Harmonic Mixture Fréchet Distribution: Properties and Applications to Lifetime Data

In this study, we propose a four-parameter probability distribution called the harmonic mixture Fréchet. Some useful expansions and statistical properties such as moments, incomplete moments, quantile functions, entropy, mean deviation, median deviation, mean residual life, moment-generating function, and stress-strength reliability are presented. Estimators for the parameters of the harmonic mixture Fréchet distribution are derived using the estimation techniques such as the maximum-likelihood estimation, the ordinary least-squares estimation, the weighted least-squares estimation, the Cramér–von Mises estimation, and the Anderson–Darling estimation. A simulation study was conducted to assess the biases and mean square errors of the estimators. e new distribution was applied to three-lifetime datasets and compared with the classical Fréchet distribution and eight (8) other extensions of the Fréchet distribution.


Introduction
Mixture distributions have turned out to be a very exible and increasingly common class of distributions over the last two decades. ey have been applied to lifetime data in many reliability and survival analysis. Whether a sample is homogeneous or heterogeneous, the statistical analysis of lifetime datasets is a momentous task in the elds of reliability engineering and survival analysis. One of the four often used extreme value distributions (EVDs) is the Fréchet distribution. e distribution, which is also known as the EVD type II, is the inverse of the Weibull distribution. Extreme events such as annual rainfall, earthquakes, and oods are modelled using it. e probability density function (PDF) of the Fréchet distribution has a unimodal shape or a decreasing shape, which depends on the shape parameter, while its failure rate function exhibits a unimodal shape always [1]. Several extensions of the Fréchet distribution have been proposed in the literature aimed at making it more exible in modelling both monotonic and non-monotonic datasets.
(1) en, where G(x) is the survival function of the baseline distribution. Subsequently, the focus of this study was to develop an extension of the Fréchet distribution using the HMG family of distributions to widen its flexibility in analysing different types of real datasets. e PDF and CDF of the Fréchet distribution are, respectively, given as follows: and where a > 0 is a shape parameter and b > 0 is a scale parameter. e corresponding survival function of the Fréchet distribution is given as follows: Our motivation for this study includes the following: (i) Developing a heavy-tailed distribution that models lifetime datasets (ii) Developing a distribution whose probability densities exhibit a left-or right-skewed shape, a reversed J shape, or a J shape (iii) Providing a distribution that consistently offers better fits to lifetime datasets than those of other generalised distributions with the same underlying model (Fréchet distribution) (iv) Proposing a distribution that can model lifetime datasets with monotonic or non-monotonic failure rates e remaining parts of the article are organised as follows: PDF, CDF, and failure rate function of the harmonic mixture Fréchet (HMF) distribution alongside their corresponding graphical representations are presented in Section 2. In Section 3, we present some statistical properties of the distribution. e maximum-likelihood estimation, the ordinary least-squares estimation, the weighted least-squares estimation, the Cramér-von Mises estimation, and the Anderson-Darling estimation of the HMF parameters are developed in Section 4. Section 5 presents the simulation results to assess the performance of the estimators of the HMF distribution. Section 6: three applications to real datasets are illustrated to ascertain the importance of the proposed model. Lastly, the conclusions of the study are reported in section 7.

Harmonic Mixture Fréchet Distribution
e PDF of the HMF distribution is obtained by substituting equations (3) and (5) into equation (1). e PDF of the HMF distribution is given as follows: where a > 0 and α > 0 are shape parameters, b > 0 is a scale parameter, and x > 0 and 0 < θ < 1. Figure 1 shows the density plot of the HMF distribution. Varying the values of the parameters, the density exhibited various kinds of shapes. e PDF of the HMF can be leftskewed, right-skewed with different degrees of kurtosis, J shape, or reversed J shape. e corresponding CDF of the HMF distribution is obtained by substituting equation (5) into (2). e CDF of the HMF distribution is given as follows:

Quantile Function.
e quantile function of a distribution is the inverse of the CDF of the distribution. It also gives us a di erent way to describe the characteristics and shapes of a distribution. e quantile function of the HMF distribution can be expressed as follows: where q ∈ (0, 1) and Q HMF (q) x q is the quantile function.
It can be seen that the quantile function of the HMF distribution does not have a closed form. Numerical approximations will therefore be used to approximate the various values of the quantile function.
Galton's measure of skewness (GS) [20] and the Moors measure of kurtosis (MK) [21], respectively, are de ned as follows: en, Table 1 shows the results of the quantile function, Galton's measure of skewness (GS), and the Moors measure of kurtosis (MK) for various parameter values. e HMF distribution could be either moderately or strongly skewed. For some parameter values, the HMF distribution is positively skewed, while for some other parameter values, the distribution is negatively skewed.
For some parameter values, the HMF distribution is platykurtic, whereas, for others, it is leptokurtic.

Moments.
Moments are essential in statistical analysis, especially in deriving some important measures of statistical distributions [22]. Measures such as mean (μ), variance (σ 2 ), coe cient of variation (CV), skewness (CS), and kurtosis (CK) can be obtained using moments. μ, σ 2 , CV, CS, and CK, respectively, are de ned as follows: en, Proposition 1. e r th noncentral moment of the HMF distribution is given as follows: Proof. By de nition, Substituting equation (9) into equation (20), we obtain Using the identity we obtain e proof is complete. σ 2 , CV, CS, and CK for the HMF distribution using the noncentral moments for some selected parameter values are shown in Table 2. e HMF distribution could be highly skewed or moderately skewed. For some parameter values, the HMF distribution is positively skewed, while for some other parameter values, the distribution is negatively skewed. e HMF distribution is platykurtic for some parameter values and leptokurtic for some parameter values. e incomplete moments are essential in obtaining the mean deviation and the median deviation.
Proof. By definition, the incomplete moment is obtained using Substituting equation (9) into equation (26), we have which implies x � (u/b a (k + 1)) − 1/a and dx � (− du/ab a (k + 1)x − a− 1 ). When x ⟶ 0, u ⟶ ∞ and when x ⟶ y, u ⟶ (b/y) a (k + 1), we obtain Using the identity we have e proof is complete. e total variation that exists in a distribution can be measured using the mean and median deviation.

Proposition 3.
e mean deviation of the HMF distribution is given as follows: Proof. e definition of mean deviation is as follows: (32) μ 0 xf HMF (x)dx can be obtained using the first incomplete moment. e proof is complete. □ Proposition 4. e median deviation of the HMF distribution is given as follows: Proof. By the definition of median deviation, e proof is complete. e mean residual life function at time t measures the expected added lifetime that a unit has survived until the time t. is function plays a major role in survival or reliability analysis [23].

Proposition 5.
e mean residual life function of the HMF distribution is given as follows: Proof. For a nonnegative random variable X, the mean residual life is given as follows: Hence, Substituting equation (5) and t 0 xf(x)dx, which can be obtained from the first incomplete moment into equation (37), we obtain the mean life residual function of the HMF distribution.
e proof is complete.   e moment-generating function of the HMF distribution is given as follows: Proof. Using the identity we can define the moment-generating function as follows: Substituting equation (25) into equation (40), we obtain the moment-generating function of the HMF distribution. e proof is complete.
Entropy. e entropy of a random variable is used to measure the variation or uncertainty. e lower the entropy, the less the uncertainty and vice versa.

Proposition 7.
e Rényi entropy of the HMF distribution is given as follows: where Proof. By definition, We obtain the Rényi entropy of the HMF distribution by rearranging and increasing the power of the PDF of the HMF to λ and following the same procedure used to obtain the rth moments. e extent to which the strength of a system can withstand the stress it is subjected to is measured using the stress-strength reliability [24]. erefore, the stress-strength reliability can be defined as the probability that a system's strength X 1 is greater than the stress it is subjected to X 2 ; thus, R � P(X 2 < X 1 ). If X 1 ≤ X 2 , the system or component fails.

Proposition 8.
If X 1 and X 2 follow the HMF distribution, then the stress-strength reliability is given as follows: Proof. By definition, We then obtain the product of the PDF and survival function given as follows: International Journal of Mathematics and Mathematical Sciences 7

Substituting equation (47) into equation (46) and letting
Using the identity e proof is complete.

Estimation of Parameters of HMF Distribution
In this section, we obtain the estimators of the HMF distribution using five estimation methods: the maximumlikelihood estimation (MLE), the ordinary least-squares method (OLS), the weighted least-squares method (WLS), the Cramér-von Mises estimation (CVM) and the Anderson-Darling estimation (ADE).

Maximum-Likelihood Estimation.
e MLE is used to obtain estimates of the unknown parameters by maximising the likelihood function. e likelihood function of the HMF distribution is given as follows: We obtain the log-likelihood function by substituting equation (6) into (50) and taking the logarithm of the resulting equation. We have We obtain the MLE of the parameters by differentiating equation (51) with respect to (α, θ, a, b) and equating the resulting functions to zero. e resulting functions are as follows: International Journal of Mathematics and Mathematical Sciences By equating these functions to zero and solving them simultaneously using numerical methods, we obtain the maximum-likelihood estimates of the unknown parameters.

Ordinary Least Squares.
e OLS estimates of the unknown parameters of the HMF distribution, where are the order statistics of the observed sample, are obtained by minimising the function (53) We differentiate equation (53) with respect to the various parameters and equate each result obtained to zero to obtain where International Journal of Mathematics and Mathematical Sciences e OLS estimates are obtained by solving these functions simultaneously using numerical methods.

Weighted Least Squares.
e WLS estimates of the unknown parameters of the HMF distribution, where x (1) < x (2) < . . . < x (n) are the order statistics of the observed sample, are obtained by minimising the function We differentiate equation (62) with respect to the various parameters and equate each result obtained to zero to obtain Λ k (x (j) ; α, θ, a, b), (k � 1, 2, 3, 4), can be obtained through equations (58)-(61). e WLS estimates are obtained by solving these functions simultaneously by employing numerical methods.

Cramér-Von Mises Estimation.
e CVM estimates of the unknown parameters of the HMF distribution, where x (1) < x (2) < . . . < x (n) are order statistics of the observed sample, are obtained by minimising the function We differentiate equation (64) with respect to the various parameters and equate each result obtained to zero to obtain (Λ k x (j) ; α, θ, a, b), (k � 1, 2, 3, 4), can be obtained through equations (58)-(61). e CVM estimates are obtained by solving these functions simultaneously by employing numerical methods.

Anderson-Darling Estimation.
e ADE estimates of the unknown parameters of the HMF distribution, where x (1) < x (2) < . . . < x (n) are the order statistics of the observed sample, are obtained by minimising the function We differentiate equation (66) with respect to the various parameters and equate each result obtained to zero to obtain where Λ k (x (·) ; α, θ, a, b), (k � 1, 2, 3, 4), can be derived from the equations (58)-(61). e ADE estimates are derived by solving these functions simultaneously by employing numerical methods.

Monte Carlo Simulation
In this section, we perform a simulation study to assess the performance of the estimators for the parameters of the HMF distribution. ree different sets of parameter values are used together with the quantile function. e experiment is replicated one thousand times for each sample size n � 30, 80, 200, 500, 1000. e average biases (ABs) and the mean square errors (MSEs) of the MLE, OLS, WLS, CVM, and ADE are shown in Tables 3-5. e ABs and MSEs were computed using the relations as follows: en, (69) Table 3 shows the AB and MSE of the MLE, OLS, WLS, CVM, and ADE of (α, θ, a, b) � (0.1, 0.8, 2.5, 3.0) for n � 30, 80, 200, 500, 1000. e ABs and MSEs for the estimators of the parameters decrease as the sample size increases despite a few fluctuations. e MLE estimators recorded the least ABs and MSEs and thus could be considered the best estimator. Table 4 shows the ABs and MSEs of the MLE, OLS, WLS, CVM, and ADE of (α, θ, a, b) � (0.3, 0.6, 1.9, 2.5) for n � 30, 80, 200, 500, 1000. e ABs and MSEs for the estimators of the unknown parameters decrease as the sample size increases. e MLE, however, recorded the least ABs and MSEs and was consistent, thus could be adjudged the best estimator. Table 5 shows the ABs and MSEs of the MLE, OLS, WLS, CVM, and ADE of (α, θ, a, b) � (0.03, 0.42, 2.2, 2.6) for n � 30, 80, 200, 500, 1000. e ABs and MSEs for the estimators of the unknown parameters in the first and second cases showed decreasing patterns. e MLE again recorded the least ABs and MSEs and was consistent; thus, it could be adjudged the best estimator.
Based on rankings (the least ABEs and MSEs to the greatest ABEs and MSEs) in Tables 3-5, the MLE is the best estimator of the parameters of the HMF distribution.
e Anderson-Darling (AD), Kolmogorov-Smirnov (K-S), and Cramér-von Mises (CVM) tests were employed to assess the goodness of fit of the selected distributions. e distribution with the lowest Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), and Bayesian information criterion (BIC) is considered the most appropriate model for the datasets. e AIC, CAIC, and BIC, respectively, are obtained using AIC � − 2log(θ) + 2k, en, 6.1. Annual Maximum Temperature. As shown in Table 6, the least annual maximum temperature value for the location selected was 27.14, while the greatest value was 29.15. e value of the coefficient of skewness is − 0.72 and that of the coefficient of kurtosis is − 0.13. e annual maximum temperature dataset is negatively skewed and less peaked than the normal curve, thus platykurtic.
e MLEs for the models fitted and their standard errors are shown in Table 7. α and β for OLXF, θ for BRXFR, β for NEXF, θ for POF, a for WFR, and α for MOF were not significant at 5% level of significance, while all others in their respective models were significant at 5% significance level. e HMF model gives a better fit to the annual maximum temperature dataset than the other nine (9) competing models. As shown in Table 8, the HMF model had the highest log-likelihood value and the lowest AIC, CAIC, and BIC values compared with the other competing models. Additionally, the HMF model had the smallest AD, K-S, and CVM values. e fitted PDFs and CDFs of the models are, respectively, presented in Figures 3 and 4. It can be seen that the HMF model fits the annual maximum temperature dataset better.   Table 9, the least unemployment rate value in Ghana (1991-2021) was 3.49, while the greatest value was 10.46. e value of the coefficient of skewness is 0.96 and that of the coefficient of kurtosis is 0.36. is shows that the annual unemployment rate dataset is positively skewed and also less peaked than the normal curve, thus platykurtic. e MLEs for the models fitted and their standard errors are shown in Table 10. α and θ for HMF, α, β, and b for  OLXF, θ and a for BRXFR, β for NEXF, θ for POF, α, a, and b for WFR, α for MFRD, and α for MOF were not significant at 5% level of significance, while all others in their respective models were significant at 5% significance level. e HMF model gives a better fit to the annual unemployment dataset than the other nine (9) competing models. As shown in Table 11, the HMF model had the highest log-likelihood value and the lowest AIC, CAIC, and BIC values compared with the other competing models. Also, the HMF model had the smallest AD, K-S, and CVM values.   Table 12, the least survival time value was 0.08, while the greatest value was 79.05. e value of the coe cient of skewness is 3.33 and that of the coe cient of kurtosis is 16.15. e survival time dataset is then highly positively skewed and more peaked than the normal curve, thus leptokurtic. e MLEs for the models tted and their standard errors are shown in Table 13. α, θ, and a for HMF, α, β, and a for OLXF, a for BRXFR, and α and a for WFR were not signi cant at 5% level of signi cance, while all others in their respective models were signi cant at 5% signi cance level.

Bladder Cancer Survival Time. In
e HMF model gives a better t to the bladder cancer dataset than the other nine (9) competing models. As shown in Table 14, the HMF model had the highest log-likelihood value and the lowest AIC, CAIC, and BIC values compared with the other competing models. Also, the HMF model had the smallest AD, K-S, and CVM values.

Conclusion
e four-parameter harmonic mixture Fréchet distribution called the HMF distribution is presented and studied in detail. e failure rate function of the HMF distribution can be monotonically increasing, monotonically decreasing, or upside-down bathtub for a di erent combination of the parameter values. Some statistical properties such as moments, incomplete moments, quantile functions, entropy, mean deviation, median deviation, mean residual life, moment-generating function (MGF), and stress-strength reliability are presented. e maximum-likelihood estimation, the ordinary leastsquares estimation, the weighted least-squares estimation, the Cramér-von Mises estimation, and the Anderson-Darling were used to estimate the parameters of the model.
e results indicate that the maximum-likelihood estimator is the better estimator.
e new distribution was applied to three-lifetime datasets and compared with the classical Fréchet distribution and eight (8) other extensions of the Fréchet distribution and was found to provide a better t.
We are committed to providing a detailed Bayesian study for the four-parameter HMF distribution in the future.
Data Availability e study is aimed at improving methodologies, and the data used have been duly cited within the manuscript.

Conflicts of Interest
e authors declare that they have no con icts of interest.