Novel Logistic Extreme Value Distribution: Properties, Applications, and Parameter Estimation Using Classical and Machine Learning Methods

In probability distribution theory the normal and the exponential distributions are two of the most basic distribution models used commonly to explain different concepts. The exponential distribution is utilized more frequently in reliability studies because of the ease with which mathematical calculations can be made and because it postulates a constant failure rate. However, its application becomes limited in complex situations that require considering the failure rate with time. To overcome this limitation, other flexible extensions such as the Weibull and the gamma distributions have been developed for capturing more than constant failure rates. Weibull and gamma can be introduced as the models that, in fact, case extend the exponential model rates are increasing or decreasing. In addition, the semi-logistic distribution which has structure similar to Weibull distribution is another choice of life distribution modelling such data so that the variety of techniques for successfully describing life behavior under different conditions is enriched [1]. Thus, the consideration of different shapes of failure rate such as decreasing, increasing, bathtub and inverse bathtub failure rate functions in the same model increases the model’s ability to capture various lifetime behaviors. For example, such composite survival models offer remarkable flexibility and increased model fitness in the analysis of a wide range of lifetime statistics. Further, they enable more precise reasoning of results because they help establish the distribution class of the data. This can be attained by creating confidence intervals for the model parameters and therefore adding value on the stochastic nature of the data. In light of these advantageous properties, we come up with the generalized variance framework to extend the existing survival models to a wider range of applications [2].

The exponential distribution forms an indispensable part of the life-testing data and probability theory. It refers to the interval between two events of a Poisson point process which is independent and continuous with equal mean intensity. In mathematical terms, it belongs to a class of distribution touch-stone or more specifically it is the continuous probability distributions akin to the geometric probability distribution. There is a specialization called the ‘memoryless property’, by means of it, signifying that the probability of an event in the future has no relation to past events. Besides Poisson processes, the exponential distribution is possibly used in reliability theory, queuing models, and survival analysis, which facts bring it to some extent to foundational position in theoretical and applied statistics [3].

The Extreme Value Distribution (EVD) is a probability distribution commonly utilized trough reliability engineering, hydrology, business and economics, and environmental simulations to model the patterns of outlying deviations from the median value of a particular dataset. Thus, the EVD has no constant failure rate as the exponential one therefore it is more appropriate to use this distribution for modelling a system where failure probability depends on time. This non-constant failure rate provides a more realistic picture of the aging process and wherein the probability of failure either increases or decreases with the age of the system. In addition, the EVD does not satisfy the memoryless property which is a characteristic or the exponential distribution. The memoryless property further means that the probability of failure in the next interval depends only on the next time interval and not any interval elapsed. On the other hand, the EVD takes into consideration the history of the system or in other word, when computing the probability of failure, h, it takes into consideration the time the system has been in the field. Statistically, EVD is usually associated with the Gumbel, Fréchet and Weibull distributions each of which is relevant to distinct sorts of extremes. The Cumulative Distribution Function (CDF) and the Hazard Rate Function (HRF) for the EVD are different based on the particular variant applied, and in all instances represent the non-memoryless and the changing failure rate characteristics [4].

Reliability analysts find important value in the Novel Logistic Extreme Value Distribution (NLEVD) framework because it analyses multiple hazard rate patterns including the engineering and biomedical bathtub shapes. Experimental failure rate structures observed in bathtub-shaped hazard rates prove inadequate for conventional distributions like exponentials and gammas because their modelling range is limited. The NLEVD enters the field because researchers require an adaptable yet minimal model which fits multiple patterns found in actual failure behavior. The NLEVD achieves superior modelling by utilizing shape-controlling parameters to rephrase its generalized functional form which enables its ability to duplicate traditional models throughout the hazard lifecycle. Real datasets support NLEVD as an exceptional model compared to multiple established models through lower AIC, BIC, and KS statistics. The NLEVD stands as a novel distributional model in current literature because it adjusts to various hazard shapes that include both monotonic and non-monotonic patterns especially for bathtub and inverted-bathtub curve distributions. The fundamental statistical and practical advantage of this feature appears as its main strength.

Logistic distribution is a continuous univariate distribution that has a shape similar to the normal distribution and thus suitable for modelling of extreme values. It is a special case of the Tukey lambda distribution which gives an index of dispersion. Probability Density Function (PDF) and CDF of the logistic distribution have found their use in the fields like logistic regression, growth modelling, logit transformation, and artificial neural network among others. It ranges from demography, physical sciences, finance, sports analysis to computational modelling. What stands out is that higher Kurtosis of the logistic makes it more suited for data that experiences larger variability than the normal distribution for it centers its probability density on zero [5].

Let $Y$ be a nonnegative random variable following a logistic distribution characterized by the shape parameter $\theta>0$. The CDF of $Y$ is formally expressed as:

$G(y ; \theta)=\frac{1}{1+e^{-\theta y}} ; \theta>0, x \in R$       (1)

and the PDF that goes with it is

$g(y ; \theta)=\frac{\theta e^{-\theta y}}{\left(1+e^{-\theta y}\right)^2} ; \theta>0, x \in R$       (2)

Some researchers proposed the logistic-X family, a new class of continuous pdfs obtained with a logistic base random variable. In this family there are smooth variations of the PDF from the shapes resembling inverted J-shaped curve, symmetric, right skewed and left skewed. In addition, the HRF can have different behaviors, including bathtub and upside-down shapes, as well as monotonically decreasing and increasing hazard rates, which makes the family very useful for fitting more complex structures in any area of applied statistics [6].

Hussain et al. [7] introduced the lognormal modified Weibull distribution which is more flexible than the modified Weibull distribution for modelling survival data. This distribution is most effective in complex lifetime data analysis where the standard modified Weibull distribution may not capture hazard rate behaviors sufficiently well because it offers a more flexible means of data distribution.

Therefore, Hussain et al. [8] introduced the Lindley exponential power distribution, which is a more general distribution developed to offer more modelling choice for lifetime data. Compared with this distribution, it supplies more flexible hazard rate function which can well describe different shapes and behaviors for advanced reliability analysis and survival modelling. In 2015, Lemonte et al. [9] used three kinds of parameters for modelling various data types cresting the exponential distribution. This extended distribution is highly successful when used in survival analysis, reliability modelling, analyzing fatigue life data, and in hydrological modelling. It can model an increasing, decreasing, constant, bathtub, inverted bathtub (unimodal), and decreasing-increasing-decreasing hazard rate function, which indicates the capability of this model in capturing a wide range of failure behaviors in practical use [9]. Three parameters exponential expansion distribution was proposed by 2019, as a promising extended model, which sub-model includes exponential distribution, logistic exponential distribution, Marshall-Olkin exponential distribution. This distribution provides great deal of leeway in the modelling of different types of data. It has unimodal or decreasing PDF; it can have increased, decreasing, bathtub and upside-down hazard rate functions. This flexibility makes the distribution particularly useful for measuring various failure characteristics in reliability and survival investigations [10]. In 2020 they introduced the semi-logistic exponential expansion distribution which is derived from the original exponential expansion distribution. This new distribution improves flexibility in modelling of various failure behaviors by using semi-logistic function and it is more versatile in analyzing survival data and reliability problems [11].

Researchers suggested the exponential logistic survival distribution where in a single model, it can capture the usual failure rate trends including decreasing, increasing, bathtub and inverted bathtub. Second, this distribution provides the capability to show a high level of flexibility in terms of using longevity and reliability data. The major strength of the exponential logistic distribution is the availability of closed forms for PDF, survival function, hazard rate function and cumulative hazard function making this model quite distinctive from other models in the inverted and standard bathtub failure classes. These properties make it most suitable for use in theoretical work as well as in practical data analysis using statistical modes [12].

$S(y ; \alpha)=\frac{1}{1+\left(e^y-1\right)^\alpha} ; \alpha>0 ; y \geq 0$        (3)

To do this we introduce a new statistical model called the NLEVD, which has been created within the methodological framework suggested by Lan and Leemis [12]. For the purpose of increasing the flexibility of the distribution one more shape parameter was introduced into the exponential extension distribution making its potential in modelling lifespan more extended and closer to the actual data. Several theoretical characteristics of the NLEVD and a number of application examples are explored in this work. The content is structured as follows: In Section 2, a formal definition of the NLEVD is stated and major mathematical and statistical properties of the estimator are discussed. Section 3 describes the MLE method as the way of parameter estimation. Section 4 introduces another method of parameter estimation based on the Support Vector Machine (SVM) technique, and then compares it with the MLE analysis. Section 5 examines the feasibility and efficiency of the proposed NLEVD on two real-world datasets. The results of the proposed model are established by comparing the fitness/predictability of the model against that of other lifetime distributions. Section 6 presents the conclusions, the main contributions of the work, and possible directions for additional work. The present work introduces an improved version of NLEVD and improves lifetime data modelling flexibility as well as the development of statistics in reliability analysis and survival analysis.

In this section, we give a brief introduction of the SVM algorithm in conjunction with its justification. For any further reading from SVM [13, 14]. Starting with the case of just two linearly separable classes, that is a two-class problem, we first. In this case, let a dataset $\left\{\left(a_i, b_i\right)\right\}_{i=1}^l$ consisting of labeled examples, where $b_i \in\{-1,1\}$ denotes the class labels. The aim is thus to choose from the numerous possibly linear classifiers that could is a least-bound on generalization-error, while ideal represents a perfect separation of the data. This principle originates from the methodology known as Structural Risk Minimization [13]. Also showed in the study [14] that the best ways to define a hyperplane is the one which maximizes the distance between the two classes in the sense of over fitting [15]. Here the margin is equal to the perpendicular distance between the hyperplane and the closest data points belonging to different classes also known as support vectors. This chimes with the theoretical properties scaffolded by statistical learning theory, and works to maximize the margin not only to guarantee sound classification, but also to offer a rigorous method for choosing a model. When the two classes are not well separable, then the SVM aims at getting the decision hyperplane that maximizes this margin and at the same time minimizing a quantity that is proportional to the number of misclassification errors. The values of these two objectives, f and g; are maximized and minimized respectively through a user-defined positive regularizing constant $C$. Mathematically this generalized optimization problem is a quadratic programming ( QP ) problem with coefficients $\beta_i$ representing the coefficients of the linear classifier $f(y)=\operatorname{sign}\left(\sum_{i=1}^l \beta_i a_i y^T y_i+b\right)$. This QP problem is defined over the hypercube $[0, C]^l$ and the formulation of it will be given in Section 3 with Eq. (1). This technique can be generalized by creating a feature space mapping of the original set of variables $y \in R^d$ so that it can capture non-linear decision surface. In particular, for this work, the transformation is regarded as $y \in R^d \rightarrow z(y) \equiv$ $\left(\varphi_1(y), \ldots, \varphi_n(y)\right) \in R^n$, where $n$ may be infinity. $y \in R^d$ into a higher-dimensional feature space. Specifically, the transformation is defined as $y \in R^d \rightarrow z(y) \equiv$ $\left(\varphi_1(y), \ldots, \varphi_n(y)\right) \in R^n$. The linear classification problem is then carried in this transformed feature space as per the following classification function. Vapnik [13] showed that for some transformations $\varphi_1$ a solution to the classification problem has a specific form, and can be realized through kernel functions. This approach relies on the so-called kernel trick, which allows construction of non-linear decision boundaries around the features but does not require the actual computation of high dimensional feature space, so the potential increase in model flexibility is bought at a manageable increase in model complexity.

3.1 SVM-based distribution parameter estimation

• Clarification of SVM-based parameter estimation.

The Support Vector Regression model (SVM regression) functions as a supervised learning algorithm to estimate the function $f(y)$ from dataset $\left\{\left(x_i, y_i\right)\right\}_{i=1}^n$ where predictions $\hat{y}=f(y)$ must remain within $\epsilon$ boundary but adds penalties for exceeding this range. In the context of parameter estimation, each SVM model is trained separately to estimate one of the target distribution parameters $\alpha$ (shape), $\beta$ (scale), and $\mu$ (location). This implies that three distinct SVR models should be constructed, each learning a mapping from input features (e.g., temporal or contextual variables derived from sensor data) to one parameter of the distribution.

• Workflow and input/ output variables.

Let $y \in R^{n \times d}$ denote the input matrix containing $n$ observations with $d$ features, and let $\theta \in\{\alpha, \beta, \mu\}$ be the target parameter. Then:

1. Inputs: $x_i \in R^d$, the feature vector (e.g., accelerometer-derived features).
2. Outputs: $y_i=\theta_i$, the known parameter value from a fitted distribution or simulated data.
3. Model: SVR using kernel $K\left(y_i, y_j\right)$.

The SVR aims to solve the following optimization problem:

$\min _{w, b, \gamma_i, \gamma_j} \frac{1}{2}\|w\|^2+C \sum_{i=1}^n\left(\gamma_i, \gamma_j\right)$        (14)

Subject to $\left\{\begin{array}{c}y_i-\left(w^T \emptyset\left(x_i\right)+b\right) \leq \epsilon+\gamma_i \\ \left(w^T \emptyset\left(x_i\right)+b\right)-y_i \leq \epsilon+\gamma_j \\ \gamma_i, \gamma_j \geq 0\end{array}\right.$

where, $\emptyset(. )$ is the feature transformation defined by the kernel, $\epsilon$ is the tolerance margin, and $C$ is the regularization parameter.

• Kernel choice justification.

The manuscript should also justify the kernel function used in SVR. If the relationship between features and parameters is believed to be non-linear, the Radial Basis Function (RBF) kernel is generally preferred:

$K\left(y_i, y_j\right)=e^{\left(-\gamma\left\|x_i-x_j\right\|^2\right)}$       (15)

This kernel enables the model to capture complex non-linear mappings necessary for accurate parameter regression. However, if interpretability and linear associations are priorities, a linear kernel may be more appropriate.

• Inputs: Feature matrix X (n × d), parameter values y (n × 1) for $\alpha, \beta$, or $\mu$.

% Split data into training and testing sets (80/20 split)

cv = cv partition (size (X, 1), 'Holdout', 0.2);

$X_{\text {train }}$ = X(training(cv), :);

$y_{\text {train }}$  = y(training(cv));

$X_{\text {test }}$ = X(test(cv), :);

$y_{\text {test }}$ = y(test(cv));

• Train SVR model with RBF kernel

svrModel = fitrsvm (X_train, $y_{\text {train }}$,'Kernel Function', 'gaussian',' Box Constraint', 1.0,'Epsilon', 0.1);

• Predict on test data

$y_{\text {pred }}=$ predict $\left(\right.$svr Model, $\left.X_{\text {test}}\right)$.

In this section, we apply the proposed methodology using two real datasets and compare efficiency of the methods. The following data sets have been used to explain the real-life applicability of theoretical framework and comparative analysis of the performance of the proposed model. Advanced model selection and diagnostic methods as discussed in reference [23] may further enhance the performance evaluation.

7.1 Data set 1

The data set under consideration, originally analyzed by Birnbaum and Saunders [1], represents the fatigue life of 6061-T6 aluminum coupons. These coupons were cut parallel to the direction of rolling and subjected to oscillatory stress at a frequency of 18 cycles per second (cps). The data set comprises 101 observations, with each observation corresponding to the fatigue life under a maximum stress of 31,000 psi per cycle. This data set is widely recognized in reliability and survival analysis literature and serves as a benchmark for evaluating the performance of statistical models. The parameters and 95 percent confidence intervals of the NLEVD model which are maximum likelihood estimates of the given data are reported in Table 2.

Table 2. The MLE and 95% confidence interval

Accordingly, the variance-covariance matrix is obtained as the inverse of the Hessian matrix of the negative log-likelihood function, evaluated at the maximum likelihood estimates.

$\begin{aligned} {[D(A)]^{-1}=} & {\left[\begin{array}{ccc}\operatorname{Var}(\hat{\alpha}) & \operatorname{Cov}(\hat{\alpha}, \hat{\beta}) & \operatorname{Cov}(\hat{\alpha}, \hat{\mu}) \\ \operatorname{Cov}(\hat{\beta}, \hat{\alpha}) & \operatorname{Var}(\hat{\beta}) & \operatorname{Cov}(\hat{\beta}, \hat{\mu}) \\ \operatorname{Cov}(\hat{\mu}, \hat{\alpha}) & \operatorname{Cov}(\hat{\mu}, \hat{\beta}) & \operatorname{Var}(\hat{\mu})\end{array}\right] } \\ & =\left[\begin{array}{ccc}0.3233 & 0.0000 & 0.0000 \\ 0.0000 & 1.4122 & 0.0000 \\ 0.0000 & 0.0000 & 3.6231\end{array}\right]\end{aligned}$

The discussion that follows underscores some of the advantages of the proposed NLEVD with a view to underlining its versatility and usefulness in statistical analysis. Further research should be directed in translating the NLEVD for multi-parameters data model for more flexibility of use, assessment of the NLEVD for other and various datasets under the different field such as health care, engineering and the environmental science, and applying more sophisticated and advanced artificial intelligence algorithms in the way of improving parameter estimation. Furthermore, enhancing program interfaces and studying theoretical features of this distribution including asymptotic properties and connections to other distributions, would enlarge its potential and improve mathematical background. By following these directions, the NLEVD can improve its recognition as a powerful tool in various domains applicable to survival analysis and reliability theory, and stimulate further theoretical and applied research in the field of statistics.