In this paper we consider an extension of the beta regression model proposed by Ferrari and Cribari-Neto (2004). We extend their model in two different ways, first, we let the regression structure be nonlinear, second, we allow a regression structure for the precision parameter, moreover, this regression structure may also be nonlinear. Generally, the beta regression is useful to situations where the response is restricted to the standard unit interval and the regression structure involves regressors and unknown parameters. We derive general formulae for second-order biases of the maximum likelihood estimators and use them to define bias-corrected estimators. Our formulae generalizes the results obtained by Ospina et al. (2006), and are easily implemented by means of supplementary weighted linear regressions. We also compare these bias-corrected estimators with three different estimators which are also bias-free to the second-order, one analytical and the other two based on bootstrap methods. These estimators are compared by simulation. We present an empirical application

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For the first time, we propose the Weibull-geometric (WG) distribution which generalizes the extended exponential-geometric (EG) distribution introduced by Adamidis et al. [K. Adamidis, T. Dimitrakopoulou, and S. Loukas, On a generalization of the exponential-geometric distribution, Statist. Probab. Lett. 73 (2005), pp. 259-269], the exponential-geometric distribution discussed by Adamidis and Loukas [K. Adamidis and S. Loukas, A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35-42] and the Weibull distribution. We derive many of its standard properties. The hazard function of the EG distribution is monotone decreasing, but the hazard function of the WG distribution can take more general forms. Unlike the Weibull distribution, the new distribution is useful for modelling unimodal failure rates. We derive the cumulative distribution and hazard functions, moments, density of order statistics and their moments. We provide expressions for the Rényi and Shannon entropies. The maximum likelihood estimation procedure is discussed and an EM algorithm [A.P. Dempster, N.M. Laird, and D.B. Rubim, Maximum likelihood from incomplete data via the EM algorithm (with discussion), J. R. Stat. Soc. B 39 (1977), pp. 1-38; G.J. McLachlan and T. Krishnan, The EM Algorithm and Extension, Wiley, New York, 1997] is given for estimating the parameters. We obtain the observed information matrix and discuss inference issues. The flexibility and potentiality of the new distribution is illustrated by means of a real data set.

The two-parameter distribution known as exponential-Poisson (EP) distribution, which has decreasing failure rate, was introduced by Kus (2007). In this paper we generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the ith order statistic. We derive the rth raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher's information matrix. Furthermore, expressions for the Rényi and Shannon entropies are given and estimation of the stress-strength parameter is discussed. Applications using two real data sets are presented.