Statistics, Optimization & Information Computing

Confidence intervals from local minimums of objective function

Azzouz Dermoune — 2023-07-08

The weighted median plays a central role in the least absolute deviations (LAD). We propose a nonlinear regression using (LAD). Our objective function $f(a, l, s)$ is non-convex with respect to the parameters a, l, s, and is such that for each fixed l, s the minimizer of $a\to f (a, l, s)$ is the weighted median $med(x(l, s), w(l, s))$ of a sequence $x(l, s)$ endowed with the weights $w(l, s)$ (all depend on $l$, $s$). We analyse and compare theoretically the minimizers of the function $(a, l, s)\to f (a, l, s)$ and the surface $(l, s) \to f (med(x(l, s), w(l, s)), l, s)$. As a numerical application we propose to fit the daily infections of COVID 19 in China using Gaussian model. We derive confident interval for the daily infections from each local minimum.

Estimation problem for continuous time stochastic processes with periodically correlated increments

Maksym Luz — 2023-07-08

We deal with the problem of optimal estimation of the linear functionals constructed from unobserved values of a continuous time stochastic process with periodically correlated increments based on past observations of this process. To solve the problem, we construct a corresponding to the process sequence of stochastic functions which forms an infinite dimensional vector stationary increment sequence. In the case of known spectral density of the stationary increment sequence, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

Statistical inferences for the Weibull distribution under adaptive progressive type-II censoring plan and their application in wind speed data analysis

Jaber Kazempoor — 2023-07-09

This paper provides four well-known statistical inferences for the principal parameters regarding the two-parameter Weibull distribution including its hazard, quantile, and survival function based on an adaptive progressive type-II censoring plan. The statistical inferences involve the likelihood and approximate likelihood methods, the Bayesian approach, the bootstrap procedure, and a new conditional technique. To construct Bayesian point estimators and credible intervals, Markov chain Monte Carlo, Metropolis-Hastings, and Gibbs sampling algorithms were used. The Bayesian estimators are developed under conjugate and non-conjugate priors and in the presence of symmetric and asymmetric loss functions. In addition, a conditional estimation technique with interesting distributional characteristics has been introduced. The aforementioned methods are compared extensively through a series of simulations. The results of comparative study showed the superiority of the conditional approach over the other ones. Finally, the developed methods are applied to analyze well-known wind speed data.

Copula based learning for directed acyclic graphs

Russul Mohsin — 2023-08-03

We provide the learning of a DAG model arising from high dimensional random variables following both normal and non-normal assumptions. To this end, the copula function utilized connecting dependent variables. Moreover to normal copula, the three most applicable copulas have been investigated modeling all three dependence structures negative, positive, and weak kinds. The copula functions, FGM, Clayton, and Gumbel are considered coving these situations and their detailed calculations are also presented. In addition, the structure function has been exactly determined due to choosing a good copula model based on statistical software R with respect to any assumed direction among all nodes. The direction with the maximum structure function has been preferred. The corresponding algorithms finding these directions and the maximization procedures are also provided. Finally, some extensive tabulations and simulation studies are provided, and in the following to have a clear thought of provided strategies, a real world application has been analyzed.

Random forests in the zero to k inflated Power series populations

Hadi Saboori — 2023-08-03

Tree-based algorithms are a class of useful, versatile, and popular tools in data mining and machine learning.
Indeed, tree aggregation methods, such as random forests, are among the most powerful approaches to boost
the performance of predictions. In this article, we apply tree-based methods to model and predict discrete
data, using a highly flexible model. Inflation may occur in discrete data at some points. Inflation can be
at points as zero, one or the other. We may even have inflation at two points or more. We use models for
inflated data sets based on a common discrete family (the Power series models). The Power series models
are one of the most famous families used in such models. This family includes common discrete models such
as the Poisson, Negative Binomial, Multinomial, and Logarithmic series models.
The main idea of this article is to use zero to k (k = 0, 1, . . .) inflated regression models based on the family
of power series to fit decision regression trees and random forests. An important point of these models is
that they can be used not only for inflated discrete data but also for non-inflated discrete data. Indeed this
model can be used for a wide range of discrete data sets.

Analysis and Applications of Quantile Approach on Residual Extropy

Amir Hamzeh Khammar — 2023-08-03

Extropy is a measure of the uncertainty of a random variable. Motivated with the wide
applicability of quantile functions in modeling and analyzing statistical data, in this paper, we study
quantile version of the extropy from residual lifetime variable, "residual quantile extropy" in short.
Unlike the residual extropy function, the residual quantile extropy determines the quantile density
function uniquely through a simple relationship. Aging classes, stochastic orders and characterization
results are derived, using proposed quantile measure of uncertainty. We also suggest some applications
related to (n i + 1)-out-of-n systems and distorted random variables. Finally, a nonparametric
estimator for residual quantile extropy is provided. In order to evaluate of proposed estimator, we use
a simulation study.

Isomorphism Check for Two-level Multi-Stage Factorial Designs with Randomization Restrictions via an R Package: IsoCheck

Pratishtha Batra — 2023-08-05

Factorial designs are often used in various industrial and sociological experiments to identify significant factors and factor combinations that may affect the process re- sponse. In the statistics literature, several studies have investigated the analysis, con- struction, and isomorphism of factorial and fractional factorial designs. When there are multiple choices for a design, it is helpful to have an easy-to-use tool for identifying which are distinct, and which of those can be efficiently analyzed/has good theoretical properties. For this task, we present an R library called IsoCheck that checks the isomorphism of multi-stage 26n factorial experiments with randomization restrictions. Through representing the factors and their combinations as a finite projective geometry, IsoCheck recasts the problem of searching over all possible relabelings as a search over collineations, then exploits projective geometric properties of the space to make the search much more efficient. Furthermore, a bitstring representation of the factorial effects is used to characterize all possible rearrangements of designs, thus facilitating quick comparisons after relabeling.

This paper presents several detailed examples with R codes that illustrate the usage of the main functions in IsoCheck. Besides checking equivalence and isomorphism of 2^n multi-stage factorial designs, we demonstrate how the functions of the package can be used to create a catalog of all non-isomorphic designs, and good designs as per a suitably defined ranking criterion.

Gaussian quantum systems and Kahler geometrical structure

Mykola Yaremenko — 2023-08-05

In this article, we study the phase-space distribution of the quantum state as a framework to describe the different properties of quantum systems in continuous-variable systems. The natural approach to quantum systems is given the Gaussian Wigner representation, to unify the description of bosonic and fermionic quantum states, we study the structure of the Kahler space geometry as the geometry generated by three forms under the agreement conditions depended on the nature of the state bosonic or fermionic. Multi-mode light is studied, and we established that the Fock space vacuum corresponds to a certain homogeneous Gaussian state.

Strong consistency of a deconvolution estimator of cumulative distribution function

Trang Bui Thuy — 2023-08-05

We study the strong consistency of a deconvolution estimator of cumulative distribution function when the distribution of error variable is assumed to be known exactly and ordinary smooth as well as supersmooth.

Sine-Cosine Weighted Circular Distributions

Fatemeh Shahsanaei — 2023-08-05

This paper introduces a new family of multimodal and skew-symmetric circular distributions, namely, the sine-cosine weighted circular distribution. The fundamental properties of this family are examined in the context of a general case and three specific examples. Additionally, general solutions for estimating the parameters of any sine-cosine weighted circular distribution using maximum likelihood are provided. A likelihood-ratio test is performed to check the symmetry of the data. Lastly, two examples are presented that illustrate how the proposed model may be utilized to analyze two real-world case studies with asymmetric datasets.

Nonparametric tests of independence using copula-based Renyi and Tsallis divergence measures

Morteza Mohammadi — 2023-08-05

‎We introduce new nonparametric independence tests based on R\'enyi and Tsallis divergence measures and copula density function‎. ‎These tests reduce the complexity of calculations because they only depend on the copula density‎. ‎The copula density estimated using the local likelihood probit-transformation method is appropriate for the identification of independence‎. ‎Also‎, ‎we present the consistency of the copula-based R\'enyi and Tsallis divergence measures estimators that are considered as test statistics‎. ‎A simulation study is provided to compare the empirical power of these new tests with the independence test based on the empirical copula‎. ‎The simulation results show that the suggested tests outperform in weak dependency‎. ‎Finally‎, ‎an application in hydrology is presented‎.

Mean-TVaR Models for Diversified Multi-period Portfolio Optimization with Realistic Factors based on Uncertainty Theory

Khalid Belabbes — 2023-08-06

The focus of any portfolio optimization problem is to imitate the stock markets and propose the optimal solutions to dealing with diverse investor expectations. In this paper, we propose new multi-period portfolio optimization problems when security returns are uncertain variables, given by experts’ estimations, and take the Tail value at risk (TVaR) as a coherent risk measure of investment in the framework of uncertainty theory. Real- constraints, in which transaction costs, liquidity of securities, and portfolio diversification, are taken into account. Equivalent deterministic forms of mean–TVaR models are proposed under the assumption that returns and liquidity of the securities obey some types of uncertainty distributions. We adapted the Delphi method in order to evaluate the expected, the standard deviation and the turnover rates values of returns of the given securities. Finally, numerical examples are given to illustrate the effectiveness of the proposed models.

A Discrete New Generalized Two Parameter Lindley Distribution: Properties, Estimation and Applications

Manal M. Salem — 2023-08-06

In this paper, a discrete new generalized two parameter Lindley distribution is proposed. Discrete Lindley and Geometric distributions are sub-models of the proposed distribution. Its probability mass function exhibits different shapes including decreasing, unimodal and decreasing-increasing-decreasing. Our proposed distribution has only two-parameters and its hazard rate function can accommodate increasing, constant, decreasing and bathtub shapes. Moreover, this distribution can describe equi and over dispersed data. Several distributional properties are obtained and several reliability characteristics are derived such as cumulative distribution function, hazard rate function, second hazard rate function, mean residual life function, reverse hazard rate function, accumulated hazard rate function and also its order statistics. In addition, the study of the shapes of the hazard rate function is provided analytically and also by plots. Estimation of the parameters is done using the maximum likelihood method. A simulation study is conducted to assess the performance of the maximum likelihood estimators. Finally, the flexibility of the model is illustrated using three real data sets.

The Marshall-Olkin Topp-Leone Half-Logistic-G Family of Distributions with Applications

Whatmore Sengweni — 2023-08-07

A new family of distributions called the Marshall-Olkin Topp-Leone Half-Logistic-G (MO-TLHL-G) family of distributions is proposed and studied. Structural properties of the new family of distributions including moments, incomplete moments, distribution of the order statistics, and Renyi entropy are derived. The maximum likelihood estimation technique is used to estimate the model parameters. A simulation study to examine the bias and mean square error of the maximum likelihood estimators and applications to real data sets to illustrates the usefulness of the generalized distribution are given.

Variational Bayesian Inference for Exponentiated Weibull Right Censored Survival Data

Jibril Abubakar — 2023-08-07

The exponential, Weibull, log-logistic and lognormal distributions represent the class of light and heavy-tailed distributions that are often used in modelling time-to-event data. The exponential distribution is often applied if the hazard is constant, while the log-logistic and lognormal distributions are mainly used for modelling unimodal hazard functions. The Weibull distribution is on the other hand well-known for modelling monotonic hazard rates. Recently, in practice, survival data often exhibit both monotone and non-monotone hazards. This gap has necessitated the introduction of Exponentiated Weibull Distribution (EWD) that can accommodate both monotonic and non-monotonic hazard functions. It also has the strength of adapting unimodal functions with bathtub shape. Estimating the parameter of EWD distribution poses another problem as the flexibility calls for the introduction of an additional parameter. Parameter estimation using the maximum likelihood approach has no closed-form solution, and thus, approximation techniques such as Newton-Raphson is often used. Therefore, in this paper, we introduce another estimation technique called Variational Bayesian (VB) approach. We considered the case of the accelerated failure time (AFT) regression model with covariates. The AFT model was developed using two comparative studies based on real-life and simulated data sets. The results from the experiments reveal that the Variational Bayesian (VB) approach is better than the competing Metropolis-Hasting Algorithm and the reference maximum likelihood estimates.

Comparing the Accuracy of Classical and Machine Learning Methods in Time Series Forecasting: A Case Study of USA Inflation

Youness Jouilil — 2023-08-20

This paper presents a comparison of statistical classical methods and machine learning algorithms for time series forecasting notably the Exponential Smoothing, hybrid ARIMA-GARCH model, K-Nearest Neighbors (KNN), Prophet, and Long-Short Term Memory (LSTM). The data set used in this study is related to US inflation and covers the period from 1965 to 2021. The performance of the models was evaluated using different metrics especially Mean Squared Error (MSE), Mean Absolute Error (MAE), Median Absolute Error (Median AE), and Root Mean Squared Error (RMSE). The results of the numerical comparison show that the best performance was achieved by Exponential Smoothing, followed closely by KNN. The results indicate that these two models are well-suited for forecasting inflation in the US. ARIMA-GARCH, LSTM, and Prophet performed relatively poorly in comparison. Overall, the findings of this study can be useful for practitioners in choosing the most suitable method for forecasting inflation in the US in the short-term period.