The Perron-Frobenius theorem without additivity

We prove the following non-linear generalization of the Perron-Frobenius theorem. Let A:R^m₊→R^m₊ be continuous, homogeneous of degree 1 and primitive (i.e., for some integer l, x≥yA^lxA^ly); then A has a positive eigenvector x₀, unique up to multiplication by a positive scalar, and for all x0, Aⁿx/|Aⁿx| converges to x₀/|x₀|.     

Una disuguaglianza riguardante la funzione gamma incompleta

Summary For testing statistical hypotheses are considered: a test _A ² based on addition theorem, as a sum ofm tests related tom indipendent samples, each one of sizen, withg _A degrees of freedom, and a second test _T ² based on the total aggregation of them samples, withg _T degrees of freedom.Respective power functions are compared, under the same probability of the error of the first kind.An existing result is generalized and it is proved that _T ² is always more powerful for any positive integer value ofg _A–g_T.     

On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γ_t be the residual life at time t of the renewal process A={A(t),t≥0 }, under suitable condition, we prove that if Var(γ_t)=E ² (γ_t),∀t≥0, then A is a Poisson process. Secondly, we show that if Var (A(t)) is proportional to E (A(t)), then A is a Poisson process also, and Var (A(t))=E (A(t)). Received: August 1999     

Bemerkungen zu den Sätzen von Jordan

Zusammenfassung Die Aussagen der Sätze von Jordan über die Wahrscheinlichkeit, daß vonn EreignissenA ₁, ...A _n mindestensm (genaum) eintreten, werden auf den Fall abzählbar vieler EreignisseA ₁,A ₂, ... verallgemeinert.

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Summary The Jordan-Theorems concerning the probability that at leastm (exactlym) amongn possible eventsA ₁, ...,A _n occur are generalized to the case of denumerably many EventsA ₁,A ₂,...

     

Lattice structure of knowledge and agreeing to disagree

This paper is a contribution to the study of the underlying mathematical structure of common-knowledge, which gives the well-known result of Aumann about the impossibility of ‘agreeing to disagree’. We present the Bayesian subjective probability model with player's belief: i.e. a triple (? %plane1D;4AF;, μ), in which i is a player. ? is a lattice in the field of sets of a state space Ω, %plane1D;4AF;, is a correspondence assigning to each state ω a filter %plane1D;4AF;(ω) in ?, and μ is a common-prior. For this model, we impose none of the important restrictions on the information structure in the Aumann-Bacharach model: axiom of knowledge K₁. axiom of transparency K₂ and axiom of wisdom K₃. We can extend both the disagreement theorem of Aumann and the agreement theorem of Geanacoplos and Polemarchakis under the assumption that each ? is an Artinian lattice.     

Functional forms for technical change functions

In this paper we examine the four properties of technical change functions introduced by Sato [1980, 1981], paying particular attention to the composition property. We show by way of examples that two of the properties are independent, but that a third is a necessary consequence of two others. A theorem providing a necessary condition for the composition property is stated and proved. Finally, we identify the general continuous functional form of technical change functions for which the efficiency of factor i is independent of the quantities of all factors j i. Noting an unappealing characteristic of a member of this class of transformations, we propose an additional property for technical change functions and note that it is independent of the others.The refereeing process of this paper was handled through S.T. Hackman.     

Adimpact—a multi-media advertising effectiveness-measurement method

Recall and recognition are the methods which are usually used to establish if a person has been exposed to advertising or not. Recall tends to greatly underestimate exposure. Recognition, though coming much closer to actual exposure, has severe practical limitations, especially for radio commercials and in terms of the number of advertisements that can be tested at a time. Adimpact is a method offering an alternative way of establishing probable exposure to advertising, which is independent of the respondents' ability to recall or recognize the advertisements they have seen or heard.     

On the application of the Minkowski-Farkas theorem to sampling designs

In the paper, we consider the following problem: Let {π_k} be a sequence satisfying 0π_kΣ1 (k=1,…, N) and π=n.Tben, is there an unordered sampling design such that, for each k=1,…N, the inclusion probability of unit k is equal to π? It is shown that it can be solved by the straightforward application of the Minkowski-Farkas theorem.     

Zur Definition von Zufallsstichproben aus endlichen Grundgesamtheiten

Summary Suppose there is a universe which containsN elements,M of which are marked by a special property. From this universe a sample of sizen is drawn without replacement. From the Laplace-condition about the ( _n ^N ) possible outcomes of this experiment it follows that the number of marked elements in the sample is distributed hypergeometrically. It is shown that an inverse of this theorem is true, whereas the analogue inverse for the with-replacement-case is not true.Further some other evident properties of samples are derived from the Laplace-condition for both cases of the drawing scheme. For all cases of possible implications between these properties and the Laplace-condition it is decided if they are true or not.     

Scharfe untere und obere Schranken für die Wahrscheinlichkeit der Realisation vonk untern Ereignissen

Summary LetA ₁,...,A _n be events in a probability space (,A,W). We denote byL _k the event, that at leastk events among then eventsA ₁,...A _n occur, and byK _k the event, that exactlyk events occur. If only the inequalities _i W(A _i ) _i ,i=1,...,n, are known, we calculate sharp lower and upper bounds forW(L _k ) andW(K _k ). These bounds only depend onn, k and _i , _i ,i=1,...,n. They are relevant, when treating combined tests or confidence procedures.     

A long dollar fall: likelihood and consequences

• ? The dollar has tended to move in long swings over the last forty years, raising the risk that the recent decline could extend considerably further. This is not our base case, but risks do look skewed towards additional dollar weakness. Our modelling work suggests that a large further dollar slide would have significant effects on the pattern of world growth – the US and some emerging markets would gain, with other advanced economies the main losers.
• ? There have been several large multi‐year swings in the dollar over the last four decades. We identify seven such episodes since 1971 including three long declines averaging 31%, the last being in 2002‐08. Since 2017 the dollar has fallen 10%, implying a possible further considerable drop.
• ? Our dollar strength indicator, which covers a range of economic variables associated with dollar moves in the past, does not currently point to a re‐run of the dollar weakness of the 2000s. But we do expect some further near‐term dollar losses and risks to our baseline forecast look skewed to the downside, especially given the emergence of large twin deficits in the US.
• ? Should a further large dollar slump nevertheless occur, our modelling suggests large effects on the pattern of world growth. The main gainers would be commodity‐producing emerging markets (EM) benefitting from improved terms of trade, positive balance sheet and external liquidity effects and scope to ease local interest rates. Rising US yields would erode some of these gains in later years.
• ? The main initial losers would be advanced economies outside the US which would lose competitiveness. In the case of the Eurozone and Japan, undershoots of inflation targets would be likely. There could also be some other negative consequences such as stoking protectionism and creating financial bubbles in some EMs.

     

When (not) to indulge in ‘puffery’: the role of consumer expectations and brand goodwill in determining advertised and actual product quality

We analyze why some firms advertise product quality at a level different from the actual quality of a product. By considering the interacting effects of product quality and advertising, we develop a dynamic model of consumer expectations about product quality and the development of brand goodwill to determine the optimal values for the decision variables. The model parameters are determined based on prior literature and we use numerical techniques to arrive at the solution. We then derive conditions under which a firm will find it optimal to overstate or understate product quality. The results suggest that quality may be overstated in markets characterized by high price sensitivity, low quality sensitivity, low brand loyalty, and high source credibility, suggesting the need for vigilance on the part of consumers, upper level managers and regulatory authorities in such market conditions. This is important because current regulatory resources are insufficient to reduce deceptive advertising practices (Davis JJ. 1994. Ethics in advertising decision‐making: implications for reducing the incidence of deceptive advertising. Journal of Consumer Affairs 28 : 380–402). Further, the law of deceptive advertising prohibits some advertising claims on the ground that they are likely to harm consumers or competitors (Preston IL, Richards JI. 1993. A role for consumer belief in FTC and Lanham Act deceptive advertising cases. American Business Law Journal 31 : 1–29). Also, Nagler (1993. Rather bait than switch: deceptive advertising with bounded consumer rationality. Journal of Public Economics 51 : 359–378) shows that deceptive advertising causes a net social welfare loss and a public policy effectively preventing deception will improve social welfare. Copyright © 2000 John Wiley & Sons, Ltd.     

Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices

We consider n × n real symmetric random matrices n ^?1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ^?1 A ^T A with independent entries of m × n matrix A. Assuming first that the 4th cumulant (excess) κ ₄ of entries of W and A is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, ${m/n\rightarrow c\in[0,\infty)}$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class ${\mathbb{C}^5}$ ). This is done by using a simple “interpolation trick”. Then, by using a more elaborated techniques, we prove the CLT in the case of non-zero excess of entries for essentially ${\mathbb{C}^4}$ test function. Here the variance contains additional term proportional to κ ₄. The proofs of all limit theorems follow essentially the same scheme.     

A study of measuring influence of advertising and forecasting cigarette sales

This study investigated the effect of advertising on cigarette sales, particularly after 1967. Data were collected using sources of the Statistical abstract of the U.S., the Historical Statistics of the U.S. and Vital Statistics between 1955 and 1979. A multiple regression model was used to analyze the data. Cigarette consumption was used as a dependent variable. Disposable Income, Death Rate due to cancer of the respiratory system/total cancer death, advertising outlays for cigarettes: Newspapers and television advertisement/total advertising cost, cigarette-production (including long and regular sizes), sales outlets, loyalty/total loyalty, average price for cigarettes and a dummy variable were used as independent variables. Analysis revealed that there is a significant negative relationship between cigarette-consumption and total cancer death and average prices on the one hand, and a significant positive association between loyalty and cigarette consumption on the other. Although advertising expenditures are not statistically significant, increased spending on advertising has an increasing effect on cigarette-consumption.     

On the Lp error of the Grenander-type estimator in the Cox model

We consider the Cox regression model and study the asymptotic global behavior of the Grenander-type estimator for a monotone baseline hazard function. This model is not included in the general setting of Durot (2007). However, we show that a similar central limit theorem holds for L_p-error of the Grenander-type estimator. As an illustration of application of our main result, we propose a test procedure for a Weibull baseline distribution, based on the L_p-distance between the Grenander estimator and a parametric estimator of the baseline hazard. Simulation studies are performed to investigate the performance of this test.     

Internet Adoption and Advertising Expenditures on Traditional Media: An Empirical Analysis Using a Panel of Countries

The Internet can affect advertising expenditures through various channels. This paper quantifies the relationships between Internet adoption and changes in advertising expenditures on traditional offline media types. I use a panel of 11 years of data at the country level that contains information on advertising expenditures by medium and Internet penetration for more than 80 countries. I find that increases in Internet penetration are negatively correlated with changes in advertising expenditures on newspapers, magazines, and television, but I do not find conclusive results for the correlation between Internet adoption and changes in advertising expenditures on radio.     

<Emphasis Type="SmallCaps">Notes and Comments:</Emphasis> On the uniqueness of convex-ranged probabilities

Abstract In Marinacci (2000), the following theorem was proved. Theorem 1. (Marinacci (2000) Let P and Q be two finitely additive probabilities on a λ -system Σ . Suppose that P is convex-ranged and that Q is countably additive. If there exists an A ⁺ ∈ Σ with 0<P(A ⁺ )<1 such that whenever B∈ Σ , then P=Q. Mathematics Subject Classification (2000): 28A10, 91B06 Journal of Economic Literature Classification: C60, D81     

World Economic Prospects

Overview: Dollar surge brings mixed consequences

• The strengthening dollar is now becoming a significant factor for global growth and our forecasts. The tradeweighted dollar is up 2.5% over the last month and over 12% on a year ago.
• Driving the latest rise are growing expectations of US rate hikes while monetary policy in many other major economies is headed in the opposite direction.
• The beginning of ECB QE has prompted a further slide in bond yields and the euro – which at 1.06/US$ is on course to fulfill our forecast of near‐parity by year‐end. Weak data in Japan also raises the chance of a further expansion of QE there later this year.
• We remain relatively positive about the advanced economies: we forecast G7 GDP growth at 2.2% for 2015 and 2.3% next. This month we have revised up German growth for 2015 to 2.4% – a four‐year high.
• Robust US growth and a strong dollar are good news for the advanced economies. US import volume growth firmed to over 5% on the year in January, while the dollar surge potentially boosts the share of other advanced countries in this growing market.
• But for the emerging economies the picture is mixed. A stronger US may boost exports, but rising US rates are pulling capital away: there has been a slump in portfolio inflows into emergers in recent months. Emerging growth may also suffer from higher costs of dollar funding and a rising burden of dollar debt as currencies soften – the more so if US rates rise faster than markets expect.
• Moreover, emergers are also under pressure from a slowing China. Chinese import growth has been weak of late and commodity prices remain under downward pressure. A notable casualty has been Brazil, which we have downgraded again this month – GDP is expected to slump 1.1% this year.
• Emerging GDP growth overall is expected to slip to 3.7% this year, the lowest since 2009. And excluding China, emerging growth will be only 2.2% – the same as the G7 and the worst performance relative to the advanced economies since 1999.

     

Optimal design of nonlinear experiments with parameter constraints

The nonlinear regression model with N observations y _i=η(x _i,θ) +ε_i, and with the parameter θ subject to q nonlinear constraints C _j (θ)=0; j=1, …,q, is considered. As an example, the spline regression with unknown nodes is taken. Expressions for the variances (variance matrices) of the LSE are discussed. Because of the complexity of these expressions, and the singularity of the variance matrix of the LSE for θ, the optimality criteria and their properties, in particular the convexity and the equivalence theorem are considered from different aspects. Also the possibility of restriction to designs with limited values of measures of nonlinearity is mentioned. Research supported by the VEGA-grant of the Slovak grant agency No. 1/7295/20.     

Comparison of test vs. control treatments using distance optimality criterion

We consider the problem of comparison of one test treatment (τ₀) with a set of v control treatments (τ₁, τ₂, …, τ_v) using distance optimality [DS-optimality] criterion introduced by Sinha (1970) in some treatment-connected design settings. It turns out that the nature of DS-optimal designs is quite similar to that for the usual A−, D− and E− optimality criteria. However, the optimality problem is quite complicated in most situations. First we deal with the CRD model and derive DS-optimal allocations for a given set of treatments. The results are almost identical to the A-optimal allocations for such problems. Then we consider a block design set-up and examine the nature of DS-optimal designs. In the process, we introduce the method of weighted coverage probability and maximize the resulting expression to obtain an optimal design. Received: December 1999