共查询到20条相似文献,搜索用时 31 毫秒
1.
B. R. Rao 《Scandinavian actuarial journal》2013,2013(1-2):57-67
Abstract Rao [1] and simultaneously Cramér [2, 3] have shown that if f (x, θ) is the probability density function of a distribution involving an unknown parameter θ and distributed over the range α ? x ? b, where a and b are independent of θ, and if x 1 x 2 ... x n is a random sample of n independent observations from this distribution, the variance of any estimate unbiased for Ψ (θ), satisfies the inequality where E denotes mathematical expectation and is Fisher's information index about θ. In (1), equality holds if, and only if, θ* is sufficient for θ. This inequality is further generalized to the multi-parametric case. 相似文献
2.
Ibrahim A. Ahmad 《Scandinavian actuarial journal》2013,2013(3):176-181
Abstract Bhattacharyya & Roussas (1969) proposed to estimate the functional Δ = ∫ ?∞/∞ f 2(x)dx by , where is a kernel estimate of the probability density f(x). Schuster (1974) proposed, alternatively, to estimate Δ by , where F n (x) is the sample distribution function, and showed that the two estimates attain the same rate of strong convergence to Δ. In this note, two large sample properties of are presented, first strong convergence of to Δ is established under less assumptions than those of Schuster (1974), and second the asymptotic normality of established. 相似文献
3.
W. Simonsen 《Scandinavian actuarial journal》2013,2013(4):220-231
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. 相似文献
4.
5.
Ernst Zwinggi 《Scandinavian actuarial journal》2013,2013(1-2):165-170
Abstract The premium for a deferred disability pension payable continually during maximum n years may be expressed by means of the well known fundamental symbolic letters as in recent times, the exact value (1) is often approximated 1 by the expression 相似文献
6.
C. G. Esseen 《Scandinavian actuarial journal》2013,2013(2):160-170
Abstract Consider a sequence of independent random variables (r.v.) X 1 X 2, …, Xn , … , with the same distribution function (d.f.) F(x). Let E (Xn ) = 0, E , E (?(X)) denoting the mean value of the r.v. ? (X). Further, let the r.v. where have the d.f. F n (x). It was proved by Berry [1] and the present author (Esseen [2], [4]) that Φ(x) being the normal d.f. 相似文献
7.
Harald Bohman 《Scandinavian actuarial journal》2013,2013(1):43-45
Abstract Given a characteristic functionφ(t) we want to calculate the corresponding distribution function. For the sake of simplicity we will assume that the mean value of the distribution is zero, i.e. that φ'(0) =0. For these calculations we will use the following formula where The dash on the summation sign indicates that the term corresponding to k = 0 is missing. 相似文献
8.
S. Vajda 《Scandinavian actuarial journal》2013,2013(3-4):180-191
Abstract I In an earlier paper [5] we discussed the problem of finding an unbiased estimator of where p (x, 0) is a given frequency density and 0 is a (set of) parameter(s). In general, will not be an unbiased estimator of (1), when Ô is an unbiased estimate of O. In [5] it was shown that is an unbiased estimator of (1), if we define yi , as the larger of 0 and X j - c. It was emphasized that the resulting estimate may very well be zero, even when it is unreasonable to assume that the premium for a stop.loss reinsurance. defined by a frequency p (x, 0) of claims x and a critical limit c, should be zero when the critical limit has not been exceeded during the n years considered for the determination of the premium. 相似文献
9.
W. Simonsen 《Scandinavian actuarial journal》2013,2013(1-2):26-45
Abstract The problem of expressing a difference of a given order of a function in terms of successive derivatives of the function and the related problem of obtaining a manageable form of the remainder- term of a special expansion of this kind have on several occasions been treated in the literature. One of the best known results of investigations on this subject is Markoff's formula,I which may be written in a slightly modified form: where Δm0µ = [Δmxµ ]x = 0 and Δm h denotes the descending difference of order m for a table-interval of length h. 相似文献
10.
Jan Grandell 《Scandinavian actuarial journal》2013,2013(1-2):76-77
Abstract A mixture-Poisson distribution is defined by where U(x) is a distribution function concentrated on (0, ∞). This distribution has been applied as a model of the number of claims occurring in an insurance business during a certain period of time. 相似文献
11.
Per Ottestad 《Scandinavian actuarial journal》2013,2013(1-2):197-201
Asbtract The hypernormal (or Lexian) frequency function can be defined by the integral where θ(p) is the frequency (or density) function of p defined in the interval. We have, of course, that and that . 相似文献
12.
Tore Dalenius 《Scandinavian actuarial journal》2013,2013(3-4):203-213
Abstract Although most applications of stratified sampling represent sampling from a finite population, π(N), consisting of k mutually exclusive sub-populations or strata, n, (N,), it is for purposes of theoretical investigations convenient to deal with a hypothetical population n, represented by a distribution function f(y), a < y < b. This hypothetical population likewise consists of k mutually exclusive strata, πi , i = 1,.2 ... k. The mean of this population is µi being the mean of ni. By means of a random sample of n observations, ni of which are selected from πi , µ, is estimated by: being the estimate of µi . 相似文献
13.
Håkan Prawitz 《Scandinavian actuarial journal》2013,2013(3):145-156
Abstract Let Xbv (v = 1,2, ..., n) be independent random variables with the distribution functions Fbvx) and suppose . We define a random variable by where and denote the distribution function of X by F (x. 相似文献
14.
N. F. Gjeddebæk 《Scandinavian actuarial journal》2013,2013(1):135-159
§ 1. Introduction. a. Grouping. From a purely mathematical point of view practical observations are often more or less grouped, but in applied mathematics only fairly coarse groupings need be taken into account. The statures of the individuals in a population are commonly given as an example of grouped observations. A measurement of between x - unit and x + unit is referred to as x units, and in order to compensate for the inaccuracy of the estimates of the means and standard deviations calculated from such observations the use of certain corrections has been advised. The benefit hereof is, however, doubtful as usually grouping is comparatively fine and then it is not really disturbing. In the following an account will be given of some facts connected with far coarser grouping. As otherwise the subject would become too extensive, the exposition will here be confined to normally distributed observations. The problem of the “best estimates” of the mean and the standard deviation for the coarsely grouped normal observations will be solved in accordance with the principle of maximum likelihood. 相似文献
15.
J. Wolfowitz 《Scandinavian actuarial journal》2013,2013(3-4):132-151
Abstract 1. Summary of results. Let E and Eo be chance variables at least one of which is not normally distributed (throughout the present paper a chance variable which is constant with probability one will be considered to be normally distributed with variance zero), and whose distribution is otherwise unknown, except that it is known that with probability one, where 0 and p are unknown constants, . Let (u; v) be jointly normally distributed chance variables with unknown covariance matrix, distributed independently of (ε, ε0). Without loss of generality we assume that the expected values E u and E v, of u and v respectively, are both zero. Define 相似文献
16.
Harald Bergström 《Scandinavian actuarial journal》2013,2013(3-4):139-153
Abstract If X and Y are mutually independent random variables whith the d. f. 1 F 1(χ) and F 2(χ), it is known 2 that the sum X + Y has the d. f. F 2(χ), defined as the convolution where the integrals are Lebesgue-Stiltjes integrals. One uses the abbreviation More generally the sum X 1 + X 2 + … + X n of n mutually independent random variables with the d. f. 1 F 1(χ), F 2(χ) , … , F n has the d. f. 相似文献
17.
Tore Dalenius 《Scandinavian actuarial journal》2013,2013(1-2):61-70
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 Yi of stratification. By solving the equations: the conditions that these points Yi must fulfil in order to make V ( y ) a minimum are arrived at. In a second paper (2) further results were given. 相似文献
18.
N. F. Gjeddebæk 《Scandinavian actuarial journal》2013,2013(2):154-159
Abstract 1. Introduction (a) Maximum Likelihood.—In a previous paper (THIS JOURNAL, vol. XXXII, 1949, pp. 135–159) the author gave tables of the functions and where ?(x) denotes the normal law of distribution, φ(x) its integral and ?′(x) its first derivative. With the aid of these tables it is practicable to solve the maximum likelihood equations for coarsely grouped normal observations. The procedure was illustrated by examples. 相似文献
19.
Harald Cramér 《Scandinavian actuarial journal》2013,2013(1):141-180
Analysis of statistical distributions. 1. Let m and σ denote the mean and the standard deviation of a statistical variable X, and let W(x) be the probability function of that variable as defined in the first paper 1 , Art. 1. If we put (cf. I, formula (3)) F(x) is the probability function of the variable , with the mean value 0 and the standard deviation 1. Denoting by µ2, µ3, ... the moments of W(x) , taken about the mean (cf. I, Art. 7, where m is supposed to be zero), we put, following Charlier, 相似文献
20.
Harald Bergström 《Scandinavian actuarial journal》2013,2013(1-2):106-127
Abstract Introduction. In an earlier paper 1 I proved the inequality for the difference between the normal d. f. 2 Φ (χ) and the d. f. of the sum of n equally distributed random variables with the mean value O. Here σ denotes the dispersion, β3 the absolute third moment of the variable Xi and C is an absolute constant. To establish the inequality I gave an identical expansion of the convolution , when the dispersion for F(χ) was 1, and a lemma for Weierstrass' singular integral. I also remarked that this method could be used for d. f.'s in the space Rk , k> 1. In fact there is very little to be changed when I now give the generalization for the space Rk . 相似文献