Robust Statistical Modeling and Machine Learning with Applications in Data ScienceView this Special Issue
A Novel Frechet-Type Probability Distribution: Its Properties and Applications
In this article, a new lifetime model, referred to as modified Frechet–Rayleigh distribution (MFRD), is developed by accommodating an additional parameter in Rayleigh distribution on the basis of the modified Frechet method. Numerous statistical properties of the suggested model are derived and discussed. The technique of maximum likelihood (ML) estimation is adopted to get estimates of the parameters. The suggested model is very flexible and has the capability to model datasets having both monotonic and nonmonotonic failure rates. The proposed model is applied on two real datasets for checking its performance in comparison with available well-known models. The suggested model has shown outclass performance in comparison with the available versions of the Rayleigh distribution used in the literature.
The distribution has many applications. It can be defined as follows.
Let and be independent normally distributed random variables. Then, Z = square root of (X2 + Y2) has a Rayleigh distribution.
Rayleigh distribution is widely used in various fields, such as life sciences, agriculture, and biological sciences, and in the analysis of metrological characteristics, such as wind speed. The Rayleigh distribution has applications in situations where sizes of normally distributed variables are vital. Rayleigh distribution is also used in the investigation of wind velocity into its orthogonal XY coordinates. Furthermore, assuming that these components are independently and normally distributed with equal variances and means equal to zero. Thus, the variable of interest is wind speed and can be modeled by a Rayleigh distribution. According to Fernandez et al.  and Akhter et al. , it has useful applications in the field of medical sciences. In the field of engineering, it is used to find the expected lifespan of material objects, reliability investigation, service periods, and theory of communication. The density function (PDF) and distribution function (CDF) of RD are specified, respectively, as
In the literature, there are many variants of the Rayleigh model which are used to provide a better fit of data. Voda  developed generalized Rayleigh and left-truncated model, Gomes et al.  suggested the Kumaraswamy generalized Rayleigh model, Merovci  suggested the transmuted Rayleigh model, Cordeiro et al.  suggested the beta generalized Rayleigh model. In , the inverse Rayleigh (IR) model was proposed to fit survival and reliability data. Voda  obtained estimates of the parameters of the IR model by ML estimation, Mukarjee and Maitim  discussed certain important statistical properties of the IR model, and Gharraph  developed expressions of descriptive statistics of the IR model. Moreover, Soliman et al.  and Gharraph  used maximum likelihood and Bayesian estimation methods to obtain estimates of parameters.
In recent times, the literature of probability models has been updated by inserting additional parameter(s) in the existing models using existing techniques available in literature or by proposing a new method to produce a better model as compared to the baseline model. Rehman and Sajjad  developed the IR model; Ahmad et al.  suggested transmuted the IR model, beta IR model , modified inverse Rayleigh model , generalized inverted scale family models , alpha power exponentiated inverse Rayleigh model , and alpha power Rayleigh distribution . These newly proposed distributions were obtained and comparatively provided a better fit of the complex data.
The primary rationale behind producing modified Frechet–Rayleigh distribution (MFRD) is to overcome the inadequacy present in the Rayleigh model. The proposed distribution provides various shapes for hazard rate functions and shows more flexibility as compared to the Rayleigh distribution.
1.1. Research Problem
In practice, we might deal with data that exhibit monotonic or nonmonotonic hazard rate shapes. The available models may fail to be applied in such cases as in the case of life distributions. Therefore, researchers are trying to derive models that incorporate the deficiencies of these distributions. So, in this study, our aim is to propose a model that might efficiently fit such kind of data. This purpose is achieved by adding an extra parameter using generators or by the existing models; for detail, we refer to see . Therefore, the modified Frechet technique is used to derive the proposed model.
1.2. Modified Frechet (MF) Technique
Alamgir et al.  suggested a novel technique, called modified Frechet technique (MFT) for developing new efficient probability models. The CDF and PDF of the newly developed technique are specified by the expressions given below:
In equations (3) and (4), and denote the CDF and PDF of the input model, respectively. This technique is used to introduce modified Frechet–Rayleigh distribution (MFRD). The basic purpose of producing MFRD has a more flexible distribution to model life time data in comparison to other versions of Rayleigh distribution.
1.3. Modified Frechet–Rayleigh Distribution (MFRD)
This section of the article introduces modified Frechet–Rayleigh distribution (MFRD).
The CDF of RD is specified by
Definition 1. If , then its PDF is specified byThe PDF and CDF are plotted in Figure 1.
The CDF of MFRD is specified byThe hazard rate (HR) function of MFRD is defined byThe survival function (SF) of MFRD is specified byFigure 2 exhibits graphs of HF and SF of MFRD.
Lemma 1. If and is nonincreasing function, then is also a nonincreasing function.
Proof. If is differentiable and if for in , then is nonincreasing and vice versa.
Take , that is,For nonnegative and less than 1 values of α and , (9) shows that .
Hence, for , is nonincreasing function.
Lemma 2. If , is nonincreasing function, and is log-convex, then is a nonincreasing function.
Proof. If exists and , then referred as log-convex.
By taking second differential of (10), we obtainWhen α is nonnegative and less than 1and then .
Therefore, for , is log-convex .
1.4. Quantile Function (QF)
If X ∼ , then MFRD has the following QF: . This implies , where the distribution of u is uniform. The QF of MFRD is
By putting in (12), we get the median of MFRD as follows:
To find mode of MFRD, solve the following for x:
Mode of the distribution satisfies (13).
1.4.3. Rth Moment
Let X ∼ ; then, by definition,
Put in (15) and simplify; we obtain the expression below:
Insert in the above expression; after simplification, it will take the form given below:
Insert series and for ; from equation (16), we obtain the final expression as given below:
The expression of Rth moment is incomplete integral. It can be solved numerically using some numerical integration technique.
1.4.4. Moment Generating Function (MGF)
Suppose X : ; then, the MGF is specified by
Insert in (18), and after some calculation, we obtain
Using result (15) in (19), we obtain
The expression of MGF is an incomplete integral. It can be solved numerically by adopting any iterative method.
1.5. Order Statistics
Consider an order statistics taken from MFRD. Then, the PDF of ith order statistics is specified by
Insert (5) and (6) of MFRD in (21); we obtain order statistic is as follows:
By putting in (22), we have expression for first-order statistic as given below:
By substituting in (22), the expression for the nth order statistic will take the form given below:
For median’s distribution insert in (22), we have
Lemma 3. The Shannon entropy (SE) of MFRD is given by
Proof. The SE of MFRD is given as follows:Substituting in (27), the expression will be likePutting in (28), after simplification, it will take the following form:Using series representation and in (27), we get the final expression given below:The final expression for SE is an incomplete integral. It can be solved numerically using numerical integration methods or some iterative procedures.
1.6. Mean Residual Life Function
Suppose X follows MFRD. Then, of MFRD has the expression given below:where
Putting, , the above expression will take the form given below:
Substitute in (33); we obtain
Insert series and for in (31), we obtain
Substitute in (33) and simplify; we have
Putting in (37) and simplifying, we obtain
Substitute series and for in (35), to have
Put (8), (32), and (36) in (29); we obtain
The final expression of is an incomplete integral. The expression in (40) can be solved numerically using any iterative method.
1.7. Stress-Strength Parameter (SSP)
Let and be two random variables, which are independent and follow the same distribution such that and ; then, the expression for the SSP is specified as
Utilizing (5) and (6) in (41), then SSP is given as
Substitute in (42) and simplify; we obtain
Now, putting in (43), after simplification, we get the expression given below:
Put series and in (44); we get the following expression for SSP:where .
This is the required result.
1.8. Parameters’ Estimation
The joint density function has the following expression:
Taking natural logarithm of (46), we obtain
Differentiate (44) with respect to and , respectively, to have the likelihood equations as
The solution of above equations is obtained using the Newton–Raphson method. For large sample size, the ML Estimators are normally distributed, that is, , where denotes variance covariance matrix acquired by inverting Fisher information matrix as follows:
Again differentiating (45) and (46) with respect to and , respectively, we obtain
Again differentiating (45) w.r.t λ, we obtain
Asymptotic confidence intervals for parameters of MFRD are obtained by and , where denote the upper percentile of the standard normal distribution.
1.9. Simulation Study
To assess the consistency of MLE, the MSE and bias are measured based on W = 1000 simulations. Samples having sizes 50, 100, and 200 are generated from MFRD using the following expression of the quantile function:where follows uniform distribution over [0, 1]. Bias and MSE of MFRD are computed by the following expressions:where . Simulation results were obtained for various values of and . The results are shown in Table 1. The consistency pattern can easily be observed as estimates’ approach to their true values by increasing the sample size. Furthermore, the values of error measures of the parameter estimates’ decrease for all combinations of parameter values. Hence, it can be concluded that MLE procedure produces consistent estimates in estimating the parameters of MFRD. The same consistency of parameters has been observed for all choices of parameter values. It is also evident from Table 1 that bias for the estimators are decreasing with increase in the sample sizes and parameter . Therefore, we conclude that these estimators are asymptotically unbiased.
To assess the performance of the MFRD, two practical datasets are used and the results are compared using some standard model selection criteria’s. The proposed model is compared with Rayleigh distribution (RD) , modified inverse Rayleigh distribution (MIRD) , generalized inverse Rayleigh distribution (GIRD) , transmuted inverse Rayleigh distribution (TIRD) (Ahmad et al., 2014), exponentiated inverse Rayleigh distribution (EIRD) , and generalized Rayleigh distribution (GRD) .
2.1. Dataset 1
Dataset 1 was taken from Bjerkedal . The data points are given below: 0.1, 0.33, 1.08, 1.08, 1.08, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 2.54, 2.78, 2.93, 3.27, 3.42, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 07, 1.09, 1.12, 1.13, 1.15,1.36, 1.39, 1.44, 1.83, 1.95, 1.96, 1.97, 2.02, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 2.13, 1.46,1.53, 1.59, 1.6, 1.63, 1.63, 1.68, 1.71, 1.72, 1.76, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53,2.54, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55.
2.2. Dataset 2
Dataset 2 is taken from Nassar and Nada . The data points are as follows: 5.9, 20.4, 14.9, 16.2, 17.2, 7.8, 6.1, 9.2, 10.2, 9.6, 13.3, 8.5, 21.6, 18.5, 5.1, 6.7, 17, 8.6, 9.7, 39.2, 35.7, 15.7, 9.7, 10, 4.1, 36, 8.5, 8, 9.2, 26.2, 21.9, 16.7, 21.3, 35.4, 14.3, 8.5, 10.6, 19.1, 20.5, 7.1, 7.7, 18.1, 16.5, 11.9, 7, 8.6, 12.5, 10.3, 11.2, 6.1, 8.4, 11, 11.6, 11.9, 5.2, 6.8, 8.9, 7.1, 10.8.
P-value and some well-known standard model selection criteria’s are used for comparison purposes. The results are displayed in Tables 2 and 3 of dataset 1 and 2, respectively. Comparative analysis of different distributions based on two data points are shown in Figures 3 and 4, respectively.
The results of Tables 2 and 3 indicate that MFRD provides improved result in comparison to other versions of the Rayleigh model on the basis of well-known model selection criteria. Figure 5 and 6 provide PP-plots, QQ-plots, plots of PDF, and CDF for the two datasets.
In this article, we suggested a new lifetime model referred to as modified Frechet–Rayleigh distribution (MFRD) using the modified Frechet technique. Several statistical properties of the MFRD were obtained such as moments, MGF, and median in closed form, stress-strength parameter, Quantile function, mean residuals’ life function, order statistics, and the expression for entropies. The parameters of the proposed model were estimated using the MLE approach. The simulation results revealed that the estimates of the parameters of MFRD are consistent and asymptotically unbiased. The performance of MFRD was evaluated on the basis of real datasets. The results indicated better performance of MFRD as compared to various versions of Rayleigh distribution.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
All authors have equally contributed to the research design, data analysis research summary and recommendation, written manuscript, and coordinated for submission of this paper.
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