Skewed Bessel function distributions with application to rainfall data

Skewed distributions have attracted significant attention in the last few years. In this article, a skewed Bessel function distribution with the probability density function (pdf) f(x)=2 g (x) G (λ x) is introduced, where g (·) and G (·) are taken, respectively, to be the (pdf) and the cumulative distribution function of the Bessel function distribution [McKay, A.T., 1932, A Bessel function distribution, Biometrica, 24, 39–44]. Several particular cases of this distribution are identified and various representations for its moments derived. Estimation procedures by the method of maximum likelihood are also derived. Finally, an application is provided to rainfall data from Orlando, Florida.     

On a Class of Distributions Defined by the Relationship Between Their Density and Distribution Functions

Knowledge concerning the family of univariate continuous distributions with density function f and distribution function F defined through the relation f(x) = F ^α(x)(1 ? F(x))^β, α, β ? , is reviewed and modestly extended. Symmetry, modality, tail behavior, order statistics, shape properties based on the mode, L-moments, and—for the first time—transformations between members of the family are the general properties considered. Fully tractable special cases include all the complementary beta distributions (including uniform, power law and cosine distributions), the logistic, exponential and Pareto distributions, the Student t distribution on 2 degrees of freedom and, newly, the distribution corresponding to α = β = 5/2. The logistic distribution is central to some of the developments of the article.     

The Extended Skew-Exponential Power Distribution and Its Derivation

We consider an extended family of asymmetric univariate distributions generated using a symmetric density, f, and the cumulative distribution function, G, of a symmetric distribution, which depends on two real-valued parameters λ and β and is such that when β = 0 it includes the entire class of distributions with densities of the form g(z | λ) = 2 G(λ z) f(z). A key element in the construction of random variables distributed according to the family is that they can be represented stochastically as the product of two random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes, as well as extensions to the multivariate case and an explicit procedure for obtaining the moments. We give special attention to the extended skew-exponential power distribution. We derive its information matrix in order to obtain the asymptotic covariance matrix of the maximum likelihood estimators. Finally, an application to a real data set is reported, which shows that the extended skew-exponential power model can provide a better fit than the skew-exponential power distribution.     

Inference on progressive-stress model for the exponentiated exponential distribution under type-II progressive hybrid censoring

In this paper, progressive-stress accelerated life tests are applied when the lifetime of a product under design stress follows the exponentiated distribution [G(x)]^α. The baseline distribution, G(x), follows a general class of distributions which includes, among others, Weibull, compound Weibull, power function, Pareto, Gompertz, compound Gompertz, normal and logistic distributions. The scale parameter of G(x) satisfies the inverse power law and the cumulative exposure model holds for the effect of changing stress. A special case for an exponentiated exponential distribution has been discussed. Using type-II progressive hybrid censoring and MCMC algorithm, Bayes estimates of the unknown parameters based on symmetric and asymmetric loss functions are obtained and compared with the maximum likelihood estimates. Normal approximation and bootstrap confidence intervals for the unknown parameters are obtained and compared via a simulation study.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

Measuring robustness for weighted distributions: Bayesian perspective

There are many situations where the usual random sample from a population of interest is not available, due to the data having unequal probabilities of entering the sample. The method of weighted distributions models this ascertainment bias by adjusting the probabilities of actual occurrence of events to arrive at a specification of the probabilities of the events as observed and recorded. We consider two different classes of contaminated or mixture of weight functions, Γ _a ={w(x):w(x)=(1−ε)w ₀(x)+εq(x),q∈Q} and Γ _g ={w(x):w(x)=w ₀ ^1−ε (x)q ^ε(x),q∈Q} wherew ₀(x) is the elicited weighted function,Q is a class of positive functions and 0≤ε≤1 is a small number. Also, we study the local variation of ϕ-divergence over classes Γ _a and Γ _g . We devote on measuring robustness using divergence measures which is based on the Bayesian approach. Two examples will be studied.     

Exponential Family Techniques for the Lognormal Left Tail

Let X be lognormal(μ,σ²) with density f(x); let θ > 0 and define . We study properties of the exponentially tilted density (Esscher transform) f_θ(x) = e^?θxf(x)/L(θ), in particular its moments, its asymptotic form as θ→∞ and asymptotics for the saddlepoint θ(x) determined by . The asymptotic formulas involve the Lambert W function. The established relations are used to provide two different numerical methods for evaluating the left tail probability of the sum of lognormals S_n=X₁+?+X_n: a saddlepoint approximation and an exponential tilting importance sampling estimator. For the latter, we demonstrate logarithmic efficiency. Numerical examples for the cdf F_n(x) and the pdf f_n(x) of S_n are given in a range of values of σ²,n and x motivated by portfolio value‐at‐risk calculations.     

Moments of Order Statistics from Weibull Distribution in the Presence of Multiple Outliers

In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X₁, X₂, …, X_{n ? p} are independent with density function f(x) while the remaining, p observations (outliers) X_{n ? p + 1}, …, X_n are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.     

On kernel estimators of density ratio

Let f(x) and g(x) denote two probability density functions and g(x)≠0. There are two ways to estimate the density ratio f(x)/g(x). One is to estimate f(x) and g(x) first and then the ratio, the other is to estimate f(x)/g(x) directly. In this paper, we derive asymptotic mean square errors and central limit theorems for both estimators.     

On An Intriguing Distributional Identity

For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω₁ → Ω₂ a continuous, strictly increasing function, such that Ω₁∩Ω₂?(a, b), but otherwise arbitrary, we establish that the random variables F(X) ? F(g(X)) and F(g^{? 1}(X)) ? F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U ? ψ(U) = ^dψ^{? 1}(U) ? U for U ～ U(0, 1), where ψ is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation.     

On a Relation Between a Family of Distributions Attaining the Bhattacharyya Bound and That of Linear Combinations of the Distributions from an Exponential Family

Abstract

An unbiased estimation problem of a function g(θ) of a real parameter is considered. A relation between a family of distributions for which an unbiased estimator of a function g(θ) attains the general order Bhattacharyya lower bound and that of linear combinations of the distributions from an exponential family is discussed. An example on a family of distributions involving an exponential and a double exponential distributions with a scale parameter is given. An example on a normal distribution with a location parameter is also given.     

Fisher information in weighted distributions

Suppose that a density f_θ (x) belongs to an exponential family, but that inference about θ must be based on data that are obtained from a density that is proportional to W(x)f_θ(x). The authors study the Fisher information about θ in observations obtained from such weighted distributions and give conditions under which this information is greater than under the original density. These conditions involve the hazard- and reversed-hazard-rate functions.     

Distributions minimizing fisher information for scale in kolmogorov neighbourhoods

We construct those distributions minimizing Fisher information for scale in Kolmogorov neighbourhoods K_?(G) = {F|sup_x|F(x) - G(x| ? ?} of d.f.'s G satisfying certain mild conditions. The theory is sufficiently general to include those cases in which G is normal, Laplace, logistic, Student's t, etc. As well, we consider G(x) = 1 - e^-x, ? 0, and correct some errors in the literature concerning this case.     

A characterization of invariant power-series abundance distributions

Consider a population the individuals in which can be classified into groups. Let y, the number of individuals in a group, be distributed according to a probability function f(y;ø_o) where the functional form f is known. The random variable y cannot be observed directly, and hence a random sample of groups cannot be obtained. Consider a random sample of N individuals from the population. Suppose the N individuals are distributed into S groups with x₁, x₂, …, x_S representatives respectively. The random variable x, the number of individuals in a group in the sample, will be a fraction of its population counterpart y, and the distributions of x and y need not have the same functional form. If the two random variables x and y have the same functional form for their distributions, then the particular common distribution is called an invariant abundance distribution. The paper provides a characterization of invariant abundance distributions in the class of power-series distributions.     

The new unified representation of multivariate skewed distributions

There are so many proposals in construction skewed distributions, and it is worth finding an overall class which covers all of these proposals. We introduce a new unified representation of multivariate skewed distributions. We will show that this new unified multivariate form of skewed distributions includes all of the continuous multivariate skewed distributions in the literature. This new unified representation is based on the multivariate probability integral transformation and can be decomposed into one factor that is original multivariate symmetric probability density function (pdf) f on ? ^k and skewed factor defined by a pdf p on [0, 1] ^k . This decomposition leads us to prove some useful properties of this new unified form. Stochastic representations and basic properties of this new form are also investigated in this article. Our work is motivated by considering the different skewing mechanisms which lead to different skewed distributions and show that all of these common-used distributions can be viewed as a new unified form.     

FIRST-EXIT TIMES FOR POISSON SHOT NOISE

For a shot-noise process X(t) with Poisson arrival times and exponentially diminishing shocks of i.i.d. sizes, we consider the first time T _b at which a given level b > 0 is exceeded. An integral equation for the joint density of T _b and X(T _b ) is derived and, for the case of exponential jumps, solved explicitly in terms of Laplace transforms (LTs). In the general case we determine the ordinary LT of the function ? P(T _b > t) in terms of certain LTs derived from the distribution function H(x; t) = P(X(t) ≤ x), considered as a function of both variables x and t. Moreover, for G(t, u) = P(T _b > t, X(t) < u), that is the joint distribution function of sup_{0 ≤ s ≤ t} X(s) and X(t), an integro-differential equation is presented, whose unique solution is G(t, u).     

Empirical bayes rules for selecting good populations

A problem of selecting populations better than a control is considered. When the populations are uniformly distributed, empirical Bayes rules are derived for a linear loss function for both the known control parameter and the unknown control parameter cases. When the priors are assumed to have bounded supports, empirical Bayes rules for selecting good populations are derived for distributions with truncation parameters (i.e. the form of the pdf is f(x|θ)= p_i(x)c_i(θ)I_{(0, θ)}(x)). Monte Carlo studies are carried out which determine the minimum sample sizes needed to make the relative errors less than ε for given ε-values.     

Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles

Let (X, Y) be a bivariate random vector whose distribution function H(x, y) belongs to the class of bivariate extreme-value distributions. If F₁ and F₂ are the marginals of X and Y, then H(x, y) = C{F₁(x),F₂(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F₁(X)}/{log F₁(X)F₂(Y)} and W = C{F₁{(X),F₂(Y)}. Using this distribution, an algorithm to generate random variables having bivariate extreme-value distribution is présentés. Furthermore, it is shown that for any bivariate extreme-value dependence function C, the distribution of the random variable W = C{F₁(X),F₂(Y)} belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.     

On bivariate transformation of scale distributions

ABSTRACT

Elsewhere, I have promoted (univariate continuous) “transformation of scale” (ToS) distributions having densities of the form 2g(Π^?1(x)) where g is a symmetric distribution and Π is a transformation function with a special property. Here, I develop bivariate (readily multivariate) ToS distributions. Univariate ToS distributions have a transformation of random variable relationship with Azzalini-type skew-symmetric distributions; the bivariate ToS distribution here arises from marginal variable transformation of a particular form of bivariate skew-symmetric distribution. Examples are given, as are basic properties—unimodality, a covariance property, random variate generation—and connections with a bivariate inverse Gaussian distribution are pointed out.