### 26/03/2014, 11:30 — 12:30 — Room P3.10, Mathematics Building

João F. D. Rodrigues, *IN+, Center for Innovation, Technology and Policy Research, Instituto Superior Técnico*

### The Prior Uncertainty and Correlation of Statistical Economic Data

Empirical estimates of source statistical economic data such as
transaction flows, greenhouse gas emissions or employment are
always subject to measurement errors but empirical estimates of
source data errors are often missing. This paper uses concepts from
Bayesian inference and the Maximum Entropy Principle to estimate
the prior probability distribution, uncertainty and correlations of
source data when such information is not explicitly provided. In
the absence of additional information, an isolated datum is
described by a truncated Gaussian distribution, and if an
uncertainty estimate is missing, its prior equals the best guess.
When the sum of a set of disaggregate data is constrained to match
an aggregate datum, it is possible to determine the prior
correlations among disaggregate data. If aggregate uncertainty is
missing, all prior correlations are positive. If aggregate
uncertainty is available, prior correlations can be either all
positive, all negative, or a mix of both. An empirical example is
presented, which reports uncertainty and correlation priors for the
County Business Patterns database.

### 19/03/2014, 11:30 — 12:30 — Room P4.35, Mathematics Building

Ana Ferreira, *Instituto Superior de Agronomia and CEAUL*

### High quantile estimation and spatial aggregation applied to
precipitation extremes

We shall address the problem of high quantile estimation in
univariate and spatial Extreme Value Theory. Univariate methods are
well known under the Maximum Domain of Attraction Condition and the
Pareto tail approximation is the basis for many estimators. It
turns out that the Pareto tail approximation is also valid under
spatial aggregation but a spatial effect comes out. We shall
address the problem both theoretically and in practice, by
presenting a case study on 100-year return value estimation for
precipitation data collected at rain-gauge stations.

### 05/03/2014, 11:30 — 12:30 — Room P3.10, Mathematics Building

António Pacheco, *CEMAT and Instituto Superior Técnico*

### Level Crossing Ordering of Stochastic Processes

Stochastic Ordering is an important area of Applied Probability
tailored for qualitative comparisons of random variables, random
vectors, and stochastic processes. In particular, it may be used to
investigate the impact of parameter changes in important
performance measures of stochastic systems, avoiding exact
computation of those performance measures. In this respect, the
great diversity of performance measures used in applied sciences to
characterize stochastic systems has inspired the proposal of many
types of stochastic orderings.

In this talk we address the level crossing ordering, proposed by
A. Irle and J. Gani in 2001, that compares stochastic processes in
terms of the times they take to reach high levels (states). After
introducing some motivation for the use of the level crossing
ordering, we present tailored sufficient conditions for the level
crossing ordering of (univariate and multivariate) Markov and
semi-Markov processes. These conditions are applied to the
comparison of birth-and- death processes with catastrophes,
queueing networks, and particle systems.

Our analysis highlights the benefits of properly using the
sample path approach, which compares directly trajectories of the
compared processes defined on a common probability space. This
approach provides, as a by-product, the basis for the construction
of algorithms for the simulation of stochastic processes ordered in
the level crossing ordering sense. In the case of continuous Markov
chains, we resort additionally to the powerful uniformization
technique, which uniformizes the rates at which transitions take
place in the processes being compared.

*Joint work with Fátima Ferreira (CM-UTAD and Universidade
de Trás-os-Montes e Alto Douro).*

### 19/02/2014, 11:30 — 12:30 — Room P3.10, Mathematics Building

Francisco Macedo, *IST/EPFL, Portugal/Switzerland*

### A low-rank tensor method for structured large-scale Markov Chains

A number of practical applications lead to Markov Chains with
extremely large state spaces. Such an instance arises from the
class of Queuing Networks, which lead to a number of applications
of interest including, for instance, the analysis of the well-known
tandem networks. The state space of a Markov process describing
these interactions typically grows exponentially with the number of
queues. More generally, Stochastic Automata Networks (SANs) are
networks of interacting stochastic automata. The dimension of the
resulting state space grows exponentially with the number of
involved automata. Several techniques have been established to
arrive at a formulation such that the transition matrix has
Kronecker product structure. This allows, for example, for
efficient matrix-vector multiplications. However, the number of
possible automata is still severely limited by the need of
representing a single vector (e.g., the stationary vector)
explicitly. We propose the use of low-rank tensor techniques to
avoid this barrier. More specifically, an algorithm will be
presented that allows to approximate the solution of certain SAN s
very efficiently in a low-rank tensor format.

### 20/11/2013, 11:00 — 16:00 — Room P3.10, Mathematics Building

Isabel Rodrigues, *CEMAT and DM-IST*

### Regressão linear múltipla com alguns erros correlacionados: métodos clássicos e robustos

Este trabalho teve como motivação principal a análise de um conjunto de dados de medicina. Esses dados resultaram de um estudo observacional destinado a identificar os factores que influenciam o resultado de um determinado método de tratamento cirúrgico da escoliose (curvatura lateral anormal da coluna vertebral). O número de observações do conujnto de dados é 392 mas algumas dessas observações são relativas ao mesmo doente (curvas duplas). O modelo de regressão linear múltipla surge à partida como uma possibilidade para analisar estes dados mas, dado que as curvas duplas não devem ser ignoradas, a suposição de erros não correlacionados é claramente violada, o que foi de facto confirmado no diagnóstico dos resíduos do modelo usual (teste de Durbin-Watson e teste “runs”). Um modelo mais apropriado obtém-se mantendo a estrutura linear mas admitindo a existência de correlações não nulas entre os erros relativos ao mesmo doente. Para este modelo consideram-se dois métodos de estimação: mínimos quadrados generalizados com parâmetros de correlação estimados iterativamente (o que pode ser visto como uma adaptação do método de Cochrane-Orcutt para erros auto-correlacionados) e estimação de todos os parâmetros por máxima verosimilhança. Finalmente apresentam-se versões robustas destes dois métodos assim como do teste de Durbin-Watson e que foram as que permitiram analisar de forma mais satisfatória os dados referidos.

### 06/11/2013, 11:00 — 16:00 — Room P3.10, Mathematics Building

Isabel Natário, *Faculdade de Ciências e Tecnologia/UNL and CEAUL*

### Modelling of road accidents, in Lisbon, with casualities using spatial point processes

Localização de acidentes rodoviários, local da ignição de incêndios florestais, a distribuição de corais no mar das Caraíbas são exemplos de acontecimentos que ocorrem tipicamente em localizações aleatórias. Se o nosso interesse é sobre esta caraterística dos dados, os processos pontuais espaciais são os modelos estatísticos mais adequados. Num problema desta natureza, principia-se o estudo tentando estabelecer se as localizações observadas satisfazem a hipótese de aleatoriedade completa, correspondendo a uma situação de uniformidade da distribuição das localizações (regularidade), ou se pelo contrário o padrão espacial observado pode ser considerado como agregado. No primeiro caso recorre-se a uma modelação baseada em processos de Poisson de taxa proporcional à area em estudo e, no segundo caso, tem de se modelar o padrão espacial agregado através de modelos que levem em conta a dependência espacial evidenciada por esses padrões. A abordagem que se faz no contexto dos processos pontuais espaciais é tradicionalmente não paramétrica, com o recurso a estatísticas resumo, tendo mais recentemente evoluído para a utilização de modelos paramétricos flexíveis, com eventual inclusão de covariáveis, sendo a inferência feita através do método da máxima verosimilhança assistido por métodos numéricos. Neste seminário faz-se uma breve descrição dos modelos mais usuais neste tipo de modelação e para este tipo de dados, numa perspectiva essencialmente aplicada a um conjunto de dados de acidentes rodoviários com vítimas na cidade de Lisboa (projecto SACRA, PTDC/TRA/66161/2006), e recorrendo ao pacote estatístico spatstat do R-project.

### 23/10/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Raquel Gaspar, *CEMAPRE and ISEG*

### Convexity Adjustments for ATS models

### 02/10/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Manuel Cabral Morais, *CEMAT & Department of Mathematics, IST*

### Strategies to reduce the probability of a misleading signal

Standard practice in statistical process control is to run two
individual charts, one for the process mean and another one for the
process variance. The resulting scheme is known as a simultaneous
scheme and it provides a way to satisfy Shewhart's dictum that
proper process control implies monitoring both location and
dispersion.

When we use a simultaneous scheme, the quality characteristic is
deemed to be out-of-control whenever a signal is triggered by
either individual chart. As a consequence, the misidentification of
the parameter that has changed can occur, meaning that a shift in
the process mean can be misinterpreted as a shift in the process
variance and vice-versa. These two events are known as misleading
signals (MS) and can occur quite frequently.

We discuss (necessary and) sufficient conditions to achieve
values of PMS smaller than or equal to \(0.5\), explore, for
instance, alternative simultaneous Shewhart-type schemes and check
if they lead to PMS which are smaller than the ones of the popular
\((\bar{X}, S^2)\) simultaneous scheme.

### 25/09/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Daniel Schwarz, *IST and CMU*

### Price Modelling in Carbon Emission and Electricity Markets

We present a model to explain the joint dynamics of the prices of
electricity and carbon emission allowance certificates as a
function of exogenously given fuel prices and power demand. The
model for the electricity price consists of an explicit
construction of the electricity supply curve; the model for the
allowance price takes the form of a coupled forward-backward
stochastic differential equation (FBSDE) with random coefficients.
Reflecting typical properties of emissions trading schemes the
terminal condition of this FBSDE exhibits a gradient singularity.
Appealing to compactness arguments we prove the existence of a
unique solution to this equation. We illustrate the relevance of
the model at the example of pricing clean spread options, contracts
that are frequently used to value power plants in the spirit of
real option theory.

### 04/07/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Sven Knoth, *Institute of Mathematics and Statistics, Helmut Schmidt University, Hamburg, Germany*

### Incorporating parameter uncertainty into the setup of EWMA control charts monitoring normal variance

Most of the literature concerned with the design of control charts relies on perfect knowledge of the distribution for at least the good (so-called in-control) process. Some papers treated the handling of EWMA charts monitoring normal mean in case of unknown parameters - refer to Jones, Champ and Rigdon (2001) for a good introduction. In Jensen, Jones-Farmer, Champ, and Woodall (2006): “Effects of Parameter Estimation on Control Chart Properties: A Literature Review” a nice overview was given. Additionally, it was mentioned that it would be interesting and useful to evaluate and take into account these effects also for variance control charts. Here, we consider EWMA charts for monitoring the normal variance. Given a sequence of batches of size $n$, $\{X_{i j}\}$, $i=1,2,\ldots$ and $j=1,2,\ldots,n$ utilize the following EWMA control chart: \begin{align*} Z_0 & = z_0 = \sigma_0^2 = 1 \,, \\ Z_i & = (1-\lambda) Z_{i-1} + \lambda S_i^2 \,,\; i = 1,2,\ldots \,,\\ & \qquad\qquad S_i^2 = \frac{1}{n-1} \sum_{i=1}^n (X_{ij} - \bar X_i)^2 \,,\; \bar X_i = \frac{1}{n} \sum_{i=1}^n X_{ij} \,, \\ L & = \inf \left\{ i \in I\!\!N: Z_i > c_u \sigma_0^2 \right\} \,. \end{align*} The parameters $\lambda \in (0,1]$ and $c_u \gt 0$ are chosen to enable a certain useful detection performance (not too much false alarms and quick detection of changes). The most popular performance measure is the so-called Average Run Length (ARL), that is $E_{\sigma}(L)$ for the true standard deviation $\sigma$. If $\sigma_0$ has to be estimated by sampling data during a pre-run phase, then this uncertain parameter effects, of course, the behavior of the applied control chart. Typically the ARL is increased. Most of the papers about characterizing the uncertainty impact deal with the changed ARL patterns and possible adjustments. Here, a different way of designing the chart is treated: Setup the chart through specifying a certain false alarm probability such as $P_{\sigma_0}(L\le 1000) \le \alpha$. This results in a specific $c_u$. Here we describe a feasible way to determine this value $c_u$ also in case of unknown parameters for a pre-run series of given size (and structure). A two-sided version of the introduced EWMA scheme is analyzed as well.

### 27/06/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Emanuele Dolera, *Università di Modena e Reggio Emilia, Italy*

###
Reaching the best possible rate of convergence to equilibrium of
Boltzmann-equation solutions

This talk concerns a definitive answer to the problem of
quantifying the relaxation to equilibrium of the solutions to the
spatially homogeneous Boltzmann equation for Maxwellian molecules.
Under really mild conditions on the initial datum - closed to
necessity - and a weak, physically consistent, angular cutoff
hypothesis, the main result states that the total variation
distance (i.e. the ${L}^{1}$-norm in the absolutely continuous case)
between the solution and the limiting Maxwellian distribution
admits an upper bound of the form $C\mathrm{exp}(-{\Lambda}_{b}^{*}t)$,
${\Lambda}_{b}^{*}$ being the spectral gap of the linearized collision
operator and $C$ a constant depending only on the initial datum.
Hilbert hinted at the validity of this quantification in 1912,
which was explicitly formulated as a conjecture by McKean in 1966.
The main line of the new proof is based on an analogy between the
problem of convergence to equilibrium and the central limit theorem
of probability theory, as suggested by McKean.

### 12/06/2013, 11:00 — 12:00 — Room P4.35, Mathematics Building

Ana M. Bianco, *Universidad de Buenos Aires and CONICET*

###
Robust Procedures for Nonlinear Models for Full and Incomplete Data

Linear models are one of the most popular models in Statistics.
However, in many situations the nature of the phenomenon is
intrinsically nonlinear and so, linear approximations are not valid
and the data must be fitted using a nonlinear model. Besides, in
some occasions the responses are incomplete and some of them are
missing at random.

It is well known that, in this setting, the classical estimator
of the regression parameter based on least squares is very
sensitive to outliers. A family of general M-estimators is proposed
to estimate the regression parameter in a nonlinear model. We give
a unified approach to treat full data or data with missing
responses. Under mild conditions, the proposed estimators are
Fisher-consistent, consistent and asymptotically normal. To study
local robustness, their influence function is also derived.

A family of robust tests based on a Wald-type statistic is
introduced in order to check hypotheses that involve the regression
parameter. Monte Carlo simulations illustrate the finite sample
behaviour of the proposed procedures in different settings in
contaminated and uncontaminated samples.

### 31/05/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building

Isabel Silva and Maria Eduarda Silva, *Faculdade de Engenharia, Universidade do Porto, and Faculdade de Economia, Universidade do Porto*

###
An INteger AutoRegressive afternoon - Statistical analysis of
discrete valued time series

Part I: Univariate and multivariate models based on thinning

Part II: Modelling and forecasting time series of counts

Time series of counts arise when the interest lies on the number
of certain events occurring during a specified time interval. Many
of these data sets are characterized by low counts, asymmetric
distributions, excess zeros, over dispersion, etc, ruling out
normal approximations. Thus, during the last decades there has been
considerable interest in models for integer-valued time series and
a large volume of work is now available in specialized monographs.
Among the most successful models for integer-valued time series are
the INteger- valued AutoRegressive Moving Average, INARMA, models
based on the thinning operation. These models are attractive since
they are linear-like models for discrete time series which exhibit
recognizable correlation structures. Furthermore, in many
situations the collected time series are multivariate in the sense
that there are counts of several events observed over time and the
counts at each time point are correlated. The first talk introduces
univariate and multivariate models for time series of counts based
on the thinning operator and discusses their statistical and
probabilistic properties. The second talk addresses estimation and
diagnostic issues and illustrates the inference procedures with
simulated and observed data.

### 17/05/2013, 14:30 — 15:30 — Room P3.10, Mathematics Building

David Taylor, *Research and Actuarial Science Division, School of Management Studies, University of Cape Town, South Africa*

###
Mathematical Finance in South Africa

I have been involved in Math Finance university education in South
Africa since 1996. During this time I have produced numerous
graduates & grown an extensive network of industry &
academic partners. I'll talk about these experiences & take
questions.

### 16/05/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

David Taylor, *Research and Actuarial Science Division, School of Management Studies, University of Cape Town, South Africa*

###
Aggregational Gaussianity Using Sobol Sequencing In the South
African Equity Markets: Implications for the Pricing of Risk

Stylized facts of asset returns in the South African market have
received extensive attention, with multiple studies published on
non-normality of returns, heavy-tailed distributions, gain-loss
asymmetry and, particularly, volatility clustering. The one such
fact that has received only cursory attention world-wide is that of
Aggregational Gaussianity - the widely-accepted/stylized fact that
empirical asset returns tend to normality when the period over
which the return is computed increases. The aggregational aspect
arises from the \(n\)-day log-return being the simple sum of \(n\)
one-day log-returns. This fact is usually established using
Q-Q-plots over longer and longer intervals, and can be
qualitatively confirmed. However, this methodology inevitably uses
overlapping data series, especially for longer period returns. When
an alternative resampling methodology for dealing with common
time-overlapping returns data is used an alternate picture emerges.
Here we describe evidence from the South African market for a
discernible absence of Aggregational Gaussianity and briefly
discuss the implications of these findings for the quantification
of risk and to the pricing and hedging of derivative securities.

### 06/05/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

Cristina Barros, *Departamento de Engenharia do Ambiente, Escola Superior de Tecnologia e Gestão - Instituto Politécnico de Leiria*

###
Real Time Statistical Process Control of the Quantity of Product in
Prepackages

In this presentation we will describe how we developed a
methodology for the statistical quantity control processes of
prepackagers and present a number of different case studies based
on the type of product, packaging, production, filling line and
system of data acquisition. With the aim of establishing a global
strategy to control the quantity of product in prepackages, an
integrated planning model based on statistical tools was developed.
This model is able to manage the production functions concerning
the legal metrological requirements. These requirements are similar
all around the world because they are based on the recommendation
R-87: 2004 (E) from the International Organization of Legal
Metrology (OIML). Based on the principles of Statistical Process
Control a methodology to analyze in real time the quantity of
product in prepackages was proposed; routine inspections, condition
monitoring of the main components and friendly comprehension of the
outputs were taken into account. Subsequently, software of data
acquisition, registration to guarantee traceability and treatment
for decisions which can be configured for any kind of filling
process was introduced. The impacts of this system, named ACCEPT-
Computer Based Help for the Statistic Control of the Filling
Processes, at the industry is demonstrated by the large number of
companies that are using this system to control their processes. In
Portugal, more than 50 companies and thousands of operators with
very low qualifications are working every day with SPC tools and
capability analysis in order to minimize variability and waste (for
example: over filling), to ensure compliance and to guarantee the
consumers rights.

### 23/04/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Stéphane Villeneuve, *Toulouse School of Economics, University of Toulouse*

###
Corporate cash policy with liquidity and profitability risks

We develop a dynamic model of a firm facing both liquidity and
profitability concerns. This leads us to study and to solve
explicitly a bi-dimensional control problem where the two state
variables are the controlled cash reserves process and the belief
process about the firm’s profitability. Our model encompasses
previous studies and provides new predictions for corporate cash
policy. The model predicts a positive relationship between cash
holdings and beliefs about the firm’s profitability, a
non-monotonic relationship between cash holdings and the volatility
of the cash flows as well as a non-monotonic relationship between
cash holdings and the risk of profitability. This yields novel
insights on the firm’s default policy and on the relationship
between volatility of stock prices and the level of stock.

### 18/04/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Ana Subtil, *CEMAT, Instituto Superior Técnico, Universidade Técnica de Lisboa; Faculdade de Ciências, Universidade de Lisboa*

###
Using Latent Class Models to Evaluate the Performance of Diagnostic
Tests in the Absence of a Gold Standard

Diagnostic tests are helpful tools for decision-making in a
biomedical context. In order to determine the clinical relevance
and practical utility of each test, it is critical to assess its
ability to correctly distinguish diseased from non-diseased
individuals. Statistical analysis has an essential role in the
evaluation of diagnostic tests, since it is used to estimate
performance measures of the tests, such as sensitivity and
specificity. Ideally, these measures are determined by comparison
with a gold standard, i.e., a reference test with perfect
sensitivity and specificity.

When no gold standard is available, admitting the supposedly
best available test as a reference may cause misclassifications
leading to biased estimates. Alternatively, Latent Class Models
(LCM) may be used to estimate diagnostic tests performance measures
as well as the disease prevalence, in the absence of a gold
standard. The most common LCM estimation approaches are the maximum
likelihood estimation using the Expectation-Maximization algorithm
and the Bayesian inference using Markov Chain Monte Carlo methods,
via Gibbs sampling.

This talk illustrates the use of Bayesian Latent Class Models
(BLCM) in the context of malaria and canine dirofilariosis. In each
case, multiple diagnostic tests were applied to distinct
subpopulations. To analyze the subpopulations simultaneously, a
product multinomial distribution was considered, since the
subpopulations were independent. By introducing constraints, it was
possible to explore differences and similarities between
subpopulations in terms of prevalence, sensitivities and
specificities.

We also discuss statistical issues such as the assumption of
conditional independence, model identifiability, sampling
strategies and prior distribution elicitation.

### 05/04/2013, 14:30 — 15:30 — Room P3.10, Mathematics Building

Paula Brito, *Faculdade de Economia / LIAAD - INESC TEC, Universidade do Porto*

###
Taking Variability in Data into Account: Symbolic Data Analysis

Symbolic Data, introduced by E. Diday in the late eighties of
the last century, is concerned with analysing data presenting
intrinsic variability, which is to be explicitly taken into
account. In classical Statistics and Multivariate Data Analysis,
the elements under analysis are generally individual entities for
which a single value is recorded for each variable - e.g.,
individuals, described by their age, salary, education level,
marital status, etc.; cars each described by its weight, length,
power, engine displacement, etc.; students for each of which the
marks at different subjects were recorded. But when the elements of
interest are classes or groups of some kind - the citizens living
in given towns; teams, consisting of individual players; car
models, rather than specific vehicles; classes and not individual
students - then there is variability inherent to the data. To
reduce this variability by taking central tendency measures - mean
values, medians or modes - obviously leads to a too important loss
of information.

Symbolic Data Analysis provides a framework allowing
representing data with variability, using new variable types. Also,
methods have been developed which suitably take data variability
into account. Symbolic data may be represented using the usual
matrix-form data arrays, where each entity is represented in a row
and each column corresponds to a different variable - but now the
elements of each cell are generally not single real values or
categories, as in the classical case, but rather finite sets of
values, intervals or, more generally, distributions.

In this talk we shall introduce and motivate the field of
Symbolic Data Analysis, present into some detail the new variable
types that have been introduced to represent variability,
illustrating with some examples. We shall furthermore discuss some
issues that arise when analysing data that does not follow the
usual classical model, and present data representation models for
some variable types.

### 14/03/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Iryna Okhrin & Ostap Okhrin, *Faculty of Business Administration and Economics at the Europa-Universität Viadrina Frankfurt (Oder) & Wirtschaftswissenschafttliche Fakultät at the Humboldt-Universität zu Berlin*

###
Forecasting The Temperature Data & Localising Temperature Risk

Forecasting The Temperature Data:

This paper aims at describing the intraday temperature
variations which is a challenging task in modern econometrics and
environmetrics. Having a high-frequency data, we separate the
dynamics within a day and over days. Three main models have been
considered in our study. As the benchmark we employ a simple
truncated Fourier series with autocorrelated residuals. The second
model uses the functional data analysis, and is called the shape
invariant model (SIM). The third one is the dynamic semiparametric
factor model (DSFM). In this work we discuss rises and pitfalls of
all the methods and compare their in- and out-of-sample
performances.

&

Localising Temperature Risk:

On the temperature derivative market, modelling temperature
volatility is an important issue for pricing and hedging. In order
to apply the pricing tools of financial mathematics, one needs to
isolate a Gaussian risk factor. A conventional model for
temperature dynamics is a stochastic model with seasonality and
intertemporal autocorrelation. Empirical work based on seasonality
and autocorrelation correction reveals that the obtained residuals
are heteroscedastic with a periodic pattern. The object of this
research is to estimate this heteroscedastic function so that,
after scale normalisation, a pure standardised Gaussian variable
appears. Earlier works investigated temperature risk in different
locations and showed that neither parametric component functions
nor a local linear smoother with constant smoothing parameter are
flexible enough to generally describe the variance process well.
Therefore, we consider a local adaptive modelling approach to find,
at each time point, an optimal smoothing parameter to locally
estimate the seasonality and volatility. Our approach provides a
more flexible and accurate fitting procedure for localised
temperature risk by achieving nearly normal risk factors. We also
employ our model to forecast the temperature in different cities
and compare it to a model developed in Campbell and Deibol
(2005).