Some remarks concerning the quotient of sample median and sample range for a sample of size 2n+1 from a normal distribution

Consider an ordered sample ₍₁₎, ₍₂₎,…, _(2n+1) of size 2 n +1 from the normal distribution with parameters μ and . We then have with probability one
₍₁₎ < ₍₂₎ < … < (2 n +1).
The random variable
_n =_(n+1)/_(2n+1)-(1)
that can be described as the quotient of the sample median and the sample range, provides us with an estimate for μ/, that is easy to calculate. To calculate the distribution of h _n is quite a different matter***. The distribution function of h₁, and the density of h₂ are given in section 1. Our results seem hardly promising for general h_n. In section 2 it is shown that h_n is asymptotically normal.
In the sequel we suppose μ= 0 and = 1, i.e. we consider only the "central" distribution. Note that h_n can be used as a test statistic replacing Student's t. In that case the central h_n is all that is needed.     

On a simple sequential decision problem

We consider a simple sequential problem that generalizes the gamble with a fair coin. A sequence [ X _n] ot {0,1} r.v.'s is observed, and at each step the gambler can bet on either 0 or 1. The sequence ( X _n) is assumed exchangeable. Except for the case of i.i.d. r.v.'s with even probability (the fair coin), there exists a strategy such that the cumulated expected gain diverges to +∞.     

CENTRAL LIMIT THEORMS IN C[0,1] FOR A CLASS OF ESTIMATORS OF A DISTRIBUTION FUNCTION

As non–parametric estimates of an unknown distribution function (d.f.) F based on i.i.d. observations X ₁ X_n with this d.f.

are used, where H _n is a sequence of d.f.'s converging weakly to the unit mass at zero. Under regularity conditions on F and the sequence ( H _n) it is shown that √_n( F _n– F ) and √_n( R _n – F ) in C [0,1] converge in distribution to a process G with G( t ) = W° ( F ( t )), where W ° is a Brownian bridge in C [0,1]. Further the a.s. uniform convergence of R., is considered and some examples are given.     

On the asymptotically uniform distribution modulo 1 of extreme order statistics

Let (X_m)∞₁ be a sequence of independent and identically distributed random variables. We give sufficient conditions for the fractional part of rnax (X₁., X_n) to converge in distribution, as n ←∞ to a random variable with a uniform distribution on [0, 1).     

First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties

Some properties of a first-order integer-valued autoregressive process (INAR)) are investigated. The approach begins with discussing the self-decomposability and unimodality of the 1-dimensional marginals of the process {X_n} generated according to the scheme X_n=α° X _n-i +e_n, where α° X _n-1 denotes a sum of X_{n - 1}, independent 0 - 1 random variables Y^(n-1), independent of X _n-1 with Pr -( y ^{(n - 1)}= 1) = 1 - Pr ( y ^(n-i)= 0) =α. The distribution of the innovation process ( e _n) is obtained when the marginal distribution of the process ( X _n) is geometric. Regression behavior of the INAR(1) process shows that the linear regression property in the backward direction is true only for the Poisson INAR(1) process.     

Steekproeven uit de halve cauchy verdeling

Summary Let x₁…, x_n be a sample from a distribution with infinite expectation, then for n→∞ the sample average x?_n tends to +∞ with probability 1 (see [4]). Sometimes x?_n contains high jumps due to large observations. In this paper we consider samples from the “absolute Cauchy” distribution. In practice, on may consider the logarithm of the observations as a sample from a normal distribution. So we found in our simulation. After rejecting the log-normality assumption, one will be tempted to regard the extreme observations as outliers. It is shown that the discarding of the outlying observations gives an underestimation of the expectation, variance and 99 percentile of the actual distribution.     

An improved empirical Bayes test for positive exponential families

We exhibit an empirical Bayes test δ^* _n for a decision problem using a linear error loss in a class of positive exponential families. This empirical Bayes test δ^* _n possesses the asymptotic optimality, and its associated regret converges to zero with rate n ⁻¹(ln n )⁶ This rate of convergence improves the previous results in the literature in the sense that a faster rate of convergence is achieved under much weaker conditions. Examples are presented to illustrate the performance of the empirical Bayes test δ^* _n     

Large deviation principles and loglog laws for observation processes

Let ( X_n, n ≥ 1) be an i.i.d. sequence of positive random variables with distribution function H . Let φ _H := {(n, X_n ), n ≥ 1) be the associated observation process. We view φ h as a measure on E := [0, ∞) ∞ (0, φ] where φ_H (A) is the number of points of φ _H which lie in A . A family ( V_s, s> 0) of transformations is defined on E in such a way that for suitable H the distributions of ( V_sφ_H, S > 0) satisfy a large deviation principle and that a related Strassen-type law of the iterated logarithm also holds. Some consequent large deviation principles and loglog laws are derived for extreme values. Similar results are proved for φ _H replaced by certain planar Poisson processes.     

On divergence and convergence of sums of nonnegative random variables

Abstract  If X ₁, X ₂,… are exponentially distributed random variables thenσ^∞_{k= 1} X_k=∞ with probability 1 iff σ^∞_{k= 1} EX_k=∞. This result, which is basic for a criterion in the theory of Markov jump processes for ruling out explosions (infinitely many transitions within a finite time) is usually proved under the assumption of independence (see FREEDMAN (1971), p. 153–154 or BREI-MAN (1968), p. 337–338), but is shown in this note to hold without any assumption on the joint distribution. More generally, it is investigated when sums of nonnegative random variables with given marginal distributions converge or diverge whatever are their joint distributions.     

Fourier inversion and the Hausdorff distance

This paper continues research done by F.H. Ruymgaart and the author. For a function f on R ^d we consider its Fourier transform F f and the functions f_M (M>0) derived from F f by the formula f_M(x) =( F( ε_M · F f ))(− x );, where the ε_M are suitable integrable functions tending to 1 pointwise as M →∞. It was shown earlier that, relative to a metric d _H , analogous to the Hausdorff distance between closed sets, one has d _H (f_M, f) = O( M ^−½) for all f in a certain class. We now show that, for such f , the estimate O( M ^−½) is optimal if and only if f has a discontinuity point.     

On the Kuiper test for normality with mean and variance unknown

Summary A modified form of the Kuiper statistic V _n is developed for testing the composite hypothesis that a sample of size n comes from a normal population with unspecified mean and variance. Its distribution is derived using Monte Carlo methods. Power comparison with the adjusted Kuiper test proposed by L outer and K oerts [6] indicates that our test is superior with respect to certain alternatives.     

Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds

The classes of monotone or convex (and necessarily monotone) densities on     can be viewed as special cases of the classes of k - monotone densities on     . These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for which asymptotic results are known, and the class of completely monotone (∞-monotone) densities on     . In this paper we consider non-parametric maximum likelihood and least squares estimators of a k -monotone density g ₀. We prove existence of the estimators and give characterizations. We also establish consistency properties, and show that the estimators are splines of degree k −1 with simple knots. We further provide asymptotic minimax risk lower bounds for estimating the derivatives     , at a fixed point x ₀ under the assumption that     .     

An alternate development of conditioning

Abstract  The "classical" development of conditioning, due to K olmogorov , does not agree with the "practical" (more intuitive, but unrigorous) way in which probabilists and statisticians actually think about conditioning. This paper describes an alternative to the classical development. It is shown that standard concepts and results can be developed, rigorously, along lines, which correspond to the "practical" approach, and so as to include the classical material as a special case. More specifically, let Xand Y be random variables (r.v.'s) from (Ω, f, P) to ( x, f_x ) and (y. f^y.), respectively. In this paper, the fundamental concept is the conditional probability P(AX = x ), a function of xε x which satisfies a "natural" defining condition. This is used to define a conditional distribution P_y/x, as a mapping x × f^y-R such that, as a function of B, P_ylx=x,(B ) is a probability measure on f^y. Then, for a numerical r.v. Y , conditional expectation E(Y/X) is defined as a mapping x → whose value at x isE(Y/X = x) = ydP_Y/x=i(y ). Basic properties of conditional probabilities, distributions, and expectations, are derived and their existence and uniqueness are discussed. Finally, for a sub-o-algebra and a numerical r.v. Y , the classical conditional expectation E(Y) is obtained as E(Y/X) with X = i , the identity mapping from (Ω, f) to (Ω).     

On a crossroad of resampling plans: bootstrapping elementary symmetric polynomials

We investigate the validity of the bootstrap method for the elementary symmetric polynomials S ^{( k )} _n =( ⁿ _k )⁻¹Σ_{1≤ i 1}< ... < i _k ≤ n X _{i 1} ... X _{i k} of i.i.d. random variables X ₁, ..., X _n . For both fixed and increasing order k , as n→∞ the cases where μ=E X ₁[moe2]0, the nondegenerate case, and where μ=E X ₁=0, the degenerate case, are considered.     

Asymptotic distributions of φ-divergences of hypothetical and observed frequencies on refined partitions

For a wide class of goodness-of-fit statistics based on φ-divergences between hypothetical cell probabilities and observed relative frequencies, the asymptotic normality is established under the assumption n / m _n →γ∈(0,∞), where n denotes sample size and m _n the number of cells. Related problems of asymptotic distributions of φ-divergence errors, and of φ-divergence deviations of histogram estimators from their expected values, are considered too.     

A LIMIT THEOREM FOR BERNOULLI RV'S AND FELLER'S SHOE PROBLEM

Consider n sets of objects, each set consisting of m distinct types (for instance n place settings each made up of m distinct dishes and silverware pieces.) s items are drawn at random from the mn items. The distribution of the number of complete sets (each consisting of all m items) in the sample of s is asymptotically Poisson distributed with parameter (a /m )^m if s = an ^1–1 and n →∞. This fact can be interpreted in terms of a certain limit theorem for a sequence of i.i.d Bernoulli rv's.     

Characteristic properties of order statistics based on random sample size from an exponential distribution

Suppose X₁, X₂, X_m is a random sample of size m from a population with probability density function f (x), x > 0), and let X_{1, m}< × 2, m <… < X_{m, m} be the corresponding order statistics.
We assume m is an integer-valued random variable with P( m = k ) = p (1- p )^k-1, k = 1,2,… and 0 < p < 1. Two characterizations of the exponential distribution are given based on the distributional properties of X_{l, m}.     

Functional laws of the iterated logarithm for small increments of empirical processes

Let F , denote the uniform empirical distribution based on the first n ≥ 1 observations from an i.i.d. sequence of uniform (0, 1) random variables. We describe the almost sure limiting behavior of the sets of increment functions {F_n(t + h_n.) - F_n(t): 0 ≤ t ≤ 1 - h_n}, when {h_n: n ≥ 1) is a nonincreasing sequence of constants such that nh_n /log n ← 0.     

De omvang van de steekproef bij een enkelvoudig steekproefsysteem

Summary  (Sample size for a single sampling scheme).
The operating characteristic of a sampling scheme may be specified by the producers 1 in 20 risk point ( p ₁), at which the probability of rejecting a batch is 0.05, and the consumers 1 in 20 risk point ( p ₂) at which the probability of accepting a batch of that quality is also 0.05.
A nomogram is given (fig. 2) to determine for single sampling schemes and for given values of p₁ and p ₂ the necessary sample size ( n ) and the allowable number of defectives in the sample ( c ).
The nomogram may reversedly be used to determine the producers and consumers 1 in 20 risk points for a given single sampling scheme.
The curves in this nomogram were computed from a table of percentage points of the χ² distribution. For v > 30 Wilson and Hilferty's approximation to the χ² distribution was used.     

Thin layer chromatography, a case study

This paper gives an account of the collaboration between two mathematical statisticians and a toxicologist (the second author) interested in thin layer chromatography (TLC). A TLC "system" consists of a medium through which a solvent is transported. If a solution of some (toxic) sample is applied to the medium, then the components are carried forward by the solvent over different distances. Section 1 describes the concept of a data bank which provides standard values for the degrees of migration characteristic for each of m well-studied substances in each of n systems. Sections 2–5 are mainly devoted to the construction of The "best design(s)"{ j ₁*… j _k * } of k systems from the n available ones. The attention is restricted to the situation that an unidentified sample exclusively contains one of the m substances covered by the data bank and produces the scores xj … ^xj_k in the systems j,… j _krespectively. Three different approaches to the identification problem were successively considered. Each approach leads to a class of procedures and their performances. The performance of the optimum procedure can be used to define the performance of any of the ( ⁿ_k ) designs ( j ₁… j_k }. The latter performance is maximized in order to determine { j ₁*.,., j_k* }. In practice usually data is obtained for mixtures instead of single. pure substances. Section 6 gives some tentative theory for the evaluation of such data.