Cosine Kumaraswamy family of distributions Properties and applications to real-world datasets

Abstract

Keywords: Cosine Family of Distributions; Order Statistics; Maximum Likelihood Estimation; Quantile Function; Lomax Distribution

Introduction

The need for flexible probability distributions has become important in statistical research because they help describe different kinds of data more accurately and support reliable analysis in areas such as engineering, finance, and biological sciences. This interest has led to the creation of generalized families of distributions, known as generators, which extend traditional models by adding extra parameters. These added parameters allow the new distributions

to handle complex data features like heavy tails and varying failure rates. The main goal of these developments is to create models that provide a better fit for various types of data. However, it is well known that no single distribution can describe all kinds of data, so researchers continue to design new probability distributions to meet this need. In recent years, several new families of probability distributions have been developed to improve how data are modeled in different areas such as economics, engineering,

*Corresponding author. *E-mail address: bash0140@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey © Author. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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biology, environmental studies, medicine, and finance. Well-known examples include the Kumaraswamy-G family [1], Weibull-G family [2], power Lindley-G family [3], weighted exponentiated-G family [4], sine Kumaraswamy-G family [5], sine Topp-Leone-G family [6], Topp-Leone odd exponential half logistic-G family [7], extended cosine generalized family [8], Sec-G family [9], logistic cotangent exponentiated generalized family [10], sine type II Topp-Leone family [11], cosine Topp-Leone family [12], secant Kumaraswamy family [13], and exponentiated cosine Topp-Leone family [14]. Building on this foundation, this study introduces a new family of probability distributions called the cosine Kumaraswamy family. It is formed by combining the Kumaraswamy and cosine families of distributions. This integration brings together the parameter-driven flexibility of the Kumaraswamy model with the trigonometric structure of the cosine model, resulting in a more versatile framework. The new family is better able to capture variations in skewness, kurtosis, and tail behavior than many existing models. The Lomax distribution emerges as a special case of this family and showed a better fit for some real datasets than the related submodels. The cosine Kumaraswamy family increases the applicability of classical distributions and serves as a flexible tool for modeling real-world data.

(6) The survival function S(x), hazard function h(x), reverse hazard function rh(x), and the cumulative hazard function R(x) for the CK-G FOD are given in equations (7) to (10): (7) (8)

Quantile Function The quantile function of the CK-G FOD is given as:

Materials And Methods

The cumulative distribution function (cdf) and the probability density function (pdf) of the Kumaraswamy family of distributions (FOD) are expressed in equations (1) and (2) as follows: (1)

Mixture Representation The pdf and the cdf of the CK-G FOD can be expanded using power series expansion as follows: Expanding the sine function in the pdf of the proposed family using a Taylor series gives:

(2) where λ > 0 and θ > 0 are shape parameters, while g(x) and G(x) represent the pdf and cdf of a baseline distribution, respectively. The cdf of the cosine FOD is given by: (3) The corresponding pdf is: (4)

By substituting equations (1) and (2) into equations (3) and (4), we get the cdf and pdf of the cosine Kumaraswamy (CK)-G FOD as follows:

The cdf of the proposed family can also be expanded using power series expansion as follows:

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Order Statistics Let x1, x2, ..., xn be a random sample of size n with pdf f(x) and cdf F(x). The pdf of the order statistics is given by:

For the CK-G FOD, the pdf of the order statistics is given by: (16)

Parameter Estimation

Therefore, (13) Equations (12) and (13) represent the reduced forms of the new family.

Mathematical Properties

Maximum Likelihood Estimation Let x1, x2, ..., xn be an observed random sample of size n from the CK-G family, then the likelihood function can be expressed as:

Taking the natural logarithm, the log-likelihood function simplifies to

Moment Generating Function (MGF) The MGF of the CK-G FOD is given by: (19)

Since obtaining closed-form solutions for this system of equations is challenging, numerical solutions can be found using iterative methods.

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Sub-Model

We introduce the Lomax distribution, as proposed by [15], as a specific case within the CK-G distribution family. The cdf and pdf of the Lomax distribution are provided in equations (21) and (22) below.

Figure 1 illustrates that the CKL distribution’s pdf is approximately symmetric, right-skewed, and unimodal. The cdf starts at zero and approaches one, confirming its validity as a probability distribution. The functions S(x), h(x), rh(x), R(x) and the ϕ(u) of the CKL distribution are given in equations (25) to (29):

(21) (25) (22) where, α > 0 and β > 0 are the shape and scale parameters respectively Cosine Kumaraswamy – Lomax (CKL) Distribution By substituting equation (21) into equation (5), the cdf of the cosine Kumaraswamy Lomax (CKL) distribution is given by: (23)

And by substituting equations (21) and (22) into (6), we have the associated pdf given as: (24)

The pdf and cdf plots of the CKL distribution are shown in Figure 1.

The survival and hazard function plots of the CKL distribution are shown in Figure 2.

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Figure 2. Survival function and Hazard function plots of the CKL distribution.

This section shows how useful the cosine Kumaraswamy Lomax (CKL) distribution is by applying it to three real datasets. The CKL model is compared with some wellknown distributions, such as the Lomax (L) [16], Lehmann type II Lomax (LTIIL) [17], half-logistic Lomax (HLL) [18], and power Lomax (PL) [19] models. Model parameters are estimated using the maximum likelihood estimation (MLE) method. To compare the models, different goodness-of-fit criteria are used, such as the Akaike information criterion (IA), Bayesian information criterion (IB), corrected Akaike information criterion (ICA), and Hannan–Quinn information criterion (IHQ). The model with the smallest values of these measures gives the best fit to the data.

17.88, 28.92, 33.0, 41.52, 42.12, 45.6, 48.8, 51.84, 51.96, 54.12, 55.56, 67.8, 68.44, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 105.84, 127.92, 128.04, 173.4. Descriptive statistics for the first dataset are shown in Table 1, and the graphs are shown in Figure 3. The parameter estimates and goodness-of-fit results for the first dataset are shown in Table 2, with the estimated pdf, cdf, survival function, and P–P plots in Figure 4. Table 1 shows the descriptive statistics for the first dataset, which indicate moderate positive skewness and a leptokurtic shape. These features are also shown in the plots in Figure 3. The kernel density plot supports the presence of positive skewness, while the TTT plot suggests an increasing failure rate. The box plot further shows that the dataset contains some outliers. The goodness-of-fit measures in Table 2 indicate that our proposed CLK model outperforms the other distributions tested, as it has the lowest evaluation metric values. Figure 4 shows the fitted density, cdf, survival function (sf), and P–P plots of the CLK model for the dataset, indicating that the model fits the data very well.

First Data The first dataset, sourced from [20], consists of the number of million revolutions before failure recorded for 23 ball bearings during life tests. The data is as follows:

Second Data The second dataset, as discussed by [20], consists of failure times for 24 mechanical components. The observations are as follows:

Figure 2 illustrates that the hazard function of the CKL distribution exhibits an increasing hazard rate, indicating that the risk of an event occurring rises over time. The survival function, on the other hand, decreases from 1 to 0, indicating a lower probability of survival with time.

Application

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Table 2. Goodness-of-fit measures for each distribution applied to the first dataset.

Ltiil

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Figure 4. Fitted density, cdf, SF and PP plot of the CLK model to the first data.

30.94, 18.51, 16.62, 51.56, 22.85, 22.38, 19.08, 49.56, 17.12, 10.67, 25.43, 10.24, 27.47, 14.70, 14.10, 29.93, 27.98, 36.02, 19.40, 14.97, 22.57, 12.26, 18.14, 18.84 Descriptive statistics for the second dataset are shown in Table 3, and the graphs are shown in Figure 5. The parameter estimates and goodness-of-fit results for the second dataset are shown in Table 4, with the estimated pdf, cdf, survival functions, and P–P plots in Figure 6.

The descriptive statistics presented in Table 3 indicate that the dataset is highly positively skewed and leptokurtic. Figure 4 shows the presence of outliers in the box plot, while the TTT plot reveals an increasing failure rate. Additionally, the kernel density plot suggests that the dataset is right-skewed. The goodness-of-fit measures in Table 4 indicate that our proposed CKL model outperforms the other distributions,

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Table 4. Goodness-of-fit measures for each distribution applied to the second dataset

Ltiil

as evidenced by its lowest evaluation metric values. Figure 6 shows the fitted density, cdf, sf, and P–P plots of the CLK model for the dataset, indicating that the model fits the data very well.

Third Data The third dataset, sourced from [21], includes the relief times (in minutes) for 20 patients treated with an analgesic. “1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.2, 1.4, 3, 1.7, 2.3, 1.6, 2”

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Figure 6. Fitted density, CDF, SF and PP plot of the CKL model to the second data.

Descriptive statistics for the third dataset are shown in Table 5, and the graphs are shown in Figure 7. The parameter estimates and goodness-of-fit results for the third dataset are shown in Table 6, with the estimated pdf, cdf, survival function, and P–P plots in Figure 8. The descriptive statistics presented in Table 5 indicate that the dataset is highly positively skewed and leptokurtic. Figure 7 shows the presence of outliers in the box

plot, while the TTT plot reveals an increasing failure rate. Additionally, the kernel density plot suggests that the dataset is right-skewed. The goodness-of-fit results in Table 6 show that the proposed CKL model performs better than the other distributions, as it has the lowest values for the evaluation measures. Figure 8 shows the fitted density, cdf, sf, and P–P

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Table 6. Goodness-of-fit measures for each distribution applied to the third dataset

Ltiil

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Figure 8. Fitted density, cdf, SF and PP plot of the CKL model to the third data.

plots of the CLK model for the dataset, indicating that the model fits the data very well.

Results And Discussion

The cosine Kumaraswamy Lomax (CKL) distribution was applied to three real datasets from engineering and medical studies. The findings show that the CKL model is flexible and can handle different types of data. In all three cases, the CKL model performed better than the Lomax, Lehmann type II Lomax, half-logistic Lomax, and power Lomax models. The lower IA, IB, ICA, and IHQ values show that the CKL model fits the data well. Results from the analysis of the two

data indicate that the CKL model can be used in both engineering and medical studies. In engineering, it can be used to describe how long components last, helping improve maintenance and design. In medicine, it can be used to study relief times and support treatment choices.

Conclusion

This study introduced a new FOD called the cosine Kumaraswamy FOD, created by combining the Kumaraswamy and cosine families. The main properties of the family, such as the quantile function, moments, and order statistics, were derived.

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A submodel of this family, the cosine Kumaraswamy Lomax (CKL) distribution, was proposed. Its performance was tested with three real datasets from engineering and medical fields and compared with other known models. The CKL model gave the best fit in all cases, showing that it can handle different types of data well. The CKL distribution adds a useful tool to statistical modeling. Future work can build on this family to create new submodels with broader use and better performance in various areas.

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