Characterizations of the exponential distribution by some properties of the record values

A sequence {X_n, n≥1} of independent and identically distributed random variables with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max {X₁, X₂, ..., X_j?1}. We define L(n)=min {j|j>L(n?1), X_j>X_L(n?1)}, with L(0)=1. Let Z_n,m=X_L(n)?X_L(m), n>m≥0. Some characterizations of the exponential distribution are considered in terms of Z_n,m and X_L(m).     

Efficient sequential estimation in multiplicative counting processes

The problem of efficient sequential estimation is counting processes with multiplicative intensity processes is considered. A sequential version of Cramér-Rao type information inequality is obtained and all the 'efficient' triples (S, f, g) are characterized: the variance of an unbiased estimator f for g attains the lower bound under a sampling plan S. Applications to Poisson processes, Markov processes, birth and death processes and Markov branching processes with immigration are also considered.     

Record values of exponentially distributed random variables

A sequence {X_n, n≥1} of independent and identically distributed random variables with absolutely continuous (with respect to Lebesque measure) cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max(X₁,…,X_j?1), j>1. Let {X_L(n), n≥0} with L(o)=1 be the sequence of such record values and Z_n,n?1=X_L(n)–X_L(n?1). Some properties of Z_n,n?1 are studied and characterizations of the exponential distribution are discussed in terms of the expectation and the hazard rate of z_n,n?1.     

On the number of near-records after the maximum observation in a continuous sample

{X_n, n≥1} are independent and identically distributed random variables with continuous distribution function F(x). For j=1,…,n, X_j is called a near-record up to time n if X_j ∈ (M_na, M_n], where M_n = max_1≤j≤n {X_j} and a is a positive constant. Let Z_n(a) denote the number of near-records after, and including the maximum observation of the sequence. In this paper, the distributional results of Z_n(a) are considered and its asymptotic behaviours are studied.     

Estimation of a multivariate mean with constraints on the norm

Estimation of the mean θ of a spherical distribution with prior knowledge concerning the norm ||θ|| is considered. The best equivariant estimator is obtained for the local problem ||θ|| = λ₀, and its risk is evaluated. This yields a sharp lower bound for the risk functions of a large class of estimators. The risk functions of the best equivariant estimator and the best linear estimator are compared under departures from the assumption ||θ|| = λ₀.     

Multi - sample test of equal gamma distribution scale parameters in presence of unknown common shape parameter

Shiue and Bain proposed an approximate F statistic for testing equality of two gamma distribution scale parameters in presence of a common and unknown shape parameter. By generalizing Shiue and Bain's statistic we develop a new statistic for testing equality of L >= 2 gamma distribution scale parameters. We derive the distribution of the new statistic ESP for L = 2 and equal sample size situation. For other situations distribution of ESP is not known and test based on the ESP statistic has to be performed by using simulated critical values. We also derive a C(α) statistic CML and develop a likelihood ratio statistic, LR, two modified likelihood ratio statistics M and MLB and a quadratic statistic Q. The distribution of each of the statistics CML, LR, M, MLB and Q is asymptotically chi-square with L - 1 degrees of freedom. We then conducted a monte-carlo simulation study to compare the perfor- mance of the statistics ESP, LR, M, MLB, CML and Q in terms of size and power. The statistics LR, M, MLB and Q are in general liberal and do not show power advantage over other statistics. The statistic CML, based on its asymptotic chi-square distribution, in general, holds nominal level well. It is most powerful or nearly most powerful in most situations and is simple to use. Hence, we recommend the statistic CML for use in general. For better power the statistic ESP, based on its empirical distribution, is recommended for the special situation for which there is evidence in the data that λ₁ < … < λ_L and n₁ < … < n_L, where λ₁ …, λ_L are the scale parameters and n₁,…, n_L are the sample sizes.     

Spectral analysis with replicated time series

A doubly stochastic process {x(b,t);b?B,t?Z} is considered, with (B,β,P_β) being a probability space so that for each b, {X(b,t);t ? Z} is a stationary process with an absolutely continuous spectral distribution. The population spectrum is defined as f(ω) = E_B[Q(b,ω)] with Q(b,ω) being the spectral density function of X(b,t). The aim of this paper is to estimate f(ω) by means of a random sample b₁,…,b_r from (B,β,P_β). For each b₁? B, the processes X(b₁,t) are observed at the same times t=1,…,N. Thus, r time series (x(b₁,t)} are available in order to estimate f(ω). A model for each individual periodogram, which involves f(ω), is formulated. It has been proven that a certain family of linear stationary processes follows the above model In this context, a kernel estimator is proposed in order to estimate f(ω). The bias, variance and asymptotic distribution of this estimator are investigated under certain conditions.     

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ _n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.     

Estimation bayesienne des effets dans le modele a effets aleatoires de classification double avec interaction

The problem of estimating the effects in a balanced two-way classification with interaction \documentclass{article}\pagestyle{empty}\begin{document}$i = 1, \ldots ,I;j = 1, \ldots ,J;k = 1, \ldots ,K$\end{document} using a random effect model is considered from a Bayesian view point. Posterior distributions of r_i, c_j and t_ij are obtained under the assumptions that r_i, c_j, t_ij and e_ijk are all independently drawn from normal distributions with zero meansand variances \documentclass{article}\pagestyle{empty}\begin{document}$\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document} respectively. A non informative reference prior is adopted for \documentclass{article}\pagestyle{empty}\begin{document}$\mu ,\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document}. Various features of thisposterior distribution are obtained. The same features of the psoterior distribution for a fixed effect model are also obtained. A numerical example is given.     

Some characterizations of geometric tail distributions based on record values

Let R_j be the jth upper record value from an infinite sequence of independent identically distributed positive integer valued random variables. We show that their common distribution must have geometric tail if R_j+k?R_j and R_j are partially independent for some j≥1 and k≥1 or if E(R_j+2?R_j+1| R_j) is a constant. Three versions of partial independence, each of which provides a characterization of the geometric tail are presented.     

Comparison of location parameters of two exponential distributions when scale parameters are different and unknown

Letx _i(1)≤x _i(2)≤…≤x _i(r_i) be the right-censored samples of sizesn _i from theith exponential distributions $\sigma _i^{ - 1} exp\{ - (x - \mu _i )\sigma _i^{ - 1} \} ,i = 1,2$ where μ_i and σ_i are the unknown location and scale parameters respectively. This paper deals with the posteriori distribution of the difference between the two location parameters, namely μ₂-μ₁, which may be represented in the form $\mu _2 - \mu _1 \mathop = \limits^\mathcal{D} x_{2(1)} - x_{1(1)} + F_1 \sin \theta - F_2 \cos \theta $ where $\mathop = \limits^\mathcal{D} $ stands for equal in distribution,F _i stands for the central F-variable with [2,2(r _i?1)] degrees of freedom and $\tan \theta = \frac{{n_2 s_{x1} }}{{n_1 s_{x2} }}, s_{x1} = (r_1 - 1)^{ - 1} \left\{ {\sum\limits_{j = 1}^{r_i - 1} {(n_i - j)(x_{i(j + 1)} - x_{i(j)} )} } \right\}$ The paper also derives the distribution of the statisticV=F ₁ sin σ?F ₂ cos σ and tables of critical values of theV-statistic are provided for the 5% level of significance and selected degrees of freedom.     

RELATIONS BETWEEN ERGODICITY AND MEAN DRIFT FOR MARKOV CHAINS1

If {X_n} is an irreducible aperiodic Markov chain on a state apace denote the mean one step change of position, or “drift”, of {X_n} at j. The main result of this note is to show that, when |µ(j)| is bounded, {X_n} admits a stationary distribution {π_j}if and only if for some N > 0 and some state i, lim inf ∑when this holds, the limit infimum is in fact . Many of the known sufficient or necessary criteria for the existence of a stationary distribution can then be derived easily from this and related results.     

Estimation in a truncated bivariate poisson distribution

Moment estimators for parameters in a truncated bivariate Poisson distribution are derived in Hamdan (1972) for the special case of λ₁ = λ₂, Where λ₁, λ₂ are the marginal means. Here we derive the maximum likelihood estimators for this special case. The information matrix is also obtained which provides asymptotic covariance matrix of the maximum likelihood estimators. The asymptotic covariance matrix of moment estimators is also derived. The asymptotic efficiency of moment estimators is computed and found to be very low.     

Estimation under modified Weibull distribution based on right censored generalized order statistics

In this paper, the maximum likelihood (ML) and Bayes, by using Markov chain Monte Carlo (MCMC), methods are considered to estimate the parameters of three-parameter modified Weibull distribution (MWD(β, τ, λ)) based on a right censored sample of generalized order statistics (gos). Simulation experiments are conducted to demonstrate the efficiency of the proposed methods. Some comparisons are carried out between the ML and Bayes methods by computing the mean squared errors (MSEs), Akaike's information criteria (AIC) and Bayesian information criteria (BIC) of the estimates to illustrate the paper. Three real data sets from Weibull(α, β) distribution are introduced and analyzed using the MWD(β, τ, λ) and also using the Weibull(α, β) distribution. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic, {AIC and BIC} to emphasize that the MWD(β, τ, λ) fits the data better than the other distribution. All parameters are estimated based on type-II censored sample, censored upper record values and progressively type-II censored sample which are generated from the real data sets.     

Sampling experiments and the quandt statistic

The distribution of the Quandt likelihood ratio λ_t. for a two-phase regression, has yet to be determined. In particular it is known that-2 log λ_t. is not distributed as chi-square (Quandt; 1960), for unknown switch point.

In this paper we describe sampling experiments which suggest that-log λ_t, has a Pearson Type III distribution, The parameters of the distribution appear to depend not only on the values of the x-vector (Feder; 1968) but also its dimension k.     

Box-Cox transformed linear models: A parameter-based asymptotic approach

A Box-Cox transformed linear model usually has the form y(λ) = μ + β₁x₁ +… + β_px_p + oe, where y(λ) is the power transform of y. Although widely used in practice, the Fisher information matrix for the unknown parameters and, in particular, its inverse have not been studied seriously in the literature. We obtain those two important matrices to put the Box-Cox transformed linear model on a firmer ground. The question of how to make inference on β = (β₁,…,β_p)^T when λ; is estimated from the data is then discussed for large but finite sample size by studying some parameter-based asymptotics. Both unconditional and conditional inference are studied from the frequentist point of view.     

Deux méthodes d'estimation pour les paramètres de processus moyenne mobile spatiaux

We consider moving average processes, {X_s, s ∈ ??}, where ?? is a triangular lattice in the plane R^2. To estimate the parameters of such processes, Adjengue & Moore (1993) have considered likelihood and gaussian pseudo-likelihood methods. We consider here two other methods. The first one is based on the estimation of the correlations and the relation between these correlations and the parameters of the process. The second relies on a linear approximation of the process. The asymptotic properties of the proposed estimators are analyzed and compared. A simulation study allows us to compare the estimators for fixed sample sizes.     

Best quadratic unbiased estimation of the variance matrix in normal regression

In 1973 Balestra examined the linear model y=XB+u, where u is a normally distributed disturbance vector, with variance matrix Ω. Ω has spectral decomposition \(\sum\limits_{i = 1}^r {\lambda _i M_i } \) , and the matrices M_i are known. Estimation of ω is thus equivalent with estimation of the λ_i. Balestra presented the best quadratic unbiased estimator of λ_i. In the present paper a derivation will be given which is based on a procedure developed by this writer (1980).     

A study of generalized skew-normal distribution

Following the paper by Genton and Loperfido [Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math. 57 (2005), pp. 389–401], we say that Z has a generalized skew-normal distribution, if its probability density function (p.d.f.) is given by f(z)=2φ _p (z; ξ, Ω)π (z?ξ), z∈? ^p , where φ _p (·; ξ, Ω) is the p-dimensional normal p.d.f. with location vector ξ and scale matrix Ω, ξ∈? ^p , Ω>0, and π is a skewing function from ? ^p to ?, that is 0≤π (z)≤1 and π (?z)=1?π (z), ? z∈? ^p . First the distribution of linear transformations of Z are studied, and some moments of Z and its quadratic forms are derived. Next we obtain the joint moment-generating functions (m.g.f.’s) of linear and quadratic forms of Z and then investigate conditions for their independence. Finally explicit forms for the above distributions, m.g.f.’s and moments are derived when π (z)=κ (α′z), where α∈? ^p and κ is the normal, Laplace, logistic or uniform distribution function.