{\phi}_{MLE}+1\right) \overline{X}}}{2
\overline{X}}\] where \(\widehat{\phi}_{MLE}\) is the solution of
the nonlinear equation \[\psi\left(\widehat{\phi}_{MLE}\right)+\frac{1}{\widehat{\lambda}_{MLE}+\widehat{\phi}_{MLE}}-
\log \left( \widehat{\lambda}_{MLE} \right) - \frac{1}{n} \sum_{i=1}^{n}
\log \left(x_{i}\right)=0,\] with \(\psi(x)= \cfrac{\mathrm{d}\log\ (
\Gamma(x))}{\mathrm{d} x}\).
- MLE\(_{c}\)
\(\begin{array}{l} \widehat{\lambda}_{MLE_c}= \cfrac{d +\sqrt{d^2 + \frac{4(1+a_1)\cdot a_0 (a_0 - 1) }{\left(\sum X_{i} / n\right)-a_1}}}{2(a_1+1)}, \label{eq:ClosedFormLambda} \quad\quad \end{array}\)\(\begin{array}{l} \widehat{\phi}_{MLE_c}= a_1 \widehat{\lambda}- a_0,\label{eq:ClosedFormPhi} \end{array}\)
where \(\begin{array}{l} d = a_0 +\frac{(1+a_1 )(1-a_0 )-1}{\sum X_{i}/n-a_1 }, \quad a_1 = \cfrac{\sum\limits_{i}^{n} X_{i} \log (X_{i})}{\sum\limits_{i}^{n} \log (X_{i})}~ \text{and}~ a_0 = \cfrac{\sum\limits_{i}^{n} \frac{X_{i} \log (X_{i})}{1+X_{i}}+n }{\sum\limits_{i}^{n} \log (X_{i})}. \end{array}\)
- MME
\(\begin{array}{l} \widehat{\lambda}_{MME}=\cfrac{-\widehat{\phi}_{MME}(\overline{X}-1)+\sqrt{\left[\widehat{\phi}_{MME}(\overline{X}-1)\right]^{2}+4 \overline{X} \widehat{\phi}_{MME}\left(\widehat{\phi}_{MME}+1\right)}}{2 \overline{X}}, \\ \\ \widehat{\phi}_{MME}=\cfrac{- g \left(\overline{X}, S^{2}\right)+\sqrt{\left[g \left(\overline{X}, S^{2}\right)\right]^{2}+16 S^{2}\left[S^{2}+(\overline{X}+1)^{2}\right] \overline{X}^{3}}}{2 S^{2}\left[S^{2}+(\overline{X}+1)^{2}\right]},\\ \text{where } g \left(\overline{X}, S^{2}\right)=S^{4}-\overline{X}\left(\overline{X}^{3}+2 \overline{X}^{2}+\overline{X}-4 S^{2}\right),\quad S^{2}=\frac{1}{n} \sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}. \end{array}\)
- MME\(_{m}\)
\(\begin{array}{l} \widehat{\lambda}_{MME_{m}}=\cfrac{\overline{X^*}(1-\overline{X^{*}})}{\overline{X}\cdot \overline{X^{*}}-(1-\overline{X^{*}})}, \\ \widehat{\phi}_{MME_{m}}=\cfrac{(1-\overline{X^{*}})^{2}}{\overline{X}\cdot \overline{X^{*}}-(1-\overline{X^{*}})},\quad\text{where } X^{*}=\cfrac{1}{1+X}. \end{array}\)
Each estimation method can be implemented by choosing
est\(\_\)method of WL
function, among "MLE", "MLEc",
"MME", and "MMEm". The default refers to
est\(\_\)method = "MLEc". If point
estimation is solely of interest, one can separately obtain point
estimates with the following codes:
data(fail_fiber) # fail_fiber is included in WLinfer package.
MLEc_WL(fail_fiber) # Closed form MLE-like estimator
MME_WL(fail_fiber) # Method of moment estimator
MMEm_WL(fail_fiber) # Modified method of moment estimator
MLE_WL(fail_fiber,init =1) # Initial value for phi is required.We use fail\(\_\)fiber data set (Bader and Priest 1982) which is included in the
WLinfer package. This data set consists of 65 observations of
the strength measurement of carbon fiber.
Bias correction
Unless sample size is sufficient for achieving consistency, bias correction is highly recommended since all the aforementioned estimators are upwardly biased in the case of small samples (Kim and Jang 2020; Mazucheli, Louzada, and Ghitany 2013; Wang and Wang 2017). To this end, the WLinfer package provides three bias correction methods: Cox and Snell’s method (Cox and Snell 1968; Cordeiro and Klein 1994) for MLE and MLE\(_c\), Firth’s method (Firth 1993) for MLE, and the bootstrap method (Efron 1979; Efron and Tibshirani 1986) for MLE\(_c\). After a series of tedious calculations, correction formulas are given as follows:
Cox and Snell’s method
\(\widehat{\boldsymbol\theta}_{CMLE}=\widehat{\boldsymbol{\theta}}_{M L E}-\left(\begin{array}{c} \delta(\widehat{\lambda}) \\ \delta(\widehat{\phi}) \end{array}\right),\)
\(\widehat{\boldsymbol\theta}_{CMLE_{c}}=\widehat{\boldsymbol{\theta}}_{M L E_{c}}-\left(\begin{array}{c} \delta(\widehat{\lambda}) \\ \delta(\widehat{\phi}) \end{array}\right),\)
where \(\delta(\widehat{\lambda}) = \cfrac{1}{n}\cdot \frac{ N_{1} \big\rvert_{\boldsymbol\theta=\widehat{\boldsymbol\theta}_{MLE_{c}s}}}{D \big\rvert_{\boldsymbol\theta=\widehat{\boldsymbol\theta}_{MLE_{c}s}}} ~\text{and}~ \delta(\widehat{\phi}) = \cfrac{1}{n}\cdot \frac{ N_{2} \big\rvert_{\boldsymbol\theta=\widehat{\boldsymbol\theta}_{MLE_{c}s}} }{D \big\rvert_{\boldsymbol\theta=\widehat{\boldsymbol\theta}_{MLE_{c}s}}}\) with
\(\begin{aligned} D =& \left[ \left( \psi^{\prime} (\phi) - \frac{1}{(\lambda+ \phi)^2} \right) \left( \frac{\phi+1}{\lambda^2} - \frac{1}{(\lambda+\phi)^2} \right) - \left( \frac{1}{(\lambda+\phi)^2} + \frac{1}{\lambda} \right)^2 \right]^2,\\ \end{aligned}\) \(\begin{aligned} N_{1} =& \left( \frac{1}{(\lambda+\phi)^2} - \psi' (\phi) \right) \left[ \left( \frac{1}{(\lambda+\phi)^2} - \psi' (\phi) \right) \left( \frac{\phi + 1}{\lambda^3} - \frac{1}{(\lambda+ \phi)^3} \right) + \left( \frac{1}{\lambda} + \frac{1}{(\lambda + \phi)^2} \right) \left( \frac{1}{2 \lambda^2} + \frac{1}{(\lambda+\phi)^3} \right) \right]\\ &+ 2 \left( \frac{1}{\lambda} + \frac{1}{(\lambda+\phi)^2} \right) \left[ \left( \frac{1}{(\lambda+\phi)^2} - \psi' (\phi) \right) \left( \frac{1}{2 \lambda^2} + \frac{1}{(\lambda + \phi)^3} \right) - \left( \frac{1}{\lambda} + \frac{1}{(\lambda + \phi)^2} \right) \frac{1}{(\lambda + \phi)^3} \right] \\ &+ \left( \frac{1}{(\lambda + \phi)^2} - \frac{\phi + 1}{\lambda^2} \right) \left[ \left( \psi' (\phi) - \frac{1}{(\lambda+\phi)^2} \right) \frac{1}{(\lambda + \phi)^3} + \left( \frac{1}{\lambda} + \frac{1}{(\lambda + \phi)^2} \right) \left( \frac{1}{(\lambda + \phi)^3} +\frac{\psi'' (\phi)}{2} \right) \right],\\ \end{aligned}\)\(\begin{aligned} N_{2} =& \left( \frac{1}{(\lambda + \phi)^2} - \psi' (\phi) \right) \left[ \left( \frac{\phi+1}{\lambda^2} - \frac{1}{(\lambda + \phi)^2} \right) \left( \frac{1}{2 \lambda^2} + \frac{1}{(\lambda + \phi)^3} \right) + \left( \frac{1}{\lambda} + \frac{1}{(\lambda + \phi)^2} \right) \left( \frac{1}{(\lambda + \phi)^3} - \frac{\phi+1}{\lambda^3} \right) \right] \\ &- 2 \left( \frac{1}{\lambda} + \frac{1}{(\lambda + \phi)^2} \right) \left[ \left( \frac{1}{\lambda} + \frac{1}{(\lambda + \phi)^2} \right) \left( \frac{1}{2 \lambda^2} + \frac{1}{( \lambda + \phi)^3} \right) + \left( \frac{\phi+1}{\lambda^2} - \frac{1}{(\lambda + \phi)^2} \right) \frac{1}{(\lambda + \phi)^3} \right]\\ &+ \left( \frac{\phi+1}{\lambda^2} - \frac{1}{(\lambda+ \phi)^2} \right) \left[ \left( \frac{1}{(\lambda +\phi)^2} - \frac{\phi+1}{\lambda^2} \right) \left( \frac{1}{(\lambda + \phi)^3} + \frac{\psi'' (\phi)}{2} \right) - \left( \frac{1}{\lambda} + \frac{1}{(\lambda+\phi)^2} \right) \frac{1}{(\lambda + \phi)^3} \right]. \end{aligned}\)
Firth’s method
The estimator corrected by Firth’s method is obtained by solving modified likelihood equations given by \[\left(\begin{array}{l} \partial l / \partial \lambda \\ \partial l / \partial \phi \end{array}\right)-A \operatorname{vec}\left(K^{-1}\right)=\boldsymbol{0},\] where
\(\begin{aligned} A &= n\left[\begin{array}{cccc} \frac{\phi+1}{\lambda^{3}}-\frac{1}{(\lambda+\phi)^{3}}\quad&-\frac{1}{2 \lambda^{2}}-\frac{1}{(\lambda+\phi)^{3}} & \quad-\frac{1}{2 \lambda^{2}}-\frac{1}{(\lambda+\phi)^{3}} & -\frac{1}{(\lambda+\phi)^{3}}\quad\quad\\ -\frac{1}{2 \lambda^{2}}-\frac{1}{(\lambda+\phi)^{3}} & -\frac{1}{(\lambda+\phi)^{3}} & -\frac{1}{(\lambda+\phi)^{3}} & -\frac{1}{(\lambda+\phi)^{3}}-\frac{1}{2} \psi^{\prime \prime}(\phi) \end{array}\right],\\ K &= n\left[\begin{array}{cc} \frac{\phi+1}{\lambda^{2}}-\frac{1}{(\lambda+\phi)^{2}} & -\frac{1}{\lambda}-\frac{1}{(\lambda+\phi)^{2}} \\ -\frac{1}{\lambda}-\frac{1}{(\lambda+\phi)^{2}} & -\frac{1}{(\lambda+\phi)^{2}}+\psi^{\prime}(\phi) \end{array}\right]. \end{aligned}\)Bootstrap method \[\widehat{\boldsymbol{\theta}}_{B O O T}=\widehat{\boldsymbol{\theta}}-\widehat{\text {Bias}}=2 \widehat{\boldsymbol{\theta}}-\frac{1}{B} \sum_{b}^{B} \widehat{\boldsymbol{\theta}}^{* b},\] where \(\widehat{\boldsymbol{\theta}}^{* b}\)s are bootstrap replications.
For bias correction, the ‘bias\(\_\)cor’ argument should be
specified since the default is bias\(\_\)cor=NULL. Note, unlike Cox
and Snell’s method that works with both MLE and MLE\(_c\), Firth’s method and the bootstrap
method are only applicable to MLE and MLE\(_c\), respectively. If other estimators are
chosen with these correction methods, a default setting would be
automatically adopted. The number of bootstrap iterations can be
specified with boot_iter, while the default is 1000.
WL(fail_fiber, est_method = "MLE",bias_cor = "firth") # Firth's method WL(fail_fiber, est_method = "MLEc",bias_cor = "coxsnell") # Cox and Snell's method WL(fail_fiber, est_method = "MLEc",bias_cor = "boots") # The bootstrap methodInterval estimation
Asymptotic confidence intervals
MLE, MLE\(_{c}\), MME, and MME\(_{m}\) for the parameters of WL distribution achieve consistency and asymptotic normality. Thus, if the sample size is large enough, confidence intervals can be straightforwardly obtained with estimated asymptotic variances. \[\widehat{\lambda}\pm z_{\alpha/2}\cdot \frac{\widehat{\sigma_{1}}}{\sqrt{n}}\quad\text{and}\quad\widehat{\phi}\pm z_{\alpha/2}\cdot \frac{\widehat{\sigma_{2}}}{\sqrt{n}}\]
One problem is that the lower bound of estimated intervals can be negative. A confidence interval with negative lower bound may not be useful since the parameters for WL distribution must be positive. Therefore, in case of negative lower bounds, confidence intervals for \(\log (\boldsymbol{\theta})\) are first obtained by the delta method and then scaled back. That is, \[\widehat{\lambda}\cdot\operatorname{exp}\left(\pm z_{\alpha/2}\cdot\frac{\widehat{\sigma_{1}}}{\sqrt{n}\cdot\widehat{\lambda}}\right)\quad\text{and} \quad \widehat{\phi}\cdot\operatorname{exp}\left(\pm z_{\alpha/2}\cdot\frac{\widehat{\sigma_{2}}}{\sqrt{n}\cdot\widehat{\phi}}\right).\]
Bootstrap confidence intervals
When asymptotic distribution is not available or the sample size is insufficient, the bootstrap confidence interval can be a good alternative. The basic bootstrap confidence interval is simply given by \[\left(2\widehat{\lambda}-\widehat{\lambda}^{*}_{(1-\alpha/2)}\quad,\quad 2\widehat{\lambda}-\widehat{\lambda}^{*}_{(\alpha/2)}\right)\quad \text{and} \quad \left(2\widehat{\phi}-\widehat{\phi}^{*}_{(1-\alpha/2)}\quad,\quad 2\widehat{\phi}-\widehat{\phi}^{*}_{(\alpha/2)}\right),\] where \(\hat{\theta}^{*}_{(p)}\) denotes \(\left(100\cdot p\right)\%\) percentile of bootstrap realizations. Readers are referred to Chapter 5 of (Davison and Hinkley 1997) for more details. As in the asymptotic case, if the lower confidence limit is negative, confidence interval for \(\log (\boldsymbol{\theta})\) is first computed and scaled back.
The default arguments of WL function for interval
estimation are CI\(\_\)method="asymp",
CI\(\_\)scale="normal",
CI\(\_\)side="two", and
CI\(\_\)alpha=0.05. If any bias
correction method is used, CI\(\_\)method="boot" is
automatically chosen. CI\(\_\)scale="exp" is recommended
in the case of a negative lower confidence limit and one-sided intervals
can be obtained by choosing CI\(\_\)side="one".
WL(fail_fiber,est_method="MLEc")$CI_list #asymptotic CI for MLEc
WL(fail_fiber,est_method="MLE")$CI_list #asymptotic CI for MLE
# Bootstrap CI for MLEc corrected by Cox and Snell's method
WL(fail_fiber,est_method = "MLEc", bias_cor = "coxsnell")$CI_listLikelihood ratio test (LRT)
Let \(X_{1}, \cdots, X_{n}\) be a
random sample from \(f(x \mid \theta),
\boldsymbol{\theta} \in \Theta,\) where \(\boldsymbol{\theta}=\left(\lambda,\phi\right)^{t}\),
\(\Theta \subset
\mathbb{R}^{+}\times\mathbb{R}^{+}\) and \(f(\boldsymbol{x}\mid\theta)\) is the pdf of
WL distribution. To test \(H_{0}:
\boldsymbol{\theta} \in \Theta_{0}\) vs. \(H_{1}: \boldsymbol{\theta} \in
\Theta_{0}^{c},\) we reject \(H_{0}\) if \[2
\log (R_{n}) =2 \log \left( \frac{\sup _{\Theta}
l_{n}(\boldsymbol{\theta} \mid \boldsymbol{x})}{\sup _{\Theta_{0}}
l_{n}(\boldsymbol{\theta} \mid \boldsymbol{x})} \right) >
\chi_{\alpha}^{2}(2),\text{ where } l_{n}(\boldsymbol{\theta} \mid
\boldsymbol{x}) = \prod\limits_{i}
f(x_{i}\mid\boldsymbol{\theta}).\] The LRT is automatically
implemented in the WL function unless
wilks\(\_\)test="FALSE" is selected.
The significance level and types of null hypotheses are set by
wilks\(\_\)alpha and
wilks\(\_\)side. The default setting
is wilks\(\_\)alpha=0.05 and
wilks\(\_\)side="two".
The LRT can be separately conducted with the wilks.test
function apart from other inferences. This function is especially useful
when testing several possible values or regions. By specifying arguments
estimator and side, both simple and composite
null hypotheses can be tested.
H\(_{0}\): (\(\lambda,\phi\)) = (\(\lambda_{0},\phi_{0}\)) vs H\(_{1}\): not H\(_{0}\).
wilks.test(fail_fiber,estimator = MME_WL(fail_fiber),side = "two")H\(_{0}\): \(\lambda\geq \lambda_{0}\) and \(\phi\geq\phi_{0}\) vsH\(_{1}\): not H\(_{0}\).
wilks.test(fail_fiber,estimator=c(1,1),side="less")
Illustration - WL function
In this section, we provide an example for illustrating how to
conduct the aforementioned steps of statistical inference from the
beginning using the WL function. The WL
function returns an S3 object of class ‘WL’ which is assigned to
fiber in this example. For this object of class ‘WL’,
summary and plot functions are provided.
> data("fail_fiber")
> fiber = WL(fail_fiber,dist_test = "ks", est_method ="MLEc",
wilks_alpha=0.05, wilks_side="two")
> summary(fiber)
Data: fail_fiber
Data summary:
Mean: 2.244, Variance: 0.1728
Min 1st Qu Median 3rd Qu Max
1.339 1.931 2.272 2.558 3.174
Kolmogorov-Smirnov test for alpha=0.05
D = 0.0723, p-value: 0.8864
>> This data follows the weighted Lindley distribution with estimated parameters.
Estimation method (bias correction): MLEc(None)
lambda: 12.9318, phi: 28.3329
Variance of lambda & phi:
Var(lambda)= 5.1126, Var(phi)= 25.2938
Two-sided confidence interval for 95%
(Asymptotic method & Normal scaled )
CI for lambda: (8.5001, 17.3635)
CI for phi: (18.4757, 38.1902)
Two-sided Wilks' theorem test for estimated parameters
X = 0.0024, p-value: 0.9988
>>The null hypothesis cannot be rejected.The simple descriptive statistic is provided by
summary(fiber), followed by the result of the goodness of
fit test. This data set can be assumed to originate from WL distribution
since the null hypothesis of the KS test is not rejected. We chose
MLE\(_c\) without bias correction since
the sample size of 65 is moderate, which allows us to use asymptotic
variance to construct the confidence interval.
Variance of lambda & phi in the output means estimated
asymptotic variances obtained by replacing \(\left(\lambda,\phi\right)\) with \(\left(\widehat{\lambda},\widehat{\phi}\right)\)
in the variance formulas. Finally, the result of LRT with \(H_{0}:\text{ }\lambda =
\widehat{\lambda}_{MLE_{c}}\text{ and }\phi =
\widehat{\phi}_{MLE_{c}}\) is given.
To obtain some helpful plots, we use plot function. Four
plots are returned by plot(fiber) as per Figure 1: the histogram with estimated density
function, the boxplot for detecting outliers, QQ plot, and contour plot
with point estimates. The first three plots provide insight on how much
WL distribution is suitable for the given data, along with the goodness
of fit tests. The last contour plot provides the surface of log
likelihood near point estimates. Note that the density function in the
histogram and the quantiles for the QQ plot are calculated based on
estimated parameters.
Summary
We presented an R package WLinfer that implements a goodness of fit test, several types of point estimation, bias correction, interval estimation, and the likelihood ratio test, and provide some useful plots. We supply a set of simple codes and an illustrative example of how to apply this package in practice. This package could practically assist practitioners, removing the need to make codes from scratch.
Acknowledgments
The corresponding author’s research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education(2018R1D1A1B07045603) and a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (2021R1A4A5032622).