An extremal property of the generalized arcsine distribution

The main result of the paper is the following characterization of the generalized arcsine density p _γ (t) = t ^γ?1(1 ? t) ^γ?1/B(γ, γ)   with ${t \in (0, 1)}$ and ${\gamma \in(0,\frac12) \cup (\frac12,1)}$ : a r.v. ξ supported on [0, 1] has the generalized arcsine density p _γ (t) if and only if ${ {\mathbb E} |\xi- x|^{1-2 \gamma}}$ has the same value for almost all ${x \in (0,1)}$ . Moreover, the measure with density p _γ (t) is a unique minimizer (in the space of all probability measures μ supported on (0, 1)) of the double expectation ${ (\gamma-\frac12 ) {\mathbb E} |\xi-\xi^{\prime}|^{1-2 \gamma}}$ , where ξ and ξ′ are independent random variables distributed according to the measure μ. These results extend recent results characterizing the standard arcsine density (the case ${\gamma=\frac12}$ ).     

{\varvec{P}}-value model selection criteria for exponential families of increasing dimension

Let $\mathcal{M }_{\underline{i}}$ be an exponential family of densities on $[0,1]$ pertaining to a vector of orthonormal functions $b_{\underline{i}}=(b_{i_1}(x),\ldots ,b_{i_p}(x))^\mathbf{T}$ and consider a problem of estimating a density $f$ belonging to such family for unknown set ${\underline{i}}\subset \{1,2,\ldots ,m\}$ , based on a random sample $X_1,\ldots ,X_n$ . Pokarowski and Mielniczuk (2011) introduced model selection criteria in a general setting based on p-values of likelihood ratio statistic for $H_0: f\in \mathcal{M }_0$ versus $H_1: f\in \mathcal{M }_{\underline{i}}\setminus \mathcal{M }_0$ , where $\mathcal{M }_0$ is the minimal model. In the paper we study consistency of these model selection criteria when the number of the models is allowed to increase with a sample size and $f$ ultimately belongs to one of them. The results are then generalized to the case when the logarithm of $f$ has infinite expansion with respect to $(b_i(\cdot ))_1^\infty $ . Moreover, it is shown how the results can be applied to study convergence rates of ensuing post-model-selection estimators of the density with respect to Kullback–Leibler distance. We also present results of simulation study comparing small sample performance of the discussed selection criteria and the post-model-selection estimators with analogous entities based on Schwarz’s rule as well as their greedy counterparts.     

On generalized transforms of distribution functions

In this note we discuss the following problem. LetX andY to be two real valued independent r.v.'s with d.f.'sF and ?. Consider the d.f.F*? of the r.v.X oY, being o a binary operation among real numbers. We deal with the following equation: $$\mathcal{G}^1 (F * \phi ,s) = \mathcal{G}^2 (F,s)\square \mathcal{G}^3 (\phi ,s)\forall s \in S$$ where \(\mathcal{G}^1 ,\mathcal{G}^2 ,\mathcal{G}^3 \) are real or complex functionals, т another binary operation ands a parameter. We give a solution, that under stronger assumptions (Aczél 1966), is the only one, of the problem. Such a solution is obtained in two steps. First of all we give a solution in the very special case in whichX andY are degenerate r.v.'s. Secondly we extend the result to the general case under the following additional assumption: $$\begin{gathered} \mathcal{G}^1 (\alpha F + (1 - \alpha )\phi ,s) = H[\mathcal{G}^i (F,s),\mathcal{G}^i (\phi ,s);\alpha ] \hfill \\ \forall \alpha \in [0,1]i = 1,2,3 \hfill \\ \end{gathered} $$ .     

An additive property of weak records from geometric distributions

Let $\{W_m\}{_{m\ge 1}}$ be the sequence of weak records from a discrete parent random variable, $X$ , supported on the non-negative integers. We obtain a new characterization of geometric distributions based on an additive property of weak records: $X$ follows a geometric distribution if and only if for certain integers, $n,\, s\ge 1, W_{n+s}\stackrel{d}{=}W_n+W^{\prime }_s$ , with $W^{\prime }_s$ independent of $W_n$ and $W^{\prime }_s\stackrel{d}{=} W_s$ .     

A new non-linear AR(1) time series model having approximate beta marginals

We consider the mixed AR(1) time series model $$X_t=\left\{\begin{array}{ll}\alpha X_{t-1}+ \xi_t \quad {\rm w.p.} \qquad \frac{\alpha^p}{\alpha^p-\beta ^p},\\ \beta X_{t-1} + \xi_{t} \quad {\rm w.p.} \quad -\frac{\beta^p}{\alpha^p-\beta ^p} \end{array}\right.$$ for ?1 < β ^p ≤ 0 ≤ α ^p  < 1 and α ^p ? β ^p  > 0 when X _t has the two-parameter beta distribution B₂(p, q) with parameters q > 1 and ${p \in \mathcal P(u,v)}$ , where $$\mathcal P(u,v) = \left\{u/v : u < v,\,u,v\,{\rm odd\,positive\,integers} \right\}.$$ Special attention is given to the case p = 1. Using Laplace transform and suitable approximation procedures, we prove that the distribution of innovation sequence for p = 1 can be approximated by the uniform discrete distribution and that for ${p \in \mathcal P(u,v)}$ can be approximated by a continuous distribution. We also consider estimation issues of the model.     

The distribution of the maximum of a first order autoregressive process: the continuous case

We give the cumulative distribution function of M _n , the maximum of a sequence of n observations from an autoregressive process of order 1. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. When the correlation is positive, $$P \left( M_n \leq x \right)\ =a_{n,x},$$ where $$a_{n,x}= \sum_{j=1}^\infty \beta_{jx}\ \nu_{jx}^{n} = O \left( \nu_{1x}^{n}\right),$$ where {?? _jx } are the eigenvalues of a non-symmetric Fredholm kernel, and ?? _1x is the eigenvalue of maximum magnitude. When the correlation is negative $$P \left( M_n \leq x \right)\ =a_{n,x} +a_{n-1,x}.$$ The weights ?? _jx depend on the jth left and right eigenfunctions of the kernel. These are given formally by left and right eigenvectors of an infinite Toeplitz matrix whose eigenvalues are just {?? _jx }. These results are large deviations expansions for extremes, since the maximum need not be standardized to have a limit. In fact, such a limit need not exist. The use of the derived expansion for P(M _n ?? x) is illustrated using both simulated and real data sets.     

Distributions determined by conditioning on a single order statistic

LetX ₁,X ₂, …,X _n(n ? 2) be a random sample on a random variablex with a continuous distribution functionF which is strictly increasing over (a, b), ?∞ ?a <b ? ∞, the support ofF andX _1:n ?X _2:n ? … ?X _n:n the corresponding order statistics. Letg be a nonconstant continuous function over (a, b) with finiteg(a ⁺) andE {g(X)}. Then for some positive integers, 1 <s ?n $$E\left\{ {\frac{1}{{s - 1}}\sum\limits_{i - 1}^{s - 1} {g(X_{i:n} )|X_{s:n} } = x} \right\} = 1/2(g(x) + g(a^ + )), \forall x \in (a,b)$$ iffg is bounded, monotonic and \(F(x) = \frac{{g(x) - g(a^ + )}}{{g(b^ - ) - g(a^ + )}},\forall x \in (a,b)\) . This leads to characterization of several distribution functions. A general form of this result is also stated.     

Consistency of a nonparametric estimation of a density functional

LetX be a random variable with distribution functionF and density functionf. Let ? and ψ be known measurable functions defined on the real lineR and the closed interval [0, 1], respectively. This paper proposes a smooth nonparametric estimate of the density functional \(\theta = \int\limits_R \phi (x) \psi \left[ {F (x)} \right]f^2 (x) dx\) based on a random sampleX ₁, ...,X _n fromF using a kernel functionk. The proposed estimate is given by \(\hat \theta = (n^2 a_n )^{ - 1} \mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^n \phi (X_i ) \psi \left[ {\hat F (X_i )} \right]k\left[ {(X_i - X_j )/a_n } \right]\) , where \(\hat F(x) = n^{ - 1} \mathop \sum \limits_{i = 1}^n K\left[ {(x - X_i )/a_n } \right]\) with \(K (w) = \int\limits_{ - \infty }^w {k (u) } du\) . The estimate \(\hat \theta \) is shown to be consistent both in the weak and strong sense and is used to estimate the asymptotic relative efficiency of various nonparametric tests, with particular reference to those using the Chernoff-Savage statistic.     

On the maxima of heterogeneous gamma variables with different shape and scale parameters

In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let \(X_{1}\) and \(X_{2}\) be two independent gamma random variables with \(X_{i}\) having shape parameter \(r_{i}>0\) and scale parameter \(\lambda _{i}\) , \(i=1,2\) , and let \(X^{*}_{1}\) and \(X^{*}_{2}\) be another set of independent gamma random variables with \(X^{*}_{i}\) having shape parameter \(r_{i}^{*}>0\) and scale parameter \(\lambda _{i}^{*}\) , \(i=1,2\) . Denote by \(X_{2:2}\) and \(X^{*}_{2:2}\) the corresponding maxima, respectively. It is proved that, among others, if \((r_{1},r_{2})\) majorize \((r_{1}^{*},r_{2}^{*})\) and \((\lambda _{1},\lambda _{2})\) weakly majorize \((\lambda _{1}^{*},\lambda _{2}^{*})\) , then \(X_{2:2}\) is stochastically larger that \(X^{*}_{2:2}\) in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature.     

Characterization of the exponential distribution function by Properties of the differenceX k+s∶n−X k∶n of order statistics

We considern independent and identically distributed random variables with common continuous distribution functionF concentrated on (0, ∞). LetX _1∶n≤X_2∶n...≤X_n∶n be the corresponding order statistics. Put $$d_s \left( x \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - P\left( {X_{s:n - k} \geqslant x} \right), x \geqslant 0,$$ and $$\delta _s \left( {x, \rho } \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - e^{ - \rho \left( {n - k} \right)x} ,\rho > 0,x \geqslant 0.$$ Fors=1 it is well known that each of the conditions d₁(x)=O ?x≥0 and δ₁ (x, p) = O ?x≥0 implies thatF is exponential; but the analytic tools in the proofs of these two statements are radically different. In contrast to this in the present paper we present a rather elementary method which permits us to derive the above conclusions for somes, 1≤n —k, using only asymptotic assumptions (either forx→0 orx→∞) ond _s(x) and δ₁ (x, p), respectively.     

On a simple characterisation of threshold graphs

In their very interesting paper “Set-packing problems and threshold graphs” [1] V. Chvatal and P. L. Hammer have shown that the constraints $$\begin{gathered} \sum\limits_{j = 1}^n {a_{ij} x_j \leqslant 1(i = 1,2, . . . , m)} \hfill \\ x_j \in (0,1)(j = 1,2, . . . , n) \hfill \\ \end{gathered} $$ are equivalent to the only inequality $$\begin{gathered} \sum\limits_{j - 1}^n {c_j x_j \leqslant d} \hfill \\ x_j \in (0,1)(j = 1,2, . . . , n) \hfill \\ \end{gathered} $$ if and only if the intersection graph associated with the matrix (a _ij ) — see § 1 — is a threshold graph i.e. a graph none of whose induced subgraphs are isomorphic to 2K ₂,P ₄,C ₄: As Chvatal and Hammer have shown [1], threshold graphs can be characterised in many different ways; the main result of this paper is to give a new, very simple characterisation which will enable us to test whether a graph is a threshold by a simple inspection of its incidence matrix.     

D-optimal chemical balance weighing designs with autoregressive errors

In this paper, we consider the estimation problem of individual weights of three objects. For the estimation we use the chemical balance weighing design and the criterion of D-optimality. We assume that the error terms ${\varepsilon_{i},\ i=1,2,\dots,n,}$ are a first-order autoregressive process. This assumption implies that the covariance matrix of errors depends on the known parameter ρ. We present the chemical balance weighing design matrix ${\widetilde{\bf X}}$ and we prove that this design is D-optimal in certain classes of designs for ${\rho\in[0,1)}$ and it is also D-optimal in the class of designs with the design matrix ${{\bf X} \in M_{n\times 3}(\pm 1)}$ for some ρ ≥ 0. We prove also the necessary and sufficient conditions under which the design is D-optimal in the class of designs ${M_{n\times 3}(\pm 1)}$ , if ${\rho\in[0,1/(n-2))}$ . We present also the matrix of the D-optimal factorial design with 3 two-level factors.     

Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices

We consider n × n real symmetric random matrices n ^?1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ^?1 A ^T A with independent entries of m × n matrix A. Assuming first that the 4th cumulant (excess) κ ₄ of entries of W and A is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, ${m/n\rightarrow c\in[0,\infty)}$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class ${\mathbb{C}^5}$ ). This is done by using a simple “interpolation trick”. Then, by using a more elaborated techniques, we prove the CLT in the case of non-zero excess of entries for essentially ${\mathbb{C}^4}$ test function. Here the variance contains additional term proportional to κ ₄. The proofs of all limit theorems follow essentially the same scheme.     

Unbiased estimates for a lognormal regression problem and a nonparametric alternative

Given a normal sample with means \({{\bf x}_{1}^{\prime} {\bf \varphi}, \ldots, {\bf x}_{n}^{\prime} {\bf \varphi}}\) and variance v, minimum variance unbiased estimates are given for the moments of L, where log L is normal with mean \({{\bf x}^{\prime} {\bf \varphi}}\) and variance v. These estimates converge to wrong values if the normality assumption is false. In the latter case estimates based on any M-estimate of \({{\bf \varphi}}\) are available of bias \({O\left(n^{-1}\right)}\) and \({O\left(n^{-2}\right)}\). More generally, these are given for any smooth function of \({\left({\bf \varphi}, F\right)}\), where F is the unknown distribution of the residuals. The regression functions need not be linear.     

Estimation of location of discontinuity in a density

Here we propose a few estimators of θ, in addition to those studied in Goria (1978), the point of discontinuity of the probability density $$f(x,\theta ) = \frac{1}{{2\Gamma (\alpha )}}e^{ - |x - \theta |} |x - \theta |^{\alpha - 1} ,$$ for $$0< \alpha< 1, - \infty< x< \infty , - \infty< \theta< \infty .$$ We establish the consistency and the optimality of the Bayes and the maximum probability estimators. Despite their nice properties, these estimators are not easy to compute in this case and their effective computation depends on the knowledge of the exponent α. Hence, we propose another class of estimators, dependent upon the spacings of the observations, computable without actual knowledge of the value of α as long as it is known that α < α₀ < 1: we show that these estimators converge at the best possible rate. We further demonstrate, using a modified version of the maximum probability estimator's technique, that the tails of the density do not substantially effect their efficiency. Finally a bivariate family of densities, having a ridge dependent on the parameter θ, is considered and it is shown that this family exhibits features similar to the univariate case, and thus, the necessary modifications of the arguments of the univariate case are utilized for the estimation of θ in this bivariate example.     

Tests of significance for the mean of a finite lot

Let ξ₁, ξ₂, ξ₃, ... be independent identically distributed random variables each with normal distribution with mean μ and variance σ². Tests for the process mean μ are well-known elements of statistical analysis: the Gauß test under known process variance σ², Student’st-test under unknown process variance σ². Let the process be partitioned in lots (ξ₁, ..., ξ _N ), (ξ _N+1, ..., ξ_2N ), ... of sizeN. Consider (ξ₁, ..., ξ _N ) as a stochastic representative of this lot sequence and let the lot be characterized by the lot mean $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } $ . The lot mean can be considered as a parameter of the joint conditional distribution function of the lot variables under $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$ . The present paper investigates the analogies of the Gauß test and Student’st-test for the lot situation, i.e. tests of significance for the lot meanz under known and unknown process variance σ². This approach is of special interest for the statistical control of product quality in situations where the quality of a lot of items 1, 2, ...,N with quality characteristics ξ₁, ξ₂, ..., ξ _N is identified with the lot average $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$ .