Good estimators are characterized as robust, unbiased, efficient, and consistent. However, the commonly used estimators are weak or lack one or more of these properties. In this article, eight (8) estimators for statistical and geometrical estimations of symmetry/asymmetry, similarity/dissimilarity, identity/unidentity, and feature transformation were proposed following Kabirian-based optinalysis and other operations. The proposed estimators are characterized as invariant (robust) under scaling, location shift, and rotation or reflection. A computing code was written in python language for each of the proposed estimators so that peers can have working codes for application and performance evaluation.

The statistical properties of a good estimator include robustness, unbiasedness, efficiency, and consistency. However, the commonly used estimators of dispersion have lack or are weak in one or more of these properties. In this paper, I proposed statistical mirroring as a good alternative estimator of dispersion around defined location estimates or points. In the main part of the paper, attention is restricted to Gaussian distribution and only estimators of dispersion around the mean that functionalize with all the observations of a dataset were considered at this time. The different estimators were compared with the proposed estimators in terms of alternativeness, scale and sample size robustness, outlier biasedness, and efficiency. Monte Carlo simulation was used to generate artificial datasets for application. The proposed estimators (of statistical meanic mirroring) turn out to be suitable alternative estimators of dispersion that is less biased (more resistant) to contaminations, robust to scale and sample size, and more efficient to a random distribution of variable than the standard deviation, variance, and coefficient of variation. However, statistical meanic mirroring is not suitable with a mean (of a normal distribution) close to zero, and on a scale below ratio level.

In this paper a more flexible extension of the Fréchet distribution is introduced. The new distribution is defined by means of the stochastic representation as the quotient of two independent random variables, a Fréchet distribution and the power of a random variable with uniform distribution in the interval (0,1). We will call this new extension the Slash Fréchet distribution and one of its main characteristics is that its tails are heavier than the Fréchet distribution. The general density of this distribution and some basic properties are determined. Its moments, skewness coefficients and kurtosis are calculated. In addition, the estimation of the model parameters is obtained by the method of moments and maximum likelihood. Finally, two applications with real data are performed by fitting the new model and comparing it with the Fréchet distribution.

The normal and Poisson distribution assumptions in the normal-Poisson mixed effects regression model are often too restrictive for many real count data. Several works have independently relaxed the Poisson conditional distribution assumption for counts or the normal distribution assumption for random effects. This work couples some recent advances in these two regards to develop a skew t–discrete gamma regression model in which the count outcomes have full dispersion flexibility and random effets can be skewed and heavy tailed. Inference in the model is achieved by maximum likelihood using pseudo-adaptive Gaussian quadature. The use of the proposal is demonstrated on a popular owl sibling negotiation data. It appears that, for this example, the proposed approach outperforms models based on normal random effects and the Poisson or negative binomial count distribution.

The chain ladder method is a popular technique to estimate the future reserves needed to handle claims that are not fully settled. Since the predictions of the aggregate portfolio (consisting of different subportfolios) in general differ from the sum of the predictions of the subportfolios, a general multivariate chain ladder (GMCL) method has already been proposed. However, the GMCL method is based on the seemingly unrelated regression (SUR) technique which makes it very sensitive to outliers. To address this issue a robust alternative is introduced which estimates the SUR parameters in a more outlier resistant way. With the robust methodology it is possible to detect which claims have an abnormally large influence on the reserve estimates. We introduce a simulation design to generate artificial multivariate run-off triangles based on the GMCL model and illustrate the importance of taking into account contemporaneous correlations and structural connections between the run-off triangles. By adding contamination to these artificial datasets, the sensitivity of the traditional GMCL method and the good performance of the robust GMCL method is shown. From the analysis of a portfolio from practice it is clear that the robust GMCL method can provide better insight in the structure of the data.

Today, non-expensive remote sensing (RS) data from different sensors and platforms can be obtained at short intervals and be used for assessing several kinds of forest characteristics at the level of plots, stands and landscapes. Methods such as composite estimation and data assimilation can be used for combining the different sources of information to obtain up-to-date and precise estimates of the characteristics of interest. In composite estimation a standard procedure is to assign weights to the different individual estimates inversely proportional to their variance. However, in case the estimates are correlated, the correlations must be considered in assigning weights or otherwise a composite estimator may be inefficient and its variance be underestimated. In this study we assessed the correlation of plot level estimates of forest characteristics from different RS datasets, between assessments using the same type of sensor as well as across different sensors. The RS data evaluated were SPOT-5 multispectral data, 3D airborne laser scanning data, and TanDEM-X interferometric radar data. Studies were made for plot level mean diameter, mean height, and growing stock volume. All data were acquired from a test site dominated by coniferous forest in southern Sweden. We found that the correlation between plot level estimates based on the same type of RS data were positive and strong, whereas the correlations between estimates using different sources of RS data were not as strong, and weaker for mean height than for mean diameter and volume. The implications of such correlations in composite estimation are demonstrated and it is discussed how correlations may affect results from data assimilation procedures.

Location-and-scale transformation of a random variable underpins normal distribution, but it is however fundamentally incorrect for scale estimation such as relative dispersion. In this paper, a parametrized alternative to a normal distribution, called scaloc-normal distribution, is proposed that efficiently works and is fundamentally correct with absolute and relative dispersion estimators. The Monte Carlos simulation experiment was used to generate a total of 600,000 artificial datasets in 600 different simulation scenarios from loc-normal (normal) and scaloc-normal distributions. The absolute and relative dispersion were estimated and compared from the two distributions. The results show that scaloc-normal distribution is a good parametrized alternative to loc-normal distribution, fundamentally correct and efficient with both standard deviation and coefficient of variation. The key statistical advancement from loc-normal to scaloc-normal distribution is its fundamental correctness (i.e., scale-invariant property) with an efficient relative estimator of dispersion (i.e., coefficient of variation). Parametrically, the loc-normal and scaloc-normal distributions are very different, but both have linear transformations.