Statistical Inference for Topp–Leone-generated Family of Distributions Based on Records
- 10.2991/jsta.d.190306.008How to use a DOI?
- Topp–Leone-generated family of distributions; ML estimators; Bayesian inference; symmetric loss function; asymmetric loss function; Bayes estimators; reliability function
In this paper, we consider a general family of distributions generated by Topp–Leone distribution (known as TL family of distributions) proposed by Rezaei et al. . We consider the problem of estimation of the shape parameter, scale parameter, and reliability function based on record data from TL family of distributions. We derive the maximum likelihood estimator (MLE) for shape parameter, scale parameter, and reliability function. We have also obtained UMVUE (uniformly minimum-variance unbiased estimator) for reliability function when scale parameter is known. A Bayesian study is carried out under symmetric and asymmetric loss functions in order to find the Bayes estimators for unknown parameters and reliability function. Further, we have predicted future record values using Bayesian approach. A numerical comparison of various estimators is also reported.
- © 2019 The Authors. Published by Atlantis Press SARL.
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
Topp–Leone (TL) distribution was first introduced by Topp and Leone . Later Nadarajah and Kotz  discussed this distribution elaborately by obtaining explicit algebraic expressions such as hazard rate function and moment, and so on. The density and distribution function (df) of TL distribution is given by
Since its emergence, many authors have studied different properties of TL distribution. We mention reliability measures and stochastic orderings Ghitany et al. ; distributions of sums, products, and ratios Zhou et al. ; behavior of kurtosis Kotz and Seier ; record values Zghoul ; moments of order statistics Genc .
Though probability distributions are very useful in practical problems but in some situations the available distributions do not support our problem appropriately. Then it becomes necessary to either define a new distribution or modify some existing distributions, so that they can be useful for various practical problems. This modification of probability distribution gives boost to generalization of distribution. From the last couple of years, we see that several authors have proposed various generated family of distributions. TL distribution is very useful and widely applicable distribution. But due to fact that it has only one parameter and its support is restricted to , it is not flexible. It cannot be used for lifetime modeling. So, the generalization of TL distribution is needed. One of the generalization of TL distribution is discussed by Al-Shomrani et al. . They have considered as the baseline df in TL distribution and obtained moments and hazard rate of the new TL family of distributions. Rezaei et al.  also generalized this TL distribution by using as the baseline distribution called TL-generated (TLG) family of distributions. The authors have explained some special cases of this distribution and also derived expressions of maximum likelihood estimators (MLEs) for unknown parameters.
In this paper, we consider TLG family of distributions proposed by Rezaei et al. . For this generated family of distribution, we consider baseline distribution where denotes an unknown scale parameter. Several well-known distributions can be used for the baseline distribution, for example, exponential distribution with df , Rayleigh distribution with df , and so on.
The density and df of TLG family of distributions is given by
Nowadays several researchers are interested in study of record data (extreme values) because of its application in various fields, such as in sports, the longest winning streak of a team, the highest runs of a player, lowest run given by a bowler in an over. In field of marketing; lowest stock market figure, minimum cost of a certain product in market. Medical sciences; most number of people affected by a disease at a particular place, and so on. In all these field of research, record data is widely used.
Chandler  introduced the idea of record values and studied some of its basic properties. After that many authors have worked in this field and gave their valuable inputs. For excellent understanding of records, one may refer to books written by Ahsanullah , Ahsanullah , and Arnold . For application of record values in various disciplines, one may refer to Minimol and Thomas , Ahsanullah , itekhan2016umvu, Bdair and Raqab , MirMostafaee et al. , Ahsanullah and Nevzorov , Arshad and Jamal , Arshad and Baklizi , Anwar , and Arshad and Jamal . Now we discuss the mathematical definition of records and its distribution.
Let be a sequence of independent and identically distributed (iid) random variables with an absolutely continuous df and probability density function (pdf) . An observation is called a lower record if its value precedes all previous observations, that is, is a lower record if for every . Let be lower records and let denote the observed values of , respectively. The density of record is given by
The joint density of and lower record is given by
The joint density of is given by
The remainder of the paper is as follows. In Section 2, we derive the expressions for finding out the MLEs for the unknown parameters. An example for finding out MLE is also provided. In Section 3, uniformly minimum-variance unbiased estimator (UMVUE) of the reliability function is derived when the scale parameter is known. In Section 4, a Bayesian study is carried out for obtaining the Bayes estimators for scale parameter, shape parameter, and reliability function under symmetric (squared error) and asymmetric (LINEX and entropy) loss functions. In Section 5, we provide Bayesian prediction interval for future records. Finally, in Section 6, a numerical study is provided to illustrate the results.
2. MAXIMUM LIKELIHOOD ESTIMATION
The likelihood function based on the lower records observed form TLG family of distributions is given by
Now taking log both sides, we get
Differentiating Eq. (3) with respect to , we get
In order to find MLE of , we will equate the above equation to and we have
Similarly, differentiating Eq. (3) with respect to and equating to , we get
The MLE of is a solution of the Eqs. (4) and (5). Because of the nonlinear nature of these equations, it is very cumbersome to obtain the numerical values of unknown parameters explicitly. So, we will use numerical computation techniques to obtain the MLEs for both the parameters and the reliability function, based on lower records obtained from TLG family of distributions. The corresponding MLE of the reliability function is obtained, after replacing and , respectively, by their MLEs and , obtained after solving Eqs. (4) and (5), that is, the MLE of the reliability function is given by
Let TLG family of distributions has baseline distribution as exponential distribution with df
For MLE of , Eq. (7) has to be solved, then MLE of can be obtained from Eq. (6), after putting value of obtained from Eq. (7). The numerical computation of the MLE of and and is illustrated in Section 6 (see Example 6.1).
3. UMVUE OF RELIABILITY FUNCTION
In this section, we derive the UMVUE of when the scale parameter is known (WLOG, assume ). For this, we need the following lemma. The proof of lemma is straightforward and is omitted. This lemma can be obtained from the Lemma 3.1 of Khan and Arshad .
Let be the first n lower records having joint pdf given in Eq. (2). Define . Then, for , the conditional distribution of given
Now we shall derive the UMVUE of . Since is a complete sufficient statistic for , it follows from the Lehmann–Scheff theorem that the UMVUE of can be obtained aswhere
Using Lemma 3.1, we have
The UMVUE of is
Example 2.1 continued The UMVUE of for the TL-Exp distribution is
4. BAYESIAN ESTIMATION
In this section, we consider the problem of estimation under Bayesian view point. For this, we consider one symmetric and two asymmetric loss functions. Under these loss functions, Bayes estimators for both the parameters and reliability function are obtained. Squared error loss function is taken as symmetric loss function, it gives equal weight to overestimation as well as underestimation. For asymmetric loss function, linear exponential (LINEX) loss function is used, which was proposed by Varian  (also see Zellner ) and entropy loss function is also taken, which was proposed by James and Stein .
In TLG family of distributions, it is not possible to find a mathematically tractable continuous joint prior distribution for both unknown parameters and . To choose a joint prior distribution for that incorporate uncertainty about both unknown parameters, we adopt the method proposed by Soland . This method is also used by several researchers (see Asgharzadeh and Fallah ).
Assume that the scale parameter is restricted to a finite number of values with prior probabilities , respectively, that is, the prior distribution for is given by
Further, we are assuming that the conditional prior distribution for given has gamma distribution with parameters and , that is,
The joint density of records is given by
The joint prior distribution can be obtained by multiplying and . Then the joint posterior distribution for is given by
Now, we will first solve the denominator integral of above equation, that is,
The marginal posterior density of is
Now we will derive the Bayes estimators of unknown quantities under various loss functions.
4.1. Squared Error Loss Function
The squared error loss function is defined aswhere is decision space and is the parameter space. Clearly, the Bayes estimator under squared error loss function is the posterior mean, then the Bayes estimator for is
Similarly, the Bayes estimator for is given by
The Bayes estimator for reliability function is
4.2. Entropy Loss Function
The entropy loss function is given by
The Bayes estimator under this loss function is
So, Bayes estimator of is
Similarly, the Bayes estimator for is
The Bayes estimator of reliability function is given by
Using the binomial expansion in the above expression, we have
4.3. LINEX Loss
The LINEX loss function iswhere is the parameter of loss function. The Bayes estimator under this loss function is
The Bayes estimator for is
The Bayes estimator for is
The Bayes estimator of reliability function is given by
To solve this, we will use exponential series expansion
5. PREDICTION INTERVAL
In this section, we will predict the future lower record while already having for . For this problem, we will use Bayesian procedure and Markovian property of record statistics. The conditional distribution of given is obtained by using Markovian property (see Arnold et al. ).where . For TLG family of distribution, with pdf given by Eq. (1), the function is given by
The Bayes predictive density function of given is given by
Let and be two constants such that
Using Eq. (17), we obtain two-sided predictive bounds for as , that is,
We are considering here a special case when , which is of our interest practically because after getting records we want the next record . The predictive survival function of is given as
Here we are assuming the case when the scale parameter is known (WLOG, ). For this case, predictive survival function can be written as
6. NUMERICAL COMPUTATIONS
In this section, a simulation study is conducted to illustrate all the estimation and prediction methods described in the preceding sections. We consider exponential distribution with dfas a special case for the baseline df in the model (1), named TL-Exp distribution.
We generate lower records of size from TL-Exp distribution for and . The lower record values are
The MLE for and are and , respectively, obtained by solving nonlinear Eqs. (6) and (7), in R software by Newton–Raphson method. Using these estimates we get the MLE of reliability function at as and as . Here we assume that scale parameter , takes finite values as , with equal probability 0.1 for each .
For obtaining the Bayes estimators for different parameters, first it is necessary to obtain the hyper-parameters for each The hyper parameters can be obtained based on the expected value of the reliability function conditional on , using
For the two values of and , the values of and for each value of can be obtained numerically from Eq. (19). A nonparametric approach can be used to estimate any two different values of the reliability function and (see Martz and Waller ). In this case, we use and . These two values are substituted into Eq. (19), where and are solved numerically for each , using the Newton–Raphson method. After that, posterior probabilities are calculated for each , and presented in Table 1. The MLEs, Bayes estimators, and reliability function (for different ) are also calculated and presented in Tables 2 and 3.
Prior information and posterior probabilities.
ML, maximum likelihood estimator; BS, Bayes estimator under squared error loss; BE, Bayes estimator under entropy loss; BL, Bayes estimator under LINEX loss.
Estimates of and
UMVUE, uniformly minimum-variance unbiased estimator; ML, maximum likelihood estimator; BS, Bayes estimator under squared error loss; BE, Bayes estimator under entropy loss; BL, Bayes estimator under LINEX loss.
Estimates of reliability for different .
Using the prediction procedure described in Section 5, the prediction interval for the next lower record is .
The mean squared error (MSEs) and risks of estimators and reliability function are compared according to following steps:
Samples of lower records with different size of are generated from the TL-Exp distribution for and different .
The values of and for a given value of are obtained using the procedure discussed.
Estimates of , and are obtained.
Above steps are repeated 10,000 times to evaluate the MSEs of these estimates and also estimated risks are compared under different loss functions using
From Tables 4 through 6, we observe that Bayes estimates for asymmetric loss functions are performing better than Bayes estimates for symmetric loss function and MLEs. UMVUE of reliability function is better than Bayes estimates of reliability and MLE. From Tables 7 and 8, comparison of risk for Bayes estimates of , and reliability function can be seen, and it is clear that, estimators for asymmetric loss function are again performing better than estimators for symmetric loss function.
MLE, maximum likelihood estimation; UMVUE, uniformly minimum-variance unbiased estimator; MSE, mean squared error.
MSE of the MLEs and UMVUE for .
MSE, mean squared error.
MSEs of the Bayes estimates of and .
MSE, mean squared error.
MSEs of the estimates of .
ER, estimated risk.
Estimated risks for Bayes estimates of and .
ER, estimated risk.
Estimated risk for Bayes estimates of .
The authors are thankful for all the valuable suggestions provided by the editor and anonymous referees, which have improved the original manuscript.
Cite this article
TY - JOUR AU - Mohd Arshad AU - Qazi Azhad Jamal PY - 2019 DA - 2019/04/22 TI - Statistical Inference for Topp–Leone-generated Family of Distributions Based on Records JO - Journal of Statistical Theory and Applications SP - 65 EP - 78 VL - 18 IS - 1 SN - 2214-1766 UR - https://doi.org/10.2991/jsta.d.190306.008 DO - 10.2991/jsta.d.190306.008 ID - Arshad2019 ER -