On a New Bimodal Normal Family

Recently, in order to get closer agreement at the extremes, skewed distributions are playing an important role in various research studies. The generalized logistic distribution (GLD) of Type II, which is indexed by one shape parameter, is introduced here to extend the scope of this distribution in some asymmetrical studies. Several properties of this distribution in relation to other probability distributions are stated. Furthermore, the maximum-likelihood (ML) method and an approximate ML method are used to derive the point estimators of the parameters based on progressive Type II censoring. A wide range of sample sizes and progressive-censoring schemes are considered in a simulation study to see the performance of estimates of location and scale parameters of the Type II GLD. The coverages probability of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. To improve the coverage probabilities, we suggest the use of unconditional simulated percentage points for the construction of confidence intervals. Two numerical examples are presented to illustrate the methods of estimation discussed here.

BackgroundWhen boosting algorithms are used for building survival models from high-dimensional data, it is common to fit a Cox proportional hazards model or to use least squares techniques for fitting semiparametric accelerated failure time models. There are cases, however, where fitting a fully parametric accelerated failure time model is a good alternative to these methods, especially when the proportional hazards assumption is not justified. Boosting algorithms for the estimation of parametric accelerated failure time models have not been developed so far, since these models require the estimation of a model-specific scale parameter which traditional boosting algorithms are not able to deal with.ResultsWe introduce a new boosting algorithm for censored time-to-event data which is suitable for fitting parametric accelerated failure time models. Estimation of the predictor function is carried out simultaneously with the estimation of the scale parameter, so that the negative log likelihood of the survival distribution can be used as a loss function for the boosting algorithm. The estimation of the scale parameter does not affect the favorable properties of boosting with respect to variable selection.ConclusionThe analysis of a high-dimensional set of microarray data demonstrates that the new algorithm is able to outperform boosting with the Cox partial likelihood when the proportional hazards assumption is questionable. In low-dimensional settings, i.e., when classical likelihood estimation of a parametric accelerated failure time model is possible, simulations show that the new boosting algorithm closely approximates the estimates obtained from the maximum likelihood method.

In this paper, a new family of continuous distributions with positive support is introduced. This family is generated by a truncation of the family of univariate symmetrical distributions. In this new family of distributions, general properties, such as moments, asymmetry and kurtosis coefficients, are derived. Particular cases of interest based on the normal, logistic, Laplace and Cauchy models are discussed in depth. The estimation of parameters is carried out by applying moments and maximum likelihood methods. Also, a simulation study was conducted to illustrate the good performance of estimators. An application to the Survival Times (in days) of Guinea Pigs dataset is included, where the special cases of distributions in this family are fitted. The option which provides the best fit is ultimately chosen. An R package, called “tpn”, has been implemented, which includes the relevant cases of interest in this family.

This study investigates the performance of four parameter estimation methods for the Laplace distribution: Method of Moments (MM), Maximum Likelihood Estimation (MLE), Minimum Chi-Square Estimation using equiprobable cells (MCE-EQ), and Minimum Chi-Square Estimation using Representative Points (MCE-RP). Through comprehensive Monte Carlo simulations with sample sizes ranging from 50 to 400, we compare the root mean squared error (RMSE) of the location (μ) and scale (b) parameter estimates. Our results demonstrate that while MLE remains robust for location estimation, the MCE-RP method consistently outperforms other estimators—including MLE—for the scale parameter, particularly in small to moderate samples. The use of Representative Points, which provide an optimal discretization of the distribution, significantly enhances estimation precision. These findings are especially relevant for medical research, where accurate estimation of variability—such as in biomarker concentration levels or physiological response times—is critical for reliable sample size determination, risk assessment, and clinical decision-making. MCE-RP thus offers a superior, reliable estimator for the Laplace scale parameter, with direct implications for improving statistical inference in applied biomedical studies. Purpose: The purpose of this research is to empirically evaluate and compare the finite-sample performance of four estimation methods for the Laplace distribution’s parameters, with a focus on the novel application of Representative Points in minimum chi-square estimation. This work seeks to bridge the gap between theoretical estimation methods and practical applications, providing applied researchers with a more robust estimation tool when modeling data with Laplace characteristics, such as those commonly encountered in medical and biomedical studies. Methods: We conducted an extensive Monte Carlo simulation study to compare the four estimation methods: MM, MLE, MCE-EQ, and MCE-RP. For each method, we generated independent and identically distributed samples from a standard Laplace distribution (μ=0, b=1) with sample sizes n = 50, 100, 200, and 400. Each scenario was replicated 1,000 times. The performance of each estimator was assessed using the root mean squared error (RMSE) for both μ and b. The MCE-RP method utilized pre-computed Representative Points for the standard Laplace distribution, which were transformed according to preliminary MLE estimates to form an optimal cell structure for chi-square minimization. All nonlinear optimizations required for MCE-EQ and MCE-RP were implemented programmatically. Results: The simulation results indicate that MLE performs best for estimating the location parameter μ across all sample sizes. However, for the scale parameter b, the MCE-RP method consistently yields lower RMSE values compared to MLE, MM, and MCE-EQ. In many cases, particularly for smaller samples, the RMSE of MCE-RP is approximately half that of MLE for b. The advantage of MCE-RP is evident across varying numbers of Representative Points (m = 5, 10, 15, 20), with optimal performance often observed at m = 10 or 15. These findings confirm that MCE-RP provides a more precise and reliable estimator for the scale parameter, making it particularly advantageous in small-sample settings. Contribution: This paper contributes to the statistical methodology for the Laplace distribution by introducing and validating the use of Representative Points within a minimum chi-square estimation framework. The key contributions are: (1) demonstrating that MCE-RP significantly outperforms established methods for estimating the scale parameter; (2) providing empirical evidence that RP-based discretization enhances estimation efficiency, especially in finite samples; (3) offering practical guidance for applied researchers in fields such as medical statistics, where accurate scale estimation is crucial for variability assessment, power analysis, and reliable inference; and (4) laying a methodological foundation for extending the RP approach to other location-scale distributions.

Aim of study: The aim of this work was to model diameter distributions of Quercus suber stands. The ultimate goal was to construct models enabling the development of more affordable forest inventory methods. This is the first study of this type on cork oak forests in the area.Area of study: The area of study is “Los Alcornocales” Natural Park (Cádiz-Málaga, Spain).Material and methods: The diameter distributions of 100 permanent plots were modelled with the two-parameter Weibull function. Distribution parameters were fitted with the non-linear regression, maximum likelihood, moment and percentile-based methods. Goodness of fit with the different methods was compared in terms of number of plots rejected by the Kolmogorov-Smirnov test, bias, mean square error and mean absolute error. The scale and shape parameters in the Weibull function were related to the stand variables by using the parameter prediction model.Main results: The best fitting was obtained with the non-linear regression approach, using as initial values those obtained by maximum likelihood method, the percentage of rejections by the Kolmogorov-Smirnov test was 2% of the total number of cases. The scale parameter (b) was successfully modelled in terms of the quadratic mean diameter under cork (R2 adj = 0.99). The shape parameter (c) was modelled by using maximum diameter, minimum diameter and plot elevation (R2 adj = 0.40).Research highlights: The proposed model diameter distribution can be a highly useful tool for the inventorying and management of cork oak forests.Key words: maximum likelihood method; moment method; non linear regression approach; parameter prediction model; percentile method; scale parameter; shape parameter.

The robust properties of the simultaneous estimators of location and scale parameters (μ*, σ*) proposed by Brown and Hwang are studied. As a pair of simultaneous M estimators of location and scale, their asymptotic efficiencies (0.650 for μ* and 0.541 for σ*) are higher than those for median (0.637) and median absolute deviation (0.368) under the normal distribution. Simulation indicates that the distributions of and are much flatter than those based on the sample mean and the sample standard deviation under the normal distribution when the sample size is small.

Using the results of Chow (Ph.D. dissertation, Texas A & M Univ., 1978) on the optimal placement of knots in the approximation of functions by piecewise polynomials, we present an algorithm for the computation of optimal designs for certain time series models considered by Eubank, Smith and Smith (Ann. Statist., 9 (1981), pp. 486–493), (Tech. Rep. 150, Southern Methodist Univ., 1981). The ideas underlying this algorithm form a unified approach to the computation of optimal spacings for the sample quantiles used in the asymptotically best linear unbiased estimator of a location or scale parameter.

Recently, many researchers have been searching for a new family of distributions that improves the fit to a specific dataset and has attractive properties. In this paper, we present a flexible method for constructing continuous distributions by controlling their reliability. Our new technique incorporates a single parameter into the chosen continuous distribution, but it is a novel method and independent of any previously used methods. We present two special cases of this family. The new modified family offers significant improvements in fit and applicability in many fields. Furthermore, the maximum likelihood method is used to estimate the parameters in the proposed family. Real data analysis is performed to demonstrate the suitability of the new family.

Estimation of common location and scale parameters in nonregular cases

Consider an i.i.d. sample X^*_1,X^*_2,...,X^*_n from a location-scale family, and assume that the only available observations consist of the partial maxima (or minima)sequence, X^*_{1:1},X^*_{2:2},...,X^*_{n:n}, where X^*_{j:j}=max{X^*_1,...,X^*_j}. This kind of truncation appears in several circumstances, including best performances in athletics events. In the case of partial maxima, the form of the BLUEs (best linear unbiased estimators) is quite similar to the form of the well-known Lloyd's (1952, Least-squares estimation of location and scale parameters using order statistics, Biometrika, vol. 39, pp. 88-95) BLUEs, based on (the sufficient sample of) order statistics, but, in contrast to the classical case, their consistency is no longer obvious. The present paper is mainly concerned with the scale parameter, showing that the variance of the partial maxima BLUE is at most of order O(1/log n), for a wide class of distributions.

The Indonesian Government introduced rice farm insurance scheme (Asuransi Usaha Tani Padi – AUTP program) to protect rice farm from loss caused by flood, drought and pest and disease infestations. It has been discussed the method to calculate pure premium of the rice farm insurance scheme in Indonesia with assume that the distribution of rice yield data is normal distribution, gamma distribution, or normal mixture distribution. The normal distribution can be used for the case of distribution of rice yield data in the form of symmetry. The gamma distribution can be used for the case of distribution of rice yield data in the form of skewed to the right or positively skewed. In addition, the normal and gamma distributions are categorized as unimodal distributions. It has also been discussed the method of calculating pure premium of the rice farm insurance scheme in Indonesia with the assumption that the rice yield data is normal mixture distribution. The normal mixture distribution can be categorized as multimodal distribution. Characteristic of the normal mixture distribution is suitable for rice yield data in Indonesia that contain rice yield data from several provinces. On the other hand it is also known that the 4-parameter beta mixture distribution can be applied to rice yield data in Indonesia. This distribution is more flexible than the normal mixture distribution in terms of the tail shape of the distribution. In this paper, pure premium calculation method of the AUTP program is formulated with assume that the distribution of rice yield data is the 4-parameter beta mixture distribution.Monte Carlo simulation is used to evaluate performance of the method. The Monte Carlo simulation results show that the proposed method has higher accuracy and precision when increasing sample size. The method is applied to the rice productivity data in several provinces in Indonesia for the period 1970 to 2016.

In this paper, we introduce two new families of distributions containing unimodal as well as bimodal and trimodal distributions. The models extend the normal model to trimodal symmetric and asymmetric situations, and typically involve fewer parameters to be estimated than the mixtures of normal distributions. Statistical properties of these new families are studied in detail. The problem of estimating parameters is addressed by considering the maximum likelihood method and Fisher information matrices are derived. A small Monte Carlo simulation study is conducted to examine the performance of the obtained estimators. The methodology developed is illustrated with three real data applications.

We proposed a new family of distributions called the Topp Leone exponentiated-G family of distributions with two extra positive shape parameters, which generalizes and also extends the Topp Leone-G family of distributions. We derived some mathematical properties of the proposed family including explicit expressions for the quantile function, ordinary and incomplete moments, generating function and reliability. Some sub-models in the new family were discussed. The method of maximum likelihood was used to estimate the parameters of the sub-model. Further, the potentiality of the family was illustrated by fitting two real data sets to the mentioned sub-models.

We introduce a new family of distributions using intervened power series distribution. It is a rich family containing distributions such as Marshall–Olkin extended families of distributions, families of distributions generated through truncated negative binomial, and families of distributions generated through truncated binomial distribution. Also, the proposed distribution is a generalization of the families of distributions generated through zero truncated Poisson distribution. As a particular case, we study exponential intervened Poisson (EIP) distribution. This is a generalization of exponential-Poisson distribution. The shape properties of the pdf and hazard rate are proved. The model identifiability is established. The expression for moments, mean deviation, distribution of order statistics, and residual life function of EIP distribution are derived. The unknown parameters of the distribution are estimated using the method of maximum likelihood. The existence and uniqueness of maximum likelihood estimates (MLEs) are proved. Simulation study is carried out to show the performance of MLEs. EIP distribution is fitted to two real data sets and it is shown that the distribution is more appropriate than other competitive models. The adequacy of the EIP model for the data set is established using parametric bootstrap approach.

<abstract><p>This paper proposes an inference approach based on a pivotal quantity under the adaptive progressive Type-II censoring scheme. To exemplify the proposed methodology, an extensively employed distribution, a Pareto distribution, is utilized. This distribution has limitations in estimating confidence intervals for unknown parameters from classical methods such as the maximum likelihood and bootstrap methods. For example, in the maximum likelihood method, the asymptotic variance-covariance matrix does not always exist. In addition, both classical methods can yield confidence intervals that do not satisfy nominal levels when a sample size is not large enough. Our approach resolves these limitations by allowing us to construct exact intervals for unknown parameters with computational simplicity. Aside from this, the proposed approach leads to closed-form estimators with properties such as unbiasedness and consistency. To verify the validity of the proposed methodology, two approaches, a Monte Carlo simulation and a real-world data analysis, are conducted. The simulation testifies to the superior performance of the proposed methodology as compared to the maximum likelihood method, and the real-world data analysis examines the applicability and scalability of the proposed methodology.</p></abstract>