Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability, ISSN 1350-7265, 2013, Volume 19, Issue 3, pp. 1006 - 1027

...}} \ {\mathrm{X}}_{\mathrm{i}}$ and ${\mathrm{V}}_{\mathrm{n}}^{2}={\mathrm{\Sigma }}_{\mathrm{i}=1}^{\mathrm{n}} \ {\mathrm{X}}_{\mathrm{i}}^{2}$ . A Cramér type moderate deviation for the maximum of the self-normalized sums max 1...

Mathematical moments | Mathematical theorems | Random variables | Statistics | Independent random variables | Maximum of self-normalized sums | STATISTICS & PROBABILITY | maximum of self-normalized sums | independent random variables | INDEPENDENT RANDOM-VARIABLES | PROBABILITIES

Mathematical moments | Mathematical theorems | Random variables | Statistics | Independent random variables | Maximum of self-normalized sums | STATISTICS & PROBABILITY | maximum of self-normalized sums | independent random variables | INDEPENDENT RANDOM-VARIABLES | PROBABILITIES

Journal Article

Electronic Journal of Probability, ISSN 1083-6489, 01/2009, Volume 14, pp. 1181 - 1197

.... In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that P(max(1 <= k <= n) S-k >= x V-n)/(1 - Phi(x...

Moderate deviation | Self-normalized sum | Large deviation | THEOREM | moderate deviation | self-normalized sum | STATISTICS & PROBABILITY | INDEPENDENT RANDOM-VARIABLES | PROBABILITIES

Moderate deviation | Self-normalized sum | Large deviation | THEOREM | moderate deviation | self-normalized sum | STATISTICS & PROBABILITY | INDEPENDENT RANDOM-VARIABLES | PROBABILITIES

Journal Article

FILOMAT, ISSN 0354-5180, 2019, Volume 33, Issue 8, pp. 2471 - 2488

For many forensic traces (inks, fibres, paints, ...) the chemical characterization of dyes or pigments is currently performed with reliable techniques that...

MATHEMATICS | MOMENT INEQUALITIES | RATES | MATHEMATICS, APPLIED | self-normalized products of sums of partial sums | MAXIMUM | rho(-)-mixing random variables | almost sure central limit theorem | CENTRAL-LIMIT-THEOREM

MATHEMATICS | MOMENT INEQUALITIES | RATES | MATHEMATICS, APPLIED | self-normalized products of sums of partial sums | MAXIMUM | rho(-)-mixing random variables | almost sure central limit theorem | CENTRAL-LIMIT-THEOREM

Journal Article

07/2013

...}r type moderate deviation for the maximum of the self-normalized sums $\max_{1\leq k\leq n}S_k/V_n$ is obtained. In particular, for identically distributed...

Journal Article

The Annals of Probability, ISSN 0091-1798, 10/2002, Volume 30, Issue 4, pp. 1576 - 1604

We consider the supremum W of self-normalized empirical processes indexed by unbounded classes of functions F...

Integers | Cauchy Schwarz inequality | Generating function | Laplace transformation | Mathematical inequalities | Mathematical moments | Random variables | Induction assumption | Distribution functions | Differential variability inequalities | Maximal inequalities | Self-normalized sums | Moderate deviations | Large deviations | Empirical processes | Logarithmic Sobolev inequalities | Concentration inequalities | large deviations | LIMIT-THEOREMS | logarithmic Sobolev inequalities | STATISTICS & PROBABILITY | concentration inequalities | MODEL | moderate deviations | maximal inequalities | self-normalized sums | empirical processes | 60E15 | 60F10 | 62E20 | 62F05

Integers | Cauchy Schwarz inequality | Generating function | Laplace transformation | Mathematical inequalities | Mathematical moments | Random variables | Induction assumption | Distribution functions | Differential variability inequalities | Maximal inequalities | Self-normalized sums | Moderate deviations | Large deviations | Empirical processes | Logarithmic Sobolev inequalities | Concentration inequalities | large deviations | LIMIT-THEOREMS | logarithmic Sobolev inequalities | STATISTICS & PROBABILITY | concentration inequalities | MODEL | moderate deviations | maximal inequalities | self-normalized sums | empirical processes | 60E15 | 60F10 | 62E20 | 62F05

Journal Article

The Annals of Probability, ISSN 0091-1798, 10/1973, Volume 1, Issue 5, pp. 788 - 809

If $X_i$ are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of $S_n(2) = (\sum^n_{i=1} X_i)/(\sum^n_{j=1} X_j^2...

Statistical variance | Zero | Absolute convergence | Numerical analysis | Local maximum | Positive laws | Eigenfunctions | Laplace transformation | Random variables | Distribution functions | stable laws | Limit theorems | domains of attraction | 60F0F | 60G50 | maxima of i.i.d. characteristic function

Statistical variance | Zero | Absolute convergence | Numerical analysis | Local maximum | Positive laws | Eigenfunctions | Laplace transformation | Random variables | Distribution functions | stable laws | Limit theorems | domains of attraction | 60F0F | 60G50 | maxima of i.i.d. characteristic function

Journal Article

The Annals of statistics, ISSN 0090-5364, 2013, Volume 41, Issue 6, pp. 2786 - 2819

We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors...

Error rates | Null hypothesis | Approximation | Non Gaussianity | Inference | Mathematical vectors | Random variables | Covariance matrices | Mathematical maxima | Estimators | Anti-concentration | High dimensionality | Stein method | Maximum of vector sums | Slepian | Dantzig selector | TESTS | maximum of vector sums | MODELS | high dimensionality | DENSITY-ESTIMATION | LASSO | STATISTICS & PROBABILITY | anti-concentration | 62F40 | 62E17

Error rates | Null hypothesis | Approximation | Non Gaussianity | Inference | Mathematical vectors | Random variables | Covariance matrices | Mathematical maxima | Estimators | Anti-concentration | High dimensionality | Stein method | Maximum of vector sums | Slepian | Dantzig selector | TESTS | maximum of vector sums | MODELS | high dimensionality | DENSITY-ESTIMATION | LASSO | STATISTICS & PROBABILITY | anti-concentration | 62F40 | 62E17

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2015, Volume 2015, Issue 1, pp. 1 - 13

By using some inequalities and properties of martingale differences, we investigate the moment of maximum normed randomly weighted sums of martingale differences under some weakly conditions...

60F25 | random weighted | Analysis | Mathematics, general | Mathematics | maximum normed | Applications of Mathematics | 60F15 | martingale differences | COMPLETE CONVERGENCE | MATHEMATICS | MATHEMATICS, APPLIED | TAIL PROBABILITIES | QUADRATIC FORMS | INDEPENDENT RANDOM-VARIABLES | Norms | Stochasticity | Martingales | Inequalities | Sums

60F25 | random weighted | Analysis | Mathematics, general | Mathematics | maximum normed | Applications of Mathematics | 60F15 | martingale differences | COMPLETE CONVERGENCE | MATHEMATICS | MATHEMATICS, APPLIED | TAIL PROBABILITIES | QUADRATIC FORMS | INDEPENDENT RANDOM-VARIABLES | Norms | Stochasticity | Martingales | Inequalities | Sums

Journal Article

Electronic Journal of Statistics, ISSN 1935-7524, 2016, Volume 10, Issue 1, pp. 1001 - 1063

Uniform and nonuniform Berry Esseen (BE) bounds of optimal orders on the rate of convergence to normality in the delta method for vector statistics are...

Cramér’s tilt | Noncentral Hotelling’s statistic | Rates of convergence | Berry–Esseen bound | Maximum likelihood estimators | Nonlinear statistics | Exponential inequalities | Sphericity test | Canonical correlation | Delta method | Pearson’s correlation coefficient | Noncentral student’s statistic | canonical correlation | TESTS | nonlinear statistics | Berry-Esseen bound | noncentral Student's statistic | STATISTICS & PROBABILITY | Cramer's tilt | FUNCTIONAL DATA | PROBABILITY-INEQUALITIES | rates of convergence | sphericity test | SUMS | DISTRIBUTIONS | NORMAL APPROXIMATION | noncentral Hotelling's statistic | COVARIANCE-MATRIX | delta method | maximum likelihood estimators | BANACH-SPACES | exponential inequalities | Pearson's correlation coefficient | INDEPENDENT RANDOM-VARIABLES | CENTRAL-LIMIT-THEOREM

Cramér’s tilt | Noncentral Hotelling’s statistic | Rates of convergence | Berry–Esseen bound | Maximum likelihood estimators | Nonlinear statistics | Exponential inequalities | Sphericity test | Canonical correlation | Delta method | Pearson’s correlation coefficient | Noncentral student’s statistic | canonical correlation | TESTS | nonlinear statistics | Berry-Esseen bound | noncentral Student's statistic | STATISTICS & PROBABILITY | Cramer's tilt | FUNCTIONAL DATA | PROBABILITY-INEQUALITIES | rates of convergence | sphericity test | SUMS | DISTRIBUTIONS | NORMAL APPROXIMATION | noncentral Hotelling's statistic | COVARIANCE-MATRIX | delta method | maximum likelihood estimators | BANACH-SPACES | exponential inequalities | Pearson's correlation coefficient | INDEPENDENT RANDOM-VARIABLES | CENTRAL-LIMIT-THEOREM

Journal Article

The Annals of statistics, ISSN 0090-5364, 2015, Volume 43, Issue 6, pp. 2451 - 2483

Consider d dependent change point tests, each based on a CUSUM-statistic. We provide an asymptotic theory that allows us to deal with the maximum over all test...

Change point analysis | High-dimensional ARMA/GARCH | Spatial econometrics | Extreme value distribution | Weakly dependent high-dimensional time series | spatial econometrics | THEOREM | MAXIMUM | STATISTICS & PROBABILITY | extreme value distribution | high-dimensional ARMA/GARCH | COVARIANCE STRUCTURE | STATIONARY-SEQUENCES | NORMALIZED SUMS | MODELS | MATRICES | weakly dependent high-dimensional time series | TIME-SERIES | AUTOREGRESSIVE PROCESSES | 62M10 | 62G32 | 60K35 | 60F05

Change point analysis | High-dimensional ARMA/GARCH | Spatial econometrics | Extreme value distribution | Weakly dependent high-dimensional time series | spatial econometrics | THEOREM | MAXIMUM | STATISTICS & PROBABILITY | extreme value distribution | high-dimensional ARMA/GARCH | COVARIANCE STRUCTURE | STATIONARY-SEQUENCES | NORMALIZED SUMS | MODELS | MATRICES | weakly dependent high-dimensional time series | TIME-SERIES | AUTOREGRESSIVE PROCESSES | 62M10 | 62G32 | 60K35 | 60F05

Journal Article

Bernoulli, ISSN 1350-7265, 11/2016, Volume 22, Issue 4, pp. 2325 - 2371

.... As an application, we study "self-normalised" versions of Xt, that is, Xt after division by SuP(0~~ Dominance | Distributional representation | Domain of attraction to normality | Maximal jump process | Lévy process | Relative stability | RATIOS | LIMITS | LAW | domain of attraction to normality | STABILITY | BEHAVIOR | STATISTICS & PROBABILITY | TIME | SUMS | relative stability | distributional representation | NORMALITY | Levy process | maximal jump process | VARIABLES | dominance ~~

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 6/2018, Volume 31, Issue 2, pp. 1142 - 1165

We provide an improved version of the Darling–Erdős theorem for sums of i.i.d. random variables with mean zero and finite variance...

Multidimensional version | Darling–Erdős theorem | Extreme value distribution | Strong invariance principle | Probability Theory and Stochastic Processes | Mathematics | Integral test | Statistics, general | Hartman–Wintner LIL | Double truncation | 60F17 | 60F15 | LAW | MAXIMUM | STATISTICS & PROBABILITY | NORMALIZED SUMS | Darling-Erdos theorem | IID RANDOM-VARIABLES | Hartman-Wintner LIL | HILBERT-SPACE

Multidimensional version | Darling–Erdős theorem | Extreme value distribution | Strong invariance principle | Probability Theory and Stochastic Processes | Mathematics | Integral test | Statistics, general | Hartman–Wintner LIL | Double truncation | 60F17 | 60F15 | LAW | MAXIMUM | STATISTICS & PROBABILITY | NORMALIZED SUMS | Darling-Erdos theorem | IID RANDOM-VARIABLES | Hartman-Wintner LIL | HILBERT-SPACE

Journal Article

Journal of Statistical Computation and Simulation, ISSN 0094-9655, 05/2020, Volume 90, Issue 7, pp. 1251 - 1266

During the past 30 years, many statistics have been proposed for tackling change-point problems. However, the null distributions of these statistics are not...

sufficient conditions | Gumbel approximation | empirical saddlepoint approximation | divergence measure | Rayleigh approximation | Beta approximation | change-point | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MAXIMUM | STATISTICS & PROBABILITY | SUM | VARIANCE | Approximation | Probability distribution functions | Mathematical analysis | Empirical analysis

sufficient conditions | Gumbel approximation | empirical saddlepoint approximation | divergence measure | Rayleigh approximation | Beta approximation | change-point | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MAXIMUM | STATISTICS & PROBABILITY | SUM | VARIANCE | Approximation | Probability distribution functions | Mathematical analysis | Empirical analysis

Journal Article

Methodology and computing in applied probability, ISSN 1387-5841, 2007, Volume 9, Issue 2, pp. 225 - 242

This paper provides an overview of results pertaining to moment convergence for certain ratios of random variables involving sums, order statistics and extreme terms in the sense of modulus...

Secondary 60G50 | Economics general | Domain of attraction of a stable law | 62P05 | Limit theorems | Moments | Statistics, general | Statistics | Order statistics | Electronic and Computer Engineering | Dominance of summands | Life Sciences, general | 62G30 | 62G20 | 62G32 | Business/Management Science, general | Functions of regular variation | Sum of . random variables | Primary 60F99 | Sumof i.i.d. random variables | domain of attraction of a stable law | order statistics | sum of i.i.d. random variables | LAW | functions of regular variation | BEHAVIOR | STATISTICS & PROBABILITY | MAXIMUM MODULUS | SUM | limit theorems | RANDOM-VARIABLES | SAMPLE | moments | dominance of summands | SQUARES | Studies | Mathematical models | Convergence | Asymptotic properties | Computation | Criteria | Random variables | Extreme values | Sums

Secondary 60G50 | Economics general | Domain of attraction of a stable law | 62P05 | Limit theorems | Moments | Statistics, general | Statistics | Order statistics | Electronic and Computer Engineering | Dominance of summands | Life Sciences, general | 62G30 | 62G20 | 62G32 | Business/Management Science, general | Functions of regular variation | Sum of . random variables | Primary 60F99 | Sumof i.i.d. random variables | domain of attraction of a stable law | order statistics | sum of i.i.d. random variables | LAW | functions of regular variation | BEHAVIOR | STATISTICS & PROBABILITY | MAXIMUM MODULUS | SUM | limit theorems | RANDOM-VARIABLES | SAMPLE | moments | dominance of summands | SQUARES | Studies | Mathematical models | Convergence | Asymptotic properties | Computation | Criteria | Random variables | Extreme values | Sums

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 4/2003, Volume 16, Issue 2, pp. 377 - 389

In this paper we inverstigate the strong approximation of a linear process with long memory to a Gaussian process. The results are then applied to derive the...

strong approximation | the law of the iterated logarithm | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | long memory process | Darling–Erdős type theorem | Strong approximation | The law of the iterated logarithm | Long memory process | Darling-Erd″os type theorem | GAUSSIAN VARIABLES | LAW | MAXIMUM | ERDOS THEOREM | STATISTICS & PROBABILITY | ITERATED LOGARITHM | EMPIRICAL PROCESS | Darling-Erdos type theorem | PARTIAL SUMS | RANGE DEPENDENCE | MOVING AVERAGES | LIMIT-THEOREMS

strong approximation | the law of the iterated logarithm | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | long memory process | Darling–Erdős type theorem | Strong approximation | The law of the iterated logarithm | Long memory process | Darling-Erd″os type theorem | GAUSSIAN VARIABLES | LAW | MAXIMUM | ERDOS THEOREM | STATISTICS & PROBABILITY | ITERATED LOGARITHM | EMPIRICAL PROCESS | Darling-Erdos type theorem | PARTIAL SUMS | RANGE DEPENDENCE | MOVING AVERAGES | LIMIT-THEOREMS

Journal Article

Journal of the American Statistical Association, ISSN 1537-274X, 2018, Volume 113, Issue 524, pp. 1813 - 1827

... under the independent and identically distributed assumption on the noise, or the weighted sum of independent chi-squared random variables obtained under nonindependent innovations...

Weak (V)ARMA models | Goodness-of-fit test | Box-Pierce and Ljung-Box portmanteau tests | Quasi-maximum likelihood estimation | Self-normalization | Box–Pierce and Ljung–Box portmanteau tests | STRUCTURAL VARMA MODELS | TIME-SERIES MODELS | ESTIMATORS | STATISTICS & PROBABILITY | INFERENCE | HETEROSKEDASTICITY | Autoregressive moving-average models | Economic models | Computer simulation | Asymptotic properties | Noise | Independent variables | Innovations | Regression analysis | Statistical tests | Averages | Diagnostic systems | Normalization | Random variables | Chi-Square Test

Weak (V)ARMA models | Goodness-of-fit test | Box-Pierce and Ljung-Box portmanteau tests | Quasi-maximum likelihood estimation | Self-normalization | Box–Pierce and Ljung–Box portmanteau tests | STRUCTURAL VARMA MODELS | TIME-SERIES MODELS | ESTIMATORS | STATISTICS & PROBABILITY | INFERENCE | HETEROSKEDASTICITY | Autoregressive moving-average models | Economic models | Computer simulation | Asymptotic properties | Noise | Independent variables | Innovations | Regression analysis | Statistical tests | Averages | Diagnostic systems | Normalization | Random variables | Chi-Square Test

Journal Article

Journal of Computational and Graphical Statistics, ISSN 1061-8600, 07/2015, Volume 24, Issue 3, pp. 846 - 865

We propose sequential Monte Carlo-based algorithms for maximum likelihood estimation of the static parameters in hidden Markov models with an intractable...

Maximum likelihood estimation | Approximate Bayesian computation | STATISTICS & PROBABILITY | PARTICLE FILTER | STOCHASTIC VOLATILITY

Maximum likelihood estimation | Approximate Bayesian computation | STATISTICS & PROBABILITY | PARTICLE FILTER | STOCHASTIC VOLATILITY

Journal Article

IET Communications, ISSN 1751-8628, 8/2015, Volume 9, Issue 12, pp. 1501 - 1509

Over the past few decades, frequency hopping (FH) techniques have attracted considerable interest in both military and commercial communications. However,...

Research Articles | ALPHA-STABLE PROCESS | PERFORMANCE ANALYSIS | SPREAD-SPECTRUM SYSTEM | PARTIAL-BAND INTERFERENCE | JAMMING INTERFERENCE | MULTIPLE-ACCESS SYSTEMS | SOFT DECISION RECEIVER | HOPPED NONCOHERENT MFSK | MAXIMUM-LIKELIHOOD RECEIVER | RAYLEIGH-FADING CHANNELS | ENGINEERING, ELECTRICAL & ELECTRONIC | Degradation | Fading | Retarding | Frequency hopping | Interference | Military communications | Channels

Research Articles | ALPHA-STABLE PROCESS | PERFORMANCE ANALYSIS | SPREAD-SPECTRUM SYSTEM | PARTIAL-BAND INTERFERENCE | JAMMING INTERFERENCE | MULTIPLE-ACCESS SYSTEMS | SOFT DECISION RECEIVER | HOPPED NONCOHERENT MFSK | MAXIMUM-LIKELIHOOD RECEIVER | RAYLEIGH-FADING CHANNELS | ENGINEERING, ELECTRICAL & ELECTRONIC | Degradation | Fading | Retarding | Frequency hopping | Interference | Military communications | Channels

Journal Article

Journal of the Royal Statistical Society. Series B (Statistical Methodology), ISSN 1369-7412, 3/2013, Volume 75, Issue 2, pp. 247 - 276

The inspection of residuals is a fundamental step for investigating the quality of adjustment of a parametric model to data. For spatial point processes, the...

Ergodic theory | Spatial points | Central limit theorem | Ergodic measures | Definiteness | Cubes | Mathematical vectors | Covariance matrices | Parametric models | Point estimators | Quadrat counting test | Campbell theorem | Central limit theorem for spatial random fields | Georgii Nguyen–Zessin formula | Maximum pseudolikelihood estimate | Georgii Nguyen-Zessin formula | PATTERNS | STATISTICS & PROBABILITY | ESTIMATOR | RANDOM-FIELDS | MODELS | POISSON PROCESSES | MAXIMUM PSEUDOLIKELIHOOD | Georgii NguyenZessin formula | CENTRAL-LIMIT-THEOREM | Studies | Statistical analysis | Equilibrium | Asymptotic methods | Errors | Null hypothesis | Asymptotic properties | Empirical equations | Consistency | Inspection | Inverse | Counting | Statistics | Statistics Theory | Mathematics

Ergodic theory | Spatial points | Central limit theorem | Ergodic measures | Definiteness | Cubes | Mathematical vectors | Covariance matrices | Parametric models | Point estimators | Quadrat counting test | Campbell theorem | Central limit theorem for spatial random fields | Georgii Nguyen–Zessin formula | Maximum pseudolikelihood estimate | Georgii Nguyen-Zessin formula | PATTERNS | STATISTICS & PROBABILITY | ESTIMATOR | RANDOM-FIELDS | MODELS | POISSON PROCESSES | MAXIMUM PSEUDOLIKELIHOOD | Georgii NguyenZessin formula | CENTRAL-LIMIT-THEOREM | Studies | Statistical analysis | Equilibrium | Asymptotic methods | Errors | Null hypothesis | Asymptotic properties | Empirical equations | Consistency | Inspection | Inverse | Counting | Statistics | Statistics Theory | Mathematics

Journal Article

Physics in Medicine and Biology, ISSN 0031-9155, 05/2010, Volume 55, Issue 9, pp. 2693 - 2708

Following the assembly of a triple-modality SPECT-CT-OT small animal imaging system providing intrinsically co-registered projection data of all three...

TRANSPORT | SPECTROSCOPY | DIFFUSE OPTICAL TOMOGRAPHY | MAXIMUM | ENGINEERING, BIOMEDICAL | IMAGE-RECONSTRUCTION | IN-VIVO | ALGORITHM | CO-CALIBRATION | ITERATIVE RECONSTRUCTION | RADIOLOGY, NUCLEAR MEDICINE & MEDICAL IMAGING | GEOMETRY | Tomography, Emission-Computed, Single-Photon - methods | Tomography, Emission-Computed, Single-Photon - instrumentation | Systems Integration | Fluorescence | Tomography, Optical - methods | Tomography, X-Ray Computed - methods | Image Processing, Computer-Assisted - methods | False Positive Reactions | Tomography, Optical - instrumentation | Animals | Bayes Theorem | Phantoms, Imaging | Tomography, X-Ray Computed - instrumentation

TRANSPORT | SPECTROSCOPY | DIFFUSE OPTICAL TOMOGRAPHY | MAXIMUM | ENGINEERING, BIOMEDICAL | IMAGE-RECONSTRUCTION | IN-VIVO | ALGORITHM | CO-CALIBRATION | ITERATIVE RECONSTRUCTION | RADIOLOGY, NUCLEAR MEDICINE & MEDICAL IMAGING | GEOMETRY | Tomography, Emission-Computed, Single-Photon - methods | Tomography, Emission-Computed, Single-Photon - instrumentation | Systems Integration | Fluorescence | Tomography, Optical - methods | Tomography, X-Ray Computed - methods | Image Processing, Computer-Assisted - methods | False Positive Reactions | Tomography, Optical - instrumentation | Animals | Bayes Theorem | Phantoms, Imaging | Tomography, X-Ray Computed - instrumentation

Journal Article

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