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In statistical hypothesis testing, the error exponent of a hypothesis testing procedure is the rate at which the probabilities of Type I and Type II decay
Error exponents in hypothesis testing
Error_exponents_in_hypothesis_testing
Concept in computing
information theory, the error exponent of a channel code or source code over the block length of the code is the rate at which the error probability decays
Error_exponent
Statistical testing method
the Renyi divergence. The first test known to achieve this optimal error exponent was based on maximum mean discrepancy. A/B testing Universal hypothesis
Two-sample_hypothesis_testing
Arithmetic operation
denoted bn, is an operation involving two numbers: the base, b, and the exponent or power, n. When n is a positive integer, exponentiation corresponds to
Exponentiation
Concise notation for large or small numbers
always written as a terminating decimal). The integer n is called the exponent and the real number m is called the significand or mantissa. The term "mantissa"
Scientific_notation
Mathematical expression with disputed status
leave it undefined or return errors, depending on the context. Many widely used formulas involving natural-number exponents require 00 to be defined as
Zero_to_the_power_of_zero
Computer approximation for real numbers
of these errors is a topic in numerical analysis; see also Accuracy problems. To multiply, the significands are multiplied while the exponents are added
Floating-point_arithmetic
Error-correcting codes
and 81 = 34 with positive exponents, but typically this isn't used. The logarithm of the inverted roots corresponds to the error locations (right to left
Reed–Solomon_error_correction
Scientific study of digital information
between information theory and hypothesis testing, as it is the optimal error exponent in asymmetric hypothesis testing. Directed information, I ( X n → Y
Information_theory
e-folding Elimination half-life Error exponent Euler's formula Euler's identity e (mathematical constant) Exponent Exponent bias Exponential (disambiguation)
List_of_exponential_topics
Establishes the limits to possible data compression
δ can be made arbitrarily small, by making n larger. Channel coding Error exponent Noisy-channel coding theorem Shen, A. and Uspensky, V.A. and Vereshchagin
Shannon's source coding theorem
Shannon's_source_coding_theorem
Strategies to make sure approximate calculations stay close to accurate
fields for the exponent and significand lengths and error information is carried in a single bit, the ubit, representing possible error in the least significant
Floating-point error mitigation
Floating-point_error_mitigation
Computational error due to rounding numbers
In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic
Round-off_error
Class of mathematical root-finding algorithm
Since n(d + 1) evaluations over n iterations give an error exponent of (d + 1)n, the exponent for one function evaluation is d + 1 d + 1 {\displaystyle
Householder's_method
IEEE standard for floating-point arithmetic
followed by w exponent bits that describe the exponent offset by a bias, and p − 1 bits that describe the significand. The width of the exponent field for
IEEE_754
Facts provided or learned about something or someone
measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include
Information
Parameter describing physics near critical points
Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal
Critical_exponent
64-bit computer number format
rounding error when rounding a number to the nearest representable one (the machine epsilon) is therefore 2−53. The 11 bit width of the exponent allows
Double-precision floating-point format
Double-precision_floating-point_format
17th-century conjecture proved by Andrew Wiles in 1994
able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was considered exceedingly difficult or unachievable
Fermat's_Last_Theorem
16-bit computer number format
Hitachi's HD61810 DSP of 1982 (a 4-bit exponent and a 12-bit mantissa), the top 16 bits of a 32-bit float (8 exponent and 7 mantissa bits) called a bfloat16
Half-precision floating-point format
Half-precision_floating-point_format
Information-theoretical limit on transmission rate in a communication channel
Bandwidth (computing) Bandwidth (signal processing) Bit rate Code rate Error exponent Nyquist rate Negentropy Redundancy Sender, Data compression, Receiver
Channel_capacity
American information theorist
also established the binary erasure channel and proposed list decoding of error-correcting codes as an alternative to unique decoding. Peter Elias was a
Peter_Elias
Mathematical function, denoted exp(x) or e^x
expression. It is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718, the base, is raised. There are several
Exponential_function
Type of mathematical proposition
In elementary number theory, the lifting-the-exponent lemma (or LTE lemma) provides several formulas for computing the p-adic valuation ν p {\displaystyle
Lifting-the-exponent_lemma
Russian mathematician (1935–2017)
Levenshtein (1989), "On the Straight-Line Bound for the Undetected Error Exponent", Problemy Peredachi Informatsii, 25 (1): 33–37 VI Levenshtein, Perfect
Vladimir_Levenshtein
Theorem of probability theory
in the area of statistical hypothesis testing, which connects the error exponents in hypothesis testing with the Kullback–Leibler divergence. This result
Stein's_lemma
Theorem about the power of the likelihood ratio test
recollections about the movie we saw. It may have been Buster Keaton. Error exponents in hypothesis testing F-test Lemma Wilks' theorem Neyman, J.; Pearson
Neyman–Pearson_lemma
Standard defining codes for currencies
currencies Production Mint Designers Coining Milling Hammering Cast Metals Errors Collection Coin collecting Coins Commemorative coins Bullion coins Grading
ISO_4217
Construct in mathematics
is revealed by plotting the Lyapunov exponent, as shown by the example below. The Lyapunov exponent is the error growth-rate of a given sequence. First
Multibrot_set
Relative weight based on mass and height
average height, while the exponent of 2.5 is a compromise between the exponent of 2 in the traditional formula for BMI and the exponent of 3 that would be expected
Body_mass_index
Prime number of the form 2^n – 1
that n should be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89. Mersenne primes were studied in antiquity
Mersenne_prime
Thermodynamic quantity
is denoted by γ (gamma) for an ideal gas or κ (kappa), the isentropic exponent for a real gas. The symbol γ is used by aerospace and chemical engineers
Heat_capacity_ratio
Limit on data transfer rate
(AEP). Another style can be found in information theory texts using error exponents. Both types of proofs make use of a random coding argument where the
Noisy-channel_coding_theorem
Quantum information paradigm
in coherent states (technically, there is a 6 dB improvement in the error exponent ). A key feature of quantum illumination is that the entanglement between
Quantum_illumination
Class of mathematical expression
or negative), a fixed-precision significand and an integer exponent. Numbers whose exponent is too large to represent instead "overflow" to positive or
Division_by_zero
Statistical test that compares goodness of fit
factor Johansen test Model selection Vuong's closeness test Sup-LR test Error exponents in hypothesis testing King, Gary (1989). Unifying Political Methodology :
Likelihood-ratio_test
Algorithm to compute rounding error
normalized exponents e a ≥ e b {\displaystyle e_{a}\geq e_{b}} Outputs rounded sum s = a ⊕ b {\displaystyle s=a\oplus b} and exact error t = a + b −
2Sum
Error-detecting code for detecting data changes
A cyclic redundancy check (CRC) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to digital
Cyclic_redundancy_check
Performing order of mathematical operations
expression has the value 1 + (2 × 3) = 7, and not (1 + 2) × 3 = 9. When exponents were introduced in the 16th and 17th centuries, they were given precedence
Order_of_operations
Mathematical function, inverse of an exponential function
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the
Logarithm
Probability distribution
\sigma ^{2}} . It follows that the normal distribution is stable (with exponent α = 2 {\textstyle \alpha =2} ). If X k ∼ N ( m k , σ k 2 ) {\textstyle
Normal_distribution
Optimization theory in computing
that the incrementing is a probabilistic event. To save space, only the exponent is kept. For example, in base 2, the counter can estimate the count to
Approximate counting algorithm
Approximate_counting_algorithm
Theorem pertaining to the ontology of quantum mechanics
; Nussbaum, Michael; Wilde, Mark M. (2024-06-05). "On the optimal error exponents for classical and quantum antidistinguishability". Letters in Mathematical
Pusey–Barrett–Rudolph_theorem
Decimal representation of real numbers in computing
of these errors is a topic in numerical analysis; see also Accuracy problems. To multiply, the significands are multiplied, while the exponents are added
Decimal_floating_point
Distinguished surfaces of dynamic trajectories
stretching tends to grow rapidly, it is more convenient to work with its growth exponent ( log δ t 0 t 1 ) / ( t 1 − t 0 ) {\displaystyle (\log {\delta
Lagrangian_coherent_structure
Error correction code
0111=\alpha ^{-5}.} The exponents of α {\displaystyle \alpha } correspond to the error locations. There is no need to calculate the error values in this example
BCH_code
Functional relationship between two quantities
in the other quantity proportional to the change raised to a constant exponent: one quantity varies as a power of another. The change is independent of
Power_law
Volunteer project using software to search for Mersenne prime numbers
introduce errors into the LL calculation. The program verifies that the round-off error is no greater than 0.4 every 128 iterations, or if the exponent being
Great Internet Mersenne Prime Search
Great_Internet_Mersenne_Prime_Search
Middle quantile of a data set or probability distribution
the location parameter. The median of a power law distribution x−a, with exponent a > 1 is 21/(a − 1)xmin, where xmin is the minimum value for which the
Median
Sports formula
fix this routine error, statisticians have performed numerous searches to find the ideal exponent. If using a single-number exponent, 1.83 is the most
Pythagorean_expectation
Algorithm for public-key cryptography
private key exponent. The public key consists of the modulus n and the public exponent e. The private key consists of the private exponent d, which must
RSA_cryptosystem
Numbers significantly larger than those used regularly
reciprocal, one billionth, is 1.0×10−9. Sometimes the letter e replaces the exponent, for example 1 billion may be expressed as 1e9 instead of 1.0×109. googol
Large_numbers
Method in computer arithmetic
significands (the non-exponent part of the floating-point number) to a single exponent, rather than single significand being assigned its own exponent. BFP can be
Block_floating_point
First edition of the IEEE 754 floating-point standard
for the final result, to minimise round-off errors: the standard only specifies minimum precision and exponent requirements for such formats. The x87 80-bit
IEEE_754-1985
Field of mathematics and science based on non-linear systems and initial conditions
divergent orbits, but does not actually compute any exponent. In this PRL paper instead the lyapunov exponent is explicit. George D. Birkhoff, Dynamical Systems
Chaos_theory
Upper bound on rounding error in floating-point arithmetic
magnitude. Since machine epsilon is a bound for relative error, it suffices to consider numbers with exponent e = 0 {\displaystyle e=0} . It also suffices to consider
Machine_epsilon
Special function defined by an integral
_{n=0}^{N-1}{\frac {n!}{(-x)^{n}}}+O(N!x^{-N})\right)} The relative error of the approximation above is plotted on the figure to the right for various
Exponential_integral
Variant of floating-point numbers in computers
format are: a variable-width storage format for both the significand and exponent, and a u-bit, which determines whether the unum corresponds to an exact
Unum_(number_format)
Radical policies of the Communist Party of Yugoslavia during World War II
Leftist errors (Serbo-Croatian: leva/lijeva skretanja, лева/лијева скретања) were a term used by the Communist Party of Yugoslavia (CPY) to describe radical
Leftist_errors
Method for signed number representation
means that inverting the leading (high-order) bit of the exponent will not convert the exponent to correct two's complement notation. Offset binary is often
Offset_binary
Formula for lattice energy
r0 = distance between closest cation [ +ve ] and anion [ -ve ] n = Born exponent, typically a number between 5 and 12, determined experimentally by measuring
Born–Landé_equation
Branch of elementary mathematics
exponentiation with a fractional exponent is to perform two separate calculations: one exponentiation using the numerator of the exponent followed by drawing the
Arithmetic
Root-finding algorithm
{\textstyle E_{x}=e_{x}+B} is the "biased exponent", where B = 127 {\displaystyle B=127} is the "exponent bias" (8 bits) M x = m x × L {\textstyle M_{x}=m_{x}\times
Fast_inverse_square_root
Matrix operation generalizing exponentiation of scalar numbers
this one-parameter subgroup. More generally, for a generic t-dependent exponent, X(t), d d t e X ( t ) = ∫ 0 1 e α X ( t ) d X ( t ) d t e ( 1 − α ) X
Matrix_exponential
Tradition of the Dhrupad music genre of Northern India
The Dagar vani is a tradition of the classical dhrupad genre of Hindustani classical music spanning 20 generations, tracing back to Swami Haridas (15th
Dagar_vani
Discrete Fourier transform algorithm
inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1/n factor, any FFT algorithm can easily be adapted for it. Fast
Fast_Fourier_transform
128-bit computer number format
The IEEE 754 standard specifies a binary128 as having: Sign bit: 1 bit Exponent width: 15 bits Significand precision: 113 bits (112 explicitly stored)
Quadruple-precision floating-point format
Quadruple-precision_floating-point_format
Mathematical function
flat-top and Gaussian fall-off can be taken by raising the content of the exponent to a power P {\displaystyle P} : f ( x ) = A exp ( − ( ( x − x 0 ) 2
Gaussian_function
Concept in information theory
likelihoods for the test data, which leads to a lower perplexity value. The exponent, − 1 N ∑ i = 1 N log b q ( x i ) {\textstyle -{\frac {1}{N}}\sum _{i=1}^{N}\log
Perplexity
Concept in computer science
and linear output size) and with exponentially small (with linear exponent) error probability. Also, this construction is effective in that given an
BPP_(complexity)
Topics referred to by the same term
power of an exponent Rigorous Approach to Industrial Software Engineering, a set of tools for software development raise, a PL/SQL error-handling command
Raise
Constant value used in mathematics
32 × 5, the prime 3 appears with an even exponent, and the prime 5 is congruent to 1 mod 4, so its exponent can be odd. Landau's theorem states that if
Landau–Ramanujan_constant
Digit necessary to represent a quantity
the greatest exponent value (the leftmost significant digit/figure), while the least significant digit is the one with the lowest exponent value (the rightmost
Significant_figures
Urdu feminist poetry
Rekhti (Urdu: ریختی, Hindi: रेख़्ती; [ˈɾeːxtiː]; Rextī), is a form of Urdu feminist poetry. A genre developed by male poets, it uses women's voices to
Rekhti
Method to maintain data backups
generation preceding it. Using a larger exponent leads to a more uniform distribution of generations, whereas a smaller exponent leads to a distribution with more
Backup_rotation_scheme
French academic (born 1954)
Venkat; Baccelli, Francois (2015). "Information theoretic capacity and error exponents of stationary point processes with additive displacement noise". Advances
François_Baccelli
Ratio of the desired signal to the background noise
strength, reducing the noise level, filtering out unwanted noise, or using error correction techniques. SNR also determines the maximum possible amount of
Signal-to-noise_ratio
Growth of quantities at rate proportional to the current amount
exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential
Exponential_growth
Computer floating-point unit
data registers (a 64-bit mantissa plus a sign bit, and a 15-bit signed exponent). It allows seven different modes of numeric representation, including
Motorola_68881
Special function occurring in problems possessing elliptic symmetry
{\displaystyle \mu } is a complex number, the Floquet exponent (or sometimes Mathieu exponent), and P {\displaystyle P} is a complex valued function
Mathieu_function
Number functioning as an exponent
to represent a number on a different scale, functioning similarly to an exponent in mathematics. A scale factor is used when a real-world set of numbers
Scale factor (computer science)
Scale_factor_(computer_science)
Exponential function of an exponential function
nearest integer the values of a double exponential function with middle exponent 2. Ionaşcu and Stănică describe some more general sufficient conditions
Double_exponential_function
engineering research firm Exponent. The initial study commissioned by Toyota beginning in December 2009 concluded "Exponent has so far been unable to
2009–2011 Toyota vehicle recalls
2009–2011_Toyota_vehicle_recalls
American musical duo
Beaver & Krause were an American musical duo comprising Paul Beaver and Bernie Krause. Their 1967 album The Nonesuch Guide to Electronic Music was a pioneering
Beaver_&_Krause
Algorithms for calculating square roots
maximum absolute error of 1.2 at a=100, and maximum relative error of 30% at S=1 and 10. To divide by 10, subtract one from the exponent of a, or figuratively
Square_root_algorithms
French novelist (1821–1880)
8 May 1880) was a French novelist. He has been considered the leading exponent of literary realism in his country and abroad. He is known especially for
Gustave_Flaubert
Argentine filmmaker (born 1963)
screenwriter, who lives and works primarily in France. He is one of the primary exponents of New French Extremity, associated its graphic nature of unsimulated
Gaspar_Noé
Measure of linear correlation
2 , {\displaystyle T=D(D^{\mathsf {T}}D)^{-{\frac {1}{2}}},} where an exponent of −+1⁄2 represents the matrix square root of the inverse of a matrix.
Pearson correlation coefficient
Pearson_correlation_coefficient
19th century musician of Indian classical music (1801 - 1890)
Qutub Bakhsh, more commonly known as Tanras Khan (c. 1801 – c. 1890), was an Indian musician of the Hindustani Classical tradition known for being a luminary
Tanras_Khan
Rules defining correctly structured Java programs
for interacting with the system, containing standard input, output, and error streams, system properties and environment variables, time, and more. java
Java_syntax
Floating-point number formats
format with 16 unused bits inserted between the exponent and significand fields, and values with exponent zero and bit 63 one are normalized values). The
Extended_precision
Idea that small causes can have large effects
point x, but it requires one positive Lyapunov exponent. In addition to a positive Lyapunov exponent, boundedness is another major feature within chaotic
Butterfly_effect
} If there is any significant difference between the Lyapunov exponents then as an error vector evolves forward in time, any displacement in the direction
Chaotic_mixing
Statistical law in machine learning
magnitude smaller in scale. Whereas previous works generally found the scaling exponent to scale like L ∝ D − α {\displaystyle L\propto D^{-\alpha }} , with α
Neural_scaling_law
Geometric progression ratio that provides a constant rate of return over the time period
annualized growth rates on quarterly or monthly values. The numerator of the exponent would be the value of 4 in the case of quarterly, and 12 in the case of
Compound_annual_growth_rate
Mathematical concept
{\displaystyle a/b^{q}} . The proof below illustrates also why Hölder conjugate exponent is the only possible parameter that makes Young's inequality hold for all
Young's inequality for products
Young's_inequality_for_products
Product of numbers from 1 to n
more quickly than exponential growth. Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can
Factorial
Mathematical approximation of a function
accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a
Taylor_series
Method to detect power-law scaling in time series
autocorrelation function) or 1/f noise. The obtained exponent is similar to the Hurst exponent, except that DFA may also be applied to signals whose
Detrended fluctuation analysis
Detrended_fluctuation_analysis
Mathematical fallacy
which does not equal 3 + 4 = 7. In this example, the error is being committed with the exponent n = 1/2. When p {\displaystyle p} is a prime number
Freshman's_dream
ERROR EXPONENT
ERROR EXPONENT
Boy/Male
American, Anglo, Australian, British, Chinese, Christian, English, French, Indian, Scottish, Teutonic
Maker of Arrows; Arror Featherer
Boy/Male
Tamil
King of terror
Girl/Female
Australian, British, English, Gaelic, Irish
Terror; Lovers
Boy/Male
Christian & English(British/American/Australian)
Wanderering Noble
Female
Arthurian
, error for Nineve (q.v.).
Boy/Male
Anglo Saxon
Terror.
Male
Swiss
, axe, or, terror.
Male
English
Scottish surname transferred to forename use, from a place name possibly ERROL means "to wander."Â
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Error-less
Boy/Male
Anglo Saxon
Terror.
Male
Swiss
, axe, or, terror.
Boy/Male
American, Anglo, British, Christian, Danish, English, German, Jamaican, Latin, Scottish
To Wander; Nobleman; Leader; Earl; Wanderer
Boy/Male
Shakespearean
The Comedy of Errors' A merchant.
Boy/Male
Hindu
King of terror
Boy/Male
Anglo Saxon
Terror.
Boy/Male
Anglo Saxon
Terror.
Boy/Male
English American German Latin Scottish
Army commander. Army, weald power. Also can be a, meaning nobleman. Famous bearer: Australian...
Boy/Male
Arabic, Hindu, Indian, Muslim, Tamil
Terror; Kings; Royal
Girl/Female
Shakespearean
The Comedy of Errors' Adriana's servant.
Girl/Female
Hindu, Indian
Without Error
ERROR EXPONENT
ERROR EXPONENT
Boy/Male
Biblical
Their hill.
Girl/Female
Tamil
Goddess Durga, A melody in classical music
Girl/Female
Tamil
Dhiyanshi | தீயாஂஷீÂ
Part of a divine power
Girl/Female
Hindu, Indian, Marathi
Skillful; Clever
Boy/Male
English
Dispenser; provider.
Boy/Male
English
Leader.
Girl/Female
American, Australian, British, Christian, English, French, German, Greek, Jamaican
Work Strength; Of a Thousand Saints; English Cognate of Melisande; Highborn Power; Strong Work; Industrious; Brave; Strong Worker
Girl/Female
Arabic, Muslim
Soft; Delicate
Boy/Male
Indian, Telugu
Father of Sky
Surname or Lastname
English
English : topographic name from Middle English hauk, hauek ‘hawk’ + ley(e) ‘open country’, ‘grassland’, ‘field’, or a habitational name from Hawkesley Hall in King’s Norton, Worcestershire, named from the Old English personal name Heafoc or Old English heafoc ‘hawk’, ‘clearing’ + lēah ‘wood’, ‘clearing’.
ERROR EXPONENT
ERROR EXPONENT
ERROR EXPONENT
ERROR EXPONENT
ERROR EXPONENT
n.
Liability to error.
n.
A departing or deviation from the truth; falsity; false notion; wrong opinion; mistake; misapprehension.
n.
Exemption from error.
n.
Misguidance; error.
n.
An error; a mistake.
a.
Liable to error; fallible.
n.
One who encourages and propagates error; one who holds to error.
n.
A wandering or deviation from the right course or standard; irregularity; mistake; inaccuracy; something made wrong or left wrong; as, an error in writing or in printing; a clerical error.
n.
Mistake; error; fault.
n.
The difference between the observed value of a quantity and that which is taken or computed to be the true value; -- sometimes called residual error.
n.
The difference between the approximate result and the true result; -- used particularly in the rule of double position.
n.
One who leads into error.
n.
A fault of a player of the side in the field which results in failure to put out a player on the other side, or gives him an unearned base.
n.
The difference between an observed value and the true value of a quantity.
n.
A mistake in the proceedings of a court of record in matters of law or of fact.
a.
Full of error; wrong.
n.
A wandering; a roving or irregular course.
n.
A moral offense; violation of duty; a sin or transgression; iniquity; fault.
n.
An error in pleading.
n.
Erroneous thought; mistaken opinion; error.