We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large-deviations approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and Varadhan [European J. Combin. 32 (2011) 1000-1017]. The theory explains a host of difficulties encountered by applied workers: many distinct models have essentially the same MLE, rendering the problems ``practically'' ill-posed. We give the first rigorous proofs of ``degeneracy'' observed in these models. Here, almost all graphs have essentially no edges or are essentially complete. We supplement recent work of Bhamidi, Bresler and Sly [2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2008) 803-812 IEEE] showing that for many models, the extra sufficient statistics are useless: most realizations look like the results of a simple Erdős-Rényi model. We also find classes of models where the limiting graphs differ from Erdős-Rényi graphs. A limitation of our approach, inherited from the limitation of graph limit theory, is that it works only for dense graphs.
In the present work, we have investigated the problem of estimating parameters of several exponential distributions with ordered scale parameters under the linex loss function. We have considered estimating ordered scale parameters when the location parameters are known and unknown. For every case, we consider a class of equivariant estimators, and sufficient condition is obtained under which this class of estimators improves upon the usual estimator. Using this result, we have shown that the restricted maximum likelihood estimator is inadmissible. Finally, for every case, we conduct a simulat…
In many families of distributions, maximum likelihood estimation is intractable because the normalization constant for the density which enters into the likelihood function is not easily available. The score matching estimator of Hyvärinen (2005) provides an alternative where this normalization constant is not required. The corresponding estimating equations become linear for an exponential family. The score matching estimator is shown to be consistent and asymptotically normally distributed for such models, although not necessarily efficient. Gaussian linear concentration models are examples …
We propose a novel approach for density estimation with exponential families for the case when the true density may not fall within the chosen family. Our approach augments the sufficient statistics with features designed to accumulate probability mass in the neighborhood of the observed points, resulting in a non-parametric model similar to kernel density estimators. We show that under mild conditions, the resulting model uses only the sufficient statistics if the density is within the chosen exponential family, and asymptotically, it approximates densities outside of the chosen exponential f…
An Accurate Standard Error Estimation for Quadratic Exponential Logistic Regressions by Applying Generalized Estimating Equations to Pseudo-Likelihoods
Chen and Lio (Computational Statistics and Data Analysis 54: 1581-1591, 2010) proposed five methods for estimating the parameters of generalized exponential distribution under progressive type-I interval censoring scheme. Unfortunately, among them, the proposed EM algorithm is incorrect. Here, we propose the correct EM algorithm and compare its performance with the maximum likelihood estimators and that proposed by Chen and Lio (2010) in a simulation study.
We study the existence, strong consistency and asymptotic normality of estimators obtained from estimating functions, that are p-dimensional martingale transforms. The problem is motivated by the analysis of evolutionary clustered data, with distributions belonging to the exponential family, and which may also vary in terms of other component series. Within a quasi-likelihood approach, we construct estimating equations, which accommodate different forms of dependency among the components of the response vector and establish multivariate extensions of results on linear and generalized linear mo…
This paper is devoted to study a new three- parameters model called the Exponential Flexible Weibull extension (EFWE) distribution which exhibits bathtub-shaped hazard rate. Some of it's statistical properties are obtained including ordinary and incomplete moments, quantile and generating functions, reliability and order statistics. The method of maximum likelihood is used for estimating the model parameters and the observed Fisher's information matrix is derived. We illustrate the usefulness of the proposed model by applications to real data.
This paper introduces a new three-parameters model called the Weibull-G exponential distribution (WGED) distribution which exhibits bathtub-shaped hazard rate. Some of it's statistical properties are obtained including quantile, moments, generating functions, reliability and order statistics. The method of maximum likelihood is used for estimating the model parameters and the observed Fisher's information matrix is derived. We illustrate the usefulness of the proposed model by applications to real data.
Le Cam's method (or the two-point method) is a commonly used tool for obtaining statistical lower bound and especially popular for functional estimation problems. This work aims to explain and give conditions for the tightness of Le Cam's lower bound in functional estimation from the perspective of convex duality. Under a variety of settings it is shown that the maximization problem that searches for the best two-point lower bound, upon dualizing, becomes a minimization problem that optimizes the bias-variance tradeoff among a family of estimators. For estimating linear functionals of a distri…
Targeted maximum likelihood estimation (TMLE) is a general method for estimating parameters in semiparametric and nonparametric models. Each iteration of TMLE involves fitting a parametric submodel that targets the parameter of interest. We investigate the use of exponential families to define the parametric submodel. This implementation of TMLE gives a general approach for estimating any smooth parameter in the nonparametric model. A computational advantage of this approach is that each iteration of TMLE involves estimation of a parameter in an exponential family, which is a convex optimizati…
In this article we propose a locally adaptive strategy for estimating a function from its Exponential Radon Transform (ERT) data, without prior knowledge of the smoothness of functions that are to be estimated. We build a non-parametric kernel type estimator and show that for a class of functions comprising a wide Sobolev regularity scale, our proposed strategy follows the minimax optimal rate up to a $\log{n}$ factor. We also show that there does not exist an optimal adaptive estimator on the Sobolev scale when the pointwise risk is used and in fact the rate achieved by the proposed estimator…
Estimating the contact angle of a virus infected saliva droplet is seen to be an important area of research as it presents an idea about the drying time of the respective droplet and in turn of the growth of the underlying pandemic. In this paper we extend the data presented by Balusamy, Banerjee and Sahu [Lifetime of sessile saliva droplets in the context of SARS-CoV-2, Int. J. Heat Mass Transf. 123, 105178 (2021)], where the contact angles are fitted using a newly proposed half-circular wrapped-exponential model, and a sequential confidence interval estimation approach is established which l…
In \cite{KumarS15J2}, it was shown that a generalized maximum likelihood estimation problem on a (canonical) $α$-power-law model ($\mathbb{M}^{(α)}$-family) can be solved by solving a system of linear equations. This was due to an orthogonality relationship between the $\mathbb{M}^{(α)}$-family and a linear family with respect to the relative $α$-entropy (or the $\mathscr{I}_α$-divergence). Relative $α$-entropy is a generalization of the usual relative entropy (or the Kullback-Leibler divergence). $\mathbb{M}^{(α)}$-family is a generalization of the usual exponential family. In this paper, we …
In this paper, we introduce a new distribution generated by Lindley random variable which offers a more flexible model for modelling lifetime data. Various statistical properties like distribution function, survival function, moments, entropy, and limiting distribution of extreme order statistics are established. Inference for a random sample from the proposed distribution is investigated and maximum likelihood estimation method is used for estimating parameters of this distribution. The applicability of the proposed distribution is shown through real data sets.
We introduce a new estimator SMUCE (simultaneous multiscale change-point estimator) for the change-point problem in exponential family regression. An unknown step function is estimated by minimizing the number of change-points over the acceptance region of a multiscale test at a level α. The probability of overestimating the true number of change-points K is controlled by the asymptotic null distribution of the multiscale test statistic. Further, we derive exponential bounds for the probability of underestimating K. By balancing these quantities, αwill be chosen such that the probability of co…
To perform a queuing analysis or design in a communications context, we need to estimate the values of the input parameters, specifically the mean of the arrival rate and service time. In this paper, we propose an approach for estimating the arrival rate of Poisson processes and the average service time for servers under the assumption that the service time is exponential. In particular, we derive sample size (i.e., the number of i.i.d. observations) required to obtain an estimate satisfying a pre-specified relative accuracy with a given confidence level. A remarkable feature of this approach …
The area-characteristic, maximum possible earthquake magnitude $T_M$ is required by the earthquake engineering community, disaster management agencies and the insurance industry. The Gutenberg-Richter law predicts that earthquake magnitudes $M$ follow a truncated exponential distribution. In the geophysical literature several estimation procedures were proposed, see for instance Kijko and Singh (Acta Geophys., 2011) and the references therein. Estimation of $T_M$ is of course an extreme value problem to which the classical methods for endpoint estimation could be applied. We argue that recent …
Simple exponential smoothing is widely used in forecasting economic time series. This is because it is quick to compute and it generally delivers accurate forecasts. On the other hand, its multivariate version has received little attention due to the complications arising with the estimation. Indeed, standard multivariate maximum likelihood methods are affected by numerical convergence issues and bad complexity, growing with the dimensionality of the model. In this paper, we introduce a new estimation strategy for multivariate exponential smoothing, based on aggregating its observations into s…
Simultaneous concurrence of extreme values across multiple climate variables can result in large societal and environmental impacts. Therefore, there is growing interest in understanding these concurrent extremes. In many applications, not only the frequency but also the magnitude of concurrent extremes are of interest. One way to approach this problem is to study the distribution of one climate variable given that another is extreme. In this work we develop a statistical framework for estimating bivariate concurrent extremes via a conditional approach, where univariate extreme value modeling …
We observe that approximate copies of the function $Λ_{n}:\mathbb{R}^{n}\rightarrow (0,\infty )$ defined by \begin{equation*} Λ_{n}(x)=\exp \left( -x_{1}-π\sum_{i=2}^{n}x_{i}^{2}\right) \end{equation*} appear in the tails of a large class of functions, with properties related to coordinate independence, convexity, homotheticity, and homogeneity. The function $Λ_{n}$ is an entropy maximizer (on a half-space) that is uniquely determined by a homogeneity condition together with rotational invariance about the $x_{1}$ direction and its behavior near the origin. These results are connected to the l…
