Monte Carlo comparison of estimation methods for the two-parameter lognormal distribution

. In the paper we compare performance of estimation methods for the two-parameter lognormal distribution via the Monte Carlo simulation. The comparison of performances is made with respect to their biases, variances, root mean square error. The methods are applied on real data set representing experimentally obtained values of ultimate tensile strength of material.


Introduction
It is well known that the lognormal distribution is frequently used in areas where the data tend to be right skewed, such as physics, economics, biostatistics, survival analysis, wireless communication, quality control, reliability analysis, material properties, fatigue life of materials, strength of some materials, etc. Various applications of the lognormal distribution are described in [1][2][3][4].
The aim of this paper is to present methods for estimation of the parameters of lognormal distribution. Here, three estimation methods are used, namely the maximum likelihood method (MLM), the method of moments (MOM) and the Finney method (UMVUE). The performance of these methods is compared using the Monte Carlo simulation study. The comparison of the estimation methods is based on the biases and the root mean square error (RMSE). For the purpose of simulation, mathematical software MATLAB R2017b is used.
The rest of the paper is organized as follows: in Section 2 definition and characteristics of the lognormal distribution are summarized. In Section 3 the estimation methods are briefly introduced. In Section 4 we present results of the Monte Carlo simulation study. The real data application is provided in Section 5 and finally conclusion appears in Section 6.

Lognormal distribution
The random variable X is said to have the 2-parameter lognormal distribution LN(  with parameters  and   if its probability density function (PDF)   The mean ) (X E and the variance ) ( X Var are given by The lognormal distribution is a right skewed distribution with a long tail. Figure 1 illustrates the PDF curves for different parameter values.

Fig. 1. PDF of the lognormal distribution for different parameter values
There is a close relationship between the lognormal and the normal distribution. If X is the random variable that has lognormal distribution LN(  , then the random variable Y = ln X has normal distribution N(  ).

Methods of estimation
In this section, we briefly describe the methods for estimation of parameters  and   of the lognormal distribution LN(  

Maximum likelihood method
The maximum likelihood method (MLM) is one of widely used parameter estimation methods. It is defined as follows: Let  be a realization of a random sample of size n from the lognormal distribution LN(  ). The likelihood function is given by Taking the natural logarithm of the likelihood function (6) we obtain The MLM estimates  and 2  of parameters and   which maximize (7) are given by  , respectively, are given by where  and 2  are given by equations (12), (13).

Finney method
Finney [5] defined the series and obtained the uniformly minimum variance unbiased estimators (UMVUE) M of the mean ) (X E and Vˆof the variance Depending on n and s 2 , 6 to 10 terms must be evaluated to achieve stability. Thus, the estimates  and 2  of the parameters  and    are  ,

Simulation study
Numerical simulations provide many benefits when modelling in science and technology. They are used for checking various scenarios, for comparison of different approaches or methods, or in first steps of designing devices (see, for example, [6][7][8][9]).
We use the Monte Carlo simulation for comparison of the performances of the chosen estimation methods for the lognormal distribution parameters. In simulation we consider respectively. For comparing the performances of the methods we consider the bias given by respectively, and the sample root mean square error (RMSE) given by respectively.
Ideal value of RMSE is close to zero. The methods with smaller RMSE are preferred. When RMSE values of two methods are close to each other, the method with less bias is preferred.   -It can be seen that in all considered cases each method has negative bias for parameter   

Real data example
Engineering products are often subject to loads when they are used in specific applications. The mechanical properties of metals determine the range of applicability and establish the expected service life. Additionally, mechanical properties provide knowledge how materials deform (elongate, compress, twist), or break as a function of applied load, time, temperature, and other conditions. The most common mechanical properties, useful for industry, are ultimate tensile strength, ductility, hardness, impact resistance, and fracture toughness. Characteristics that indicate the elastic or inelastic behaviour of a material under pressure from tensile tests are ultimate tensile strength, yield stress, ductility, toughness and elongation. The maximum stress the material can withstand before fracture is known as ultimate tensile strength (UTS). The UTS represents maximal stress (load) which material withstands without fracture, to original cross section area before testing. Chemical composition, microstructure of material and others influence mechanical properties of material (see, e.g. [10,11]). In this section we apply estimation methods, described in Section 3, to find estimates of parameters of the lognormal distribution LN(  ) for data set representing UTS. As a testing material, common steel 1.0036 was used. The tensile tests were conducted on shredder machine ZDM 30. There were used round specimens with specified dimension according standard STN EN ISO 6892-1. The specimen was axially loaded; the stress (load) was being increased at a uniform rate until the fracture of the specimen. Stress and elongation were recorded during test.
The following data set represents the UTS (in MPa) from 20 tests of experimental material.  Table 5 gives the descriptive statistics for the tensile strength data. The value of indicates that the empirical distribution is skewed to the right. Values in Table 6 indicate that all three methods provide similar results. The UMVUE is only slightly worse than the MLM and MOM. Figure 2 shows the fitted PDF versus the histogram of the tensile strength for all three estimation methods.

Conclusion
In the paper, three methods for parameter estimation of lognormal distribution were compared via the Monte Carlo simulation study. To compare the methods, we used bias and root mean square error. According to results, we may conclude that the maximum likelihood method (MLM) and the Finney method (UMVUE) are suitable for estimation of both,  and   , parameters of lognormal distribution. To estimate parameter , MLM can be preferred.
For parameter   estimation, UMVUE provides better results when   is small  