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The application of compound distributions has recently increased due to the flexibility in fitting actual data in various fields such as economics, insurance, etc. Poisson-half-logistic distribution is one of these distributions with an increasing-constant hazard rate that can be used in parallel systems and complementary risk models. Because of the complexity of the form of this distribution, it is not possible to obtain classical parameter estimates (such as MLE) by the analytical method for the location and scale parameters. We present a simple way of deriving explicit estimators by approximating the likelihood equations appropriately. This paper presents the AMLE (Approximate Maximum Likelihood Estimator) method to estimate the location and scale parameters. Using simulation, we show that this method is as efficient as the maximum likelihood estimators (MLEs). We obtain the variance of estimators from the inverse of the observed Fisher information matrix, and we see that when sample size increases, bias and variance of these estimators, and hence MSEs of parameters, decrease. Some pivotal quantities are proposed for finding confidence intervals for location and scale parameters based on asymptotic normality. From the coverage probability, the MLEs do not work well, especially for the small sample sizes; thus, simulated percentiles based on the Monte Carlo method are used to improve the coverage probability. Finally, we present a numerical example to illustrate the methods of inference developed here. [ABSTRACT FROM AUTHOR]

In this paper, we derive some estimators of the scale parameter of the exponentiated half-logistic distribution based on the multiply Type-I hybrid censoring scheme. We assume that the shape parameter λ is known. We obtain the maximum likelihood estimator of the scale parameter σ. The scale parameter is estimated by approximating the given likelihood function using two different Taylor series expansions since the likelihood equation is not explicitly solved. We also obtain Bayes estimators using prior distribution. To obtain the Bayes estimators, we use the squared error loss function and general entropy loss function (shape parameter q = -0.5, 1.0). We also derive interval estimation such as the asymptotic confidence interval, the credible interval, and the highest posterior density interval. Finally, we compare the proposed estimators in the sense of the mean squared error through Monte Carlo simulation. The average length of 95% intervals and the corresponding coverage probability are also obtained. [ABSTRACT FROM AUTHOR]

The efficiency and accuracy of estimating the Hurst exponent have been two inevitable considerations. Recently, an efficient implementation of the maximum likelihood estimator (MLE) (simply called the fast MLE) for the Hurst exponent was proposed based on a combination of the Levinson algorithm and Cholesky decomposition, and furthermore the fast MLE has also considered all four possible cases, including known mean, unknown mean, known variance, and unknown variance. In this paper, four cases of an approximate MLE (AMLE) were obtained based on two approximations of the logarithmic determinant and the inverse of a covariance matrix. The computational cost of the AMLE is much lower than that of the MLE, but a little higher than that of the fast MLE. To raise the computational efficiency of the proposed AMLE, a required power spectral density (PSD) was indirectly calculated by interpolating two suitable PSDs chosen from a set of established PSDs. Experimental results show that the AMLE through interpolation (simply called the interpolating AMLE) can speed up computation. The computational speed of the interpolating AMLE is on average over 24 times quicker than that of the fast MLE while remaining the accuracy very close to that of the MLE or the fast MLE. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of estimating the scale parameter of the inverse Rayleigh distribution based on general progressively Type-II censored samples and progressively Type-II censored samples. The pivotal quantity method is used to derive the estimator of the scale parameter. Besides, considering that the maximum likelihood estimator is tough to obtain for this distribution, we derive an explicit estimator of the scale parameter by approximating the likelihood equation with Taylor expansion. The interval estimation is also studied based on pivotal inference. Then we conduct Monte Carlo simulations and compare the performance of different estimators. We demonstrate that the pivotal inference is simpler and more effective. The further application of the pivotal quantity method is also discussed theoretically. Finally, two real data sets are analyzed using our methods. [ABSTRACT FROM AUTHOR]

This paper deals with the estimation of R = P[Y < X] when X, and Y are two independent Weibull distributions with different scale parameters, but having the same shape parameter. The maximum likelihood estimator, and the approximate maximum likelihood estimator of R are proposed. We obtain the asymptotic distribution of the maximum likelihood estimator of R. Based on the asymptotic distribution, the confidence interval of R can be obtained. We also propose two bootstrap confidence intervals. We consider the Bayesian estimate of R, and propose the corresponding credible interval for R. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes. [ABSTRACT FROM AUTHOR]

A hybrid censoring scheme is mixture of Type-I and Type-II censoring schemes. Based on hybrid censored samples, this paper deals with the inference on $$R = P(X>Y)$$ , when X and Y are two independent Weibull distributions with different scale parameters, but having the same shape parameter. The maximum likelihood estimator (MLE), and the approximate MLE of R are obtained. The asymptotic distribution of the MLE of R is obtained. Based on the asymptotic distribution, the confidence interval of R is constructed. Two bootstrap confidence intervals are also proposed. We consider the Bayesian estimate of R, and propose the corresponding credible interval for R. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes. [ABSTRACT FROM AUTHOR]

Type-II progressively hybrid censoring is a mixture of Type-II progressive and hybrid censoring schemes. In this paper, we consider the estimation ofwhenXandYare two independent Weibull distributions with different scale parameters, but having the same shape parameter, and the data obtained from both distributions are Type-II progressively hybrid censored. Different methods for estimatingare applied. The maximum likelihood estimator and the approximate maximum likelihood estimator ofare proposed. The observed Fisher information matrix is computed and it is used to construct an asymptotic confidence interval for. Also, two bootstrap confidence intervals are proposed. Bayes estimate ofand the associated credible interval are also obtained under the assumption of independent suitable priors. Monte Carlo simulations are performed to compare the efficiency of the proposed estimators. One data analysis has been presented for illustrative purposes. [ABSTRACT FROM AUTHOR]

Recently, Gupta and Gupta [Analyzing skewed data by power-normal model,Test17 (2008), pp. 197–210] proposed the power-normal distribution for which normal distribution is a special case. The power-normal distribution is a skewed distribution, whose support is the whole real line. Our main aim of this paper is to consider bivariate power-normal distribution, whose marginals are power-normal distributions. We obtain the proposed bivariate power-normal distribution from Clayton copula, and by making a suitable transformation in both the marginals. Lindley–Singpurwalla distribution also can be used to obtain the same distribution. Different properties of this new distribution have been investigated in detail. Two different estimators are proposed. One data analysis has been performed for illustrative purposes. Finally, we propose some generalizations to multivariate case also along the same line and discuss some of its properties. [ABSTRACT FROM PUBLISHER]

This paper shows that discretization after the application of Itô formula in the Girsanov likelihood produces estimators of the drift which have faster rates of convergence than the Euler estimator for stationary ergodic diffusions and is free of approximating the stochastic integral. The discretization schemes are related to the Hausdorff moment problem. Interalia I introduce a new stochastic integral which will be of independent interest. I show strong consistency, asymptotic normality and a Berry-Esseen bound for the corresponding approximate maximum likelihood estimators of the drift parameter from high frequency data observed over a long time period. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of estimating stress-strength reliability for inverseWeibull lifetime models having the same shape parameters but different scale parameters. We obtain the maximum likelihood estimator and its asymptotic distribution. Since the classical estimator doesn't hold explicit forms, we propose an approximate maximum likelihood estimator. The asymptotic confidence interval and two bootstrap intervals are obtained. Using the Gibbs sampling technique, Bayesian estimator and the corresponding credible interval are obtained. The Metropolis-Hastings algorithm is used to generate random variates. Monte Carlo simulations are conducted to compare the proposed methods. Analysis of a real dataset is performed. [ABSTRACT FROM AUTHOR]

In this paper, some estimators of the scale parameter of the half-logistic distribution are derived under the multiply Type-II hybrid censoring scheme. We propose three methods. First, we obtain the maximum likelihood estimator of the scale parameter of the half-logistic distribution. Second, we propose some approximate maximum likelihood estimators of the scale parameter by employing three different types of Taylor series expansions. Lastly, we obtain the Bayes estimators by employing some prior distributions and loss functions. The interval estimations such as the asymptotic confidence interval and the credible interval and the highest posterior density interval are obtained. To assess the performance of the proposed estimators, we obtain the mean squared error, average length of 95% interval and coverage probability through Monte Carlo simulation and applies to the real dataset. • The multiply Type-II hybrid censoring scheme (MTHCS-II) is considered. • Some estimators of the scale parameter are obtained under MTHCS-II. • The interval estimations are also derived under MTHCS-II. • The proposed estimators are compared through Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

The progressive censoring scheme has received a considerable amount of attention in the last 15 years. During the last few years, joint progressive censoring scheme has gained some popularity. Recently, the authors Mondal and Kundu ("A New Two Sample Type‐II Progressive Censoring Scheme," Communications in Statistics‐Theory and Methods) introduced a balanced two‐sample type II progressive censoring scheme and provided the exact inference when the two populations are exponentially distributed. In this article, we consider the case when the two populations follow Weibull distributions with the common shape parameter and different scale parameters. We obtain the maximum likelihood estimators of the unknown parameters. It is observed that the maximum likelihood estimators cannot be obtained in explicit forms; hence, we propose approximate maximum likelihood estimators, which can be obtained in explicit forms. We construct the asymptotic and bootstrap confidence intervals of the population parameters. Further, we derive an exact joint confidence region of the unknown parameters. We propose an objective function based on the expected volume of this confidence region, and using that, we obtain the optimum progressive censoring scheme. Extensive simulations have been performed to see the performances of the proposed method, and one real data set has been analyzed for illustrative purposes. [ABSTRACT FROM AUTHOR]

This paper studies the Bayes estimator, the maximum likelihood estimator and the approximate likelihood estimator of the scale parameter for the Marshall-Olkin exponential distribution under the progressive type-II censored sample. All the estimators, Bayes estimator, maximum likelihood estimator and approximate likelihood estimator are presented and derived in simple forms. It observed that the Bayes estimator and the maximum likelihood estimator can not be solved analytically, hence it is solved numerically. Finally the comparison method is presented in order to compare the performance between these estimators.

Spatial autocorrelation is a parameter of importance for network data analysis. To estimate spatial autocorrelation, maximum likelihood has been popularly used. However, its rigorous implementation requires the whole network to be observed. This is practically infeasible if network size is huge (e.g., Facebook, Twitter, Weibo, WeChat, etc.). In that case, one has to rely on sampled network data to infer about spatial autocorrelation. By doing so, network relationships (i.e., edges) involving unsampled nodes are overlooked. This leads to distorted network structure and underestimated spatial autocorrelation. To solve the problem, we propose here a novel solution. By temporarily assuming that the spatial autocorrelation is small, we are able to approximate the likelihood function by its first-order Taylor’s expansion. This leads to the method of approximate maximum likelihood estimator (AMLE), which further inspires the development of paired maximum likelihood estimator (PMLE). Compared with AMLE, PMLE is computationally superior and thus is particularly useful for large-scale network data analysis. Under appropriate regularity conditions (without assuming a small spatial autocorrelation), we show theoretically that PMLE is consistent and asymptotically normal. Numerical studies based on both simulated and real datasets are presented for illustration purpose. [ABSTRACT FROM AUTHOR]

Two estimation problems are studied based on the general progressively censored samples, and the distributions from the inverted scale family (ISF) are considered as prospective life distributions. One is the exact interval estimation for the unknown parameter θ , which is achieved by constructing the pivotal quantity. Through Monte Carlo simulations, the average 90 % and 95 % confidence intervals are obtained, and the validity of the above interval estimation is illustrated with a numerical example. The other is the estimation of R = P (Y < X) in the case of ISF. The maximum likelihood estimator (MLE) as well as approximate maximum likelihood estimator (AMLE) is obtained, together with the corresponding R-symmetric asymptotic confidence intervals. With Bootstrap methods, we also propose two R-asymmetric confidence intervals, which have a good performance for small samples. Furthermore, assuming the scale parameters follow independent gamma priors, the Bayesian estimator as well as the HPD credible interval of R is thus acquired. Finally, we make an evaluation on the effectiveness of the proposed estimations through Monte Carlo simulations and provide an illustrative example of two real datasets. [ABSTRACT FROM AUTHOR]