Some simple estimators for the two-parameter gamma distribution

ABSTRACT

In this paper, we propose two new simple estimation methods for the two-parameter gamma distribution. The first one is a modified version of the method of moments, whereas the second one makes use of some key properties of the distribution. We then derive the asymptotic distributions of these estimators. Also, bias-reduction methods are suggested to reduce the bias of these estimators. The performance of the estimators are evaluated through a Monte Carlo simulation study. The probability coverages of confidence intervals are also discussed. Finally, two examples are used to illustrate the proposed methods.     

Efficient closed-form maximum a posteriori estimators for the gamma distribution

We proposed a new class of maximum a posteriori estimators for the parameters of the Gamma distribution. These estimators have simple closed-form expressions and can be rewritten as a bias-corrected maximum likelihood estimators presented by Ye and Chen [Closed-form estimators for the gamma distribution derived from likelihood equations. Am Statist. 2017;71(2):177–181]. A simulation study was carried out to compare different estimation procedures. Numerical results revels that our new estimation scheme outperforms the existing closed-form estimators and produces extremely efficient estimates for both parameters, even for small sample sizes.     

An extension of the generalized linear failure rate distribution

In this article, we introduce a new extension of the generalized linear failure rate (GLFR) distributions. It includes some well-known lifetime distributions such as extension of generalized exponential and GLFR distributions as special sub-models. In addition, it can have a constant, decreasing, increasing, upside-down bathtub (unimodal), and bathtub-shaped hazard rate function (hrf) depending on its parameters. We provide some of its statistical properties such as moments, quantiles, skewness, kurtosis, hrf, and reversible hrf. The maximum likelihood estimation of the parameters is also discussed. At the end, a real dataset is given to illustrate the usefulness of this new distribution in analyzing lifetime data.     

Parameter and quantile estimation for the three-parameter gamma distribution based on statistics invariant to unknown location

The three-parameter gamma distribution is widely used as a model for distributions of life spans, reaction times, and for other types of skewed data. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter gamma distribution, which avoids the problem of unbounded likelihood, based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared to other prominent methods in terms of bias and mean squared error. Finally, we present two illustrative examples.     

Topp–Leone generated family of distributions: Properties and applications

We introduce a new family of distributions based on a one-parameter distribution exhibiting bathtub-shaped hazard rates. We study the mathematical properties of the family and estimate its parameters by the method of maximum likelihood. Finally, the usefulness of the family is illustrated using a real dataset.     

An extended Gompertz-Makeham distribution with application to lifetime data

In this paper, we propose an extension of the Gompertz-Makeham distribution. This distribution is called the transmuted Gompertz-Makeham (TGM). The new model which can handle bathtub-shaped, increasing, increasing-constant and constant hazard rate functions. This property makes TGM is useful in survival analysis. Various statistical and reliability measures of the model are obtained, including hazard rate function, moments, moment generating function (mgf), quantile function, random number generating, skewness, kurtosis, conditional moments, mean deviations, Bonferroni curve, Lorenz curve, Gini index, mean inactivity time, mean residual lifetime and stochastic ordering; we also obtain the density of the ith order statistic. Estimation of the model parameters is justified by the method of maximum likelihood. An application to real data demonstrates that the TGM distribution can provides a better fit than some other very well known distributions.     

The beta generalized gamma distribution

For the first time, a new five-parameter distribution, called the beta generalized gamma distribution, is introduced and studied. It contains at least 25 special sub-models such as the beta gamma, beta Weibull, beta exponential, generalized gamma (GG), Weibull and gamma distributions and thus could be a better model for analysing positive skewed data. The new density function can be expressed as a linear combination of GG densities. We derive explicit expressions for moments, generating function and other statistical measures. The elements of the expected information matrix are provided. The usefulness of the new model is illustrated by means of a real data set.     

Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data

In this article, we investigate the potential usefulness of the three-parameter transmuted generalized exponential distribution for analyzing lifetime data. We compare it with various generalizations of the two-parameter exponential distribution using maximum likelihood estimation. Some mathematical properties of the new extended model including expressions for the quantile and moments are investigated. We propose a location-scale regression model, based on the log-transmuted generalized exponential distribution. Two applications with real data are given to illustrate the proposed family of lifetime distributions.     

Modified skew-slash distribution

Abstract

In this work, we introduce a new skewed slash distribution. This modification of the skew-slash distribution is obtained by the quotient of two independent random variables. That quotient consists on a skew-normal distribution divided by a power of an exponential distribution with scale parameter equal to two. In this way, the new skew distribution has a heavier tail than that of the skew-slash distribution. We give the probability density function expressed by an integral, but we obtain some important properties useful for making inferences, such as moment estimators and maximum likelihood estimators. By way of illustration and by using real data, we provide maximum likelihood estimates for the parameters of the modified skew-slash and the skew-slash distributions. Finally, we introduce a multivariate version of this new distribution.     

The exponentiated transmuted Weibull geometric distribution with application in survival analysis

This article introduces a new generalization of the transmuted Weibull distribution introduced by Aryal and Tsokos in 2011. We refer to the new distribution as exponentiated transmuted Weibull geometric (ETWG) distribution. The new model contains 22 lifetime distributions as special cases such as the exponentiated Weibull geometric, complementary Weibull geometric, exponentiated transmuted Weibull, exponentiated Weibull, and Weibull distributions, among others. The properties of the new model are discussed and the maximum likelihood estimation is used to evaluate the parameters. Explicit expressions are derived for the moments and examine the order statistics. To examine the performance of our new model in fitting several data we use two real sets of data, censored and uncensored, and then compare the fitting of the new model with some nested and nonnested models, which provides the best fit to all of the data. A simulation has been performed to assess the behavior of the maximum likelihood estimates of the parameters under the finite samples. This model is capable of modeling various shapes of aging and failure criteria.     

A multivariate pareto distribution

In this paper, we introduce a new multivariate pareto (MVP) distribution with many interesting properties. we extend the results of characterization of univariate and bivariate pareto distributions given by Krishnaji (1970) and veenus and Nair (1994) respectively. We also extend the property of dullness of univariate pareto distribution given by Talwalkar (1980) to the multivariate pareto case. We obtain the maximum likelihood estimate (MLE) of the parameters and their asymptotic multivariate normal (AMVN) distrioutions. We propose large sample studentized test for testing independence and identical marginals of the components.     

An Application of EM Algorithm to a Change-Point Problem

We consider the problem of change-point in a classical framework while assuming a probability distribution for the change-point. An EM algorithm is proposed to estimate the distribution of the change-point. A change-point model for multiple profiles is also proposed, and EM algorithm is presented to estimate the model. Two examples of Illinois traffic data and Dow Jones Industrial Averages are used to demonstrate the proposed methods.     

Estimation of parameters of a right truncated exponential distribution

The maximum likelihood, moment and mixture of the estimators are for samples from the right truncated exponential distribution. The estimators are compared empirically when all the parameters are unknown; their bias and mean square error are investigated with the help of numerical technique. We have shown that these estimators are asymptotically unbiased. At the end, we conclude that mixture estimators are better than the maximum likelihood and moment estimators.     

Multivariate extension of modified Sarhan-Balakrishnan bivariate distribution

Recently Kundu and Gupta [2010, Modified Sarhan-Balakrishnan singular bivariate distribution, Journal of Statistical Planning and Inference, 140, 526-538] introduced the modified Sarhan-Balakrishnan bivariate distribution and established its several properties. In this paper we provide a multivariate extension of the modified Sarhan-Balakrishnan bivariate distribution. It is a distribution with a singular part. Different ageing and dependence properties of the proposed multivariate distribution have been established. The moment generating function, the product moments can be obtained in terms of infinite series. The multivariate hazard rate has been obtained. We provide the EM algorithm to compute the maximum likelihood estimators and an illustrative example is performed to see the effectiveness of the proposed method.     

The exponentiated generalized gamma distribution with application to lifetime data

A four-parameter extension of the generalized gamma distribution capable of modelling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in lifetime data analysis and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma and generalized Rayleigh, among others. We derive two infinite sum representations for its moments. We calculate the density of the order statistics and two expansions for their moments. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is obtained. Finally, a real data set from the medical area is analysed.     

Using the singly truncated normal distribution to analyze satellite data

This paper proposes the singly truncated normal distribution as a model for estimating radiance measurements from satellite-borne infrared sensors. These measurements are made in order to estimate sea surface temperatures which can be related to radiances. Maximum likelihood estimation is used to provide estimates for the unknown parameters. In particular, a procedure is described for estimating clear radiances in the presence of clouds and the Kolmogorov-Smirnov statistic is used to test goodness-of-fit of the measurements to the singly truncated normal distribution. Tables of quantile values of the Kolmogorov-Smirnov statistic for several values of the truncation point are generated from Monie Carlo experiment Mnally a numerical emample using satetic data is presented to illustrate the application of the procedures.     

The McDonald Gompertz distribution: Properties and applications

This article introduces a five-parameter lifetime model called the McDonald Gompertz (McG) distribution to extend the Gompertz, generalized Gompertz, generalized exponential, beta Gompertz, and Kumaraswamy Gompertz distributions among several other models. The hazard function of new distribution can be increasing, decreasing, upside-down bathtub, and bathtub shaped. We obtain several properties of the McG distribution including moments, entropies, quantile, and generating functions. We provide the density function of the order statistics and their moments. The parameter estimation is based on the usual maximum likelihood approach. We also provide the observed information matrix and discuss inferences issues. The flexibility and usefulness of the new distribution are illustrated by means of application to two real datasets.     

Estimations of the parameter of a Dirichlet distribution using residual allocation model representations and sampling properties

Estimating the parameter of a Dirichlet distribution is an interesting question since this distribution arises in many situations of applied probability. Classical procedures are based on sample of Dirichlet distribution. In this paper we exhibit five different estimators from only one observation. They are based either on residual allocation model decompositions or on sampling properties of Dirichlet distributions. Two ways are investigated: the first one uses fragments’ size and the second one uses size-biased permutations of a partition. Numerical computations based on simulations are supplied. The estimators are finally used to estimate birth probabilities per month.     

Exact confidence region for the parameters of the gamma distribution based on two independent samples

The gamma distribution has been discussed by many authors. This article proposes an exact confidence region for the parameters of a two-parameter gamma distribution. The result is based on the fact that the percentiles of the F-distribution, with equal degrees of freedom k, are monotonic in k.     

The beta generalized exponential distribution

A new distribution called the beta generalized exponential distribution is proposed. It includes the beta exponential and generalized exponential (GE) distributions as special cases. We provide a comprehensive mathematical treatment of this distribution. The density function can be expressed as a mixture of generalized exponential densities. This is important to obtain some mathematical properties of the new distribution in terms of the corresponding properties of the GE distribution. We derive the moment generating function (mgf) and the moments, thus generalizing some results in the literature. Expressions for the density, mgf and moments of the order statistics are also obtained. We discuss estimation of the parameters by maximum likelihood and obtain the information matrix that is easily numerically determined. We observe in one application to a real skewed data set that this model is quite flexible and can be used effectively in analyzing positive data in place of the beta exponential and GE distributions.