Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

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).     

A New Test for New Better Than Used in Expectation Lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. A popular nonparametric model of aging is new better than used in expectation (NBUE), when M(t) ? M(0) for all t ? 0. The exponential distribution lies at the boundary. There is a large literature on testing exponentiality against NBUE alternatives. However, comparisons of tests have been made only for alternatives much stronger than NBUE. We show that a new Kolmogorov-Smirnov type test is much more powerful than its competitors in most cases.     

MRL and MPL Functions of Parallel Systems with INID Components under Double Monitoring

This article investigates parallel systems with n independent but not identically distributed (INID) components. Under the condition that, at time t ₁ (t ₁ > 0) the total number of failures of the components is r (r < n), and at time t ₂ (t ₂ ≥ t ₁) the sys-tem is still working or it has failed, the mean residual life (MRL) function of the parallel system and the mean past lifetime (MPL) function of the components are conducted. Some representations and corresponding properties of the MRL function and the MPL function under several specific conditions are obtained.     

Bayesian and Robust Bayesian analysis under a general class of balanced loss functions

For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form L_{r, w, d0}(q, d)=wr(d₀, d)+ (1-w) r(q, d){L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}, as well as the weighted version q(q) L_{r, w, d0}(q, d){q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}, where ρ(θ, δ) is an arbitrary loss function, δ ₀ is a chosen a priori “target” estimator of q, w ? [0,1){\theta, \omega \in[0,1)}, and q(·) is a positive weight function. we develop Bayesian estimators under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω > 0 by relating such estimators to Bayesian solutions under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections between optimal actions derived under balanced and unbalanced losses.     

The Uniform Asymptotics of the Overshoot of a Random Walk with Light-Tailed Increments

Let {S _n : n ≥ 0} be a random walk with light-tailed increments and negative drift, and let τ(x) be the first time when the random walk crosses a given level x ≥ 0. Tang (2007 Tang , Q. ( 2007 ). The overshoot of a random walk with negative drift . Statist. Probab. Lett. 77 : 158 – 165 .[Crossref], [Web of Science ®] , [Google Scholar]) obtained the asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, which is uniform for y ≥ f(x) for any positive function f(x) → ∞ as x → ∞. In this article, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for 0 ≤ y ≤ N for any positive number N will be given. Using the above two results, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for y ≥ 0, is presented.     

First-exit times for compound poisson processes for some types of positive and negative jumps

We consider the one-sided and the two-sided first-exit problem for a compound Poisson process with linear deterministic decrease between positive and negative jumps. This process (X(t)) _t≥0 occurs as the workload process of a single-server queueing system with random workload removal, which we denote by M/G _u /G _d /1, where G _u (G _d ) stands for the distribution of the upward (downward) jumps; other applications are to cash management, dams, and several related fields. Under various conditions on G _u and G _d (assuming e.g. that one of them is hyperexponential, Erlang or Coxian), we derive the joint distribution of τ _y =inf{t≥0|X(t)?(0,y)}, y>0, and X(τ _y ) as well as that of T=inf{t≥0|X(t)≤0} and X(T). We also determine the distribution of sup{X(t)|0≤t≤T}.     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

Strassen type limit points for moving averages of a wiener process

Let {W(s); 8 ≥ 0} be a standard Wiener process, and let β_N = (2a_N (log (N/a_N) + log log N)^-1/2, 0 < α_N ≤ N < ∞, where α_N↑ and α_N/N is a non-increasing function of N, and define γ_N(t) = β_N[W(n_N + ta_N) ? W(n_N)), 0 ≥ t ≥ 1, with n_N = N – a_N. Let K = {x ? C[0,1]: x is absolutely continuous, x(0) = 0 and }. We prove that, with probability one, the sequence of functions {γ_N(t), t ? [0,1]} is relatively compact in C[0,1] with respect to the sup norm ||·||, and its set of limit points is K. With a_N = N, our result reduces to Strassen's well-known theorem. Our method of proof follows Strassen's original, direct approach. The latter, however, contains an oversight which, in turn, renders his proof invalid. Strassen's theorem is true, of course, and his proof can also be rectified. We do this in a somewhat more general context than that of his original theorem. Applications to partial sums of independent identically distributed random variables are also considered.     

On moments of the supremum of normed weighted averages

Let {X, X_n; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (a_nk; 1 ≤ k ≤ n, n ≥ 1); a_nk, ? R and sup_{n, k} |a_n,k| < ∞}. Set S_n( A ) = ∑ⁿ_k=1a_{n, k}X_k for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{sup_{n ≥ 1}, |S_n( A )|/n^1/r}^p < ∞ or E{sup_{n ≥ 2} |S_n( A )|/2n log n)^1/2}^p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).     

On the Exponential Decay Rate of the Tail of a Discrete Probability Distribution

Abstract

We give a sufficient condition for the exponential decay of the tail of a discrete probability distribution π = (π _n ) _n≥0 in the sense that lim _n→∞(1/n) log∑ _i>n π _i  = ?θ with 0 < θ < ∞. We focus on analytic properties of the probability generating function of a discrete probability distribution, especially, the radius of convergence and the number of poles on the circle of convergence. Furthermore, we give an example of an M/G/1 type Markov chain such that the tail of its stationary distribution does not decay exponentially.     

A Nonrandomized,Nonconservative Version of the Fisher Exact Test

The Fisher exact test has been unjustly dismissed by some as ‘only conditional,’ whereas it is unconditionally the uniform most powerful test among all unbiased tests, tests of size α and with power greater than its nominal level of significance α. The problem with this truly optimal test is that it requires randomization at the critical value(s) to be of size α. Obviously, in practice, one does not want to conclude that ‘with probability x the we have a statistical significant result.’ Usually, the hypothesis is rejected only if the test statistic's outcome is more extreme than the critical value, reducing the actual size considerably.

The randomized unconditional Fisher exact is constructed (using Neyman–structure arguments) by deriving a conditional randomized test randomizing at critical values c(t) by probabilities γ(t), that both depend on the total number of successes T (the complete-sufficient statistic for the nuisance parameter—the common success probability) conditioned upon.

In this paper, the Fisher exact is approximated by deriving nonrandomized conditional tests with critical region including the critical value only if γ (t) > γ₀, for a fixed threshold value γ₀, such that the size of the unconditional modified test is for all value of the nuisance parameter—the common success probability—smaller, but as close as possible to α. It will be seen that this greatly improves the size of the test as compared with the conservative nonrandomized Fisher exact test.

Size, power, and p value comparison with the (virtual) randomized Fisher exact test, and the conservative nonrandomized Fisher exact, Pearson's chi-square test, with the more competitive mid-p value, the McDonald's modification, and Boschloo's modifications are performed under the assumption of two binomial samples.     

A lower bound for the transition density function of a stochastic differential equation

Let W_t be a one-dimensional Brownian motion on the probability space (Ω,F,P), and let dx_t = a(x_t)dt + b(x_t)dw_t, b²(x) > 0, be a one-dimensional Ito stochastic differential equation. For a(x) = a₀ + a₁x + … + a_nxⁿ on a bounded interval we obtain a lower bound for p(t,x,y), the transition density function of the homogeneous Markov process x_t, depending directly on the coefficients a₀,a₁, …, a_n, and b(x).