The regression problem involving non-random variates in the case of stratified sample from normal parent populations with varying regression coefficients

Abstract

1. Introduction.

A sample of N independently observed points (x_{o 1} | x₁₁ , x₂₁ ,... , x_pt ), i = 1, 2, ... , N≥ p is given, where xk, k = 1, 2, ... , p are known, possibly choosable, non-random variates. Suppose now that, for any fixed values of x₁ , x₂ ..., x_p the random variable _o is normally distributed with the mean and the variance λ_o α _x and λ_o are unknown parameters, not involving xk, the regression coefficients and the residual variance of the parent population respectively.     

On minimum variance stratification for estimating the mean of a Weibull population

Abstract

The problem of dividing the frequency function of the Weibull distribution into L(L = 1, ... ,6) strata for the purpose of estimating the population mean under optimum allocation from a stratified random sample is considered. The optimum points of stratification (y ¹,..., y^L-1 ) determining the minimum variance of the estimator are obtained. The variance of the sampling units in each stratum and the variance of the estimate are also given.     

Consistency and asymptotic normality of maximum likelyhood estimators

Abstract

Let the random variable X denote the time taken in completion of a process. For a fixed a, if the observed value of X is less than a, the X is observable, but if X is greater than a, the process is tampered with and is accelerated or decelerated at time a by some unknown factor α, and Y=a+α(X-a) is observed. If the experimenter has only partial control over the experiment, it may be difficult to get several observations on Y corresponding to the same a value. Thus we have a set of independent but not identically distributed observations. The large sample behavior of m.l.e. of the unknown parameters based on tampered random variables Y ^b1 , ..., Y ^bn is studied. If X follows an exponential distribution with mean (1/--), ... the consistency and asymptotic normality of the m.l.e. of α and -- is established under mild conditions on a ^b1, a ^b2, ... The conditions needed for establishing the consistency of m.l.e. of lX are given when X follows a uniform distribution U(O, --) or when X has any known distributional form     

Über das Theorem von Tchebycheff

Abstract

Eine Grösse X hänge in der Weise vom Zufall ab, dass sie verschiedene Werte x ₁, x ₂, … XII annimmt, je nachdem das Ereignis E ₁, oder das Ereignis E ₂ oder … oder das Ereignis E _n eintritt, wofür die Wahrscheinlichkeiten p ₁, p ₂, … p _n bestehen sollen; p ₁ + p ₂ + … + p _n = 1. Mann nennt X eine von Zufall abhängige Grösse oder X eine variable Grösse mit dem Wertevorrat (x ₁, x ₂, … x _n), wobei jedem einzelnen dieser Werte eine bestimmte Wahrscheinlichkeit zukommt. Zwei Grössen X, Y mit den Individualwerten x ₁, x ₂, … x _n; y ₁, y ₂, … y _m heissen unabhängig von einander, wenn die Wahrscheinlichkeit p _i von x _i dieselbe bleibt, welches auch der Wert von Y sei, und wenn auch die Wahrscheinlichkeit q _p von y _p dieselbe bleibt, welchen Wert auch X annehmen möge.     

The Property Tax as a Tax on Value: Deadweight Loss

Consider an atomistic developer who decides when and at what density to develop his land, under a property value tax system characterized by three time-invariant tax rates: τ_V, the tax rate on pre-development land value; τ_S, the tax rate on post-development residual site value; and τ_K, the tax rate on structure value. Arnott (2005) identified the subset of property value tax systems that are neutral. This paper investigates the relative efficiency of four idealized, non-neutral property value tax systems [(i) “Canadian' property tax system: τ_V = 0, τ _S = τ_K; (ii) simple property tax system: τ_V = τ _S = τ_K; (iii) residual site value tax system: τ_K = 0,τ _V = τ_S; (iv) two-rate property tax system: τ_V = τ _S > τ_K > 0] under the assumption of a constant rental growth rate. JEL Code: H2     

Sur une loi de probabilité considérée par J. F. Steffensen

Abstract

Dans ce même périodique, vous avez considéré¹, à la page 7, la loi de probabilité de deux variables aléatoires X, Y,² où la probabilité élémentaire ?(x, y) dx dy pour que X et Y soient respectivement compris entre x et x + dx, y et y + dy, est de la forme où K, a ₁, a ₂, b ₁, b ₂ sont des constantes. Nous nous proposons, dans ce qui suit, d'apporter quelques compléments à votre exposé.     

On the solution of a maximum-likelihood equation of the negative binomial distribution

Summary

In a paper in Biometrika, Anscombe (1950) considered the question of solving the equation with respect to x. Here “Log” denotes the natural logarithm, while N _s , where N _k >0 and N _s =0 for s>k, denotes the number of items ?s in a sample of independent observations from a population with the negative binomial distribution and m denotes the sampling mean: it can in the case k ? 2 be shown that the equation (*) has at least one root. In vain search for “Gegenbeispiele”, Anscombe was led to the conjecture (l.c., 367) that (*) has no solution, if m ² > 2S, and a unique solution, if k ? 2 and m ² < 2S. In the latter case, x equals the maximum-likelihood estimate of the parameter ?.

In the present paper it will, after some preliminaries, be shown that the equation (*) has no solution, if k=l, or if k?2 and m ² ? 2S, whereas (*) has a unique solution, if k ? 2 and m ² < 2S.     

A note on the extended elementary renewal theorem

Abstract

Let X ₁ ^(µ), X ₂ ^(µ), ... be an infinite sequence of independent and identically distributed random variables defined on the whole real axis and with EX₁ ^(µ) = µ > 0. Put M_n ^(µ) = max (S₀ ^(µ), S₁ ^(µ), ..., S_n ^(µ) , where S_n ^(µ) = X₁ ^(µ) + ... + X_n ^(µ) for n = 1 , 2, ... and S₀ ^(µ) = 0, and define      

Optimal stratification according to the method of least squares

Abstract

A one-dimensional random variable X is given. We have L points, µ₁, µ₂, …, µ _L , and define the random variable Z = min_{µ h} | X — µ _h |, that is the distance to the nearest of the L points µ₁, …, µ _L . We want to find that set of points µ _h for which the function has a minimum. As we shall see in section 2, this problem is equivalent to finding L strata with the set of points of stratification x ₁, x ₂, …, x _L?1 that makes a minimum. w_h is the probability mass and σ² _h the variance of the hth stratum. By differentiation of φ with respect to x_h one can show [3] that a necessary condition for minimum is where µh is the mean of the hth stratum. In section 2 we obtain this condition in another way, which at the same time gives a method of finding the points µh and x_h .     

On the asymptotic normality of some unbiased estimates

Abstract

Let X _m(n) =(X _j , n, ..., X _{j m,n} ) be a subset of observations of a sample X_n = (X_1n X _2n ... , X _nn ). Here the X_jn 'S in Xn are not necessarily independent or identically distributed, and m(n) mayor may not tend to infinity as n tends to infinity. Suppose the joint density function h_n =h_n (x _m (n); θ) of the X _jn 's in Xm(n) is completely specified except the values of the parameters in the parameter vector θ = (θ1 θ2, ... , θ _k ), where θ belongs to a non-degenerate open subset H of the k-dimensional Euclidean space Rk and k?m(n).     

Analytical studies in stop-loss reinsurance

Abstract

Introduction.

Consider a unit of risk, say the whole portfolio of an office, or a comprehensive contract of a branch of casualty insurance, which can give rise to a variety of total amounts of claims during a chosen period, say one year. The total claims of the years i =- 1, 2, ... will be denoted by x ₁. They follow some frequency distribution and we assume that during the years considered they are independent from year to year and subject to the same parent distribution. This means, implicitly, that the volume of business and the value of money have remained unaltered and this assumption will be made, since the adjustments otherwise needed are technically trivial and we are not dealing here with the commercial aspect (dif. ficult though it may be of solution) arising out of changes in monetary value. The frequency distribution mentioned can then be regarded as given by a sample from a population whose probability distribution is given by p (x), say, so that      

Holm's probability-paper tests in relation to the Kolmogorov—Smirnov one-sample test

Abstract

In [5] S. Holm proposed teststatistics for testing simple hypotheses by means of the probability paper for distribution functions (d.f.) of the form F ₀(x) = Φ[(x - μ₀)/σ₀], where μ₀ is location parameter, σ₀ scale parameter, and Φ is an absolutely continuous distribution function with Φ(0) = 1/2. If μ₀ and (σ₀ are known, the hypothesis H ₀ is:

• H ₀: H(x) = F ₀(x) = Φ[(x?μ₀)/σ₀],

while the three possible alternatives are

• H ₁: H(x) > F ₀(x)

• H ₂: H(x) < F ₀(x)

• H ₃: H(x) ≠ F ₀(x).

     

Remark on a theorem by Gurland

Abstract

Let X ₁,X ₂,...,X _n be a random sample of size from a distribution with probability density function p(x|θ), where the unknown parameter θ belongs to a non-degenerate interval I. The unknown true value of θ will be denoted by θ₀.     

On a rank-order test for the equality of probability of an event

Abstract

The following situation is considered. A fixed number (= n) or sequence of independent trials T ₁ T ₂,…, T _n is given, and in each of these an event E mayor may not occur, It is further observed that the event E occurs a total of k times amongst the n trials T _i , (i = l,…, n). It is then required to test the hypothesis H ₀ that the probability of the occurrence of E is constant from trial to trial, i.e. H ₀ is the hypothesis: p ₁ = p ₂ = ? = p _n = p, if p _n (i = 1, …, n) represents the probability that E occurs on the ith trial.     

A characterization of geometric distributions by distributional properties of order statistics

Abstract

Let X ₁, X ₂ be independent identically distributed positive integer valued random variables. H the X _i 's have a geometric distribution, then the conditional distribution of R = max(X ₁, X ₂)-min(X ₁, X ₂), given R > 0, is the same as the distribution of X ₁. This property is shown to characterize the geometric distribution.     

Sur le calcul numérique des moments ordinaires et des moments composés d'une distribution statistique

Abstract

Considérons une distribution statistique (empirique ou théorique) x _v (v = 1, 2 ... n) désignant les valeurs que peut assumer une variable fortuite une-dimensionelle x, et P_v (v= 1,2 ... n) désignant les fréquences observées (absolues ou relatives) ou bien les probabilités des valeurs x_v .     

Bestimmung einer Verteilung durch ihre ersten Momente

Abstract

Die Frage, wie weit die Werte einer Verteilungsfunktion V (x) durch ihre ersten Momente M_v =∫x^v dV(x) (v=0, 1, 2, ... m) bestimmt werden, ist, zumindest für ungerade m, durch die klassischen Arbeiten von Tchebychef vollständig erledigt worden. Man kann seine Ergebnisse durch einen einfaehen Zusatz fur den Fall gerader m ergänzen (§ 7 der vorl. Arbeit).     

The prohlem of optimum stratification in a special type of design

Abstract

The concept of optimum stratification was introduced in a paper published in this journal 1950 (1), where as well a technique was devised to determine the condition for optimum stratification. This technique is based upon the representation of the population by a density function ? (y). The variance V ( y ) of the estimate y for a specific type of allocation of the sample to the k strata is then dealt with as a function of the points Y_i of stratification. By solving the equations: the conditions that these points Y_i must fulfil in order to make V ( y ) a minimum are arrived at. In a second paper (2) further results were given.     

On the relations between differences and differential coefficients

Abstract

In a recent paper¹ I have expressed a doubt as to the method followed by Markoff,² in deriving the remainder-terms for the developments of ?⁽ⁿ⁾ (x) in terms of Δ ⁿ?(x) , Δ ⁿ⁺¹ ?(x), ... and of Δ ⁿ?(x) in terms of ?⁽ⁿ⁾ (x), ?⁽ⁿ⁺¹⁾ (x), ‥ .I have since then realized that I have done Markoff something less tha,n justice, and that his line of argument, after filling up a slight gap in one of his proofs, is not only sound but easily accessible to certain generalizations which I proceed to give.