A Logical,Simple Method for Solving the Problem of Properly Indexing Social Security Benefits

Abstract

The problem of allocating responsibility for risk among members of a portfolio arises in a variety of financial and risk-management contexts. Examples are particularly prominent in the insurance sector, where actuaries have long sought methods for distributing capital (net worth) across a number of distinct exposure units or accounts according to their relative contributions to the total “risk” of an insurer’s portfolio. Although substantial work has been done on this problem, no satisfactory solution has yet been presented for the case of inhomogeneous loss distributions— that is, losses X ～ F _{X| λ} such that F _X|tλ (X) ≠ F _{tX| λ} (X) for some t > 0. The purpose of this article is to show that the value-assignment method of nonatomic cooperative games proposed in 1974 by Aumann and Shapley may be used to solve risk-allocation problems involving losses of this type. This technique is illustrated by providing analytical solutions for a useful class of multivariatenormal loss distributions.     

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).

     

On the convergence rate of fixed-width sequential confidence intervals

Abstract

Let X _f1, X _f2, ... be a sequence of i.i.d. random variables with mean µ and variance σ²∈ (0, ∞). Define the stopping times N(d)=min {n:n ^?1 Σ ⁿ _i=1} (X _i&#x2212;X _n)²+n ^?1?nd ²/a ²}, d>0, where X n =n ^?1 Σ ⁿ _i=1} Xi and (2π)^?½ ∫ ^a _?a exp (?u ²/2) du=α ∈(0,1). Chow and Robbins (1965) showed that the sequence I_n,d =[X_n ?d, X _n + d], n=1,2, ... is an asymptotic level -α fixed-width confidence sequence for the mean, i.e. lim_d→0 P(µ∈I_N(d),d )=α. In this note we establish the convergence rate P(µ∈I_N(d),d )=α + O(d½?δ) under the condition E|X₁|^3+?+5/(28) < ∞ for some δ ∈ (0, ½) and ??0. The main tool in the proof is a result of Landers and Rogge (1976) on the convergence rate of randomly selected partial sums.     

The skewed multifractal random walk with applications to option smiles

Abstract

We generalize the construction of the multifractal random walk (MRW) due to Bacry et al (Bacry E, Delour J and Muzy J-F 2001 Modelling financial time series using multifractal random walks Physica A 299 84) to take into account the asymmetric character of financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. We compute the leading behaviour of q-moments of the process, which behave as power laws of the time lag with an exponent ζ _q =p?2p(p?1)λ² for even q=2p, as in the symmetric MRW, and as ζ _q =p + 1?2p ²λ²?α (q=2p + 1), where λ and α are parameters. We show that this extended model reproduces the ‘HARCH’ effect or ‘causal cascade’ reported by some authors. We illustrate the usefulness of this ‘skewed’ MRW by computing the resulting shape of the volatility smiles generated by such a process, which we compare with approximate cumulant expansion formulae for the implied volatility. A large variety of smile surfaces can be reproduced.     

Economies of scale in innovations with block-busters

Abstract

Are large-scale research programmes that include many projects more productive than smaller ones with fewer projects? This problem of economies of scale is relevant for understanding recent mergers, in particular in the pharmaceutical industry. We present a quantitative theory based on the characterization of distributions of discounted sales S resulting from new innovations. Assuming that these complementary cumulative distributions have fat tails with approximate power law structure S ^?α, we demonstrate that economies of scales are realized if and only if α<1. ‘Economies of scale’ is here understood according to the criterion that the probability to earn more than any fixed factor proportional to its size N is larger for the merged company C=A+B of size N _A+N _B than for any of the two component companies A and B with size N _A and N _B. In essence, the mechanism underlying the ‘economies of scale’ is that a very large payoff from a successful project can pay for all of the losing projects. Some empirical evidence suggests that α?2/3 for the pharmaceutical industry. This could provide a simple rationalization for recent mergers or alternatively for portfolio diversification since the same effect could also be achieved in part if each firm held shares in all of its competitors.     

Short note: Stein's two-stage procedure for the intra-class model

In his nice paper (Mykhopadhyay, 1982) as well as in his significant monograph (Mykhopadhyay & Solanky, 1994) N. Mykhopadhyay considers the following application of STEIN's two-stage procedure: Suppose that (X ₁,..., X_n ) ^T , n = 1, 2,..., is n-dimensional normal with mean vector µ = µ l and dispersion matrix Σ _n =σ ²(ρ_ij ) with ρ_ij = 1, ρ_ij = ρ ^*, i ≠ j = 1,..., n where (µ, Σ, ρ) ∈ ? × ?⁺ × (-1, 0); this is called the intra-class model. For given d > 0 and α ∈ (0, 1) one wants to construct a (sequential) confidence interval I for µ having width 2d and confidence coefficient at least (1 - α). It is claimed that where N is determined, according to Stein's two-stage procedure (Stein, 1945), as where m ? 2 is the first stage sample size and denotes the sample variance, fulfills this aim.     

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).     

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.     

On an analogue of Cramér-Rao's inequality

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.     

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.     

The asymptotic relative efficiencies of tests and the derivatives of their power functions

Abstract

For comparing two consistent tests of a simple null hypothesis H ₀ : θ = θ₀ against a given alternative hypothesis H ₀ : θ = θ₁, the measure most frequently used is the asymptotic relative efficiency (ARE), due to Pitman (1948). The ARE is defined as the limit of the reciprocal of the ratio of sample sizes required to attain the same power. The limit is taken as sample sizes tend to infinity and simultaneously θ₁ → θ₀, this being necessary to keep the powers of the tests bounded away from 1.     

Exact moments of order statistics from a power-function distribution

Summary

In this note a problem on exact moments of order statistics from a power-function distribution is considered. The characteristic function of the kth order statistic is obtained and moments about the origin of the kth order statistic are expressed in terms of gamma functions. An exact expression for the covariance of any two order statistics Y_i < Y_j is obtained in terms of beta and gamma functions. Various recurrence relations between the expected values of order statistics are also obtained.     

The Problem of Optimum Stratification

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, π_i , i = 1,.2 ... k. The mean of this population is µ_i being the mean of ni. By means of a random sample of n observations, n_i of which are selected from π_i , µ, is estimated by: being the estimate of µ_i .     

Large Sample Behavior of the CTE and VaR Estimators under Importance Sampling

Abstract

The α-level value at risk (Var) and the α-level conditional tail expectation (CTE) of a continuous random variable X are defined as its α-level quantile (denoted by q_α ) and its conditional expectation given the event {X > q_α }, respectively. Var is a popular risk measure in the banking sector, for both external and internal reporting purposes, while the CTE has recently become the risk measure of choice for insurance regulation in North America. Estimation of the CTE for company assets and liabilities is becoming an important actuarial exercise, and the size and complexity of these liabilities make inference procedures with good small sample performance very desirable. A common situation is one in which the CTE of the portfolio loss is estimated using simulated values, and in such situations use of variance reduction techniques such as importance sampling have proved to be fruitful. Construction of confidence intervals for the CTE relies on the availability of the asymptotic distribution of the normalized CTE estimator, and although such a result has been available to actuaries, it has so far been supported only by heuristics. The main goal of this paper is to provide an honest theorem establishing the convergence of the normalized CTE estimator under importance sampling to a normal distribution. In the process, we also provide a similar result for the Var estimator under importance sampling, which improves upon an earlier result. Also, through examples we motivate the practical need for such theoretical results and include simulation studies to lend insight into the sample sizes at which these asymptotic results become meaningful.     

A note on different models of stochastic processes dealt with in the collective theory of risk

Abstract

1. For the definition of general processes with special regard to those concerned in Collective Risk Theory reference is made to Cramér (Collective Risk Theory, Skandia Jubilee Volume, Stockholm, 1955). Let the independent parameter of such a process be denoted by τ, with the origin at the point of departure of the process and on a scale independent of the number of expected changes of the random function. Denote with p(τ, n)dt the asymptotic expression for the conditional probability of one change in the random function while the parameter passes from τ to τ + dτ: relative to the hypothesis that n changes have occurred, while the parameter passes from 0 to τ. Assume further—unless the contrary is stated—that the probability of more than one change, while the parameter passes from τ to τ + dτ, is of smaller order than dτ.     

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.     

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.     

A Note on s. or n.s. cPp

Abstract

1. In a s. or n.s. cPp (stationary or non-stationary compound Poisson process) the probability for occurrence of m events, while the parameter (one-or more-dimensional) passes from zero to τ ₀ as measured on an absolute scale (the τ-scale), is defined as a mean of Poisson probabilities with intensities, which are distributed with distribution functions defining another random process, called the primary process with respect to the s. or n.s cPp. The stationarity (in the weak sence) and the non-stationarity of the primary process imply the same properties of the s. or n.s. cPp.     

The relative maxima of the likelihood function. II

Abstract

In [1] it was shown that under mild conditions, with probability approaching unity as n increases the likelihood function has a relative maximum in a fixed open interval centered at the true value of the parameter. This paper strengthens the result by having the length of the interval approach zero as n increases. An application is given.