Weighted generalized Quasi Lindley distribution: Di ﬀ erent methods of estimation, applications for Covid-19 and engineering data

: Recently, a new lifetime distribution known as a generalized Quasi Lindley distribution (GQLD) is suggested. In this paper, we modiﬁed the GQLD and suggested a two parameters lifetime distribution called as a weighted generalized Quasi Lindley distribution (WGQLD). The main mathematical properties of the WGQLD including the moments, coe ﬃ cient of variation, coe ﬃ cient of skewness, coe ﬃ cient of kurtosis, stochastic ordering, median deviation, harmonic mean, and reliability functions are derived. The model parameters are estimated by using the ordinary least squares, weighted least squares, maximum likelihood, maximum product of spacing’s, Anderson-Darling and Cramer-von-Mises methods. The performances of the proposed estimators are compared based on numerical calculations for various values of the distribution parameters and sample sizes in terms of the mean squared error (MSE) and estimated values (Es). To demonstrate the applicability of the new model, four applications of various real data sets consist of the infected cases in Covid-19 in Algeria and Saudi Arabia, carbon ﬁbers and rain fall are analyzed for illustration. It turns out that the WGQLD is empirically better than the other competing distributions considered in this study.


Introduction
Recently, some researchers in statistics have been interested in generating new flexible statistical models based on different techniques such as the weight distributions which are commonly used in many fields of life situations such as sciences, ecology, biostatistics, medicine, engineering, pharmacy and environment and so on.
The concept of weighted distribution is suggested by [15] to study how verification methods can affect the form of the distribution of recorded observations. Later, the weighted distributions are unified and formulated by [31] in general terms in connection with modeling statistical data.
Suppose X is a non-negative random variable with a probability density function (pdf) f (x). Let W(x) be a non negative weight function, then the probability density function of the weighted random variable X w is given by: . (1.1) If the weight function has the form W(x) = x λ , the resulting distribution is known as a size biased distribution of order λ with pdf given by: . (

1.2)
If λ =1 or 2, the yielded distributions are known as the length biased and area biased distributions, respectively.
For example, the length-biased Suja distribution is proposed by [5] as a generalization of the Suja distribution. The size biased Ishita distribution is offered by [6] as a new modification of the Ishita distribution. The Marshall-Olkin length-biased exponential distribution is suggested by [18]. The length-biased weighted generalized Rayleigh distribution with its properties is proposed by [2]. The weighted Lomax distribution is introduced by [23]. Other types of distributions are also suggested based on other procedures as [8] for the exponentiated new Weibull-Pareto distribution. The Topp-Leone Mukherjee-Islam distribution is offered by [7]. Also, see [27,16,17,26].
To the best of our knowledge, the use of the weighted method to extend the generalized Quasi Lindley distribution introduced by [9] is still unexplored in the literature. In this study, the weighted generalized Quasi Lindley distribution is proposed. Indeed, the importance of the suggestion of the WGQLD arises from the fact it is a modification of the well known Quasi Lindley distribution which is considered by many researchers in different life situations.
The layout of this paper is organized as follows. Section 2 concerns with the pdf and cumulative distribution function (cdf) of the WGQLD and its shapes. Moment generating function and moments includes the rth moment, variance, the coefficient of skewness, kurtosis, and variation are presented in Section 3 theoretically and supported by some simulations. The distribution of order statistics, median deviations and harmonic mean are presented in Section 4. The stochastic ordering, reliability, hazard, reversed hazard rate and odds functions are given in Section 5. Bonferroni and Lorenz curves as well as the Gini index are provided in Section 6. In Section 7, different methods of estimation for the distribution parameters are discussed including maximum likelihood, maximum product of spacing's, ordinary least squares, weighted least squares, Cramer-von-Mises, and Anderson-Darling methods. A simulation study is conducted to compare the performance of the proposed estimators in Section 8. Illustrative examples of real data applications are given in Section 9. The paper is concluded with some suggestions for future works in Section 10.

The WGQLD distribution
This section introduces the pdf and cdf of the WGQLD along with the shapes of the model. The probability density function of the generalized Quasi Lindley distribution is given by: x ≥ 0, α > −1, θ ≥ 0, (2.1) and the corresponding cdf is: With reference to Eq 1.1, and in this work without loss of generality we considered the weight function as W(x) = x given that the mean of the GQLD is E(X) = 2(2+α) θ(1+α) . Hence, the weighted generalized Quasi Lindley distribution can be obtained by substituting f GQLD (x; θ, α), E(X) and W(x) in Eq 1.1 to get:    Based on Figures 1 and 2, it can be observed that the WGQLD is a unimodel and positively skewed. Also, it is approximately symmetric as in Figure 2 when θ = 1 for some parameters. The curve of the distribution function is more flat for α = 1, θ = 0.25 and when α = θ = 1. than other values.
The corresponding cdf equation of the WGQLD is given by +xθ[24 + xθ(12 + xθ(4 + xθ))] (2.4) Figure 3 reveals that the cdf plots are approach 1 when θ = 3 (a) faster than that of α = 1 (b). In comparing Figure 3      : Let X ∼ f WGQLD (x, θ, α), then the rth moment of X about the origin is: Based on Eq 7, it is simple to have the first, second, third and fourth moments of the WGQLD, respectively, as
(3.5) Therefore, the variance of the WGQLD can be obtained as The distribution shape analysis can be performed by studying the coefficient of skewness, coefficient of kurtosis, and coefficient of variation. For the WGQLD, these coefficients, respectively, are given by:
It is of interest to note here that the coefficient of skewness, coefficient of variation and coefficient of kurtosis are free of θ. Table ( 1) presents some values of the mean, standard deviation, coefficient of variation, coefficient of skewness and coefficient of kurtosis of WGQLD for various values of the parameters α and θ. It can be noted that the mean and the standard deviation are decreasing when values of α and θ are increasing for fixed values of θ and α, respectively. The results of simulation emphasize that the coefficient of skewness, coefficient of variation and coefficient of kurtosis don't depend on θ and increasing when α values are increasing.
The moment generating function (MGF) of the WGQLD is given in the following theorem.

Theorem 2.
: Let X ∼ f WGQLD (x, θ, α), then the MGF of X about the origin is: Proof. : To prove the MGF of the WGQLD, let

Order statistics
Let X 1 , X 2 , . . . , X m be a random sample of size m from a distribution with pdf f (x) and cdf F(x). The pdf of the ith order statistic X ( j:m) for j = 1, 2, . . . , m are defined by [14] as Based on Eq (4.1), the pdf of the ith order statistic, X (i:m) , from the WGQLD, will be Based on Eq (18), the pdf of the minimum and maximum order statistics, respectively, are given by

Harmonic mean
Let .
Therefore, the harmonic mean of the WGQLD is given by (4.5) Table 2 presents some values of the harmonic mean of WGQLD for various values of the parameters θ and α. Based on Table 2, we can conclude that: the harmonic mean values are decreasing in α when θ = 5 and are decreasing in θ when α = 2. Also, in general the harmonic mean values when θ < α are larger than the case of θ > α.

Stochastic ordering
The stochastic ordering is an important tool in finance and reliability theory to evaluate the comparative behaviour of the models or systems. Let the random variables X and Y having the probability density functions, cumulative distribution functions and survival functions f (x), f (y), F(x), F(y), F(x) = 1 − F(x), and F(y) = 1 − F(y), respectively, then X ≤ Y in: All these stochastic orders defined above are related to each other and [29] showed the following relation is hold.
Theorem 3. : Let the random variables X and Y be independent follow the pdf f X (x, θ, α) and f Y (x, µ, ω), respectively. If θ > µ and α > ω, then X ≤ LRO Y, X ≤ HRO Y, X ≤ MRLO Y and X ≤ S O Y.
, and log( Taking the derivative of the last equation with respect to x yields
(5.2) Figure 5, reveals that the reliability of the WGQLD are decreasing while the hazard rate function is increasing for α = 1, 2, . . . , 5. The reversed hazard rate and odds functions for the WGQLD distribution, respectively, are defined as As it is shown in Figure 6, the reversed hazard function is decreasing taking inverse J shape while the odds functions is increasing with J shape.

Methods of estimation
In this section, we consider six methods of estimation for estimating the unknowns parameters α and θ of the WGQLD distribution. These methods include the (1) maximum likelihood (ML) method, (2) method of maximum product of spacings, (3) ordinary least square method, (4) weight least square method, (5) method of Cramer-Von-Mises, and (6) method of Anderson-Darling.

Maximum likelihood estimation
First of all, we investigate the ML estimates (MLEs) of θ and α. Let X 1 , X 2 . . . , X n be a random sample of size n selected from the WGQLD. The likelihood function is given by: and the log-likelihood function Ξ = ln L (x; θ, α) is: The derivatives of Ξ with respect to θ and α are: Since there is no closed form for these equations, then the MLEsθ andα of θ and α, respectively, can be solved simultaneously using a numerical method as Newton Raphson method.

Methods of least squares
The least square methods are introduced by [28] to estimate the parameters of beta distribution. Let X i:n be the ith order statistic of the random sample X 1 , X 2 , . . . , X n with distribution function F(x), then a main result in probability theory indicates that F(X i:n ) ∼ Beta(i, n − i + 1). Moreover, we have and Var[F(X i:n )] = i(n − i + 1) (n + 1) 2 (n + 2) .
Using the expectations and variances, we obtain two variants of the least squares methods.

Methods of minimum distances
Here, we use two popular methods based on the minimization of test statistics between the theoretical and empirical cumulative distribution functions. The methods are Cramer-von-Mises method and the method of Anderson-Darling (for more details see [13] and [24]).

Method of Anderson-Darling
The Anderson-Darling (AD) estimates of the WGQLD distribution parameters θ and α denoted bŷ θ AD andα AD can be obtained by minimizing the following function with respect to θ and α, or by solving the following two equations F(x n−i+1:n |α, λ) = 0, and where ∆ 1 (x i:n |θ, α) and ∆ 2 (x i:n |θ, α) are specified in 7.6 and 7.7, respectively.

Simulation
This section compares the performances of the proposed estimators of the WGQLD parameters α and θ. This

Application on real data
In this section, we use data sets to illustrate the usefulness of the WGQL model, where four different data sets are used related to the environment, engineering and two medical data sets are considered. We compare the WGQL distribution to some well known distributions of two parameters as the generalized Quasi Lindley distribution, Quasi Lindley distribution, two-parameter Sujatha distribution, and the Pareto distribution.
The first data set is taken from [22] represents the 100 annual maximum precipitation (inches) for one rain gauge in Fort Collins, Colorado, from 1900 through 1999. The data are given below: Data Set 1: 239,232,434,85,302,174,170,121,193,168,148,116,132,132,144,183,223,96,298,97,116,146,84,230,138,170,117,115,132,125,156,124,189,193,71,176,105,93,354,60,151,160,219,142,117,87,223,215,108,354,213,306,169,184,71,98,96,218,176,121,161,321,102,269,98,271,95,212,151,136,240,162,71,110,285,215,103,443,185,199,115,134,297,187,203,146,94,129,162,112,348,95,249,103,181,152,135,463,183,241 5. The second data set from [25] consists of 100 observations on breaking stress of carbon fibers (in Gba). The data are as follows: The box and TTT plots for the above data are given in Figures 7 and 8, respectively.
To choose the best model fitting , we considered Akaike information criterion (AIC) introduced by [1], Baysian information criterion (BIC) proposed by [30], Hannan Quinn Information Criterion (HQIC) suggested by [21], Consistent Akaike Information Criterion (CAIC) by [10] Based on the results reported in Tables 9, 10, 11 and 12, we observe that the WGQLD provides the better fit with the smallest values of AIC, AICc, BIC, HQIC and K-S with maximum P-values as compared to its competitive models considered in this study. Figures 9, 10, 11 and 12 support this claim.

Conclusions
In this article, we proposed the WGQLD distribution along with some of its properties such as, stochastic ordering, Median deviation, Harmonic mean, some plots of the pdf and cdf, Bonferroni and Lorenz curves and Gini index moments, coefficient of variation, coefficient of skewness and coefficient of kurtosis. Also, the hazard rate function, reliability function, reversed hazard rate and odds functions are presented. The maximum likelihood estimates is computed as well as the maximum product of spacing's, ordinary least squares, weighted least squares, Cramer-von-Mises, and Anderson-Darling methods are obtained. The results show that the best method of estimation is the MLE method. Applications of various real data sets are analyzed for illustration. It is proved that the WGQLD is empirically better than other competitors models considered in this study including the base GQLD. Therefore, in the future, the authors intend to investigate the performance of different estimators of the WGQLD based on ranked set sampling method and its modifications, see [3,4,19,20,32].