A simple proof of the optimality of the best N-policy in the M/G/l queueing control problem with removable server

Abstract  An M/G/l queueing system with removable server is considered. The following costs are incurred: a holding cost of h per unit time per customer in the system, a cost of r_l(r₂) per unit time when the service mechanism is on (off) and a fixed cost of K₁,(K₂ ,) for turning it on (off).
In this paper we shall give a very simple proof for the well known and intuitively obvious fact that the best N-policy is optimal for the average cost criterion among the class of all policies by first proving that the average cost formula of that policy and its relative cost function satisfy the optimality equation for the average cost criterion. The optimality of the best N-policy is then an immediate consequence.     

A central limit theorem for sums of correlated products

Consider a sequence of random points placed on the nonnegative integers with i.i.d. geometric (1/2) interpoint spacings y _i . Let x _i denote the numbers of points placed at integer i . We prove a central limit theorem for the partial sums of the sequence x ₀ y ₀, x ₁ y ₁, . . . The problem is connected with a question concerning different bootstrap procedures.     

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.     

Substituut-variabelen in correlatieberekingen

The recently repeated assertion that in correlation analysis it makes little difference whether one variable (x₂) is used instead of another one (x₃), provided the coefficient of correlation (r₂₃) between x₂ and x₃ is high, is scrutinized.
To that purpose the ranges of coefficients of correlation with respect to the substitute variable are expressed in formula 3. Moreover, by way of example, extreme values of coefficients of simple correlation (r₁₃ and r₃₄), of multiple correlation (R_1.34 and R_3.14) and of regression (α₁₃ and α₁₄, α₃₁ and α₃₄) relating to the substitute variable, are calculated on the basis of empirical values of coefficients of simple correlation relating to the substituted and the remaining variables.
The outcome of those calculations are summarized in the tables 1 and 3, and in the graph.
Table 1 presents ranges of r₁₃ for given values of r₁₂ and r₂₃, table 3 shows extreme values of coefficients of single and multiple correlation and regression in case an additional variable x₄ is introduced and r₁₂, r₁₄, r₂₄ and r₂₃ are given. The graph shows an ellipse as the boundary of the inner closed domain of compatible values of r₁₃ and r₃₄.
Those results clearly indicate the need for caution in substituting one variable by another.     

A test for equality of two exponential distributions

Abstract Let X ₁., X _n1 and Y ₁., Y _n1, be two independent random samples from exponential populations. The statistical problem is to test whether or not two exponential populations are the same, based on the order statistics X _[1],. X _[r1] and Y [₁],. Y _[rs] where 1 r₁ n ₁ and 1 r₂ n ₂. A new test is given and an asymptotic optimum property of the test is proved.     

Het verdelen van steekproeven over subpopulaties bij accountantscontroles

Summary  "Stratificationprocedures for a typical auditing problem".
During the past ten years, much experience was gained in The Netherlands in using random sampling methods for typical auditing problems. Especially, a method suggested by VAN. HEERDEN [2] turned out to be very fruitful. In this method a register of entries is considered to be a population of T guilders, if all entries total up to T guilders. The sample size n ₀ is determined in such a way that the probability β not to find any mistake in the sample, if a fraction p ₀ or more of T is incorrect, is smaller than a preassigned value β₀. So n ₀ should satisfy (l- p )ⁿ⁰≤β₀ for p ≥ p ₀. A complication arises if it is not possible to postpone sampling until the whole population T is available. One then wants to take samples from a population which is growing up to T . Suppose one is going to take samples n _i from e.g. r subpopulations

Using the minimax procedure, it is shown, that in this case one should choose the sizes n _i equal to ( T _i/ T ) n ₀. The minimax-value of the probability not to find any incorrect guilder in the r samples, taken together is equal to β₀.     

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.     

On certain type of Queues in series

Summary This paper studies the steady-state behaviour of a queueing system consisting of three servers S₁, S₂ and S₃ . Two types of units each requiring two phases of service arrive and form two queues Q₁ and Q₂ in-front of the two servers S₁ and S₂, respectively, which attend to their first phase of service in order of arrival. On completion of the first phase the units discharged from these servers from a single queue Q₃ in-front of S₃ for the phase 2 service. The various stochastic processes namely, the interrarrival times and the service times, have been assumed to be distributed exponentially. We derive for this queueing process an expression for the mean queue length and some particular cases thereof.     

The Efficiency of Some Distribution-Free Tests

Laten T₁ en T₂ twee toetsen zijn voor dezelfde hypothese θ=θ₀betreffende de waarde van een parameter θ, Zij verder de onbetrouwbaarheidsdrempel van beide toetsen gelijk aan α en het onderscheidingsvermogen tegen de alternatieve hypothese θ=θ₁ geliik aan 1-β. Indien toets T₁ nu n₁ waarnemingen vergt en toets T₂n₂ waarnemingen, dan wordt de relatieve doeltreffendheid (Eng.: efficiency) van toets T₁ ten opzichte van toets T₂ (als toetsen voor θ=θ₀ tegen θ=θ₁ gegeven door: e = n₂/n₁. Indien men de waarde van θ₁ op een bepaalde wijze naar θ₀ laat convergeren bij toenemende n₁, is het in vele gevallen, door gebruik te maken van een stelling van α en β Deze limiet-waarde wordt de asymptotische relatieve doeltreffendheid (volgens Pitman) genoemd. In dit artikel wordt een overzicht gegeven van hetgeen bekend is over de asymptotische relatieve doeltreffendheid van een aantal verdelingsvrije toetsen ten opzichte van de corresponderende standaardtoetsen.
De conclusie van de schrijver is, dat men bij het gebruik van verdelingsvrije methoden met een hoge doeltreffendheid (bijv. de symmetrietoets en de twee-steek-proeven-toets van Wilcoxon, de toets van Kruskal voor k steekproeven en de methode van m rangschikkingen) slechts zeer weinig informatie kan verliezen en dat zelfs het gebruik van minder doeltreffende verdelingsvrije methoden gerechtvaardigd kan zijn.     

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.     

Covariance formulas via marginal martingw

Prenctice and Cai recently introduced and studied the function C defined as the covariance function of the two marginal counting process martingales of a pair of dependent survival times (T₁, T₂ ). They show that the function C together with the marginal distributions determines the joint survival function F of (T₁, T₂ ). In this note we show how the key characterizing equation of Prentice and Cai yields a formula for the covariance of T₁ and T₂ in termsof the marginal mean residual life functions and C. The resulting formula generalizes a formula for the variance of a one-dimensional random variable Tdueto Pyke (1965). We also explore several generalizations of the covariance formula, and obtain a valid k-dimensional version of the Prentice and Cai formula.     

Selection from Logistic populations

Assume k ( k ≥ 2) independent populations π₁, π₂μ_k are given. The associated independent random variables X_i,( i = 1,2,… k ) are Logistically distributed with unknown means μ₁, μ₂, μ_k and equal variances. The goal is to select that population which has the largest mean. The procedure is to select that population which yielded the maximal sample value. Let μ₍₁₎≤μ₍₂₎≤…≤μ_(k) denote the ordered means. The probability of correct selection has been determined for the Least Favourable Configuration μ₍₁₎=μ₍₂₎==μ_{(k – 1)}=μ_(k)–δ where δ > 0. An exact formula for the probability of correct selection is given.     

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.     

Some problems in actuarial finance involving sums of dependent risks

A trend in actuarial finance is to combine technical risk with interest risk. If Y_t , t = 1, 2, denotes the timevalue of money (discount factors at time t ) and X_t the stochastic payments to be made at time t , the random variable of interest is often the scalar product of these two random vectors V = X_t Y_t . The vectors X and Y are supposed to be independent, although in general they have dependent components. The current insurance practice based on the law of large numbers disregards the stochastic financial aspects of insurance. On the other hand, introduction of the variables Y ₁, Y ₂, to describe the financial aspects necessitates estimation or knowledge of their distribution function.
We investigate some statistical models for problems of insurance and finance, including Risk Based Capital/Value at Risk, Asset Liability Management, the distribution of annuities, cash flow evaluations (in the framework of pension funds, embedded value of a portfolio, Asian options) and provisions for claims incurred, but not reported (IBNR).     

Uitbijternegerende schatters bij duplobepalingen

Outlyer-ignoring estimators for measurement in duplo.
By hypothesis a measurement u is the sum of two independent random variables, the normal random variable with expectation μ, and standard error σ, and a random error φ:

Basically two independent measurements u₁ and u₂ over u are to give the estimate x=1/2(u₁+ u₂) over μ.
However, to reduce the effect of the error φ on a final estimate of μ, one adds, according to a common practice, a third or even a fourth measurement u₃, u₄, in the case that the basic pair differs by more than a number A. For this extended set of measurements two outlyer-ignoring estimator y and z of μ are defined, and investigated against three specifications fo the error φ. Also an outlyer-ignoring estimate of σ is considered, and its application is illustrated by an example.     

Diversification for general copula dependence

We generalize the extreme value analysis for Archimedean copulas (see Alink , Löwe and Wüthrich , 2003) to the non-Archimedean case: Assume we have d ≥2 exchangeable and continuously distributed risks X ₁,…, X _d . Under appropriate assumptions there is a constant q _d such that, for all large u , we have . The constant q _d describes the asymptotic dependence structure. Typically, q _d will depend on more aspects of this dependence structure than the well-known tail dependence coefficient.     

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}.     

Inequalities relating maximal moments to other measures of dispersion

Let X , X ₁, ..., X_k be i.i.d. random variables, and for k ∈ N let D_k ( X ) = E ( X ₁ V ... V X _{k +1}) − EX be the k th centralized maximal moment. A sharp lower bound is given for D ₁( X ) in terms of the Lévy concentration Q_l ( X ) = sup _{x ∈ R} P ( X ∈[ x , x + l ]). This inequality, which is analogous to P. Levy's concentration-variance inequality, illustrates the fact that maximal moments are a gauge of how much spread out the underlying distribution is. It is also shown that the centralized maximal moments are increased under convolution.     

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 Bayesian selection of the best normal population using theKullback–Leibler divergence measure

In this paper, we use the Bayesian approach to study the problem of selecting the best population among k different populations π₁, ..., π_k (k≥2) relative to some standard (or control) population π₀. Here, π₀ is considered to be the population with the desired characteristics. The best population is defined to be the one which is closest to the ideal population π₀ . The procedure uses the idea of minimizing the posterior expected value of the Kullback–Leibler (KL) divergence measure of π _i from π₀. The populations under consideration are assumed to be multivariate normal. An application to regression problems is also presented. Finally, a numerical example using real data set is provided to illustrate the implementation of the selection procedure.