# Category Archives: Stimatore consistente in media quadratica

### Stimatore consistente in media quadratica

Giorgio E. Maria Giovanna Ranalli. Download PDF. A short summary of this paper. La stima per regressione generalizzata assume che la relazione tra le variabili di interesse e le variabili ausiliarie sia ben descritta da un modello di regressione lineare. Sebbene sia possibile utilizzare diverse misure di distanza, gli stimatori calibrati corrispondenti sono asintoticamente equivalenti allo stimatore per regressione generalizzata basato su di un modello lineare che impiega le stesse variabili ausiliarie utilizzate nella calibrazione.

Con riferimento all'approccio dianzi descritto, alcuni avanzamenti metodologici consistono nel determinare i coefficienti di regressione attraverso il metodo dei minimi quadrati di tipo ridge.

Montanari e Ranalli a fanno ricorso a modelli a effetti misti nello studio della relazione fra stimatore per regressione generalizzata e stimatore ottimo Montanari, Breidt e Opsomer considerano per primi un modello di tipo nonparametrico per la costruzione di uno stimatore della media basato su polinomi locali in un approccio assistito dal modello.

Wu e Sitter considerano sia modelli di regressione nonlineare che lineari generalizzati nel contesto della calibrazione. Montanari e Ranalli c generalizzano ulteriormente questo approccio introducendo la calibrazione rispetto ad un modello nonparametrico e considerando metodi quali reti neurali e polinomi locali per ottenere i valori predetti da impiegare nei vincoli di calibrazione.

Un aspetto critico sia degli stimatori basati su metodi nonparametrici diversi dai polinomi locali e dalle splines penalizzate che di quelli calibrati su un modello nonlineare -sia esso parametrico, lineare generalizzato, nonparametrico -consiste nel fatto che i pesi di riporto all'universo che si ottengono dipendono dalla variabile di interesse modellizzata e, quindi, non sono gli stessi per tutte le variabili di interesse. Il lavoro si articola nel modo seguente.

Nel paragrafo 2 viene introdotta la notazione impiegata e rivista la calibrazione in senso classico e quella rispetto al modello; vengono considerati sia modelli parametrici che nonparametrici. Nel paragrafo 4 viene affrontato il problema della calibrazione rispetto a molte variabili. Il paragrafo 5 riporta i risultati di uno studio di simulazione condotto per analizzare il comportamento degli stimatori proposti nei campioni di ampiezza finita.

In questo approccio possono essere impiegati modelli lineari, nonlineari e lineari generalizzati. Si noti che impiegando modelli diversi per f p si ottengono stimatori diversi. In questo caso il numero dei vincoli di calibrazione si limita ad uno per ciascun set di pesi.Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising.

If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy. See our Privacy Policy and User Agreement for details. Published on Jan 26, Una efficace presentazione sugli stimatori delle mie bravissime studentesse di 5A Turismo. SlideShare Explore Search You.

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### Bias of an estimator

Home Explore. Successfully reported this slideshow. We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime. Upcoming SlideShare. Like this presentation? Why not share! Embed Size px. Start on. Show related SlideShares at end. WordPress Shortcode. Published in: Science.

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Full Name Comment goes here. Are you sure you want to Yes No. Anna Nessuno. No Downloads. Views Total views. Actions Shares. No notes for slide. Si tratta dunque di una funzione di tutte le possibili ennuple campionarie che si possono estrarre da una popolazione.Esercizio 1. Tipo materna elementare media secondaria tot ni Statistica n.

I distributori hanno uguale capienza, vengono riforniti uno alla volta e richiedono rifornimenti con uguale frequenza. X F X 0 0. Il voto medio riportato al primo appello risulta pari a Al secondo appello il voto medio risulta pari a Al terzo appello si osserva uno s. Ubicazione n. Classi di addetti Aziende N. X n X 15 13 15 12 15 10 Classi di reddito milioni Frequenze relative fino a 10 0. Calcolare media, s. Commentare i risultati. X ni 15 13 12 11 10 10 8 6 6 5 4. X n X 35 29 25 28 11 8.

Classe di superficie Numero di aziende 33 43 12 10 2. Esercizio 2. Si determinino i due valori mancanti X5 e Y4. Aziende di credito Amministrazioni postali Totale 78 Scientific Research An Academic Publisher.

Methods in literature consider stationary models in explaining the underlying data generating process. However, stationarity is arguably a very strong assumption in many real-world applications as process characteristics evolve over time. Reviewed literature reveals that the use of one model may not be appropriate to model a non-stationary series and as such various change-point estimation methods have been proposed. However, they are limited in different ways and their suitability depends on the underlying assumptions.

Statistical research works have shown that with time, the underlying data generating processes undergo occasional sudden changes . In its simplest form, change-point detection is the name given to the problem of estimating the point at which the statistical properties of a sequence of observations change .

The overall behavior of observations can change over time due to internal systemic changes in distribution dynamics or due to external factors.

Time series data entail changes in the dependence structure and therefore modelling non-stationary processes using stationary methods to capture their time-evolving dependence aspects will most likely result in a crude approximation as abrupt changes fail to be accounted for .

The process X is assumed to be piece-wise stationary implying that some characteristics of the process change abruptly at unknown points in time. The corresponding segments are then said to be homogeneous within but each of the subsequent segments is heterogeneous in characteristics.

Parametric tests for change point are mainly based on the likelihood ratio statistics and estimation based on the maximum likelihood method whose general results can be found in . Detection of change points is critical to statistical inference as a near perfect translation to reality is sought through model selection and parameter estimation. Parametric methods assume models for a given set of empirical data.

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Within a parametric setting change, points can be attributed to change in the parameters of the underlying data distribution. Generally, change point methods can be compared based on general characteristics and properties such as test size, power of the test or the rate of convergence to estimate the correct number of change point and the change-point locations.

Change point problems can be classified as off-line which deals with only a fixed sample or on-line which considers new information as it observed. Off-line change point problems deal with fixed sample sizes which are first observed and then detection and estimation of change points are done. Since this pioneering work, methodologies used for change point detection have been widely researched on with methods extending to techniques for higher order moments within time series data.

Ideally, it is desired to test how many change points are present within a given set of data and to estimate the parameters associated with each segment. The null hypothesis of no change against the alternative that there exists a time when the distribution characteristics of the series changed is then tested. Stationarity in the strict sense, implies time-invariance of the distribution underlying the process.

Then the change point problem is to test the hypotheses about the population parameter s. At a given level of significance, if the null hypothesis is rejected, then the process X is said to be locally piecewise-stationary and can be approximated by a sequence of stationary processes that may share certain features such as the general functional form of the distribution F. Many authors such as  -  have considered both parametric and non-parametric methods of change point detection in time series data.

Ideally, change points cannot be assumed to be known in advance hence the need for various methods of detection and estimation. This paper is organized as follows: Section 2 gives an overview of the change point estimator based on a pseudo-distance measure.

Section 3 provides key results for consistency of the estimator. Section 4 provides an application of the change point estimator to the shape and scale parameters of the generalized Pareto distribution. Section 5 gives an application of the estimator and consistency is shown through simulations. Finally 6 provides concluding remarks. Definition 2.

Assumption 1. These assumptions ensure the existence of the integrals. More generally a divergence measure is a function of two probability density or distribution functions, which has non-negative values and takes the value zero only when the two arguments distributions are the same.

A divergence measure grows larger as two distributions are further apart. Hence, a large divergence implies departure from the null hypothesis. This may result to an erratic behavior of the test statistic  due to instability of the estimators of the parameters. A minimal requirement for a good statistical decision rule is its increasing reliability with increasing sample sizes .

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The likelihood function can be expressed as.In statisticsan L-estimator is an estimator which is a linear combination of order statistics of the measurements which is also called an L-statistic. This can be as little as a single point, as in the median of an odd number of valuesor as many as all points, as in the mean. The main benefits of L-estimators are that they are often extremely simple, and often robust statistics : assuming sorted data, they are very easy to calculate and interpret, and are often resistant to outliers.

They thus are useful in robust statistics, as descriptive statisticsin statistics educationand when computation is difficult. However, they are inefficientand in modern times robust statistics M-estimators are preferred, though these are much more difficult computationally. In many circumstances L-estimators are reasonably efficient, and thus adequate for initial estimation. A basic example is the median.

These are both linear combinations of order statistics, and the median is therefore a simple example of an L-estimator. A more detailed list of examples includes: with a single point, the maximum, the minimum, or any single order statistic or quantile ; with one or two points, the median; with two points, the mid-rangethe rangethe midsummary trimmed mid-range, including the midhingeand the trimmed range including the interquartile range and interdecile range ; with three points, the trimean ; with a fixed fraction of the points, the trimmed mean including interquartile mean and the Winsorized mean ; with all points, the mean.

Note that some of these such as median, or mid-range are measures of central tendencyand are used as estimators for a location parametersuch as the mean of a normal distribution, while others such as range or trimmed range are measures of statistical dispersionand are used as estimators of a scale parametersuch as the standard deviation of a normal distribution. L-estimators can also measure the shape of a distribution, beyond location and scale. For example, the midhinge minus the median is a 3-term L-estimator that measures the skewnessand other differences of midsummaries give measures of asymmetry at different points in the tail.

Sample L-moments are L-estimators for the population L-moment, and have rather complex expressions. L-moments are generally treated separately; see that article for details. L-estimators are often statistically resistanthaving a high breakdown point. This is defined as the fraction of the measurements which can be arbitrarily changed without causing the resulting estimate to tend to infinity i.

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Not all L-estimators are robust; if it includes the minimum or maximum, then it has a breakdown point of 0. These non-robust L-estimators include the minimum, maximum, mean, and mid-range. The trimmed equivalents are robust, however. Robust L-estimators used to measure dispersion, such as the IQR, provide robust measures of scale.

In practical use in robust statisticsL-estimators have been replaced by M-estimatorswhich provide robust statistics that also have high relative efficiencyat the cost of being much more computationally complex and opaque. However, the simplicity of L-estimators means that they are easily interpreted and visualized, and makes them suited for descriptive statistics and statistics education ; many can even be computed mentally from a five-number summary or seven-number summaryor visualized from a box plot.

L-estimators play a fundamental role in many approaches to non-parametric statistics. Though non-parametric, L-estimators are frequently used for parameter estimationas indicated by the name, though they must often be adjusted to yield an unbiased consistent estimator.

The choice of L-estimator and adjustment depend on the distribution whose parameter is being estimated. For example, when estimating a location parameterfor a symmetric distribution a symmetric L-estimator such as the median or midhinge will be unbiased. However, if the distribution has skewsymmetric L-estimators will generally be biased and require adjustment. For example, in a skewed distribution, the nonparametric skew and Pearson's skewness coefficients measure the bias of the median as an estimator of the mean.

When estimating a scale parametersuch as when using an L-estimator as a robust measures of scalesuch as to estimate the population variance or population standard deviationone generally must multiply by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.

L-estimators can also be used as statistics in their own right — for example, the median is a measure of location, and the IQR is a measure of dispersion.In statisticsthe bias or bias function of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated.

## L-estimator

An estimator or decision rule with zero bias is called unbiased. In statistics, "bias" is an objective property of an estimator.

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Bias can also be measured with respect to the medianrather than the mean expected valuein which case one distinguishes median -unbiased from the usual mean -unbiasedness property. Bias is a distinct concept from consistency. Consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased; see bias versus consistency for more. All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators with generally small bias are frequently used.

When a biased estimator is used, bounds of the bias are calculated. A biased estimator may be used for various reasons: because an unbiased estimator does not exist without further assumptions about a population; because an estimator is difficult to compute as in unbiased estimation of standard deviation ; because an estimator is median-unbiased but not mean-unbiased or the reverse ; because a biased estimator gives a lower value of some loss function particularly mean squared error compared with unbiased estimators notably in shrinkage estimators ; or because in some cases being unbiased is too strong a condition, and the only unbiased estimators are not useful.

These are all illustrated below. In a simulation experiment concerning the properties of an estimator, the bias of the estimator may be assessed using the mean signed difference. The sample variance of a random variable demonstrates two aspects of estimator bias: firstly, the naive estimator is biased, which can be corrected by a scale factor; second, the unbiased estimator is not optimal in terms of mean squared error MSEwhich can be minimized by using a different scale factor, resulting in a biased estimator with lower MSE than the unbiased estimator.

Concretely, the naive estimator sums the squared deviations and divides by n, which is biased. Conversely, MSE can be minimized by dividing by a different number depending on distributionbut this results in a biased estimator.

Suppose X 1If the sample mean and uncorrected sample variance are defined as. Then, the previous becomes:. The ratio between the biased uncorrected and unbiased estimates of the variance is known as Bessel's correction. That is, when any other number is plugged into this sum, the sum can only increase. This is in fact true in general, as explained above. A far more extreme case of a biased estimator being better than any unbiased estimator arises from the Poisson distribution.Minimize baths and the amount of water you use for each.

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## Consistenza (statistica)

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