1 Introduction
Statistical models are very useful in describing and predicting realworld phenomena. Recent developments focus on defining new families that extend wellknown distribution and at the same time providing greater flexibility in modelling data in practice. Many well known lifetime distributions for modelling lifetime data such as exponential, Gamma, Weibull, etc. have been extensively studied.
Let
is a random variable taking values
. So the distribution ofmay be absolutely continuous or discrete. The probability density function (PDF) and the cumulative distribution function (CDF) of the Lindley distribution [see, Lindley (1958)] are given by
(1.1) 
and
(1.2) 
respectively.
The exponential distribution is closed in form to the Lindley distribution given in . The PDF and the CDF of the exponential distribution are given by
(1.3) 
and
(1.4) 
Many of the mathematical properties (e.g., the mode of the distribution, moments, skewness and kurtosis measures, cumulants, failure rate and mean residual life, mean deviation, entropies, etc.) are more flexible than those of the exponential distribution. For the exponential distribution, some of the properties are constant, usually not appropriate assumptions in reality whereas for the Lindley distribution there is scope to vary [see, Ghitany et al. (2008)].
The Lindley distribution is one way to describe the lifetime of a process or device. It can be used in a wide variety of fields, including biology, engineering and medicine. Ghitany et al. (2008) fitted this distribution to the waiting times for getting service of bank customers data. Ghitany et al. (2011) stated that it is especially useful for modeling in mortality studies. Mazucheli and Achcar (2011) discussed the applications of the Lindley distribution to competing risk lifetime data. Mukherjee and Maiti (2014) has used this distribution for constructing an acceptance sampling plan for the variable. Maiti et al. (2014) applied it in the context of describing a new process capability index.
It has been generalized by a host of authors. To mention a few, Zakerzadeh and Dolati (2009), Bakouch et al. (2012), Shanker and Ghebretsadik (2013), Elbatal et al. (2013), Ghitany et al. (2013) among others. Bouchahed and Zeghdoudi (2018) has proposed a new and unified approach in generalizing the Lindley’s distribution. They investigated some structural properties like moments, skewness, kurtosis, median, mean deviations, Lorenz curve, entropies and limiting distribution of extreme order statistics; reliability properties like reliability function, hazard rate, stressstrength reliability, stochastic ordering; and estimation methods like the method of moment and maximum likelihood. We call the proposed distribution of Bouchahed and Zeghdoudi (2018) as the one parameter polynomial exponential (OPPE) distribution.
The PDF of a random variable of the OPPE distribution can be written as
(1.5) 
where, ,
The distribution can also be written as
(1.6)  
where
is the PDF of a gamma distribution with shape parameter
and scale parameter and ’s are nonnegative constants. The distribution is a finite mixture of gamma distributions.The CDF is given by
(1.7) 
where .
Some special cases are as follows:

gives the exponential distribution,

gives the Lindley distribution,

gives the Akash distribution [c.f. Shanker (2015)],

gives the Aradhana distribution [c.f. Shanker (2016b)],

gives the Sujatha distribution [c.f. Shanker (2016e)],

gives the lengthbiased Lindley distribution [c.f. Ayesha (2017)],

gives the Amarendra distribution [c.f. Shanker (2016a)],

gives the Devya distribution [c.f. Shanker (2016c)],

gives the Shambhu distribution [c.f. Shanker (2016d)].
Statisticians are most of the times interested about inferring the parameter(s) involved in the distribution. Maximum likelihood estimator (MLE) and Bayes estimate of the parameter has been focused by the authors. Hardly any unbiased estimator of the parameter has been studied so far and finding out uniformly minimum variance unbiased estimator (UMVUE) of the parameter seems to be intractable and consequently the comparison with any unbiased class of estimators is not being made. However, instead of studying the estimators of the parameter(s), we have scope to find out unbiased estimator of the PDF and the CDF as well as biased estimator of the same and comparison between the estimators could be made. That is why we have shifted our focus from estimation of parameter(s) to estimation of the PDF and the CDF.
The estimators for the PDF, CDF or both can be used to estimate various functions, like differential entropy, Rényi entropy, KullbackLeibler divergence, Fisher information, reliability function, cumulative residual entropy, the quantile function, Bonferroni curve, Lorenz curve, probability weighted moments, hazard rate function, mean deviation about mean etc.
There are few works available relating to estimation of the PDF and the CDF of different probability distributions. References include; Asrabadi (1990), Dixit and Jabbari (2010) and Dixit and Jabbari (2011)  Pareto distribution; Alizadeh et al. (2015), Mukherjee et al. (2016)  generalized exponential distribution; Bagheri et al. (2014)  generalized exponentialPoisson distribution; Bagheri et al. (2016b)  Weibull extension model; Bagheri et al. (2016a)  exponentiated Gumbell distribution; Jabbari and Jabbari (2010)  exponentiated Pareto distribution; Maiti and Mukherjee (2018)  Lindley distribution; Tripathi et al. (2017b)  Generalized Logistic distribution; Tripathi et al. (2017a)  exponentiated moment exponential distribution; Mukherjee and Maiti (2019)  lognormal distribution.
Organization of the article is as follows. Section 2 deals with MLE of the PDF and the CDF of the OPPE distribution. Section 3 is devoted to finding out the UMVUE of the PDF and the CDF and their MSEs (in this case the variances). Particular case like lengthbiased Lindley and Sujatha distribution have been discussed in section 4. In section 5, simulation study results are reported and comparisons are made. Reallife data sets have been analyzed in section 6. In section 7, concluding remarks are made based on the findings of this article.
2 MLE of the PDF and the CDF
Let be random sample of size drawn from the PDF in . Here we try to find the MLE of which is denoted as . The loglikelihood of is given by
Now,
(2.8) 
Since, the MLE of is not of a closed form expression, we have to solve (2) numerically to obtain the MLE of . Theoretical expressions for the MSE of the MLEs are not available. MSE will be studied through simulation.
3 UMVUE of the PDF and the CDF
In this section, we obtain the UMVUE of the PDF and the CDF of the OPPE distribution. Also, we obtain the MSEs of these estimators.
Theorem 3.1.
Let . Then the distribution of is
with and .
Proof.
The mgf of is
Hence, the distribution of is
where . ∎
Lemma 3.1.
The conditional distribution of given is
where
and
with
Proof.
∎
Theorem 3.2.
Let be given. Then
(3.9)  
is UMVUE for and
(3.10)  
is UMVUE for , where is an incomplete beta function and .
Proof.
4 Particular case
In this section, we have studied in detail of the lengthbiased Lindley and Sujatha distribution that are particular cases of the OPPE distribution. The estimators of the PDF and the CDF are explicitly written, and their MSEs are compared. Another particular case of the OPPE distribution is the Lindley distribution. The estimation of the PDF and the CDF of the Lindley distribution has been studied in detail in Maiti and Mukherjee (2018).
4.1 Lengthbiased Lindley distribution
Substituting in and , we can have the PDF and the CDF of the lengthbiased Lindley distribution respectively.
The PDF is
(4.13) 
and the CDF is
(4.14) 
4.1.1 MLE of the PDF and the CDF
If we substitute in we will get the given expression as
Solving numerically the above mentioned equation, we can get the MLE of the lengthbiased Lindley distribution.
4.1.2 UMVUE of the PDF and the CDF
In this section, we obtain the UMVUE of the PDF and the CDF of the lengthbiased Lindley distribution. Also, we obtain the MSEs of these estimators.
To derive the UMVUE of the PDF and the CDF, we substitute in Theorem .
Theorem 4.1.
Let be given. Then the UMVUE of the PDF is
and the UMVUE of the CDF is
The MSE of the UMVUE of the PDF is given by
(4.15)  
Using Theorem 3.1, in , we can get the value of the MSE of UMVUE of the PDF.
And the MSE of the UMVUE of the CDF is given by
(4.16)  
Similarly, using Theorem 3.1, in , we can get the value of the MSE of UMVUE of the CDF.
Theoretical graph of the MSE of the UMVUE of the lengthbiased Lindley distribution is presented in Figure 1.
4.2 Sujatha distribution
By substituting in and , we will get the PDF and the CDF of the Sujatha Distribution respectively.
The PDF is
(4.17) 
and the CDF is
(4.18) 
4.2.1 MLE of the PDF and the CDF
If we substitute in we will get the given expression as
Solving numerically the above mentioned equation, we can get the MLE of the Sujatha distribution.
4.2.2 UMVUE of the PDF and the CDF
In this section, we obtain the UMVUE of the PDF and the CDF of the Sujatha distribution. Also, we obtain the MSEs of these estimators.
To derive the UMVUE of the PDF and the CDF, we use Theorem 3 and replace .
Theorem 4.2.
Let be given. Then the UMVUE of the PDF is
and the UMVUE of the CDF is
The MSE of the UMVUE of the PDF is given by
(4.19)  
Using Theorem 3.1, in , we can get the value of the MSE of UMVUE of the PDF.
And the MSE of the UMVUE of the CDF is given by
(4.20)  
Similarly, using Theorem 3.1, in , we can get the value of the MSE of UMVUE of the CDF.
Theoretical graph of the MSE of the UMVUE of the Sujatha distribution is presented in Figure 2.
5 Simulation
Direct application of Monte Carlo Simulation technique fails for generating random samples from the unified generalized Lindley, since the equation
cannot be explicitly solved in . On the other hand, one can use the fact that the distribution is a mixture of gamma distributions given in .
For the OPPE distribution the generation of random sample is distributed in the following algorithm:

Generate

If ,then set ,
where and if , then set , where
A simulation is carried out with repetitions. We choose , and for both distribution. We compute MSE of the MLE and UMVUE of the PDF and the CDF. From Figures 3 and 4, it is clear that MSE decreases with increasing sample size that shows the consistency property of the estimators.
6 Data Analysis
In this section, we provide the analysis of two real data sets for comparing the performances of MLE and UMVUE for the PDF and the CDF. Table 1 represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Elbatal et al. (2013). Table 2 represents the failure times of the air conditioning system of an airplane and it is obtained from Linhart and Zucchini (1986).
We fit lengthbiased Lindley distribution in data setI (Table 1). The corresponding graph of histogram and the estimated PDF and the CDF has been shown in Figure 5. From Figure 5, we observe that the lengthbiased Lindley distribution shows good fit for data setI.
Sujatha distribution have been fitted to data setII (Table 2). Here, for computational ease, we have divided the whole data set by . The graph of histogram and the estimated PDF and the CDF has been shown in Figure 6. From Figure 6, we observe that the Sujatha distribution shows good fit for data setII.
0.1  0.33  0.44  0.56  0.59  0.72  0.74  0.77  0.92 

0.93  0.96  1  1  1.02  1.05  1.07  1.07  1.08 
1.08  1.08  1.09  1.12  1.13  1.15  1.16  1.2  1.21 
1.22  1.22  1.24  1.3  1.34  1.36  1.39  1.44  1.46 
1.53  1.59  1.6  1.63  1.63  1.68  1.71  1.72  1.76 
1.83  1.95  1.96  1.97  2.02  2.13  2.15  2.16  2.22 
2.3  2.31  2.4  2.45  2.51  2.53  2.54  2.54  2.78 
2.93  3.27  3.42  3.47  3.61  4.02  4.32  4.58  5.55 
23  261  87  7  120  14  62  47  225  71 
246  21  42  20  5  12  120  11  3  14 
71  11  14  11  16  90  1  16  52  95 
Negative loglikelihood  

Estimators  Lengthbiased Lindley distribution 
MLE  95.81244 
UMVUE  95.7132 
Negative loglikelihood  

Estimators  Sujatha distribution 
MLE  15.10749 
UMVUE  15.44566 
Table 3 and 4 give the estimate of the negative loglikelihood values. Lower the value of negative loglikelihood indicates the better fit. From table 3 and 4, we see that the UMVUE and MLE is better in a negative loglikelihood sense, respectively.
7 Concluding Remarks
Two estimators  MLE and UMVUE, has been found out for the PDF and the CDF of the lengthbiased Lindley and Sujatha distribution. The estimators have been compared theoretically as well as through simulation study in MSE sense. UMVUE is better for the PDF and the CDF in MSE sense for the lengthbiased Lindley and Sujatha distribution.
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