# Harmonic Mixture Fréchet Distribution: Properties and Applications to Lifetime Data

Authors: Selasi Kwaku Ocloo; Lewis Brew; Suleman Nasiru; Benjamin Odoi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

---

## Abstract

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. The new distribution was applied to three-lifetime datasets and compared with the classical Fréchet distribution and eight (8) other extensions of the Fréchet distribution.


---

## Body

## 1. Introduction

Mixture distributions have turned out to be a very flexible and increasingly common class of distributions over the last two decades. They 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 fields of reliability engineering and survival analysis. One of the four often used extreme value distributions (EVDs) is the Fréchet distribution. The 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 floods are modelled using it. The 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 flexible in modelling both monotonic and non-monotonic datasets. Some extensions of the Fréchet distribution include the Burr X Fréchet (BRXFR) [2], the odd Lomax Fréchet (OLXF) [3], the Poisson–Fréchet (POF) [4], the new exponential-X Fréchet (NEXF) [5], the Weibull Fréchet (WFR) [6], extended Poisson Fréchet distribution (P-BX-Fr) [7], the Burr XII Fréchet (BrXIIFr) [8], the modified Fréchet–Rayleigh distribution (MFRD) [9], truncated Weibull Fréchet distribution (TWFr) [10, 11], the Marshall–Olkin Fréchet distribution (MOF) [12], the gamma extended Fréchet distribution (GEF) [13], the Lehmann type II Fréchet Poisson distribution (LFP) [14], the exponential transmuted Fréchet distribution (ETF) [15], the modified Fréchet (MF) [16], the generalised truncated Fréchet generated family distributions (TGFr-G) [17], and the double truncated transmuted Fréchet distribution (DTTF) [18]. Not long ago, [19] proposed a new family of mixture distribution using the weighted harmonic means of two survival functions and called it the harmonic mixture-G (HMG) family. According to [19], the PDF and cumulative distribution function (CDF), respectively, are given as follows:(1)fHmx=gxG¯α−1xα1−θ+θG¯α−1x1−θ1−G¯α−1x2,x∈R,α≥0,0≤θ≤1.Then,(2)FHmx=1−G¯αx1−θ1−G¯α−1x,x∈R,α≥0,0≤θ≤1,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. The PDF and CDF of the Fréchet distribution are, respectively, given as follows:(3)fFrx;a,b=abax−a−1e−b/xa,x>0,a>0,b>0,and(4)FFrx;a,b=exp−bxa,x>0,where a>0 is a shape parameter and b>0 is a scale parameter.The corresponding survival function of the Fréchet distribution is given as follows:(5)SFrx;a,b=1−exp−bxa,x>0,a>0,b>0.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 ratesThe 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 Section2. In Section 3, we present some statistical properties of the distribution. The 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.


## 2. Harmonic Mixture Fréchet Distribution

The PDF of the HMF distribution is obtained by substituting equations (3) and (5) into equation (1). The PDF of the HMF distribution is given as follows:(6)fHMFx=α1−θabax−a−1e−b/xa1−e−b/xaα−1+θabax−a−1e−b/xa1−e−b/xa2α−21−θ1−1−e−b/xaα−12,where a>0 and α>0 are shape parameters, b>0 is a scale parameter, and x>0 and 0<θ<1.Figure1 shows the density plot of the HMF distribution. Varying the values of the parameters, the density exhibited various kinds of shapes. The PDF of the HMF can be left-skewed, right-skewed with different degrees of kurtosis, J shape, or reversed J shape.Figure 1
PDF plot of the HMF distribution.The corresponding CDF of the HMF distribution is obtained by substituting equation (5) into (2). The CDF of the HMF distribution is given as follows:(7)FHMFx=1−1−e−b/xaα1−θ1−1−e−b/xaα−1,x>0.The failure rate function of the HMF distribution is given as follows:(8)hHMFx=abax−a−1e−b/xa1−e−b/xaα−11−e−b/xaαα1−θ+θ1−e−b/xaα−11−θ1−1−e−b/xaα−1,x>0.Figure2 shows the plot of the failure rate function of HMF distribution. For various values of the parameters, the plots exhibited various desirable shapes, such as decreasing, increasing, and unimodal.Figure 2
Failure rate function plots of the HMF distribution.Lemma 1.
The PDF of the HMF distribution, expressed in mixture form, is given as follows:(9)fHMFx=∑i=0∞∑j=0i∑k=0∞ϖijkabax−a−1e−b/xak+1,where ϖijk=α1−θωijk+θωijk∗,ωijk=−1j+ki+1ijα−1j+1kθi,ωijk∗=−1j+ki+1ijα−1j+2kθi,x>0,a>0,b>0,α>0, and0<θ<1.Proof.
For any real non-integerβ>0, the series expansion for 1−x−β, for x<1, is as follows:(10)1−x−β=∑i=0∞−1i−βixi.
Since0<1−e−b/xa<1, using the series expansion equation in equation (10) twice, we obtain(11)1−θ1−1−e−b/xaα−1−2=∑i=0∞∑j=0i−1ji+1ijθi1−e−b/xajα−1.
Substituting equation (11) into equation (6) and applying equation (10) severally yield(12)fHMFx=α1−θabax−a−1∑i=0∞∑j=0i∑k=0∞−1j+ki+1ijα−1j+1kθie−b/xak+1+θabax−a−1∑i=0∞∑j=0i∑k=0∞−1j+ki+1ijα−1j+2kθie−b/xak+1.
Hence,(13)fHMFx=∑i=0∞∑j=0i∑k=0∞ϖijkabax−a−1e−b/xak+1.
The proof is complete.


## 3. Statistical Properties

The moment-generating function (MGF), stress-strength reliability, entropy, mean deviation, median deviation, mean residual life, and other statistical properties are discussed in this section.


### 3.1. Quantile Function

The quantile function of a distribution is the inverse of the CDF of the distribution. It also gives us a different way to describe the characteristics and shapes of a distribution.The quantile function of the HMF distribution can be expressed as follows:(14)1−q1−θ+θ1−e−b/xqaα−1−1−e−b/xqaα=0,where q∈0,1 and QHMFq=xq 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 defined as follows:(15)GS=Q3/4−2Q1/2+Q1/4Q3/4−Q1/4.Then,(16)MK=Q7/8−Q5/8+Q3/8−Q1/8Q3/4−Q1/4.Table1 shows the results of the quantile function, Galton’s measure of skewness (GS), and the Moors measure of kurtosis (MK) for various parameter values. The 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.Table 1
HMF quantiles for some parameter values.
qα  = 0.09, θ  = 0.5,α  = 15.5, θ  = 0.95,α  = 3.5, θ  = 0.5,α  = 0.009, θ  = 0.08,α  = 50.5, θ  = 0.99,a = 0.5, b = 0.5a = 1.5,b = 0.2a = 1.1,b = 2.5a = 1.1, b = 0.13a = 1.5, b = 0.20.10.15960.09630.89380.21680.09190.250.54440.11591.24180.50820.10180.351.05870.12481.46360.76330.10550.452.01440.13231.70251.09930.10840.52.79500.13591.83391.31260.10980.658.19000.14642.31422.28200.11380.7519.83430.15422.76913.51630.11670.8566.98740.16413.49076.25530.12040.9165.50840.17124.10569.51970.1230GS0.7667−0.04190.22460.4652−0.0752MK4.94571.28301.37722.02281.3645For some parameter values, the HMF distribution is platykurtic, whereas, for others, it is leptokurtic.


### 3.2. Moments

Moments are essential in statistical analysis, especially in deriving some important measures of statistical distributions [22]. Measures such as mean μ, variance σ2, coefficient of variation (CV), skewness (CS), and kurtosis (CK) can be obtained using moments. μ, σ2, CV, CS, and CK, respectively, are defined as follows:(17)μ=μ1′,σ2=μ2′−μ2,CV=μ2′−μ2μ,CS=EX−μ3EX−μ23/2,=μ3′−3μμ2′+2μ3μ2′−μ23/2.Then,(18)CK=EX−μ4EX−μ22,=μ4′−4μμ3′+6μ2μ2′−3μ4μ2′−μ22.Proposition 1.
Therth noncentral moment of the HMF distribution is given as follows:(19)μr′=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/aΓ1−ra,r<a,r=1,2,….Proof.
By definition,(20)μr′=EXr=∫0∞xrfHMFxdx.
Substituting equation (9) into equation (20), we obtain(21)EXr=∫0∞xr∑i=0∞∑j=0i∑k=0∞ϖijkabax−a−1e−b/xak+1dx,=aba∑i=0∞∑j=0i∑k=0∞ϖijk∫0∞xr−a−1e−b/xak+1dx.
Letu=b/xak+1, which implies x=u/bak+1−1/a and dx=−du/abak+1x−a−1. When x⟶0,u⟶∞ and when x⟶∞,u⟶0, we obtain(22)μr′=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/a∫0∞u−r/ae−udx.
Using the identity(23)ΓS=∫0∞xS−1e−xdx,we obtain(24)μr′=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/aΓ1−ra.
The proof is complete.
σ2, CV, CS, and CK for the HMF distribution using the noncentral moments for some selected parameter values are shown in Table 2. The 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.
The HMF distribution is platykurtic for some parameter values and leptokurtic for some parameter values.Table 2
HMF distribution of first five moments for some parameter values.
rα  = 9, θ  = 0.5,α  = 4.0, θ  = 0.4,α  = 7.0, θ  = 0.04,α  = 10.0, θ  = 0.05,α  = 5.5, θ  = 0.005,a = 6, b = 1.5a = 10,b = 2.0a = 10.0,b = 2.5a = 12.0, b = 0.5a = 8.00, b = 0.05μ1′1.31691.91812.30200.46111.47×10−6μ2′1.74353.69295.31090.21298.31×10−8μ3′2.32027.137512.28040.09845.09×10−9μ4′3.104013.849128.46010.04553.29×10−10μ5′4.174126.979566.16780.02112.19×10−11σ29.2744×10−30.01380.01172.8679×10−48.31×10−8CV0.07310.06120.04700.0367196.10CS−0.27100.76440.6798−6.7262212.97CK14.3370−3.6506−9.1900−121.304.76×104


### 3.3. Incomplete Moments

The incomplete moments are essential in obtaining the mean deviation and the median deviation.Proposition 2.
Therth incomplete moment of the HMF distribution is given as follows:(25)mry=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/aΓ1−ra,byak+1,r<a,where Γ⋅,⋅ is the upper incomplete gamma function and r=1,2,3,….Proof.
By definition, the incomplete moment is obtained using(26)mry=EXr|X≤y=∫0yxrfHMFxdx.
Substituting equation (9) into equation (26), we have(27)mry=aba∑i=0∞∑j=0i∑k=0∞ϖijk∫0yxr−a−1e−b/xak+1dx.
Letu=b/xak+1, which implies x=u/bak+1−1/a and dx=−du/abak+1x−a−1. When x⟶0,u⟶∞ and when x⟶y,u⟶b/yak+1, we obtain(28)mry=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/a∫b/yak+1∞u−r/ae−udx.
Using the identity(29)Γa,x=∫x∞ta−1e−tdt,we have(30)mry=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/aΓ1−ra,byak+1.
The proof is complete.


### 3.4. Mean Deviation and Median Deviation

The total variation that exists in a distribution can be measured using the mean and median deviation.Proposition 3.
The mean deviation of the HMF distribution is given as follows:(31)Δ1x=2μFHMFμ−2b∑i=0∞∑j=0i∑k=0∞ϖijkk+11/aΓ1−1a,byak+1,a>1.Proof.
The definition of mean deviation is as follows:(32)Δ1x=∫0∞|x−μ|fHMFxdx,=2μFHMFμ−2∫0μxfHMFxdx.
∫0μxfHMFxdx can be obtained using the first incomplete moment.
The proof is complete.Proposition 4.
The median deviation of the HMF distribution is given as follows:(33)Δ2x=μ−2b∑i=0∞∑j=0i∑k=0∞ϖijkk+11/aΓ1−1a,byak+1.Proof.
By the definition of median deviation,(34)Δ2x=∫0∞x−MfHMFxdx,=∫0MM−xfHMFxdx+∫M∞x−MfHMFxdx,=μ−2∫0MxfHMFxdx.
The proof is complete.


### 3.5. Mean Residual Life

The mean residual life function at timet measures the expected added lifetime that a unit has survived until the time t. This function plays a major role in survival or reliability analysis [23].Proposition 5.
The mean residual life function of the HMF distribution is given as follows:(35)MHMFt=1SHMFtμ−b∑i=0∞∑j=0i∑k=0∞ϖijkk+11/aΓ1−1a,byak+1−t,a>1.Proof.
For a nonnegative random variable X, the mean residual life is given as follows:(36)MLRt=EX−t|X>t=1St∫t∞x−tfxdx,t≥0.
Hence,(37)MLRt=1Stμ−∫0txfxdx−t.
Substituting equation (5) and ∫0txfxdx, which can be obtained from the first incomplete moment into equation (37), we obtain the mean life residual function of the HMF distribution.
The proof is complete.


### 3.6. Moment-Generating Function

The moment-generating function, if it exists, is used to derive the moments of a distribution.Proposition 6.
The moment-generating function of the HMF distribution is given as follows:(38)MHMFt=∑i=0∞∑j=0i∑k=0∞∑r=0∞ϖijkk+1r/abtrr!Γ1−ra,r<a.Proof.
Using the identity(39)etX=∑r=0∞trXrr!,we can define the moment-generating function as follows:(40)MHMFt=EetX,=∑r=0∞trEXrr!,=∑r=0∞trr!μr′.
Substituting equation (25) into equation (40), we obtain the moment-generating function of the HMF distribution.
The proof is complete.


### 3.7. Entropy

The entropy of a random variable is used to measure the variation or uncertainty. The lower the entropy, the less the uncertainty and vice versa.Proposition 7.
The Rényi entropy of the HMF distribution is given as follows:(41)IRλ=11−λlogK∗∑i=0∞∑j=0i∑k=0∞∑m=0∞ψijkm∗Γλ+1aλ−1,where K∗=abaλ−1ba+11−λα1−θλ and(42)ψijkm∗=−1j+k2λ+i−1iijα−1λ+m+jkλmλ+k1−λ1+1aθi+mα1−θ−m.Proof.
By definition,(43)IRλ=11−λlog∫0∞fHMFxλdx.
We obtain the Rényi entropy of the HMF distribution by rearranging and increasing the power of the PDF of the HMF toλ and following the same procedure used to obtain the rth moments.


### 3.8. Stress-Strength Reliability

The extent to which the strength of a system can withstand the stress it is subjected to is measured using the stress-strength reliability [24]. Therefore, the stress-strength reliability can be defined as the probability that a system’s strength X1 is greater than the stress it is subjected to X2; thus, R=PX2<X1. If X1≤X2, the system or component fails.Proposition 8.
IfX1 and X2 follow the HMF distribution, then the stress-strength reliability is given as follows:(44)Rss=1−∑i=0∞∑j=0i∑k=0∞δijkk+1,where δijk=α1−θηijk+θηijk∗, ηijk=−1j+ki+22ijjα−1+2α−1kθi, and(45)ηijk∗=−1j+ki+22ijjα−1+3α−2kθi.Proof.
By definition,(46)Rss=∫0∞fHMFxFHMFxdx,=1−∫0∞fHMFxSHMFxdx.
We then obtain the product of the PDF and survival function given as follows:(47)fHMFxSHMFx=α1−θabax−a−1∑i=0∞∑j=0i∑k=o∞ηijkjα−1+2α−1ke−b/xak+1+θabax−a−1∑i=0∞∑j=0i∑k=0∞ηijkjα−1+3α−2ke−b/xak+1,where ηijk=−1j+ki+22ijθi.
Substituting equation (47) into equation (46) and letting u=b/xak+1 imply x=u/bak+1−1/a and dx=−du/abak+1x−a−1. When x⟶0,u⟶∞ and when x⟶∞,u⟶0, we have(48)Rss=1−α1−θ∑i=0∞∑j=0i∑k=o∞ηijkjα−1+2α−1k∫0∞e−uk+1du+θ∑i=0∞∑j=0i∑k=0∞ηijkjα−1+3α−2k∫0∞e−uk+1du.
Using the identity∫0∞e−udu=1, we obtain(49)Rss=1−α1−θ∑i=0∞∑j=0i∑k=o∞ηijkjα−1+2α−1k1k+1+θ∑i=0∞∑j=0i∑k=0∞ηijkjα−1+3α−2k1k+1.
The proof is complete.


## 3.1. Quantile Function

The quantile function of a distribution is the inverse of the CDF of the distribution. It also gives us a different way to describe the characteristics and shapes of a distribution.The quantile function of the HMF distribution can be expressed as follows:(14)1−q1−θ+θ1−e−b/xqaα−1−1−e−b/xqaα=0,where q∈0,1 and QHMFq=xq 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 defined as follows:(15)GS=Q3/4−2Q1/2+Q1/4Q3/4−Q1/4.Then,(16)MK=Q7/8−Q5/8+Q3/8−Q1/8Q3/4−Q1/4.Table1 shows the results of the quantile function, Galton’s measure of skewness (GS), and the Moors measure of kurtosis (MK) for various parameter values. The 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.Table 1
HMF quantiles for some parameter values.
qα  = 0.09, θ  = 0.5,α  = 15.5, θ  = 0.95,α  = 3.5, θ  = 0.5,α  = 0.009, θ  = 0.08,α  = 50.5, θ  = 0.99,a = 0.5, b = 0.5a = 1.5,b = 0.2a = 1.1,b = 2.5a = 1.1, b = 0.13a = 1.5, b = 0.20.10.15960.09630.89380.21680.09190.250.54440.11591.24180.50820.10180.351.05870.12481.46360.76330.10550.452.01440.13231.70251.09930.10840.52.79500.13591.83391.31260.10980.658.19000.14642.31422.28200.11380.7519.83430.15422.76913.51630.11670.8566.98740.16413.49076.25530.12040.9165.50840.17124.10569.51970.1230GS0.7667−0.04190.22460.4652−0.0752MK4.94571.28301.37722.02281.3645For some parameter values, the HMF distribution is platykurtic, whereas, for others, it is leptokurtic.


## 3.2. Moments

Moments are essential in statistical analysis, especially in deriving some important measures of statistical distributions [22]. Measures such as mean μ, variance σ2, coefficient of variation (CV), skewness (CS), and kurtosis (CK) can be obtained using moments. μ, σ2, CV, CS, and CK, respectively, are defined as follows:(17)μ=μ1′,σ2=μ2′−μ2,CV=μ2′−μ2μ,CS=EX−μ3EX−μ23/2,=μ3′−3μμ2′+2μ3μ2′−μ23/2.Then,(18)CK=EX−μ4EX−μ22,=μ4′−4μμ3′+6μ2μ2′−3μ4μ2′−μ22.Proposition 1.
Therth noncentral moment of the HMF distribution is given as follows:(19)μr′=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/aΓ1−ra,r<a,r=1,2,….Proof.
By definition,(20)μr′=EXr=∫0∞xrfHMFxdx.
Substituting equation (9) into equation (20), we obtain(21)EXr=∫0∞xr∑i=0∞∑j=0i∑k=0∞ϖijkabax−a−1e−b/xak+1dx,=aba∑i=0∞∑j=0i∑k=0∞ϖijk∫0∞xr−a−1e−b/xak+1dx.
Letu=b/xak+1, which implies x=u/bak+1−1/a and dx=−du/abak+1x−a−1. When x⟶0,u⟶∞ and when x⟶∞,u⟶0, we obtain(22)μr′=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/a∫0∞u−r/ae−udx.
Using the identity(23)ΓS=∫0∞xS−1e−xdx,we obtain(24)μr′=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/aΓ1−ra.
The proof is complete.
σ2, CV, CS, and CK for the HMF distribution using the noncentral moments for some selected parameter values are shown in Table 2. The 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.
The HMF distribution is platykurtic for some parameter values and leptokurtic for some parameter values.Table 2
HMF distribution of first five moments for some parameter values.
rα  = 9, θ  = 0.5,α  = 4.0, θ  = 0.4,α  = 7.0, θ  = 0.04,α  = 10.0, θ  = 0.05,α  = 5.5, θ  = 0.005,a = 6, b = 1.5a = 10,b = 2.0a = 10.0,b = 2.5a = 12.0, b = 0.5a = 8.00, b = 0.05μ1′1.31691.91812.30200.46111.47×10−6μ2′1.74353.69295.31090.21298.31×10−8μ3′2.32027.137512.28040.09845.09×10−9μ4′3.104013.849128.46010.04553.29×10−10μ5′4.174126.979566.16780.02112.19×10−11σ29.2744×10−30.01380.01172.8679×10−48.31×10−8CV0.07310.06120.04700.0367196.10CS−0.27100.76440.6798−6.7262212.97CK14.3370−3.6506−9.1900−121.304.76×104


## 3.3. Incomplete Moments

The incomplete moments are essential in obtaining the mean deviation and the median deviation.Proposition 2.
Therth incomplete moment of the HMF distribution is given as follows:(25)mry=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/aΓ1−ra,byak+1,r<a,where Γ⋅,⋅ is the upper incomplete gamma function and r=1,2,3,….Proof.
By definition, the incomplete moment is obtained using(26)mry=EXr|X≤y=∫0yxrfHMFxdx.
Substituting equation (9) into equation (26), we have(27)mry=aba∑i=0∞∑j=0i∑k=0∞ϖijk∫0yxr−a−1e−b/xak+1dx.
Letu=b/xak+1, which implies x=u/bak+1−1/a and dx=−du/abak+1x−a−1. When x⟶0,u⟶∞ and when x⟶y,u⟶b/yak+1, we obtain(28)mry=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/a∫b/yak+1∞u−r/ae−udx.
Using the identity(29)Γa,x=∫x∞ta−1e−tdt,we have(30)mry=br∑i=0∞∑j=0i∑k=0∞ϖijkk+1r/aΓ1−ra,byak+1.
The proof is complete.


## 3.4. Mean Deviation and Median Deviation

The total variation that exists in a distribution can be measured using the mean and median deviation.Proposition 3.
The mean deviation of the HMF distribution is given as follows:(31)Δ1x=2μFHMFμ−2b∑i=0∞∑j=0i∑k=0∞ϖijkk+11/aΓ1−1a,byak+1,a>1.Proof.
The definition of mean deviation is as follows:(32)Δ1x=∫0∞|x−μ|fHMFxdx,=2μFHMFμ−2∫0μxfHMFxdx.
∫0μxfHMFxdx can be obtained using the first incomplete moment.
The proof is complete.Proposition 4.
The median deviation of the HMF distribution is given as follows:(33)Δ2x=μ−2b∑i=0∞∑j=0i∑k=0∞ϖijkk+11/aΓ1−1a,byak+1.Proof.
By the definition of median deviation,(34)Δ2x=∫0∞x−MfHMFxdx,=∫0MM−xfHMFxdx+∫M∞x−MfHMFxdx,=μ−2∫0MxfHMFxdx.
The proof is complete.


## 3.5. Mean Residual Life

The mean residual life function at timet measures the expected added lifetime that a unit has survived until the time t. This function plays a major role in survival or reliability analysis [23].Proposition 5.
The mean residual life function of the HMF distribution is given as follows:(35)MHMFt=1SHMFtμ−b∑i=0∞∑j=0i∑k=0∞ϖijkk+11/aΓ1−1a,byak+1−t,a>1.Proof.
For a nonnegative random variable X, the mean residual life is given as follows:(36)MLRt=EX−t|X>t=1St∫t∞x−tfxdx,t≥0.
Hence,(37)MLRt=1Stμ−∫0txfxdx−t.
Substituting equation (5) and ∫0txfxdx, which can be obtained from the first incomplete moment into equation (37), we obtain the mean life residual function of the HMF distribution.
The proof is complete.


## 3.6. Moment-Generating Function

The moment-generating function, if it exists, is used to derive the moments of a distribution.Proposition 6.
The moment-generating function of the HMF distribution is given as follows:(38)MHMFt=∑i=0∞∑j=0i∑k=0∞∑r=0∞ϖijkk+1r/abtrr!Γ1−ra,r<a.Proof.
Using the identity(39)etX=∑r=0∞trXrr!,we can define the moment-generating function as follows:(40)MHMFt=EetX,=∑r=0∞trEXrr!,=∑r=0∞trr!μr′.
Substituting equation (25) into equation (40), we obtain the moment-generating function of the HMF distribution.
The proof is complete.


## 3.7. Entropy

The entropy of a random variable is used to measure the variation or uncertainty. The lower the entropy, the less the uncertainty and vice versa.Proposition 7.
The Rényi entropy of the HMF distribution is given as follows:(41)IRλ=11−λlogK∗∑i=0∞∑j=0i∑k=0∞∑m=0∞ψijkm∗Γλ+1aλ−1,where K∗=abaλ−1ba+11−λα1−θλ and(42)ψijkm∗=−1j+k2λ+i−1iijα−1λ+m+jkλmλ+k1−λ1+1aθi+mα1−θ−m.Proof.
By definition,(43)IRλ=11−λlog∫0∞fHMFxλdx.
We obtain the Rényi entropy of the HMF distribution by rearranging and increasing the power of the PDF of the HMF toλ and following the same procedure used to obtain the rth moments.


## 3.8. Stress-Strength Reliability

The extent to which the strength of a system can withstand the stress it is subjected to is measured using the stress-strength reliability [24]. Therefore, the stress-strength reliability can be defined as the probability that a system’s strength X1 is greater than the stress it is subjected to X2; thus, R=PX2<X1. If X1≤X2, the system or component fails.Proposition 8.
IfX1 and X2 follow the HMF distribution, then the stress-strength reliability is given as follows:(44)Rss=1−∑i=0∞∑j=0i∑k=0∞δijkk+1,where δijk=α1−θηijk+θηijk∗, ηijk=−1j+ki+22ijjα−1+2α−1kθi, and(45)ηijk∗=−1j+ki+22ijjα−1+3α−2kθi.Proof.
By definition,(46)Rss=∫0∞fHMFxFHMFxdx,=1−∫0∞fHMFxSHMFxdx.
We then obtain the product of the PDF and survival function given as follows:(47)fHMFxSHMFx=α1−θabax−a−1∑i=0∞∑j=0i∑k=o∞ηijkjα−1+2α−1ke−b/xak+1+θabax−a−1∑i=0∞∑j=0i∑k=0∞ηijkjα−1+3α−2ke−b/xak+1,where ηijk=−1j+ki+22ijθi.
Substituting equation (47) into equation (46) and letting u=b/xak+1 imply x=u/bak+1−1/a and dx=−du/abak+1x−a−1. When x⟶0,u⟶∞ and when x⟶∞,u⟶0, we have(48)Rss=1−α1−θ∑i=0∞∑j=0i∑k=o∞ηijkjα−1+2α−1k∫0∞e−uk+1du+θ∑i=0∞∑j=0i∑k=0∞ηijkjα−1+3α−2k∫0∞e−uk+1du.
Using the identity∫0∞e−udu=1, we obtain(49)Rss=1−α1−θ∑i=0∞∑j=0i∑k=o∞ηijkjα−1+2α−1k1k+1+θ∑i=0∞∑j=0i∑k=0∞ηijkjα−1+3α−2k1k+1.
The proof is complete.


## 4. Estimation of Parameters of HMF Distribution

In this section, we obtain the estimators of the HMF distribution using five estimation methods: the maximum-likelihood 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).


### 4.1. Maximum-Likelihood Estimation

The MLE is used to obtain estimates of the unknown parameters by maximising the likelihood function. The likelihood function of the HMF distribution is given as follows:(50)Lx,α,θ,a,b=∏i=1nfHMFx,α,θ,a,b.We obtain the log-likelihood function by substituting equation (6) into (50) and taking the logarithm of the resulting equation. We have(51)lx,α,θ,a,b=nlna+anlnb+−a−1∑i=1nlnxi+α−1∑i=1nln1−e−b/xia+∑i=1nlnα1−θ+θ1−e−b/xiaα−1−2∑i=1nln1−θ1−1−e−b/xiaα−1.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. The resulting functions are as follows:(52)∂l∂α=∑i=0nln1−e−b/xia+∑i=0n1−θ+θln1−e−b/xiaα1−θ+θ1−e−b/xiaα−1−∑i=0n2θln1−e−b/xia1−e−b/xiaα−11−θ+θ1−e−b/xiaα−1,∂l∂θ=−2∑i=0n1−e−b/xiaα−1−11−θ+θ1−e−b/xiaα−1−∑i=0n1−e−b/xiaα−1−αα1−θ+θ1−e−b/xiaα−1,∂l∂a=na+nlnb−∑i=0nlnxi+∑i=0nθα−1b/xialnb/xie−b/xia1−e−b/xiaα−2α1−θ+θ1−e−b/xiaα−1−α∑i=0nα−1b/xialnb/xi1−e−b/xia−∑i=0n2θα−1b/xialnb/xie−b/xia1−e−b/xiaα−21−θ+θ1−e−b/xiaα−1,∂l∂b=nab+∑i=0nθα−1b/xiaa/be−b/xia1−e−b/xiaα−2α1−θ+θ1−e−b/xiaα−1+∑i=0nα−1b/xiaa/be−b/xia1−e−b/xia−∑i=0n2θα−1b/xiaa/be−b/xia1−e−b/xiaα−21−θ+θ1−e−b/xiaα−1.By equating these functions to zero and solving them simultaneously using numerical methods, we obtain the maximum-likelihood estimates of the unknown parameters.


### 4.2. Ordinary Least Squares

The OLS estimates of the unknown parameters of the HMF distribution, wherex1<x2<…<xn are the order statistics of the observed sample, are obtained by minimising the function(53)LSα,θ,a,b=∑j=1nFHMFxj−jn+12.We differentiate equation (53) with respect to the various parameters and equate each result obtained to zero to obtain(54)∂LS∂α=∑j=1nFHMFxj−jn+1⋅Λ1xj;α,θ,a,b=0,(55)∂LS∂θ=∑j=1nFHMFxj−jn+1⋅Λ2xj;α,θ,a,b=0,(56)∂LS∂a=∑j=1nFHMFxj−jn+1⋅Λ3xj;α,θ,a,b=0,(57)∂LS∂b=∑j=1nFHMFxj−jn+1⋅Λ4xj;α,θ,a,b=0,where(58)Λ1xj;α,θ,a,b=θ−1log1−e−b/xja1−e−b/xjaαeb/xja−12θ−1+eb/xja1+θ1−e−b/xjaα−12,(59)Λ2xj;α,θ,a,b=1−e−b/xjaα1−eb/xjaα−1−11+θ1−e−b/xjaα−1−12,(60)Λ3xj;α,θ,a,b=bxjalogbxj1−e−b/xjaα×αθ−1+eb/xjaα+θ1−α1−e−b/xjaαθ−1+eb/xja1+θ1−e−b/xjaα−12,(61)Λ4xj;α,θ,a,b=abbxja1−e−b/xjaα×αθ−1+eb/xjaα+θ1−α1−e−b/xjaαθ−1+eb/xja1+θ1−e−b/xjaα−12.The OLS estimates are obtained by solving these functions simultaneously using numerical methods.


### 4.3. Weighted Least Squares

The WLS estimates of the unknown parameters of the HMF distribution, wherex1<x2<…<xn are the order statistics of the observed sample, are obtained by minimising the function(62)WLSα,θ,a,b=∑j=1nn+12n+2jn−j+1FHMFxj−jn+12.We differentiate equation (62) with respect to the various parameters and equate each result obtained to zero to obtain(63)∂WLS∂a=∑j=1nn+12n+2jn−j+1FHMFxj−jn+1⋅Λ3xj;α,θ,a,b=0,∂WLS∂a=∑j=1nn+12n+2jn−j+1FHMFxj−jn+1⋅Λ3xj;α,θ,a,b=0,∂WLS∂a=∑j=1nn+12n+2jn−j+1FHMFxj−jn+1⋅Λ3xj;α,θ,a,b=0,∂WLS∂b=∑j=1nn+12n+2jn−j+1FHMFxj−jn+1⋅Λ4xj;α,θ,a,b=0.Λkxj;α,θ,a,b, k=1,2,3,4, can be obtained through equations (58)–(61).The WLS estimates are obtained by solving these functions simultaneously by employing numerical methods.


### 4.4. Cramér–Von Mises Estimation

The CVM estimates of the unknown parameters of the HMF distribution, wherex1<x2<…<xn are order statistics of the observed sample, are obtained by minimising the function(64)CMEα,θ,a,b=112n+∑j=1nFHMFxj−2j−12n2.We differentiate equation (64) with respect to the various parameters and equate each result obtained to zero to obtain(65)∂CME∂α=∑j=1nFHMFxj−2j−12n⋅Λ1xj;α,θ,a,b=0,∂CME∂θ=∑j=1nFHMFxj−2j−12n⋅Λ2xj;α,θ,a,b=0,∂CME∂a=∑j=1nFHMFxj−2j−12n⋅Λ3xj;α,θ,a,b=0,∂CME∂b=∑j=1nFHMFxj−2j−12n⋅Λ4xj;α,θ,a,b=0,Λkxj;α,θ,a,b, k=1,2,3,4, can be obtained through equations (58)–(61).The CVM estimates are obtained by solving these functions simultaneously by employing numerical methods.


### 4.5. Anderson–Darling Estimation

The ADE estimates of the unknown parameters of the HMF distribution, wherex1<x2<…<xn are the order statistics of the observed sample, are obtained by minimising the function(66)ADEα,θ,a,b=−n−1n∑j=1n2j−1⋅logFHMFxj+log1−FHMFxn+1−j.We differentiate equation (66) with respect to the various parameters and equate each result obtained to zero to obtain(67)∂ADE∂α=∑j=1n2j−1Λ1xj;α,θ,a,bFHMFxj−Λ1xn+1j;α,θ,a,b1−FHMFxn+1−j=0,∂ADE∂θ=∑j=1n2j−1Λ2xj;α,θ,a,bFHMFxj−Λ2xn+1j;α,θ,a,b1−FHMFxn+1−j=0,∂ADE∂a=∑j=1n2j−1Λ3xj;α,θ,a,bFHMFxj−Λ3xn+1j;α,θ,a,b1−FHMFxn+1−j=0,∂ADE∂b=∑j=1n2j−1Λ4xj;α,θ,a,bFHMFxj−Λ4xn+1j;α,θ,a,b1−FHMFxn+1−j=0,where Λkx⋅;α,θ,a,b, k=1,2,3,4, can be derived from the equations (58)–(61).The ADE estimates are derived by solving these functions simultaneously by employing numerical methods.


## 4.1. Maximum-Likelihood Estimation

The MLE is used to obtain estimates of the unknown parameters by maximising the likelihood function. The likelihood function of the HMF distribution is given as follows:(50)Lx,α,θ,a,b=∏i=1nfHMFx,α,θ,a,b.We obtain the log-likelihood function by substituting equation (6) into (50) and taking the logarithm of the resulting equation. We have(51)lx,α,θ,a,b=nlna+anlnb+−a−1∑i=1nlnxi+α−1∑i=1nln1−e−b/xia+∑i=1nlnα1−θ+θ1−e−b/xiaα−1−2∑i=1nln1−θ1−1−e−b/xiaα−1.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. The resulting functions are as follows:(52)∂l∂α=∑i=0nln1−e−b/xia+∑i=0n1−θ+θln1−e−b/xiaα1−θ+θ1−e−b/xiaα−1−∑i=0n2θln1−e−b/xia1−e−b/xiaα−11−θ+θ1−e−b/xiaα−1,∂l∂θ=−2∑i=0n1−e−b/xiaα−1−11−θ+θ1−e−b/xiaα−1−∑i=0n1−e−b/xiaα−1−αα1−θ+θ1−e−b/xiaα−1,∂l∂a=na+nlnb−∑i=0nlnxi+∑i=0nθα−1b/xialnb/xie−b/xia1−e−b/xiaα−2α1−θ+θ1−e−b/xiaα−1−α∑i=0nα−1b/xialnb/xi1−e−b/xia−∑i=0n2θα−1b/xialnb/xie−b/xia1−e−b/xiaα−21−θ+θ1−e−b/xiaα−1,∂l∂b=nab+∑i=0nθα−1b/xiaa/be−b/xia1−e−b/xiaα−2α1−θ+θ1−e−b/xiaα−1+∑i=0nα−1b/xiaa/be−b/xia1−e−b/xia−∑i=0n2θα−1b/xiaa/be−b/xia1−e−b/xiaα−21−θ+θ1−e−b/xiaα−1.By equating these functions to zero and solving them simultaneously using numerical methods, we obtain the maximum-likelihood estimates of the unknown parameters.


## 4.2. Ordinary Least Squares

The OLS estimates of the unknown parameters of the HMF distribution, wherex1<x2<…<xn are the order statistics of the observed sample, are obtained by minimising the function(53)LSα,θ,a,b=∑j=1nFHMFxj−jn+12.We differentiate equation (53) with respect to the various parameters and equate each result obtained to zero to obtain(54)∂LS∂α=∑j=1nFHMFxj−jn+1⋅Λ1xj;α,θ,a,b=0,(55)∂LS∂θ=∑j=1nFHMFxj−jn+1⋅Λ2xj;α,θ,a,b=0,(56)∂LS∂a=∑j=1nFHMFxj−jn+1⋅Λ3xj;α,θ,a,b=0,(57)∂LS∂b=∑j=1nFHMFxj−jn+1⋅Λ4xj;α,θ,a,b=0,where(58)Λ1xj;α,θ,a,b=θ−1log1−e−b/xja1−e−b/xjaαeb/xja−12θ−1+eb/xja1+θ1−e−b/xjaα−12,(59)Λ2xj;α,θ,a,b=1−e−b/xjaα1−eb/xjaα−1−11+θ1−e−b/xjaα−1−12,(60)Λ3xj;α,θ,a,b=bxjalogbxj1−e−b/xjaα×αθ−1+eb/xjaα+θ1−α1−e−b/xjaαθ−1+eb/xja1+θ1−e−b/xjaα−12,(61)Λ4xj;α,θ,a,b=abbxja1−e−b/xjaα×αθ−1+eb/xjaα+θ1−α1−e−b/xjaαθ−1+eb/xja1+θ1−e−b/xjaα−12.The OLS estimates are obtained by solving these functions simultaneously using numerical methods.


## 4.3. Weighted Least Squares

The WLS estimates of the unknown parameters of the HMF distribution, wherex1<x2<…<xn are the order statistics of the observed sample, are obtained by minimising the function(62)WLSα,θ,a,b=∑j=1nn+12n+2jn−j+1FHMFxj−jn+12.We differentiate equation (62) with respect to the various parameters and equate each result obtained to zero to obtain(63)∂WLS∂a=∑j=1nn+12n+2jn−j+1FHMFxj−jn+1⋅Λ3xj;α,θ,a,b=0,∂WLS∂a=∑j=1nn+12n+2jn−j+1FHMFxj−jn+1⋅Λ3xj;α,θ,a,b=0,∂WLS∂a=∑j=1nn+12n+2jn−j+1FHMFxj−jn+1⋅Λ3xj;α,θ,a,b=0,∂WLS∂b=∑j=1nn+12n+2jn−j+1FHMFxj−jn+1⋅Λ4xj;α,θ,a,b=0.Λkxj;α,θ,a,b, k=1,2,3,4, can be obtained through equations (58)–(61).The WLS estimates are obtained by solving these functions simultaneously by employing numerical methods.


## 4.4. Cramér–Von Mises Estimation

The CVM estimates of the unknown parameters of the HMF distribution, wherex1<x2<…<xn are order statistics of the observed sample, are obtained by minimising the function(64)CMEα,θ,a,b=112n+∑j=1nFHMFxj−2j−12n2.We differentiate equation (64) with respect to the various parameters and equate each result obtained to zero to obtain(65)∂CME∂α=∑j=1nFHMFxj−2j−12n⋅Λ1xj;α,θ,a,b=0,∂CME∂θ=∑j=1nFHMFxj−2j−12n⋅Λ2xj;α,θ,a,b=0,∂CME∂a=∑j=1nFHMFxj−2j−12n⋅Λ3xj;α,θ,a,b=0,∂CME∂b=∑j=1nFHMFxj−2j−12n⋅Λ4xj;α,θ,a,b=0,Λkxj;α,θ,a,b, k=1,2,3,4, can be obtained through equations (58)–(61).The CVM estimates are obtained by solving these functions simultaneously by employing numerical methods.


## 4.5. Anderson–Darling Estimation

The ADE estimates of the unknown parameters of the HMF distribution, wherex1<x2<…<xn are the order statistics of the observed sample, are obtained by minimising the function(66)ADEα,θ,a,b=−n−1n∑j=1n2j−1⋅logFHMFxj+log1−FHMFxn+1−j.We differentiate equation (66) with respect to the various parameters and equate each result obtained to zero to obtain(67)∂ADE∂α=∑j=1n2j−1Λ1xj;α,θ,a,bFHMFxj−Λ1xn+1j;α,θ,a,b1−FHMFxn+1−j=0,∂ADE∂θ=∑j=1n2j−1Λ2xj;α,θ,a,bFHMFxj−Λ2xn+1j;α,θ,a,b1−FHMFxn+1−j=0,∂ADE∂a=∑j=1n2j−1Λ3xj;α,θ,a,bFHMFxj−Λ3xn+1j;α,θ,a,b1−FHMFxn+1−j=0,∂ADE∂b=∑j=1n2j−1Λ4xj;α,θ,a,bFHMFxj−Λ4xn+1j;α,θ,a,b1−FHMFxn+1−j=0,where Λkx⋅;α,θ,a,b, k=1,2,3,4, can be derived from the equations (58)–(61).The ADE estimates are derived by solving these functions simultaneously by employing numerical methods.


## 5. 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. Three different sets of parameter values are used together with the quantile function. The experiment is replicated one thousand times for each sample sizen=30,80,200,500,1000. The average biases (ABs) and the mean square errors (MSEs) of the MLE, OLS, WLS, CVM, and ADE are shown in Tables 3–5.Table 3
Results of simulations forα,θ,a,b=0.1,0.8,2.5,3.0.
ParameterNABMSEMLEOLSCVMADEWLSMLEOLSCVMADEWLSα300.03030.12920.17800.10120.10070.00290.09150.13710.07110.0623800.01760.05780.03990.08390.07470.00330.04320.02690.06650.05812000.00840.02440.01960.03680.03170.00200.01740.01370.02850.02375000.00510.00480.00330.01020.01390.00210.00270.00150.00760.015010000.00210.00270.00310.00410.00650.00070.00150.00200.00270.0050θ300.94950.65090.42170.61040.49678.68652.06800.76082.62431.0813800.10800.15480.25700.27720.20720.11110.52550.85370.89800.58532000.31860.08050.09070.07740.050017.05960.24520.36990.12840.05585000.01750.02070.01840.02320.02220.06630.02420.01830.02850.027410000.00380.00940.00880.01180.01550.00270.00940.00800.01470.0364a300.40580.78280.79140.78840.75531.13361.84171.88041.86971.7393800.18660.29840.29620.29370.28010.37630.71290.70230.69030.64272000.03680.11710.11950.11830.11890.03880.27450.28600.28000.28315000.00740.04820.04870.04820.04750.00360.11630.11850.11600.113110000.00260.02440.02450.02430.02240.00090.05950.05940.058930.0523b300.40530.45910.41530.27760.60060.75610.79290.63430.40201.5635800.09480.17310.18890.23440.25460.07840.32600.33590.48290.64122000.09930.07280.05460.11410.11011.21280.13840.07590.27500.26885000.00660.03120.02250.03860.04700.00340.05820.02980.08590.118210000.00240.01350.01120.01680.02230.00080.02040.01260.03520.0569Table 4
Results of simulations forα,θ,a,b=0.3,0.6,1.9,2.5.
ParameterNABMSEMLECVMADEOLSWLSMLECVMADEOLSWLSα300.03970.18040.15930.17250.14110.00520.10280.08620.09630.0716800.02520.04710.06050.06430.05470.01010.02160.03210.03560.02652000.00850.01700.02120.02260.02810.00220.00690.01020.01150.01665000.00330.00790.01040.00860.01080.00090.00360.00580.00430.008410000.00210.00280.00310.00400.00540.00070.00120.00150.00190.0030θ300.19621.89640.73410.77921.57580.115743.39022.56283.731614.0697800.12010.11950.18850.28330.47790.36470.12590.31871.52905.53062000.01370.06820.07130.05800.14010.00640.12210.11540.07131.14125000.00870.02340.03910.02930.02940.00490.02940.08470.05400.063310000.00280.01350.01440.01470.01600.00110.02010.02260.02390.0349a300.93920.59360.63020.57720.53103.18121.05881.21821.00340.9115800.08350.22230.22510.22500.20710.09410.39600.40580.40540.35082000.01940.09220.08990.09110.08720.01750.17000.16190.16600.15275000.00600.03680.03640.03660.03660.00260.06780.06640.06710.067210000.00160.01850.01850.01830.01790.00050.03420.03420.03350.0319b300.27140.79220.53530.52970.40470.24942.79161.01650.91860.6813800.07310.22480.19380.22400.39970.08410.42820.39570.44334.31322000.01550.09320.05540.10490.07780.00780.18070.10100.22590.13505000.00610.03120.03420.03530.03440.00220.06200.07400.07140.087610000.00160.01080.01140.01200.02080.00040.01920.01920.02400.0439Table 5
Results of simulations forα,θ,a,b=0.03,0.42,2.2,2.6.
ParameterNABMSEMLEADEOLSCVMWLSMLEADEOLSCVMWLSα300.03360.12540.12320.26030.22460.00600.10720.10440.22430.2184800.01520.02440.05670.08690.07790.00310.01840.04820.09110.06772000.00830.01840.01870.01840.04340.00250.01540.01590.01520.03785000.00350.00220.00610.00780.02030.00090.00160.00460.00640.023810000.00060.00200.00260.00300.00600.00010.00150.00300.00230.0051θ300.71361.15210.74020.72230.986110.75266.61913.49012.54886.1550800.03580.22230.26360.17080.23090.01250.76920.87900.27950.69472000.01120.08300.10010.07290.13850.00320.17090.28760.11520.61005000.00400.02890.02810.03370.04130.00090.04820.04160.06770.144410000.00130.01450.01320.01340.02250.00020.02340.02250.01830.0847a300.45160.69350.69640.68250.67450.82991.44561.45751.39901.3753800.12090.26030.26040.25660.24800.19290.54310.54300.52720.49902000.01820.10270.10470.10480.10330.00800.21130.21930.22000.21365000.00540.04290.04200.04210.04150.00180.09230.08810.08890.086110000.00110.02150.02120.02120.02000.00020.04620.04480.04490.0407b300.49400.30970.48630.64730.72802.28361.01650.83751.33532.1240800.05810.13000.19910.21170.19160.03530.39570.34240.39300.34852000.01240.06660.07640.07780.10060.00610.10100.13100.13410.22535000.00460.02700.02840.03180.04340.00140.07400.04680.05800.098310000.00160.02310.01130.01420.01980.00040.01920.01470.02500.0449The ABs and MSEs were computed using the relations as follows:(68)AB=1N∑i=1NVi^−V,Then,(69)MSE=1N∑i=1NVi^−V2.Table3 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. The ABs and MSEs for the estimators of the parameters decrease as the sample size increases despite a few fluctuations. The MLE estimators recorded the least ABs and MSEs and thus could be considered the best estimator.Table4 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. The ABs and MSEs for the estimators of the unknown parameters decrease as the sample size increases. The MLE, however, recorded the least ABs and MSEs and was consistent, thus could be adjudged the best estimator.Table5 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. The ABs and MSEs for the estimators of the unknown parameters in the first and second cases showed decreasing patterns. The 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 Tables3–5, the MLE is the best estimator of the parameters of the HMF distribution.


## 6. Applications

In this section, the HMF distribution is applied to three datasets to ascertain its versatility. These datasets include the annual maximum temperature of a location in the Upper East Region, Ghana (this region provides the relatively highest annual temperature values) (1970–2020), the annual unemployment rate in Ghana (1991–2021), and the survival times data of 128 bladder cancer patients. The annual maximum temperature data in degrees Celsius (°C) used in the analysis were generated from (https://www.globalclimatemonitor.org/) using latitude (10.9922) and longitude (−1.1133). The data are 27.87, 27.42, 27.7, 28.15, 27.27, 27.2, 27.14, 27.74, 27.48, 27.67, 27.82, 28.3, 27.91, 28.35, 28.26, 28.45, 28.38, 29.15, 28.57, 28.21, 28.9, 28.1, 27.65, 28.38, 27.94, 28.2, 28.28, 28.32, 28.71, 28.27, 28.2, 28.6, 28.49, 28.6, 28.55, 28.88, 28.5, 28.35, 28.17, 28.59, 28.8, 28.65, 28.51, 28.26, 28.5, 28.6, 28.59, 28.37, 28.35, 28.8, and 28.7.The annual unemployment rate data were retrieved from (https://data.worldbank.org/country/GH). The data are 3.49, 4.70, 5.27, 5.86, 6.44, 7.02, 7.61, 8.20, 10.10, 10.46, 9.50, 8.53, 7.56, 6.59, 5.62, 4.64, 4.84, 4.99, 5.22, 5.38, 5.60, 5.91, 6.20, 6.52, 6.81, 5.53, 4.22, 4.28, 4.32, 4.65, and 4.70.The bladder cancer data were used by [25, 26]. The data are 0.08, 6.97, 2.46, 9.74, 3.88, 15.96, 4.26, 79.05, 11.79, 8.37, 12.07, 2.09, 9.02, 3.64, 14.76, 5.32, 36.66, 5.41, 1.35, 18.1, 12.02, 21.73, 3.48, 13.29, 5.09, 26.31, 7.39, 1.05, 7.63, 2.87, 1.46, 2.02, 2.07, 4.87, 0.4, 7.26, 0.81, 10.34, 2.69, 17.12, 5.62, 4.4, 3.31, 3.36, 6.94, 2.26, 9.47, 2.62, 14.83, 4.23, 46.12, 7.87, 5.85, 4.51, 6.93, 8.66, 3.57, 14.24, 3.82, 34.26, 5.41, 1.26, 11.64, 8.26, 6.54, 8.65, 13.11, 5.06, 25.82, 5.32, 0.9, 7.62, 2.83, 17.36, 11.98, 8.53, 12.63, 23.63, 7.09, 0.51, 7.32, 2.69, 10.75, 4.33, 1.4, 19.13, 12.03, 22.69, 0.2, 9.22, 2.54, 10.06, 4.18, 16.62, 5.49, 3.02, 1.76, 20.28, 2.23, 13.8, 3.7, 14.77, 5.34, 43.01, 7.66, 4.34, 3.25, 2.02, 3.52, 25.74, 5.17, 32.15, 7.59, 1.19, 11.25, 5.71, 4.5, 3.36, 4.98, 0.5, 7.28, 2.64, 10.66, 2.75, 17.14, 7.93, 6.25, and 6.76.The performance of the HMF distribution is compared with the classical Fréchet distribution and eight (8) modifications of the classical Fréchet distribution. These eight (8) distributions include the Burr X Fréchet (BRXFR) [2], the odd Lomax Fréchet (OLXF) [3], the Poisson–Fréchet (POF) [4], the new exponential-X Fréchet (NEXF) [5], the Weibull Fréchet (WFR) [6], the modified Fréchet–Rayleigh distribution (MFRD) [9], the Marshall–Olkin Fréchet distribution (MOF) [12], and the modified Fréchet (MF) [16].The 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.The 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. The AIC, CAIC, and BIC, respectively, are obtained using(70)AIC=−2logθ^+2k,CAIC=AIC+2kk+1n−k−1.Then,(71)BIC=−2logθ^+klogn.


### 6.1. Annual Maximum Temperature

As shown in Table6, the least annual maximum temperature value for the location selected was 27.14, while the greatest value was 29.15. The value of the coefficient of skewness is −0.72 and that of the coefficient of kurtosis is −0.13. The annual maximum temperature dataset is negatively skewed and less peaked than the normal curve, thus platykurtic.Table 6
Descriptive statistics of annual maximum temperature.
MinimumMaximumMeanMedianCVCSCK27.1429.1528.2528.350.01614−0.721740.13901The MLEs for the models fitted and their standard errors are shown in Table7. 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.Table 7
MLEs for annual maximum temperature dataset.
ModelParametersEstimatesStandard errorsZ-statisticP valueHMFα40.0789513.132193.05200.002274∗θ0.907810.154255.88523.975×10−9∗a15.869753.078465.15512.535×10−7∗b30.249470.3419288.46972.2×10−16∗FRa45.3510994.8768649.29922.2×10−16∗b28.0734730.089386314.07112.2×10−16∗OLXFα23.1059137.983070.60830.5430β0.563030.704810.79880.4244a30.506660.9191733.18952.2×10−16∗b18.352233.923584.67742.905×10−16∗BRXFRθ2.384682.671430.89270.37204a27.481040.8142633.74972.0×10−16∗b17.113648.285282.06550.03887∗NEXFλ15.06363.10434.85251.219×10−6∗α31.45701.067329.47232.2×10−16∗β100.256997.15461.03190.3021POFθ7.342254.865061.50920.1312α51.385327.379116.96363.316×10−12∗β27.016720.4884155.31562.2×10−16∗WFRα19.9391940.0961828207.30522×10−16∗β3.98034970.45489668.75002×10−16∗a0.00110350.00106511.03600.3002b5.34370210.63438088.42352×10−16∗MFRDα2.1947×1023.2926×10−86.6655×1092.2×10−16∗λ8.0889×10−29.5331×10−484.8502.2×10−16∗MFα100.7434612.833197.85024.153×10−15∗β16.221951.991548.14543.779×10−16∗λ0.739570.134235.50963.596×10−8∗MOFα0.100040.138830.72060.47115a31.075419.764663.18240.00146∗b29.020310.7679037.79202×10−16∗∗means significant at 5% significance level.The HMF model gives a better fit to the annual maximum temperature dataset than the other nine (9) competing models. As shown in Table8, 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.Table 8
Log-likelihood and comparison criteria for annual maximum temperature dataset.
ModelℓAICCAICBICK-SADCVMHMF−28.790965.5841766.4537473.311480.0644650.339620.046964FR−42.5650789.1301589.4634892.99380.215663.18040.54316OLXF−30.924669.849270.7187777.576510.13550.960510.18141BRXFR−30.299366.598667.1092474.394070.124670.747470.14096NEXF−30.986967.973868.484473.769180.117950.722690.12837POF−40.6970987.3941887.9048293.189660.183282.62560.44571WFR−80.51591169.0318169.4529176.75910.3604212.0362.4098MFRD−95.52306195.0461195.2961198.90980.4650714.0052.9005MF−47.38782100.7756101.2862106.57110.259584.45830.80337MOF−41.8304789.6609590.171695.456420.1772.50150.41265The fitted PDFs and CDFs of the models are, respectively, presented in Figures3 and 4. It can be seen that the HMF model fits the annual maximum temperature dataset better.Figure 3
Fitted PDFs for annual maximum temperature dataset.Figure 4
Fitted CDFs for annual maximum temperature dataset.


### 6.2. Ghana Annual Unemployment Rate Data

In Table9, the least unemployment rate value in Ghana (1991–2021) was 3.49, while the greatest value was 10.46. The value of the coefficient of skewness is 0.96 and that of the coefficient of kurtosis is 0.36. This shows that the annual unemployment rate dataset is positively skewed and also less peaked than the normal curve, thus platykurtic.Table 9
Descriptive statistics of Ghana unemployment rate (1991–2021).
MinimumMaximumMeanMedianCVCSCK3.49010.4606.1545.6200.28732460.96362150.3614099The MLEs for the models fitted and their standard errors are shown in Table10. 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.Table 10
MLEs for Ghana unemployment rate dataset.
ModelParametersEstimatesStandard errorsZ-statisticP valueHMFα1.365340.974241.40140.161082θ0.316961.905370.16640.867881a3.739281.447482.58330.0009786∗b5.510650.935765.88893.887×10−9∗FRa5.446070.678388.0289.906×10−16∗b5.057760.1832427.6012.2×10−16∗OLXFα2.32562.97590.78150.4345234β2.05662.76200.74460.4565157a5.57031.68703.30200.0009601∗b3.31001.72131.92290.0544865BRXFRθ10.6357114.136570.75240.45184a1.752121.355901.29220.19628b0.620930.241762.56840.01022∗NEXFλ1.945110.868282.24020.02508∗α8.927273.798892.35000.01877∗β5.857466.741590.86890.38493POFθ3.119858.310200.37540.70734α4.979640.714036.97403.081×10−12∗β4.300572.043822.10420.03536∗WFRα13.727557.212911.90320.057015β1.970160.616043.19810.001383∗a6.542224.637951.41060.158367b0.409800.285061.43760.150555MFRDα14.50738615.3008150.94810.3431λ0.2866280.0616264.65113.302×10−6∗MFα107.733.0280×10−43.5577×1052.2×10−16∗β1.03450.140477.36471.775×10−13∗λ0.583338.6299×10−26.75951.385×10−11∗MOFα6.381829.760380.65390.5132a5.846671.235154.73362.206×10−6∗b4.163970.749575.55512.774×10−8∗∗means significant at 5% level of significance.The HMF model gives a better fit to the annual unemployment dataset than the other nine (9) competing models. As shown in Table11, 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 11
Log-likelihood and comparison criteria for Ghana unemployment rate dataset.
ModelℓAICCAICBICK-SADCVMHMF−56.37705120.7541122.2926126.49010.0581580.142510.014025FR−59.96641123.9328124.3614126.80080.151811.15220.17659OLXF−57.72038123.4408124.9793129.17670.0629410.170550.019846BRXFR−58.17907122.7959122.3581126.66010.0883620.312830.040405NEXF−58.5243123.0486123.9375127.350560.0705970.188530.024728POF58.35685122.7137123.6026127.015660.0658780.163490.018076WFR−57.86349123.727125.2655129.46290.0624010.174960.020735MFRD−63.57905131.1581131.5867134.02610.148532.20530.1855MF−58.1597122.3194123.2083128.62140.0668050.185440.022714MOF−58.446122.892123.7808127.193960.0630190.194180.022846The fitted PDFs and CDFs of the models are, respectively, presented in Figures5 and 6. As shown in Figures 3 and 6, the HMF model fits the annual unemployment rate dataset better. The OLXF, BRFXR, NEXF, POF, WFR, MF, and MOF models are alternatively good for fitting the dataset as their goodness-of-fit values are closer to that of the HMF distribution.Figure 5
Fitted PDFs for annual unemployment rate dataset.Figure 6
Fitted CDFs for annual unemployment rate dataset.


### 6.3. Bladder Cancer Survival Time

In Table12, the least survival time value was 0.08, while the greatest value was 79.05. The value of the coefficient of skewness is 3.33 and that of the coefficient of kurtosis is 16.15. The survival time dataset is then highly positively skewed and more peaked than the normal curve, thus leptokurtic.Table 12
Descriptive statistics of bladder cancer survival time.
MinimumMaximumMeanMedianCVCSCK0.08079.0509.3666.3951.122013.3256716.15373The MLEs for the models fitted and their standard errors are shown in Table13. α, θ, and a for HMF, α, β, and a for OLXF, a for BRXFR, and α and a for WFR were not significant at 5% level of significance, while all others in their respective models were significant at 5% significance level.Table 13
MLEs for bladder cancer dataset.
ModelParametersEstimatesStandard errorsZ-statisticP valueHMFα0.00694700.00568861.22120.222θ0.00113180.00311370.53550.59233a1.79427640.24628317.28543.207×10−13∗b0.14381220.08518121.68830.09135FRa0.7520800.04242417.72772.2×10−16∗b3.2582240.4074407.99681.277×10−15∗OLXFα10.9599410.713551.02300.306309β7.371786.747951.09240.274636a6.3764611.147130.57200.567303b0.593400.227812.60480.009193∗BRXFRθ10.5872043.4611093.05890.002221∗a0.0420310.0319131.31710.187818b0.1536510.0155919.85522.2×10−16∗NEXFλ0.3705970.03081312.02722.2×10−16∗α105.1600349.2872382.13360.0328743∗β12.5055583.6672653.41010.0006495∗POFθ6.4777231.3234414.89469.850×10−7∗α1.0022680.05797517.28792.2×10−16∗β0.6369160.1485694.28701.811×10−5∗WFRα9.75106618.5793150.52480.5996982β0.2545900.0673053.78260.0001552∗a4.2135790.9774560.60390.5459203b2.5171080.9359292.68940.0071576∗MFRDα0.43856080.039776711.0262.2×10−16∗λ0.04385330.003871511.3272.2×10−16∗MFα11.1132653.8317672.90030.003728∗β0.4190340.0562967.44349.815×10−15∗λ0.0942450.0144526.52116.980×10−11∗MOFα23.3226988.0535892.89590.00378∗a1.2731760.08049615.81662.2×10−16∗b0.4479300.0888375.04224.603×10−7∗∗means significant at 5% level of significance.The HMF model gives a better fit to the bladder cancer dataset than the other nine (9) competing models. As shown in Table14, 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 14
Log-likelihood and comparison criteria for bladder cancer dataset.
ModelℓAICCAICBICK-SADCVMHMF−410.0845828.1689828.4941839.5770.0365610.152540.017329FR−444.0008892.0015890.0975897.70560.14086.11820.97872OLXF−421.0557850.1114850.4366861.51960.110762.40230.44085BRXFR−415.4706836.9412837.1347845.49730.0680190.902120.14456NEXF−417.0997840.1993840.3928848.75540.0711271.03780.16461POF−427.1315860.2629860.4564868.8190.91085-36.886WFR−411.7882831.5763831.9015842.98440.0607920.533170.083999MFRD−422.7039849.4077849.5037855.11180.127163.10310.59593MF−413.8641833.7281833.9216842.28420.0753420.82320.12748MOF−422.5995851.1991851.3926859.75520.106242.71680.44012The fitted PDFs and CDFs of the models are, respectively, presented in Figures7 and 8. As shown in Figures 7 and 8, the HMF model fits the bladder cancer survival time dataset better. The BRXFR, NEXF, WFR, and MF models are alternatively good in fitting the bladder remission time dataset as their goodness-of-fit values are closer to that of the HMF distribution.Figure 7
Fitted PDFs for bladder cancer survival time dataset.Figure 8
Fitted CDFs for bladder cancer survival time dataset.


## 6.1. Annual Maximum Temperature

As shown in Table6, the least annual maximum temperature value for the location selected was 27.14, while the greatest value was 29.15. The value of the coefficient of skewness is −0.72 and that of the coefficient of kurtosis is −0.13. The annual maximum temperature dataset is negatively skewed and less peaked than the normal curve, thus platykurtic.Table 6
Descriptive statistics of annual maximum temperature.
MinimumMaximumMeanMedianCVCSCK27.1429.1528.2528.350.01614−0.721740.13901The MLEs for the models fitted and their standard errors are shown in Table7. 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.Table 7
MLEs for annual maximum temperature dataset.
ModelParametersEstimatesStandard errorsZ-statisticP valueHMFα40.0789513.132193.05200.002274∗θ0.907810.154255.88523.975×10−9∗a15.869753.078465.15512.535×10−7∗b30.249470.3419288.46972.2×10−16∗FRa45.3510994.8768649.29922.2×10−16∗b28.0734730.089386314.07112.2×10−16∗OLXFα23.1059137.983070.60830.5430β0.563030.704810.79880.4244a30.506660.9191733.18952.2×10−16∗b18.352233.923584.67742.905×10−16∗BRXFRθ2.384682.671430.89270.37204a27.481040.8142633.74972.0×10−16∗b17.113648.285282.06550.03887∗NEXFλ15.06363.10434.85251.219×10−6∗α31.45701.067329.47232.2×10−16∗β100.256997.15461.03190.3021POFθ7.342254.865061.50920.1312α51.385327.379116.96363.316×10−12∗β27.016720.4884155.31562.2×10−16∗WFRα19.9391940.0961828207.30522×10−16∗β3.98034970.45489668.75002×10−16∗a0.00110350.00106511.03600.3002b5.34370210.63438088.42352×10−16∗MFRDα2.1947×1023.2926×10−86.6655×1092.2×10−16∗λ8.0889×10−29.5331×10−484.8502.2×10−16∗MFα100.7434612.833197.85024.153×10−15∗β16.221951.991548.14543.779×10−16∗λ0.739570.134235.50963.596×10−8∗MOFα0.100040.138830.72060.47115a31.075419.764663.18240.00146∗b29.020310.7679037.79202×10−16∗∗means significant at 5% significance level.The HMF model gives a better fit to the annual maximum temperature dataset than the other nine (9) competing models. As shown in Table8, 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.Table 8
Log-likelihood and comparison criteria for annual maximum temperature dataset.
ModelℓAICCAICBICK-SADCVMHMF−28.790965.5841766.4537473.311480.0644650.339620.046964FR−42.5650789.1301589.4634892.99380.215663.18040.54316OLXF−30.924669.849270.7187777.576510.13550.960510.18141BRXFR−30.299366.598667.1092474.394070.124670.747470.14096NEXF−30.986967.973868.484473.769180.117950.722690.12837POF−40.6970987.3941887.9048293.189660.183282.62560.44571WFR−80.51591169.0318169.4529176.75910.3604212.0362.4098MFRD−95.52306195.0461195.2961198.90980.4650714.0052.9005MF−47.38782100.7756101.2862106.57110.259584.45830.80337MOF−41.8304789.6609590.171695.456420.1772.50150.41265The fitted PDFs and CDFs of the models are, respectively, presented in Figures3 and 4. It can be seen that the HMF model fits the annual maximum temperature dataset better.Figure 3
Fitted PDFs for annual maximum temperature dataset.Figure 4
Fitted CDFs for annual maximum temperature dataset.


## 6.2. Ghana Annual Unemployment Rate Data

In Table9, the least unemployment rate value in Ghana (1991–2021) was 3.49, while the greatest value was 10.46. The value of the coefficient of skewness is 0.96 and that of the coefficient of kurtosis is 0.36. This shows that the annual unemployment rate dataset is positively skewed and also less peaked than the normal curve, thus platykurtic.Table 9
Descriptive statistics of Ghana unemployment rate (1991–2021).
MinimumMaximumMeanMedianCVCSCK3.49010.4606.1545.6200.28732460.96362150.3614099The MLEs for the models fitted and their standard errors are shown in Table10. 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.Table 10
MLEs for Ghana unemployment rate dataset.
ModelParametersEstimatesStandard errorsZ-statisticP valueHMFα1.365340.974241.40140.161082θ0.316961.905370.16640.867881a3.739281.447482.58330.0009786∗b5.510650.935765.88893.887×10−9∗FRa5.446070.678388.0289.906×10−16∗b5.057760.1832427.6012.2×10−16∗OLXFα2.32562.97590.78150.4345234β2.05662.76200.74460.4565157a5.57031.68703.30200.0009601∗b3.31001.72131.92290.0544865BRXFRθ10.6357114.136570.75240.45184a1.752121.355901.29220.19628b0.620930.241762.56840.01022∗NEXFλ1.945110.868282.24020.02508∗α8.927273.798892.35000.01877∗β5.857466.741590.86890.38493POFθ3.119858.310200.37540.70734α4.979640.714036.97403.081×10−12∗β4.300572.043822.10420.03536∗WFRα13.727557.212911.90320.057015β1.970160.616043.19810.001383∗a6.542224.637951.41060.158367b0.409800.285061.43760.150555MFRDα14.50738615.3008150.94810.3431λ0.2866280.0616264.65113.302×10−6∗MFα107.733.0280×10−43.5577×1052.2×10−16∗β1.03450.140477.36471.775×10−13∗λ0.583338.6299×10−26.75951.385×10−11∗MOFα6.381829.760380.65390.5132a5.846671.235154.73362.206×10−6∗b4.163970.749575.55512.774×10−8∗∗means significant at 5% level of significance.The HMF model gives a better fit to the annual unemployment dataset than the other nine (9) competing models. As shown in Table11, 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 11
Log-likelihood and comparison criteria for Ghana unemployment rate dataset.
ModelℓAICCAICBICK-SADCVMHMF−56.37705120.7541122.2926126.49010.0581580.142510.014025FR−59.96641123.9328124.3614126.80080.151811.15220.17659OLXF−57.72038123.4408124.9793129.17670.0629410.170550.019846BRXFR−58.17907122.7959122.3581126.66010.0883620.312830.040405NEXF−58.5243123.0486123.9375127.350560.0705970.188530.024728POF58.35685122.7137123.6026127.015660.0658780.163490.018076WFR−57.86349123.727125.2655129.46290.0624010.174960.020735MFRD−63.57905131.1581131.5867134.02610.148532.20530.1855MF−58.1597122.3194123.2083128.62140.0668050.185440.022714MOF−58.446122.892123.7808127.193960.0630190.194180.022846The fitted PDFs and CDFs of the models are, respectively, presented in Figures5 and 6. As shown in Figures 3 and 6, the HMF model fits the annual unemployment rate dataset better. The OLXF, BRFXR, NEXF, POF, WFR, MF, and MOF models are alternatively good for fitting the dataset as their goodness-of-fit values are closer to that of the HMF distribution.Figure 5
Fitted PDFs for annual unemployment rate dataset.Figure 6
Fitted CDFs for annual unemployment rate dataset.


## 6.3. Bladder Cancer Survival Time

In Table12, the least survival time value was 0.08, while the greatest value was 79.05. The value of the coefficient of skewness is 3.33 and that of the coefficient of kurtosis is 16.15. The survival time dataset is then highly positively skewed and more peaked than the normal curve, thus leptokurtic.Table 12
Descriptive statistics of bladder cancer survival time.
MinimumMaximumMeanMedianCVCSCK0.08079.0509.3666.3951.122013.3256716.15373The MLEs for the models fitted and their standard errors are shown in Table13. α, θ, and a for HMF, α, β, and a for OLXF, a for BRXFR, and α and a for WFR were not significant at 5% level of significance, while all others in their respective models were significant at 5% significance level.Table 13
MLEs for bladder cancer dataset.
ModelParametersEstimatesStandard errorsZ-statisticP valueHMFα0.00694700.00568861.22120.222θ0.00113180.00311370.53550.59233a1.79427640.24628317.28543.207×10−13∗b0.14381220.08518121.68830.09135FRa0.7520800.04242417.72772.2×10−16∗b3.2582240.4074407.99681.277×10−15∗OLXFα10.9599410.713551.02300.306309β7.371786.747951.09240.274636a6.3764611.147130.57200.567303b0.593400.227812.60480.009193∗BRXFRθ10.5872043.4611093.05890.002221∗a0.0420310.0319131.31710.187818b0.1536510.0155919.85522.2×10−16∗NEXFλ0.3705970.03081312.02722.2×10−16∗α105.1600349.2872382.13360.0328743∗β12.5055583.6672653.41010.0006495∗POFθ6.4777231.3234414.89469.850×10−7∗α1.0022680.05797517.28792.2×10−16∗β0.6369160.1485694.28701.811×10−5∗WFRα9.75106618.5793150.52480.5996982β0.2545900.0673053.78260.0001552∗a4.2135790.9774560.60390.5459203b2.5171080.9359292.68940.0071576∗MFRDα0.43856080.039776711.0262.2×10−16∗λ0.04385330.003871511.3272.2×10−16∗MFα11.1132653.8317672.90030.003728∗β0.4190340.0562967.44349.815×10−15∗λ0.0942450.0144526.52116.980×10−11∗MOFα23.3226988.0535892.89590.00378∗a1.2731760.08049615.81662.2×10−16∗b0.4479300.0888375.04224.603×10−7∗∗means significant at 5% level of significance.The HMF model gives a better fit to the bladder cancer dataset than the other nine (9) competing models. As shown in Table14, 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 14
Log-likelihood and comparison criteria for bladder cancer dataset.
ModelℓAICCAICBICK-SADCVMHMF−410.0845828.1689828.4941839.5770.0365610.152540.017329FR−444.0008892.0015890.0975897.70560.14086.11820.97872OLXF−421.0557850.1114850.4366861.51960.110762.40230.44085BRXFR−415.4706836.9412837.1347845.49730.0680190.902120.14456NEXF−417.0997840.1993840.3928848.75540.0711271.03780.16461POF−427.1315860.2629860.4564868.8190.91085-36.886WFR−411.7882831.5763831.9015842.98440.0607920.533170.083999MFRD−422.7039849.4077849.5037855.11180.127163.10310.59593MF−413.8641833.7281833.9216842.28420.0753420.82320.12748MOF−422.5995851.1991851.3926859.75520.106242.71680.44012The fitted PDFs and CDFs of the models are, respectively, presented in Figures7 and 8. As shown in Figures 7 and 8, the HMF model fits the bladder cancer survival time dataset better. The BRXFR, NEXF, WFR, and MF models are alternatively good in fitting the bladder remission time dataset as their goodness-of-fit values are closer to that of the HMF distribution.Figure 7
Fitted PDFs for bladder cancer survival time dataset.Figure 8
Fitted CDFs for bladder cancer survival time dataset.


## 7. Conclusion

The four-parameter harmonic mixture Fréchet distribution called the HMF distribution is presented and studied in detail. The failure rate function of the HMF distribution can be monotonically increasing, monotonically decreasing, or upside-down bathtub for a different 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.The maximum-likelihood estimation, the ordinary least-squares estimation, the weighted least-squares estimation, the Cramér–von Mises estimation, and the Anderson–Darling were used to estimate the parameters of the model. The results indicate that the maximum-likelihood estimator is the better estimator.The 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 fit.We are committed to providing a detailed Bayesian study for the four-parameter HMF distribution in the future.



---

*Source: 6460362-2022-09-28.xml*