Half circular modified burr−III distribution, application with different estimation methods

The data related to many medical, environmental and ecological variables are often measured in terms of angles wherein its range is defined in [0,π). This type of data is referred to as axial or half circular data. Modeling based on half circular data has not received its due share of attention in statistical literature. In this paper, we introduce a new half circular distribution based on inverse stereographic projection technique on modified Burr−III distribution, called the half circular modified Burr−III (hcMB−III) distribution. The basic properties of the proposed distribution are derived. It is common observation that while estimating the parameters of a model, one usually adopts maximum likelihood estimation method as the starting point. In this paper, we consider seven frequentist methods of estimation, besides using maximum likelihood method for estimating the parameters of the hcMB−III distribution. Monte Carlo simulations are performed for investigating the performances of the considered methods in terms of their biases and mean square errors using small, medium and large sample sizes. Finally, one data set related to posterior corneal curvature of the eyes of 23 patients, is analyzed to check potentiality of the newly proposed model.


Introduction
Circular data analysis is a specific statistical branch that lies somewhere between linear data analysis and spherical data analysis. Circular or directional data is used for measuring observations arising in the fields of meteorology, biology, medical sciences etc. Circular data is measured in degrees and radians. It can also be considered as a point on a circle of unit radius, or a unit vector in a plane. For better comprehension of circular data, it can be regarded as being distributed on the circumference of a unit circle. Circular distributions are of great significance in modeling of cross-bedding data [1], studying paleo-currents [2] measuring wind directions [3], Analysis of time patterns in crime incidence [4], analyzing mother's day celebrations [5] among various fields. Other significant reviews on circular distributions with their properties can be found in [6][7][8][9][10][11][12][13].
Developing a probability density function for angles has proven to be a challenging assignment for statistician and practitioners. Numerous useful circular models possibly generated by posterior segment of the eyes of 23 patients. Finally, some concluding remarks are given in Section 7.

The hcMB−III distribution
Several lifetime models have recently been developed and utilized to model data in a variety of fields. A system of twelve kinds of distribution functions based on generating the Pearson differential equation was developed by [31]. The function of density has a variety of forms that are applicable to a wide range of applications [32]. Some recent developments in Burr family of distributions are Burr X Pareto distribution [33], Weibull Burr XII distribution [34], Burr III-Marshal Olkin-G family [35], Unit generalized log Burr XII distribution [36], Unit Burr-XII distribution [37] and Burr XII-moment exponential distribution [38]. The Burr XII distribution is a frequently used variant of the Burr distribution system. Burr −III is the inverse distribution of Burr−XII. For the purpose of statistical modeling, the Burr III distribution has been used in a variety of contexts. For applications of this distribution in various fields one can refer to [39][40][41][42][43][44][45]. The cumulative distribution function (cdf) of Burr−III distribution is where α, β are the shape parameters.
In recent past, a new generalization of the Burr−III distribution, called the modified Burr III (MB−III) distribution was proposed by [46]. The cumulative distribution function (cdf) of MB−III distribution is given by where α, β, γ are the shape parameters of MB−III distribution. Modified Burr−III distribution has attracted many researchers due to its tractable properties. [47] studied the transmuted modified Burr III. Characterization of transmuted modified Burr III distribution was done by [48]. [49] developed MBIII-G Family of distributions based on odds ratio of any baseline distribution. The application of Modified Burr III distribution in reliability analysis was done by [50]. [51] originated the McDonald modified Burr-III. [52] developed Cubic rank transmuted modified Burr III-Pareto. Moreover, [53] came up with Unit MB−III distribution. MB−III is a sub model of modified Dagum distribution by [54]. [55] proposed Extended Marshall-Olkin Burr−III distribution.
The Half circular modified burr−III (hcMB−III) distribution can be obtained by applying a transformation θ = 2tan −1 (x), θ �(0, π). Let m(θ) = tan (θ/2). By using inverse stereographic projection, the pdf of the hcMB−III distribution is given by we have j m 0 ðyÞ j¼j 1 2 sec 2 ðy=2Þ j¼ 1 1 þ cosðyÞ and f ðmðyÞÞ ¼ ab tan Consequently, the pdf of hcMB−III (α, β, γ) is given by The cdf of hcMB−III distribution is given as Since we do not decide shapes of the density and hazard rate function analytically, we plot them based on some selected parameters value to see their possible shapes. The shape of hcMB −III distribution for various values of (α, β, γ) are presented in Figs 1 and 2 demonstrates circular presentation of hcMB−III distribution and it's cumulative distribution function. Different values of parameters show the flexibility of hcMB−III distribution such as negatively

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hcMB−III dist, application with diff estimation methods skewed, symmetric and positively skewed. Therefore, the hcMB−III distribution is quite flexible and can be applied to various data sets.
Consider z y ¼ 1 þ g tan y 2 À � À � À b and the hazard function of hcMB−III distribution is defined as hðyÞ ¼ The hazard function of hcMB−III distribution for some parametric values are given below in Fig 3 shows that failure rate function can be increasing and bathtub shaped. Therefore, hcMB−III distribution can be applied to various data sets.
Three sub models of modified burr III distribution were listed by [46]. Hence, hcMB−III also have three sub models named as hc-Generalized Inverse Weibull (hc-GIW), hc-Burr III distribution and hc-Log Logistic (hc-LL) distribution. The cdfs of sub models of proposed model are listed in Table 1 along with their hazard functions.
Mode of hcMB−III distribution is derived by taking log of the probability density of hcMB−III distribution.  and to find mode put Since it is apparent that the equation has not an explicit solution in the general case. Consequently, we discuss it empirically: 1. 8 γ distribution is bimodal as α, β ! 0.

Characteristics function and properties related to trigonometric moments of hcMB−III distribution
The characteristic function of a half/semi-circular model with pdf g(θ) is defined as The characteristic function defined above also called the pth trigonometric moment. Since θ and θ + 2π represents the same direction so it is necessary to restrict p to integer value. Characteristics function of hcMB−III distribution is presented graphically in Fig 4 for α = 1.5, β = 3.5, γ = 3.5 is as follows: The trigonometric moments of the distribution are given by ϕ; ±1, ±2, ±3, � � �, where and being the p th order cosine and sine moments of the random angle θ, respectively and are required to study distribution characteristics. and Putting p = 1, we get 1st order trigonometric moments as The direction μ is called the mean direction; the mean direction of hcMB−III distribution is defined as α 1 and β 1 are defined in Eqs (13) and (14). The mean resultant length (MRL) of hcMB−III distribution is defined as and denoted by ρ r ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi α 1 and β 1 are defined above and mean resultant length (MRL) is invariant under rotation.
A useful measure of dispersion on the circle is the circular variance. Circular variance of hcMB−III distribution is defined as Where 0 � υ � 1, ρ denotes the MRL and α 1 and β 1 are defined above. Circular standard deviation of hcMB-III distribution is defined as s ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi α 1 and β 1 are defined above and circular standard deviation measures the average direction from mean direction. Circular skewness of hcMB-III distribution is defined as b � 2 is 2 nd trigonometric moment about mean and circular kurtosis measures the kurtosis of circular distribution.
Circular kurtosis of hcMB-III distribution is defined as a � 2 is 2 nd trigonometric moment about mean and circular kurtosis measures the kurtosis of circular distribution.
Using expressions in [9,56] and the first two trigonometric moments, the characteristics of stereographic hcMB−III distribution presented above are calculated numerically and are presented in the S1 Appendix, by using Mathematica 12.0 for some parametric values of hcMB−III distribution. Following results are obtained.

Maximum likelihood estimates
The method of maximum likelihood is the most frequently used method of parameter estimation. The method's success stems no doubt from its many desirable properties including consistency, asymptotic efficiency, invariance and simply its intuitive appeal. The log-likelihood function for the vector of parameters z = (α, β, γ) of the hcMB−III distribution is The resulting partial derivatives of the Eq (21) are: and The MLEs of unknown parameters cannot be derived analytically from the above normal equations because of convoluted non-linear expressions. Therefore, the iterative methods can be used to obtain the estimated values of the unknown α, β and γ simultaneously.
Under some regularity conditions for unknown parameters in the interior of parameter space but not on the boundaries, the asymptotic distribution of ffi ffi ffi n pĉ À c � � , where ψ = (α, β, γ) t , follows multivariate normal with mean vector zero and variance-covariance matrix is It can be noted that K c ð Þ ¼

Ordinary and weighted least square estimates
The least square estimators (LSE) and weighted least square estimators (WLSE) were proposed by [57] to estimate the parameters of Beta distributions. Suppose F(x (i) )denotes the distribution function of the ordered random variables X (1) < X (2) <. . .< X (n) be ordered sample of size n from hcMB-III distribution. Then, the expectation of the empirical cumulative distribution function is defined as . . . ; n: The least square estimates (LSEs) say,â LSE ,b LSE ,ĝ LSE , of α, β and γ are obtained by minimizing The variance of the empirical cumulative distribution function is defined as . . . ; n: Thus, the weighted least square estimates (WLSEs) say,â MPS ,b MPS ,ĝ MPS , of α,β and γ are obtained by minimizing by minimizing

Percentile estimates (PCE)
If the data come from a distribution function which has a closed form, then we can estimate the unknown parameters by fitting straight line to the theoretical points obtained from the distribution function and the sample percentile points. This method was originally suggested by [58,59] and it has been used for weibull distribution and for generalized exponential distribution. In this paper, we apply the same technique for the hcMB−III(α, β, γ) distribution. Let X (i) be the i th order statistic, i.e X (1) < X (2) <. . .< X (n) . If p i denotes some estimate of F (x (i) ) then the percentiles estimates,â PC ,b PC ,ĝ PC of α, β and γ can be obtained by minimizing Several estimators of p i can be used. In this paper, we consider p i = i nþ1 . [60,61] introduced the maximum product of spacings (MPS) method as an alternative to MLE for the estimation of parameters of continuous uni-variate distributions. [62] independently developed the same method as an approximation for the Kullback-Leibler measure of information. This method is constructed on a clue that differences (spacings) between the values of the cdf at consecutive data points should be identically distributed. [60] proved that this method is as efficient as the MLEs and consistent under more general conditions. The geometric mean of the differences is given as

Minimum distances estimators
This section presents three estimation methods for α, β and γ based on the minimization of the goodness−of−fit statistics. This class of statistics is based on the difference between the estimate of the cumulative distribution function and the empirical distribution function [63].

Cramér-von-Mises estimates (CVME).
To motivate our choice of Cramér-von-Mises type minimum distance estimators, [64] provided empirical evidence that the bias of the estimator is smaller than the other minimum distance estimators. Thus, the Cramér-von-Mises estimates,â CVM ,b CVM ,ĝ CVM , of α, β and γ are obtained by minimizing Anderson-Darling estimates(ADE). The Anderson-Darling(AD) test was developed by [65] as an alternative to other statistical tests for detecting sample distributions departure from normality. It is interesting to note that the Anderson-Darling test converges so quickly towards the asymptote [65]. The Anderson-Darling estimatesâ AD ,b AD ,ĝ AD , of α, β and γ are obtained by minimizing the following function

Simulation study for the comparison of different estimation methods
This section presents simulation studies by using the hcMB−III distribution to assess the performance of the above estimators discussed in the previous section and obtained numerical and graphical results. We generate N = 10, 000 samples of the size n = (25, 50, 75, 100) from hcMB−III distribution with parameter settings (α, β, γ) = {(2, 3, 4), (1, 3.5, 1), (4, 3, 4)}. The random numbers generation is obtained by its quantile function. In this simulation study, we calculate the empirical mean, bias and mean square errors (MSEs) of all estimators to compare in the terms of their biases and MSEs with varying sample size. It is noticed that 10,000 iterations are sufficiently large to have stable results. The empirical bias and MSE are calculated by (for t = α, β, γ) respectively. All results related to estimation were obtained by using software Mathematica 12.0. The results of simulations are shown in Fig 5 in  Simulation results for hcMB−III distribution in terms of mean, bias, mean square error and standard error is given to compare performance of different estimation methods for different parameter settings S2-S4 Appendices.

Application
In this section, we consider the data obtained from the images of the posterior segment of the eyes of 23 patients. This data set has also been analyzed by [22,27,28,66]. The data is also available in S1 Data. The half circular variable of our interest is the angle which measures the posterior corneal curvature defined below. Fig 6 presents an image of the posterior segment, where O is the intersection of the geometrical axis of the eye (horizontal line) with the line made between the nasal and temporal scleral spurs (vertical line). The circular plot is given in Fig 7, where it is obvious that the angles are concentrated in the first and second quadrant with range 1.76, which confirms that the data is a random sample from a half circular distribution, where θ 2 [0, π).
We compute MLEs and their standard errors (S.Es) in parenthesis for hcMB−III distribution, half-circular gamma distribution, half-circular Burr-XII distribution and sub models of hcMB −III distribution are given in Table 2. For the selection of the best fit model, we compute the estimate of likelihood ratio statistic À 2 'ðŶÞ, AIC, CAIC, BIC, Anderson-Darling (A � ), Cràmer-von-Mises (A � ) and Kolmogorov-Smirnove (K-S) statistic along with its p-value for all competing models. All models are evaluated at MLE by using FitdistrPlus, AdequacyModel, Pracma packages in R and the results are presented in Table 3. Among all other competitive models, it is noted that the hcMB−III distribution has the smallest values of À 2 'ðŶÞ, AIC, BIC and CAIC. It is also noted that hcMB−III distribution has lowest value of (A � ), (W � ) and K-S (p-value) which indicates that it best fits the given eye data set than other half-circular distribution being used for comparison. Fig 8 shows the fitted models vs hcMB−III distribution along with cdfs of all competing models. We fitted the hcMB−III distribution using the eight estimation methods. The parameter estimates for eye data set are reported in Table 4. Descriptive statistics for different estimated values of hcMB−III distribution (α, β, γ) for all estimation methods are given in Table 5 and we observe that CVME has smallest variance among all others while MPSE has smallest value of skewness. A graphical presentation of fitting above mentioned estimation methods is To check the shape of the hazard function of the data set, we have used TTT (Total time on test) plot proposed by [67] in Fig 10 indicates that the data set has increasing hazard rate. Also see [68] for more details about parameter estimation and TTT plot. We can perceive that the hcMB−III distribution is best fitted to empirical data

Conclusions
Circular data is used in measuring observations arising in the different fields of science. Due to the wide range of applications for half-circular data, it is still worth exploring it further. In this paper, a new half circular distribution is proposed based on an inverse stereographic projection technique applied on the distribution of MB−III. Various properties of suggested distribution are derived. The parameter estimates are obtained by employing the eight estimation

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hcMB−III dist, application with diff estimation methods

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hcMB−III dist, application with diff estimation methods methods. The estimation methods were compared on the basis of their Means, Average biases, MSEs and Standard errors for different parameters settings. We perform simulation studies on the basis of the graphical as well as numerical results to see the performances of the estimates of hcMB− III distribution. Different properties such as mean direction, variance, trigonometric moments, resultant length, skewness and kurtosis are calculated for all eight estimation methods. The suggested model best fits the eye data of 23 patients' posterior corneal curvature when compared to existing semi-circular models and sub models of hcMB− III distribution. The potentiality of hcMB− III distribution illustrates that it is flexible, competitive and parsimonious. Thus, it should be included in the distribution theory to facilitate researchers and practitioners dealing with angular data. Further, as perspective of future projects, we may study some rigorous issues (i)Characterization of proposed model can be done by using different methods (ii)Outliers detection. (iii)ℓ-axial half circular Modified Burr III can be studied.
(iv)Sub models of half circular distribution can be explored in detail. (v)Bayesian analysis can be performed to study complexity of the proposed model. (vi)Bi-variate case of hcMB−III distribution may also be studied.
(TIF) S1 Data. Excel file for eye data set used in data analysis. (XLSX)