No Arabic abstract
In this paper, we propose to obtain the skewed version of a unimodal symmetric density using a skewing mechanism that is not based on a cumulative distribution function. Then we disturb the unimodality of the resulting skewed density. In order to introduce skewness we use the general method which transforms any continuous unimodal and symmetric distribution into a skewed one by changing the scale at each side of the mode.
This paper considers a family of distributions constructed by a stochastic mixture of the order statistics of a sample of size two. Various properties of the proposed model are studied. We apply the model to extend the exponential and symmetric Laplace distributions. An extension to the bivariate case is considered.
A new acceptance-rejection method is proposed and investigated for the Bingham distribution on the sphere using the angular central Gaussian distribution as an envelope. It is shown to have high efficiency and to be straightfoward to use. The method can also be extended to Fisher and Fisher-Bingham distributions on spheres and related manifolds.
In this paper we propose a class of weighted rank correlation coefficients extending the Spearmans rho. The proposed class constructed by giving suitable weights to the distance between two sets of ranks to place more emphasis on items having low rankings than those have high rankings or vice versa. The asymptotic distribution of the proposed measures and properties of the parameters estimated by them are studied through the associated copula. A simulation study is performed to compare the performance of the proposed statistics for testing independence using asymptotic relative efficiency calculations.
An approximate maximum likelihood method of estimation of diffusion parameters $(vartheta,sigma)$ based on discrete observations of a diffusion $X$ along fixed time-interval $[0,T]$ and Euler approximation of integrals is analyzed. We assume that $X$ satisfies a SDE of form $dX_t =mu (X_t ,vartheta ), dt+sqrt{sigma} b(X_t ), dW_t$, with non-random initial condition. SDE is nonlinear in $vartheta$ generally. Based on assumption that maximum likelihood estimator $hat{vartheta}_T$ of the drift parameter based on continuous observation of a path over $[0,T]$ exists we prove that measurable estimator $(hat{vartheta}_{n,T},hat{sigma}_{n,T})$ of the parameters obtained from discrete observations of $X$ along $[0,T]$ by maximization of the approximate log-likelihood function exists, $hat{sigma}_{n,T}$ being consistent and asymptotically normal, and $hat{vartheta}_{n,T}-hat{vartheta}_T$ tends to zero with rate $sqrt{delta}_{n,T}$ in probability when $delta_{n,T} =max_{0leq i<n}(t_{i+1}-t_i )$ tends to zero with $T$ fixed. The same holds in case of an ergodic diffusion when $T$ goes to infinity in a way that $Tdelta_n$ goes to zero with equidistant sampling, and we applied these to show consistency and asymptotical normality of $hat{vartheta}_{n,T}$, $hat{sigma}_{n,T}$ and asymptotic efficiency of $hat{vartheta}_{n,T}$ in this case.
We introduce new shape-constrained classes of distribution functions on R, the bi-$s^*$-concave classes. In parallel to results of Dumbgen, Kolesnyk, and Wilke (2017) for what they called the class of bi-log-concave distribution functions, we show that every $s$-concave density $f$ has a bi-$s^*$-concave distribution function $F$ for $s^*leq s/(s+1)$. Confidence bands building on existing nonparametric bands, but accounting for the shape constraint of bi-$s^*$-concavity, are also considered. The new bands extend those developed by Dumbgen et al. (2017) for the constraint of bi-log-concavity. We also make connections between bi-$s^*$-concavity and finiteness of the CsorgH{o} - Revesz constant of $F$ which plays an important role in the theory of quantile processes.