On the Existence and Uniqueness of the Maximum Likelihood Estimators of Normal and Lognormal Population Parameters with Grouped Data

Lognormal distribution has abundant applications in various fields. In literature, most inferences on the two parameters of the lognormal distribution are based on Type-I censored sample data. However, exact measurements are not always attainable especially when the observation is below or above the detection limits, and only the numbers of measurements falling into predetermined intervals can be recorded instead. This is the so-called grouped data. In this paper, we will show the existence and uniqueness of the maximum likelihood estimators of the two parameters of the underlying lognormal distribution with Type-I censored data and grouped data. The proof was first established under the case of normal distribution and extended to the lognormal distribution through invariance property. The results are applied to estimate the median and mean of the lognormal population.

Association between mean residual life (MRL) and failure rate functions for continuous and discrete lifetime distributions

The purpose of this study was to correct some mistakes in the literature and derive a necessary and sufficient condition for the MRL to follow the roller-coaster pattern of the corresponding failure rate function. It was also desired to find the conditions under which the discrete failure rate function has an upside-down bathtub shape if corresponding MRL function has a bathtub shape. The study showed that if discrete MRL has a bathtub shape, then under some conditions the corresponding failure rate function has an upside-down bathtub shape. Also the study corrected some mistakes in proofs of Tang, Lu and Chew (1999) and established a necessary and sufficient condition for the MRL to follow the roller-coaster pattern of the corresponding failure rate function. Similarly, some mistakes in Gupta and Gupta (2000) are corrected, with the ensuing results being expanded and proved thoroughly to establish the relationship between the crossing points of the failure rate and associated MRL functions. The new results derived in this study will be useful to model various lifetime data that occur in environmental studies, medical research, electronics engineering, and in many other areas of science and technology.

Field data, numerical simulations and probability analyses to assess lava flow hazards at Mount Etna

Improving lava flow hazard assessment is one of the most important and challenging fields of volcanology, and has an immediate and practical impact on society. Here, we present a methodology for the quantitative assessment of lava flow hazards based on a combination of field data, numerical simulations and probability analyses. With the extensive data available on historic eruptions of Mt. Etna, going back over 2000 years, it has been possible to construct two hazard maps, one for flank and the other for summit eruptions, allowing a quantitative analysis of the most likely future courses of lava flows. The effective use of hazard maps of Etna may help in minimizing the damage from volcanic eruptions through correct land use in densely urbanized area with a population of almost one million people. Although this study was conducted on Mt. Etna, the approach used is designed to be applicable to other volcanic areas.

A Framework for Citizen-Centred Social Statistics and Analysis (Working Paper 48)

Economic policy-making has long been more integrated than social policy-making in part because the statistics and much of the analysis that supports economic policy are based on a common conceptual framework – the system of national accounts. People interested in economic analysis and economic policy share a common language of communication, one that includes both concepts and numbers. This paper examines early attempts to develop a system of social statistics that would mirror the system of national accounts, particular the work on the development of social accounts that took place mainly in the 60s and 70s. It explores the reasons why these early initiatives failed but argues that the preconditions now exist to develop a new conceptual framework to support integrated social statistics – and hence a more coherent, effective social policy. Optimism is warranted for two reasons. First, we can make use of the radical transformation that has taken place in information technology both in processing data and in providing wide access to the knowledge that can flow from the data. Second, the conditions exist to begin to shift away from the straight jacket of government-centric social statistics, with its implicit assumption that governments must be the primary actors in finding solutions to social problems. By supporting the decision-making of all the players (particularly individual citizens) who affect social trends and outcomes, we can start to move beyond the sterile, ideological discussions that have dominated much social discourse in the past and begin to build social systems and structures that evolve, almost automatically, based on empirical evidence of ‘what works best for whom’. The paper describes a Canadian approach to developing a framework, or common language, to support the evolution of an integrated, citizen-centric system of social statistics and social analysis. This language supports the traditional social policy that we have today; nothing is lost. However, it also supports a quite different social policy world, one where individual citizens and families (not governments) are seen as the central players – a more empirically-driven world that we have referred to as the ‘enabling society’.

What's in the Numbers? Child Welfare Statistics and the Children (NI) Order

This article draws attention to the importance of routinely collected administrative data as an important source for understanding the characteristics of the Northern Ireland child welfare system as it has developed since the Children (Northern Ireland) Order 1995 became its legislative base. The article argues that the availability of such data is a strength of the Northern Ireland child welfare system and urges local politicians, lobbyists, researchers, policy-makers, operational managers, practitioners and service user groups to make more use of them. The main sources of administrative data are identified. Illustration of how these can be used to understand and to ask questions about the system is provided by considering some of the trends since the Children Order was enacted. The “protection” principle of the Children Order provides the focus for the illustration. The statistical trends considered relate to child protection referrals, investigations and registrations and to children and young people looked after under a range of court orders available to ensure their protection and well-being.

The census of Iowa, as returned in the year 1873 : showing, in detail, the population, agricultural statistics, and other items of interest / published under direction of the Executive Council, 1874

This is the Census of Iowa as returned in the year 1873, showing in detail, the population, agricultural statistics, and other items of interest

The collision branching process

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

Multivariate prediction

The problem of prediction is considered in a multidimensional setting. Extending an idea presented by Barndorff-Nielsen and Cox, a predictive density for a multivariate random variable of interest is proposed. This density has the form of an estimative density plus a correction term. It gives simultaneous prediction regions with coverage error of smaller asymptotic order than the estimative density. A simulation study is also presented showing the magnitude of the improvement with respect to the estimative method.

Power variation of some integral fractional processes

We consider the asymptotic behaviour of the realized power variation of processes of the form ¿t0usdBHs, where BH is a fractional Brownian motion with Hurst parameter H E(0,1), and u is a process with finite q-variation, q<1/(1¿H). We establish the stable convergence of the corresponding fluctuations. These results provide new statistical tools to study and detect the long-memory effect and the Hurst parameter.

Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion

In this note we prove an existence and uniqueness result for the solution of multidimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter H > 1/2. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann¿Stieltjes integral.

Criteria for hitting probabilities with applications to systems of stochastic wave equations

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension k >- 1 driven by a d-dimensional spatially homogeneous additive Gaussian noise that is white in time and colored in space.

Stochastic differential equations with random coefficients

In this paper we establish the existence and uniqueness of a solution for different types of stochastic differential equation with random initial conditions and random coefficients. The stochastic integral is interpreted as a generalized Stratonovich integral, and the techniques used to derive these results are mainly based on the path properties of the Brownian motion, and the definition of the Stratonovich integral.

Large deviations for stochastic Volterra equations

This paper is devoted to prove a large-deviation principle for solutions to multidimensional stochastic Volterra equations.