key: cord-0448310-ppn7l44x
authors: Singh, Bhupendra; Agiwal, Varun; Singh, Ravindra Pratap; Tyagi, Abhishek
title: Discrete Teissier distribution: properties, estimation and application
date: 2021-10-20
journal: nan
DOI: nan
sha: 679d2115e7f52b1ba80708cf34422ea50a799a95
doc_id: 448310
cord_uid: ppn7l44x

In this article, a discrete analogue of continuous Teissier distribution is presented. Its several important distributional characteristics have been derived. The estimation of the unknown parameter has been done using the method of maximum likelihood and the method of moment. Two real data applications have been presented to show the applicability of the proposed model.

The concept of discretization generally arises from the field of survival/reliability analysis when it becomes impossible or inconvenient to measure the life length of a product/unit on a continuous scale. Few real examples where lifetime is to be recorded on a discrete scale rather than on continuous are the survival times for those suffering from the diseases like lung cancer or period from remission to relapse may be recorded as the number of days, the number of the cycle before the first failure when device work in cycle, the number of to and fro motions of a pendulum or spring device before resting, the number of times the device is switched on/off, modelling probability distribution of count data etc. These are some practical situations that have to catch the eyes of many researchers and hence motivate them to find more plausible discrete distributions to model discrete data arising from various real-life situations.

Discretization of the existing continuous distribution can be done using different methodologies [see Chakraborty, 2015] . Out of these, one widely used methodology is described as follows:

If the underlying random variable Y has the survival function (SF) ( ) ( ) Y S y P Y y   then the random variable [ ] X Y  , largest integer less than or equal to Y will have the probability mass function (PMF)

( ) ( 1); 0,1, 2,3,... X X Y y P y X y S y S y y

One of the important virtue of this methodology is that the developed discrete distribution retains the same functional form of the SF as that of its continuous counterpart. Due to this feature, many reliability characteristics of the distribution remain unchanged.

Over the last two decades, this approach has gotten a lot of attention. Using this technique, Roy (2003) gave a discretized version of the normal distribution. Following this, Roy and showed the applicability of the proposed family to count data sets.

In this paper, we have proposed the discrete analogue of the Teissier model [Teissier, 1934] , in the so-called discrete Teissier (DT) distribution using the survival discretization method.

The rest of the article is organized as follows: Section 2 introduces the one-parameter DT distribution. In Section 3 some important distributional and reliability characteristics are studied.

In section 4, we estimate the parameter of DT distribution by the method of maximum likelihood and method of the moment. Two real data illustrations are presented in Section 5. Finally, some concluding remarks are given in Section 6.

If X follows univariate continuous Teissier distribution with parameter  then its probability density function (PDF) and SF can be written as

Using the survival discretization approach (1), the discrete Teissier (DT) distribution can be obtained as ( 1) [ 

After re-parametrization exp( )    , the PMF in (4) can be written as ( 1) [ ] exp(1) (exp( ) exp( )); 0,1, 2,..., 1.

The cumulative distribution function (CDF) corresponding to PMF (5) 

The PMF plots of the DT distribution for different parametric values are shown in Figure 1 . 

The th r raw moments say / r  of the DT distribution can be obtained by using

Using Equation (7), the first four raw moments of the DT distribution are

The variance of the DT distribution is

The moment generating function (MGF) for the proposed model is

The index of dispersion (IOD) in the case of the proposed model is

The coefficient of variation (CV) for DT distribution can be obtained as

Due to the non-closure form of the above expressions, we use R software to demonstrate these characteristics numerically.  The proposed model is appropriate for modeling leptokurtic and platykurtic data sets.

 The DT distribution can be used to analyze over-dispersed, under-dispersed, and equidispersed data sets.

 As the value of  rises, the CV tends to increase. 

Entropy is a crucial measure of uncertainty and has many applications in applied fields. One of the important entropy is Rényi entropy (RE) (see, Rényi, 1961 

The SF and hazard rate function (HRF) of the DT distribution is respectively given by, 

The mean residual life (MRL) function is used extensively in a wide variety of areas, including reliability engineering, survival analysis, and biomedical research since it represents the ageing mechanism. It is well known that the MRL function characterizes the distribution function F uniquely since it contains all of the model's information. In discrete setup, the MRL, symbolized by ( ) m i , can be defined as

If Y has DT distribution with parameter  , then the MRL function of Y is

The expected inactivity time function or mean past life function (MPL), denoted by * ( ) m i , measures the time elapsed since the failure of X given that the system has failed sometime before 'i'. It has many applications in a wide variety of areas, including reliability theory and survival analysis, actuarial research, and forensic science. In discrete setup, MPL function is defined as

By replacing the CDF (6) in the expression of * ( ) m i , we can easily obtain the MPL for the proposed model.

The stress-strength (S-S * ) analysis is widely applicable in various areas including engineering, medical science, psychology etc. The probability of failure is based on the probability of S exceeding S * . Suppose that the domain of S and S * is positive, then the stress-strength reliability (R) can be computed as 

( 1)

The corresponding PMF of th r order statistics is

Particularly, by setting 1 r  and r n  in Equation (14), we can obtain the PMF of minimum

In this section, the property of infinite divisibility of the DT distribution is examined. This property is critical in the theorems of probability theory, modelling problems, and waiting time 

In this section, we address the problem of estimation through the method of maximum likelihood and the method of moment estimation.

Y be a random sample (RS) of size n with mean y , then the likelihood-function (LF) for DT distribution can be written as

The log-likelihood (LL) function can be represented as 

Taking the partial derivative of the LL function with respect to the parameter, we get the following normal-equation,

The ML estimator of  can be found by simplifying Equation (17), but unfortunately, this equation does not yield an analytical solution. Therefore, we use an iterative approach such as Newton-Raphson (NR) to calculate the estimate computationally.

In this estimation process, firstly, we equate population moment(s) to the corresponding sample moment(s) and then solve this equation for the unknown parameter(s). In our case, the concerned equation is

where y represents the mean based on the RS 1 2 , ,..., n y y y drawn from the DT distribution (5).

We can obtain the MOM estimator ˆM OM  , by solving Equation (18) for  . Since Equation (18) does not provide the MOM estimator of  in explicit form, so we can use numerical methods to compute ˆM OM  .

In this part, we use two real data sets to demonstrate the relevance of the DT distribution. The fitting capability of the proposed model has been compared to the models listed in Table 3 . This data set is modelled with DT and other competitive models listed in Table 3 . For ease of fitting, data have been divided by 10,000 and their floor values have been stored. Table 4 contains the estimated parameters and their corresponding standard errors (SEs) as well as the various fitting measures discussed earlier. From Table 4 , we conclude that the DT model is the best-performed model among others since it has the lowest values of AIC, BIC, CAIC, HQIC, and K-S test statistics with the highest p-value. We have plotted the -LL and CDF plots in Figure   3 (upper left and upper right panel). This figure not only confirms the unique existence of the MLE but also portrays that the fitted CDF closely follow the pattern of the empirical CDF for the considered data. Table 5 consists of MLE and MOM estimates with their SEs. To compare these methods, the KS statistics with associated p-values are also provided in Table 5 . From Table 5 , we can easily observe the MLE perform better as compared to MOM since MLE has lesser K-S statistics, SE and higher p-value. The second data set (II): This data set gives the survival times of a group of laboratory mice, which were exposed to a fixed dose of radiation at an age of 5 to 6 weeks [Lawless, 2011, pp. 445 ]. This group of mice lived in a conventional lab environment. The above data set is modelled with DT and DW, DR, PB, DsLi, DPL, Geo, DB, DPa models.

The estimated parameters and other fitting measures are reported in Table 6 . From the outcomes of Table 6 , we conclude that the DT distribution is the best choice among other competitive models since it has the lowest values of -LL, AIC, BIC, CAIC, HQIC, and K-S statistics with the highest P-value. Figure 3 (lower left and lower right panel) also depicts that DT distribution has a unique MLE for the given data and it is well enough to model this data. Table 7 consists of MLE and MOM estimates with their SEs, and K-S statistics with associated p-values. From Table 7 , we can easily observe the MLE perform better as compared to MOM since MLE have lesser K-S statistics, SE and higher p-value. Figure 3 . The -LL and CDFs plots for data set I and II.

In this article, a new one-parameter discrete distribution so-called discrete Teissier distribution is obtained. Its several impressive features have been discussed. The classical estimation using the method of maximum likelihood and method of moment is performed. Finally, the fitting capability of the proposed model using two real data sets is demonstrated. In future, we will develop its bivariate extension.

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The last author is thankful to the Department of Science and Technology, India for providing