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 the `WL`

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.

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 et al. 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 et al. 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 et al. 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.

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.

*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
(Mazucheli et al. 2013; Ghitany et al. 2017; Kim and Jang 2020) 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.

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
(Mazucheli et al. 2013; Wang and Wang 2017; Kim and Jang 2020). 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 method `

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, \[\widehat{\lambda}\cdot\operatorname{exp}\left(\pm z_{\alpha/2}\cdot\frac{\widehat{\sigma_{1}}}{\sqrt{n}\cdot\widehat{\lambda}}\right)\quad\text{and} \quad \widehat{\phi}\cdot\operatorname{exp}\left(\pm z_{\alpha/2}\cdot\frac{\widehat{\sigma_{2}}}{\sqrt{n}\cdot\widehat{\phi}}\right).\]

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_list
```

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")`

`WL`

functionIn 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)
: fail_fiber
Data
:
Data summary: 2.244, Variance: 0.1728
Mean1st Qu Median 3rd Qu Max
Min 1.339 1.931 2.272 2.558 3.174
-Smirnov test for alpha=0.05
Kolmogorov= 0.0723, p-value: 0.8864
D >> This data follows the weighted Lindley distribution with estimated parameters.
method (bias correction): MLEc(None)
Estimation : 12.9318, phi: 28.3329
lambda
& phi:
Variance of lambda Var(lambda)= 5.1126, Var(phi)= 25.2938
-sided confidence interval for 95%
Two& Normal scaled )
(Asymptotic method for lambda: (8.5001, 17.3635)
CI for phi: (18.4757, 38.1902)
CI
-sided Wilks' theorem test for estimated parameters
Two 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.

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.

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).

WLinfer, mle.tools, fitdistrplus, LindleyR

ActuarialScience, Distributions, Survival

This article is converted from a Legacy LaTeX article using the texor package. The pdf version is the official version. To report a problem with the html, refer to CONTRIBUTE on the R Journal homepage.

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Text and figures are licensed under Creative Commons Attribution CC BY 4.0. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".

For attribution, please cite this work as

Jang, et al., "WLinfer: Statistical Inference for Weighted Lindley Distribution", The R Journal, 2023

BibTeX citation

@article{RJ-2022-042, author = {Jang, Yu-Hyeong and Kim, SungBum and Jung, Hyun-Ju and author), Hyoung-Moon Kim (Corresponding}, title = {WLinfer: Statistical Inference for Weighted Lindley Distribution}, journal = {The R Journal}, year = {2023}, note = {https://rjournal.github.io/}, volume = {14}, issue = {3}, issn = {2073-4859}, pages = {13-19} }