The purpose of this study is to examine the finite sample aspects of estimates of the parameters of the weighted Lindley distribution derived by four estimation methods: maximum likelihood, method of moments, ordinary least squares, and weighted least squares, using Monte Carlo simulations. As a comparison criterion, bias and mean-squared error are used. In both small and large samples, the Cramer-von Mises approach is found to be very competing with the maximum likelihood method. To substantiate the conclusion, a statistical analysis of a real data set related to weather is performed.

Keywords: modified Lindley distribution, method of moments, maximum likelihood, bias, mean square error, weighted models

In modelling data related to reliability, one of the most sought out distribution is the Lindley distribution. Lindley distribution was studied by Lindley,¹ with an advantage of having a single parameter, with an increasing hazard rate function (hrf). The Lindley distribution is a linear combination of the Gamma (2, β) and exponential (β) distribution.

Let Z be a random variable following the Lindley distribution with parameter β. The probability density function (pdf) of the Lindley distribution is defined by Lindley¹ as follows

$g(z)=\frac{{\beta }^2}{1+\beta }\left(1+z\right)e^{-\beta z}$ , $z>0$ ,  $\beta >0$

The distribution function (df) of the Lindley distribution is also defined as follows

$G(z)=1-\left[1+\frac{\beta z}{1+\beta }\right]e^{-\beta z}$ , $z>0$  ,$\beta >0$  

Ghitany² discussed the application of the Lindley distribution with a real-world dataset, after which the distribution became popular. Many generalizations of the Lindley distribution were developed. We redirect the readers to Tomy³ and Chesneau⁴ to get a better understanding of the different generalizations and applications of the Lindley distribution.

Among which the modified Lindley (ML) distribution developed by Chesneau⁵ is of prime importance. Suppose Y is a random variable following the ML distribution with parameter β, then the pdf of ML distribution is defined as follows

$g\left(y\right)=\frac{\beta }{1+\beta }{e}^{-2\beta y}\left[\left(1+\beta \right)e^{\beta y}+2\beta y-1\right]$ , $y>0$ , $\beta >0$   (1)

The df of the ML distribution are also defined as follows,         

$G(y)=1-\left[1+\frac{\beta z}{1+\beta }{e}^{-\beta y}\right]e^{-\beta y}$   , $y>0$ ,   $\beta >0$

The mean and variance of ML distribution is given as follows,

Figure 1 Illustrates the plots of the ML distribution with various values of the parameter β.

Observations –

The shapes of the pdf plots are unimodal, and are decreasing in nature.

In this paper, we discuss the different methods to estimate the unknown single parameter β of the ML distribution. A simulation study is also conducted to compare the efficiency of different estimators.

The structure of the paper is as follows. Section 2 discusses the different methods of estimating β. In Section 3, we compare the efficiency of the estimates produced by the different methods of estimation. Section 4 shows the application of different estimation methods over a real-world dataset.

Inferential aspects

In this section, we concentrate on the estimation of parameter β, with an assumption that β is unknown. Here, we consider estimating the parameter of the ML distribution using the methods, which include Maximum Likelihood (ML), Method of Moment (MOM) estimation, least squares (LS) and Weighted Least squares (WLS) estimation, and Cram´er-von Mises (CVM) estimation.

ML estimation

Consider (y₁, y₂, ..., y_n) to be a random sample from the ML distribution with parameter , with pdf defined in Equation (1). The log-likelihood (LL) function is given by

The ML estimate of β, $\stackrel{⌢}{\beta }$  is defined as

The ML estimate of β can be obtained by taking the derivative of LL(β) and equating it to zero, which is as follows:

As there doesn’t exist any explicit form of β, numerical methods are used to get the estimate of.

MOM estimation

Let (y₁, y₂, ..., y_n) represent n independent ML distribution observations. Chesneau⁵ showed that the unique MoM estimate of the pdf defined in Equation, , of  is given by

$f\left(z\right)=\frac{-\left(z-1\right)+\sqrt{{(z-1)}^2}+5z}{2z}$   $z>0$ ,

LS and WLS estimation

Swain⁶ devised the method of least squares. The difference between both the vector of uniformized order statistics and the associated vector of expected values is reduced using this approach. Let (y₁, y₂, ..., y_n) represent a random sample from the ML distribution with parameters. Also, let y_(1), y_(2), … y_(n) be the order values of y₁,y₂,...,y_n in ascending order.

The LS estimate of  is defined by

After partially differentiating with regard to unknown parameter β, one can construct a nonlinear equation using the df of the ML distribution. The Monte Carlo simulation can be used to find the solution to this nonlinear equation.

The WLS function is given as follows:

where     ${\eta }_i=\frac{{\left(n+1\right)}^2\left(n+2\right)}{[i\left(n-i+1\right)]}$

The WLS estimate of β is defined by

As a result, the WLS estimate of can be derived using a process similar to the LS estimate.

CVM estimation

The CVM approach is identical to the two methods earlier described. The CVM function is described as follows:

The CVM estimate of β is defined by

As a result, the CVM estimate of follows the same approach as the WL or LS estimations.

Simulation study

We conduct a simulation analysis in this part to assess the efficiency of the ML model parameter estimates reported in Section 3. We employed the Monte Carlo technique in R software, as well as Newton's method and the BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm, developed by Broyden⁷, Fletcher⁸, Goldfarb⁹, & Shanno¹⁰.

The bias and Mean Square Error (MSE) were examined in the study,

Bias $\left(\widehat{\beta }\right)=\frac{1}{N{\displaystyle {\sum }_{j=1}^N\left({\widehat{\beta }}_j-\beta \right)}}$   and $MSE(\widehat{\beta })=\frac{1}{{\displaystyle {\sum }_{j=1}^N{\left({\widehat{\beta }}_j-\beta \right)}^2}}$ .

where, for each sample size n, $\tilde{\beta }$_J is the parameter's estimate at the j^th iteration using a specific estimation method. R software, version 4.0.5, was used for all calculations.

Tables 1–3 show the biases and mean square error (MSE) of the proposed model's parameters based on N = 5000 replicates. To further analyse the nature of the estimates, various sample sizes (n = 50, 100, 200, and 500) and values of the parameters, β= (0.5,1,1.5) are employed.

Some observations from Tables 1-3,

1. The estimators of all the parameters are positively biased, except for the WLS estimator.
2. The biases of all the estimators tend to zero for large values of the sample size n.
3. WLS has the largest MSE among the considered four estimators.
4. The CVM estimator has the least bias among all the other estimators.

Real data analysis

We use a well-known real data set to demonstrate the ML model's use in real-life circumstances. To accomplish this, we compare the estimates obtained through the different estimation methods of the ML model . Standard performance validation criteria such the goodness-of-fit testing statistic such as the Kolmogorov Smirnov (K-S), also denoted by D_n, are used to discover the best model.

The p-values for the KS test statistic are also taken into account. The best method of estimation has the smallest KS, as well as the highest p-values. We refer the readers to Kenneth and Anderson¹¹ for definitions and more insights on these attributes. The data set includes 30 measurements of precipitation in March in Minneapolis, with inch as the unit of measurement. Hinkley¹² has provided this data set.

Table 4 contains the parameter estimates obtained using various estimation methods for the ML distribution in order to complete the comparison of the estimation methods of the ML distribution.

We use the essential metrics to compare the four estimation approaches for the weighted Lindley distribution:

• The K-S test with the highest p-value (which considers the greatest difference between the theoretical and empirical distributions)

Table 4 demonstrates that the CVM approach meets the above-mentioned criteria, namely, the highest p-values for the K–S test. As a result, we can conclude that the CVM estimation approach is the best appropriate among the five estimate methods for the presented data set.

In this study, we compared the estimate of the parameters of the weighted Lindley distribution using four well-known estimation methods: maximum likelihood, method of moments, least-squares, weighted least-squares, and Cramer-von Mises, utilising extensive simulation trials. In comparison to the maximum likelihood method, simulations reveal that Cramer-von Mises is a very competing method. This is backed up by the use of a real-world climatic data set.

None.

The authors declare that there are no conflicts of interest.

1. Lindley DV. Fiducial distributions and Bayes theorem. Jour­nal of the Royal Statistical Society. 1958;20(1):102–107.
2. Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and its application. Mathematics and computers in simulation. 2008;78(4): 493–506.
3. Tomy L. A Retrospective study on Lindley distribution. Biom Biostat Int J. 2018;7(2):163–169.
4. Chesneau C, Tomy L, Jose M. A Retrospective Study on Applications of the Lindley Distribution. Journal of Non-linear modelling and Analysis. 2022;4(2):291–309.
5. Chesneau C, Tomy L, Gillariose J. A new modified Lindley distribution with properties and applications. Journal of Statistics and Management Systems. 2021;24(7):1383–1403.
6. Swain JJ, Venkatraman S, Wilson JR. Least-squares estimation of distribution functions in Johnson’s translation system. J Stat Comput Simul. 1988;29(4):271–297.
7. Broyden CG. The convergence of a class of double-rank minimization algorithms 1. general considerations. IMA J App Math. 1970;6:76–90.
8. Fletcher R. A new approach to variable metric algorithms. Comput J. 1970;13(3):317–322.
9. Goldfarb D. A family of variable-metric methods derived by variational means. Math Comput. 1970;24(109):23–26.
10. Shanno DF. Conditioning of quasi-Newton methods for function minimization. Math Comput. 1970;24(111):647–656.
11. Kenneth PB, Anderson DR. Model Selection and Multi-Model Inference. A Practical Information-Theoretic Approach. 2nd edn. 2002.
12. Hinkley D. On quick choice of power transformations. Ap­plied Statistics. 1977;26(1):67–69.