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Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be theta
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long Tail Keywords (2 words) maximum likelihood parameter space likelihood function likelihood estimation displaystyle theta |
long Tail Keywords (3 words) maximum likelihood estimation maximum likelihood estimator ell partial theta partial ell partial frac partial ell fnmathbf y theta theta mathbf y |
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wikipedia free encyclopedia displaystyle theta lefttheta ldots krightmathsf fcdot mid mathbf mathcal lntheta fnmathbf prod nfkmathsf univaryktheta hat underset operatorname argmax nmathbf nmathbb nto ell frac partial quad widehat leftwidehat rightbeginbmatrixleftfrac righttheta leftfrac dots krighttheta vdots ddots kpartial endbmatrix mathbb khtheta right htheta lefth hrtheta hrhr hast hkright phi ihitheta sigma gamma mathsf tgamma tpartial lambda leftlambda rrightmathsf xfrac nsum nln fximid gtheta alpha ghat mathrm mle xrightarrow textp textas neq leftrightarrow bigl fxmid bigr dxquad text all sup xell sqrt nleftwidehat nleft gwidehat bar lalpha ltheta ygx fyyfrac fxxgx textmletheta ijkoperatorname biggl ftheta xtpartial jpartial kbiggr bhequiv righthbiggr ijk mmathcal ihimathcal ijkleftfrac kijkjjikright kijkjjikoperatorname ipartial kfrac jfrac textmlewidehat xnfrac xnmid xoperatorname wunderset woperatorname int infty errormid wimid xmid wioperatorname htextbayesunderset wbigr htextbayes qhat ptheta ysim beginalignedhat lptheta pmathbf npyimid sum nlog pyimid leftsum rightunderset nleftlog log argmin nhtheta yiquad longrightarrow ehtheta yunderset yhtheta ydyunderset ylog pymid dyunderset dtextklptheta parallel endaligned xlog pxmid beginalignedoperatorname ptfrac binom tfrac approx ptoperatorname lpfdmathrm pbinom beginaligned pleftbinom ptp left pright prightendaligned nmu exp xmu nfximid rightn nximu lmu nbar xsum nfrac xin nsigma nxibar nxi nxixj delta iequiv imu ibigr lwidehat mathit ynfrac detmathit lefty ynmu nrightmathit nrightmathrm rho rightright cdots xmn xmmid pmfrac nprod xiprod pixibinom xmp pmxm pmlog mlog xisum mxilog pmlambda mpiright pifrac rright reta rmathbf rleftwidehat eta rin rightnabla rightmathbf rwidehat rightleftfrac rightmathsf tright leftigamma kykskmathsf trightmathbf kleftigamma kskykmathsf trightgamma kykykmathsf yknabla xksknabla yktsk skxk bkfrac ykykmathsf tykmathsf tskfrac bkskskmathsf tbkmathsf tskmathsf tbksk itheta leftmathbf icon wikimedia foundation powered mediawiki
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w httpsenwikipediaorgwindexphptitlemaximumlikelihoodestimationoldid1218599025
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wiki partialresponse maximumlikelihood
statistics
estimating
parameter
probability distribution
numerical optimization
likelihood
statistical model
observed data
point
parameters
statistical inference
differentiable
necessary conditions
ordinary least squares
linear regression
normal distribution
bayesian inference
maximum posterior estimator
prior
frequentist inference
extremum estimator
sample
joint probability distribution
parameters
parametric family
euclidean space
independent and identically distributed random variables
probability density function
measurable
estimator
sample space
sufficient but not necessary
continuous
compact
open
logarithm
loglikelihood
monotonic function
roots
hessian matrix
negative semidefinite
concavity
exponential families
logarithmically concave
restricted maximum likelihood
restrictions
vectorvalued function
constraint
constrained optimization
onetoone function
covariance matrix
positivedefinite
upper triangular matrix
transpose
lagrange multipliers
jacobian matrix
lagrange multiplier test
empirical likelihood
objective function
independent and identically distributed
asymptotics
consistency
invariance
equivariance
efficiency
cramrrao lower bound
mean squared error
asymptotic normality
consistent
converges in probability
almost surely
all models are wrong
identification
observationally equivalent
compact
level sets
neighborhood
upper semicontinuity
uniform law of large numbers
iid
stochastically equicontinuous
converges in distribution
fisher information
cramrrao bound
fisher information matrix
unbiased estimators
expansion
maximum a posteriori
bayesian estimator
maximum a posteriori estimate
machine learning
bayes theorem
cross entropy
iid
law of large numbers
expectancy
law of the unconscious statistician
logarithm
shannons entropy
german tank problem
uniform distribution
expected value
unfair coin
tails
probability mass function
binomial distribution
differentiating
bernoulli trials
independent identically distributed
strictly increasing
range
information entropy
sample mean
statistical error
least squares
nonlinear least squares
confidence interval
confidence region
bivariate
maximum likelihood
iteratively
optimization problem
descent direction
learning rate
gradient descent
newtonraphson method
score
inverse
computationally costly
berndthallhallhausman algorithm
outer product
quasinewton methods
davidonfletcherpowell formula
broydenfletchergoldfarbshanno algorithm
taylor expansion
fishers scoring
generalized linear models
stationary point
saddle point
negative definite
wellconditioned
ronald fisher
carl friedrich gauss
pierresimon laplace
thorvald n thiele
edgeworth francis y
proofs
heuristic
samuel s wilks
wilks theorem
chisquared distribution
ransac
outliers
raoblackwellization
generalized method of moments
mestimator
maximum spacing estimation
maximum entropy estimation
method of moments
method of support
minimumdistance estimation
partial likelihood methods for panel data
quasimaximum likelihood
isbn
hendry david f
ward michael d
journal of mathematical psychology
stack exchange
mcfadden daniel
cox david r
snell e joyce
journal of the royal statistical society series b
jstor
nocedal jorge
amemiya takeshi
sargan denis
wright margaret h
walter de gruyter
annals of mathematical statistics
savage leonard j
stigler stephen m
hald anders
cramer js
king gary
le cam lucien
encyclopedia of mathematics
ems press
sargent thomas
outline
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plugin
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prediction interval
tolerance interval
resampling
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testing hypotheses
1 2tails
power
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multiple comparisons
likelihoodratio test
scorelagrange multiplier
wald
specific tests
ztest normal
students ttest
ftest
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gtest
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normality shapirowilk
model selection
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1way kruskalwallis
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ordered alternative jonckheereterpstra
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arma model
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frequency domain
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fourier analysis
leastsquares spectral analysis
wavelet
whittle likelihood
survival function
kaplanmeier estimator product limit
proportional hazards models
accelerated failure time aft model
first hitting time
hazard function
nelsonaalen estimator
logrank test
applications
biostatistics
bioinformatics
clinical trials
studies
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engineering statistics
chemometrics
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probabilistic design
process
quality control
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system identification
social statistics
actuarial science
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crime statistics
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e936aadebaccbeac19b2d039e74d7a5bad37141b height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {l}}_{n}(\theta )={\mathcal {l}}_{n}(\theta ;\mathbf {y} )=f_{n}(\mathbf {y} ;\theta )\;,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc35592aa22e723a2843a60a5caee69dbb05dcf height: height attribute not set width: width attribute not set description: {\displaystyle f_{n}(\mathbf {y} ;\theta )=\prod _{k=1}^{n}\,f_{k}^{\mathsf {univar}}(y_{k};\theta )~.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/44e1b5745b0ec9d8fc7cace56114a711feb5c314 height: height attribute not set width: width attribute not set description: {\displaystyle \;{\hat {\theta }}_{n}:\mathbb {r} ^{n}\to \theta \;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c562fa434142561968142e1ab888ca810f2ea46c height: height attribute not set width: width attribute not set description: {\displaystyle \,{\widehat {\theta \,}}\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7941ed0c67598b3435e679a5695a249823059896 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {h} \left({\widehat {\theta \,}}\right)={\begin{bmatrix}\left.{\frac {\partial ^{2}\ell }{\partial \theta _{1}^{2}}}\right|_{\theta ={\widehat {\theta \,}}}&\left.{\frac {\partial ^{2}\ell }{\partial \theta _{1}\,\partial \theta _{2}}}\right|_{\theta ={\widehat {\theta \,}}}&\dots &\left.{\frac {\partial ^{2}\ell }{\partial \theta _{1}\,\partial \theta _{k}}}\right|_{\theta ={\widehat {\theta \,}}}\\\left.{\frac {\partial ^{2}\ell }{\partial \theta _{2}\,\partial \theta _{1}}}\right|_{\theta ={\widehat {\theta \,}}}&\left.{\frac {\partial ^{2}\ell }{\partial \theta _{2}^{2}}}\right|_{\theta ={\widehat {\theta \,}}}&\dots &\left.{\frac {\partial ^{2}\ell }{\partial \theta _{2}\,\partial \theta _{k}}}\right|_{\theta ={\widehat {\theta \,}}}\\\vdots &\vdots &\ddots &\vdots \\\left.{\frac {\partial ^{2}\ell }{\partial \theta _{k}\,\partial \theta _{1}}}\right|_{\theta ={\widehat {\theta \,}}}&\left.{\frac {\partial ^{2}\ell }{\partial \theta _{k}\,\partial \theta _{2}}}\right|_{\theta ={\widehat {\theta \,}}}&\dots &\left.{\frac {\partial ^{2}\ell }{\partial \theta _{k}^{2}}}\right|_{\theta ={\widehat {\theta \,}}}\end{bmatrix}}~,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/336030fbce1dc47e876d07faebf4743b2ca7311a height: height attribute not set width: width attribute not set description: {\displaystyle \theta =\left\{\theta :\theta \in \mathbb {r} ^{k},\;h(\theta )=0\right\}~,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a23ad57f96b49388f045cb8275c5cb50dd3f673d height: height attribute not set width: width attribute not set description: {\displaystyle \;h(\theta )=\left[h_{1}(\theta ),h_{2}(\theta ),\ldots ,h_{r}(\theta )\right]\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dbeb71220ef791dd441a066acadfd966b42421e0 height: height attribute not set width: width attribute not set description: {\displaystyle \,\mathbb {r} ^{k}\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cbbae3e0c661193f970cc7df5dbfedf80f967d5c height: height attribute not set width: width attribute not set description: {\displaystyle \;\mathbb {r} ^{r}~.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b height: height attribute not set width: width attribute not set description: {\displaystyle \theta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd6f3b1d08534a961486d126645be2cc080144c height: height attribute not set width: width attribute not set description: {\displaystyle ~h(\theta )=0~.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f104389d954ef1a88cf05c50c31e8063914b9c height: height attribute not set width: width attribute not set description: {\displaystyle \;h_{1},h_{2},\ldots ,h_{r}\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a930736a2af15b71bee884c3abfc15d2025c5b7b height: height attribute not set width: width attribute not set description: {\displaystyle \;h_{1},h_{2},\ldots ,h_{r},h_{r+1},\ldots ,h_{k}\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4240b5d569b53b759ffc773c3231459ca3a20f height: height attribute not set width: width attribute not set description: {\displaystyle \;h^{\ast }=\left[h_{1},h_{2},\ldots ,h_{k}\right]\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcd8908c9fa46eb979ef7b67d1bb65eb3692cbb height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} ^{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/be1c91fc4291b418e0ba168832230022db25faaf height: height attribute not set width: width attribute not set description: {\displaystyle \;\phi _{i}=h_{i}(\theta _{1},\theta _{2},\ldots ,\theta _{k})~.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4a42c672c2d696fec166838e9963b78f1b5d59b0 height: height attribute not set width: width attribute not set description: {\displaystyle \,\sigma \,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/194328bf3316d61ee485ed7cf3c3cccfd88995ac height: height attribute not set width: width attribute not set description: {\displaystyle \;\sigma =\gamma ^{\mathsf {t}}\gamma \;,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed5035aabd1f1d2699c0559343ea2b0b7915880 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{\mathsf {t}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d53b49274f43a85a5a397ebd25b9096556415c36 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\partial \ell }{\partial \theta }}-{\frac {\partial h(\theta )^{\mathsf {t}}}{\partial \theta }}\lambda =0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1edf2977a6af77d45efaef0b163098e27a292908 height: height attribute not set width: width attribute not set description: {\displaystyle h(\theta )=0\;,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6331c4a0e717b73b6647fc2357ac381f47194d height: height attribute not set width: width attribute not set description: {\displaystyle ~\lambda =\left[\lambda _{1},\lambda _{2},\ldots ,\lambda _{r}\right]^{\mathsf {t}}~} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/94b2f5f338ea81a703ffd4754cfe9f5e0f506006 height: height attribute not set width: width attribute not set description: {\displaystyle \;{\frac {\partial h(\theta )^{\mathsf {t}}}{\partial \theta }}\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/68ff47ef36c0e5c195a1cff1966285f6aba10d43 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\ell \,}}(\theta \,;x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/167c37f9f65131bd48894e79b527b0f9a030f1c6 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\ell \,}}(\theta \,;x)={\frac {1}{n}}\sum _{i=1}^{n}\ln f(x_{i}\mid \theta ),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/310cfbae14903fec61dee274d4e67992edc3ff03 height: height attribute not set width: width attribute not set description: {\displaystyle \ell (\theta )=\operatorname {\mathbb {e} } [\,\ln f(x_{i}\mid \theta )\,]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eaae56d74c5844e86caeed8ae205ff9e413bba height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {\theta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/588fa0a6c09981bb9a4c97c7dd3b62ebba2d5693 height: height attribute not set width: width attribute not set description: {\displaystyle g(\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/76dcf4c1fbdf2a75a09bf5c7139c2c18cc364e36 height: height attribute not set width: width attribute not set description: {\displaystyle \alpha =g(\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5828eb020a60c26a6986592fc1e2effcafd2bb07 height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {\alpha }}=g({\hat {\theta }})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77 height: height attribute not set width: width attribute not set description: {\displaystyle g} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/19f3cebee12186db63604f6f11c273d7acd4adc5 height: height attribute not set width: width attribute not set description: {\displaystyle f(\cdot \,;\theta _{0})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/32538e8706bc97d4406be3733abc036659e3b4b9 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\theta \,}}_{\mathrm {mle} }\ {\xrightarrow {\text{p}}}\ \theta _{0}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f39f5f9fa4f9157604a3aec38bc57fb7372e5454 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\theta \,}}_{\mathrm {mle} }\ {\xrightarrow {\text{a.s.}}}\ \theta _{0}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1e25d808c14da03c735f0a0863d08b0d9881753c height: height attribute not set width: width attribute not set description: {\displaystyle \theta \neq \theta _{0}\quad \leftrightarrow \quad f(\cdot \mid \theta )\neq f(\cdot \mid \theta _{0}).} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/8a/ee_noncompactness.svg/240px-ee_noncompactness.svg.png height: 88 width: 240 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d9110b4a94639d23daa24b3cac7449a67ace0e height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {\mathbb {p} } {\bigl [}\;\ln f(x\mid \theta )\;\in \;c^{0}(\theta )\;{\bigr ]}=1.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/617047dbdd5df4af69943f4d6e8b4c0c8801aa8f height: height attribute not set width: width attribute not set description: {\displaystyle {\bigl |}\ln f(x\mid \theta ){\bigr |} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/22160b6eab9ab09c7859b89a88da404244cd4b9d height: height attribute not set width: width attribute not set description: {\displaystyle \sup _{\theta \in \theta }\left|{\widehat {\ell \,}}(\theta \mid x)-\ell (\theta )\,\right|\ {\xrightarrow {\text{p}}}\ 0.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fda6fee5376f28abb5a6052c773017e222e3989c height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\ell \,}}(\theta \mid x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3e150c9c993a8c74368efda5190f6440eb9161 height: height attribute not set width: width attribute not set description: {\displaystyle \sup _{\theta \in \theta }\left\|\;{\widehat {\ell \,}}(\theta \mid x)-\ell (\theta )\;\right\|\ \xrightarrow {\text{a.s.}} \ 0.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4080ea085e2a72443f4eaaa9114c13dfa833a8d5 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {n}}\left({\widehat {\theta \,}}_{\mathrm {mle} }-\theta _{0}\right)\ \xrightarrow {d} \ {\mathcal {n}}\left(0,\,i^{-1}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4379acaccbd4fe05aac12faa10885671bf192409 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\alpha }}=g(\,{\widehat {\theta \,}}\,).\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3de9ba775b5e2a143e3571bbd15d48afddd88cb7 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {l}}(\alpha )=\sup _{\theta :\alpha =g(\theta )}l(\theta ).\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/26c08f8fd3471dad5e2c45c2f753ffd7c9aba4ed height: height attribute not set width: width attribute not set description: {\displaystyle y=g(x)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8963f42d4d5a3f0d9adff3b51eabb9af9a09e9a9 height: height attribute not set width: width attribute not set description: {\displaystyle f_{y}(y)={\frac {f_{x}(x)}{|g'(x)|}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab height: height attribute not set width: width attribute not set description: {\displaystyle x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f height: height attribute not set width: width attribute not set description: {\displaystyle y} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c6dbb6ff21ff22012fca066c1115ef63f4a590 height: height attribute not set width: width attribute not set description: {\displaystyle ~f(\cdot \,;\theta _{0})~,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b32cfa298811ddd0814bc5ad36407ca3acc1f293 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {n\,}}\,\left({\widehat {\theta \,}}_{\text{mle}}-\theta _{0}\right)\ \ \xrightarrow {d} \ \ {\mathcal {n}}\left(0,\ {\mathcal {i}}^{-1}\right)~,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/63f80f4a15ab7a7c663791123ee87a215222d449 height: height attribute not set width: width attribute not set description: {\displaystyle ~{\mathcal {i}}~} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/77f78bb077bde62448dcccfe59a51dcbb872c0bf height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {i}}_{jk}=\operatorname {\mathbb {e} } \,{\biggl [}\;-{\frac {\partial ^{2}\ln f_{\theta _{0}}(x_{t})}{\partial \theta _{j}\,\partial \theta _{k}}}\;{\biggr ]}~.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8340c5d355664078d79aeb28617e374438e8c60d height: height attribute not set width: width attribute not set description: {\displaystyle b_{h}\;\equiv \;\operatorname {\mathbb {e} } {\biggl [}\;\left({\widehat {\theta }}_{\mathrm {mle} }-\theta _{0}\right)_{h}\;{\biggr ]}\;=\;{\frac {1}{\,n\,}}\,\sum _{i,j,k=1}^{m}\;{\mathcal {i}}^{hi}\;{\mathcal {i}}^{jk}\left({\frac {1}{\,2\,}}\,k_{ijk}\;+\;j_{j,ik}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/95ea54922b0b2f15bb16fc33b9545388081c71f7 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {i}}^{jk}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6b28ecae0347aa808fb249d142e3eca9411974f6 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {i}}^{-1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/60eeb1b9aa84cb9d4ae9f0eb635f8d6a41a485fd height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{\,2\,}}\,k_{ijk}\;+\;j_{j,ik}\;=\;\operatorname {\mathbb {e} } \,{\biggl [}\;{\frac {1}{2}}{\frac {\partial ^{3}\ln f_{\theta _{0}}(x_{t})}{\partial \theta _{i}\;\partial \theta _{j}\;\partial \theta _{k}}}+{\frac {\;\partial \ln f_{\theta _{0}}(x_{t})\;}{\partial \theta _{j}}}\,{\frac {\;\partial ^{2}\ln f_{\theta _{0}}(x_{t})\;}{\partial \theta _{i}\,\partial \theta _{k}}}\;{\biggr ]}~.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/61527b36693d07ebb28872fb56c1026ac7ae37b4 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\theta \,}}_{\text{mle}}^{*}={\widehat {\theta \,}}_{\text{mle}}-{\widehat {b\,}}~.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc378f74580de305f39caba3b3019360dddf984 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {\mathbb {p} } (\theta \mid x_{1},x_{2},\ldots ,x_{n})={\frac {f(x_{1},x_{2},\ldots ,x_{n}\mid \theta )\operatorname {\mathbb {p} } (\theta )}{\operatorname {\mathbb {p} } (x_{1},x_{2},\ldots ,x_{n})}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f189373fe12f4bbeaf654ca92b88940d173d59 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {\mathbb {p} } (\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/101e4cee3a2156cffb8b092347ca6fb756807aea height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {\mathbb {p} } (x_{1},x_{2},\ldots ,x_{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/613e9a46e5413c860763420ac8e705fcae2e63e7 height: height attribute not set width: width attribute not set description: {\displaystyle f(x_{1},x_{2},\ldots ,x_{n}\mid \theta )\operatorname {\mathbb {p} } (\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5927797ec79a72143454376a4e66fb94564c2a84 height: height attribute not set width: width attribute not set description: {\displaystyle f(x_{1},x_{2},\ldots ,x_{n}\mid \theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3c9d320a6c5bf1e6a5705d2159a397f5cf67be height: height attribute not set width: width attribute not set description: {\displaystyle \;w_{1}\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/efa06a173896d7791a83dffb2726b21eec60e2e0 height: height attribute not set width: width attribute not set description: {\displaystyle ~\operatorname {\mathbb {p} } (w_{1}|x)\;>\;\operatorname {\mathbb {p} } (w_{2}|x)~;~} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b329bdd86e798ca2fb831eb1063ae269297264d9 height: height attribute not set width: width attribute not set description: {\displaystyle \;w_{2}\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/46b6975525cc032e58925ce180fd2a89535ff958 height: height attribute not set width: width attribute not set description: {\displaystyle \;w_{1}\,,w_{2}\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/534e83751ca421a1979939ee4da55048118c6527 height: height attribute not set width: width attribute not set description: {\displaystyle w={\underset {w}{\operatorname {arg\;max} }}\;\int _{-\infty }^{\infty }\operatorname {\mathbb {p} } ({\text{ error}}\mid x)\operatorname {\mathbb {p} } (x)\,\operatorname {d} x~} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a60af0c2391c535e77c534a1a0fc4c4513e54a64 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {\mathbb {p} } ({\text{ error}}\mid x)=\operatorname {\mathbb {p} } (w_{1}\mid x)~} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/59c1194e6e5e0c1903cd9caf79912dcc3bcfb10b height: height attribute not set width: width attribute not set description: {\displaystyle \;\operatorname {\mathbb {p} } ({\text{ error}}\mid x)=\operatorname {\mathbb {p} } (w_{2}\mid x)\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe92e564f39eeaa756853de1adc8edef09ed184 height: height attribute not set width: width attribute not set description: {\displaystyle \;w_{1}\;.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d3abe16ca3942af18b7284cb8d89dfeae5bea7f7 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {\mathbb {p} } (w_{i}\mid x)={\frac {\operatorname {\mathbb {p} } (x\mid w_{i})\operatorname {\mathbb {p} } (w_{i})}{\operatorname {\mathbb {p} } (x)}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a8946d22343b827b9f92e1bdd602bb4f99c8f379 height: height attribute not set width: width attribute not set description: {\displaystyle h_{\text{bayes}}={\underset {w}{\operatorname {arg\;max} }}\,{\bigl [}\,\operatorname {\mathbb {p} } (x\mid w)\,\operatorname {\mathbb {p} } (w)\,{\bigr ]}\;,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd776c956e546109d90b3f48d856e31efbf858f height: height attribute not set width: width attribute not set description: {\displaystyle h_{\text{bayes}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8c3d79d4ffcc602280ef0387680a4a04db75946f height: height attribute not set width: width attribute not set description: {\displaystyle \;\operatorname {\mathbb {p} } (w)\;} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/adc86adb15c404f22a05526777d6b39a12353a76 height: height attribute not set width: width attribute not set description: {\displaystyle q_{\hat {\theta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/50fe6689cb91e5fb22a2bd8b09719cc87048b6c3 height: height attribute not set width: width attribute not set description: {\displaystyle p_{\theta _{0}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/68e23e3f9a7a0f339527dab3e613447552d87125 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7330ddcc1db0ea75f949d0ec26e439b111ae4e3b height: height attribute not set width: width attribute not set description: {\displaystyle y\sim p_{\theta _{0}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b182d5f3dbad882191525f97e9570d408a8adebe height: height attribute not set width: width attribute not set description: {\displaystyle p_{\theta }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2c8b560e4cc7f5d9d1584e7361c86fb7db2f2b height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\hat {\theta }}&={\underset {\theta }{\operatorname {arg\,max} }}\,l_{p_{\theta }}(\mathbf {y} )={\underset {\theta }{\operatorname {arg\,max} }}\,p_{\theta }(\mathbf {y} )={\underset {\theta }{\operatorname {arg\,max} }}\,p(\mathbf {y} \mid \theta )\\&={\underset {\theta }{\operatorname {arg\,max} }}\,\prod _{i=1}^{n}p(y_{i}\mid \theta )={\underset {\theta }{\operatorname {arg\,max} }}\,\sum _{i=1}^{n}\log p(y_{i}\mid \theta )\\&={\underset {\theta }{\operatorname {arg\,max} }}\,\left(\sum _{i=1}^{n}\log p(y_{i}\mid \theta )-\sum _{i=1}^{n}\log p(y_{i}\mid \theta _{0})\right)={\underset {\theta }{\operatorname {arg\,max} }}\,\sum _{i=1}^{n}\left(\log p(y_{i}\mid \theta )-\log p(y_{i}\mid \theta _{0})\right)\\&={\underset {\theta }{\operatorname {arg\,max} }}\,\sum _{i=1}^{n}\log {\frac {p(y_{i}\mid \theta )}{p(y_{i}\mid \theta _{0})}}={\underset {\theta }{\operatorname {arg\,min} }}\,\sum _{i=1}^{n}\log {\frac {p(y_{i}\mid \theta _{0})}{p(y_{i}\mid \theta )}}={\underset {\theta }{\operatorname {arg\,min} }}\,{\frac {1}{n}}\sum _{i=1}^{n}\log {\frac {p(y_{i}\mid \theta _{0})}{p(y_{i}\mid \theta )}}\\&={\underset {\theta }{\operatorname {arg\,min} }}\,{\frac {1}{n}}\sum _{i=1}^{n}h_{\theta }(y_{i})\quad {\underset {n\to \infty }{\longrightarrow }}\quad {\underset {\theta }{\operatorname {arg\,min} }}\,e[h_{\theta }(y)]\\&={\underset {\theta }{\operatorname {arg\,min} }}\,\int p_{\theta _{0}}(y)h_{\theta }(y)dy={\underset {\theta }{\operatorname {arg\,min} }}\,\int p_{\theta _{0}}(y)\log {\frac {p(y\mid \theta _{0})}{p(y\mid \theta )}}dy\\&={\underset {\theta }{\operatorname {arg\,min} }}\,d_{\text{kl}}(p_{\theta _{0}}\parallel p_{\theta })\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/85f8ebef03c387fe21caec8445f68da51a84eb26 height: height attribute not set width: width attribute not set description: {\displaystyle h_{\theta }(x)=\log {\frac {p(x\mid \theta _{0})}{p(x\mid \theta )}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/25c2ae5b62f78358391aa0e06a57b37a2047ec35 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/547b3a227b1ae110fb6930d9dd0e586273e639f1 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\operatorname {\mathbb {p} } {\bigl [}\;\mathrm {h} =49\mid p={\tfrac {1}{3}}\;{\bigr ]}&={\binom {80}{49}}({\tfrac {1}{3}})^{49}(1-{\tfrac {1}{3}})^{31}\approx 0.000,\\[6pt]\operatorname {\mathbb {p} } {\bigl [}\;\mathrm {h} =49\mid p={\tfrac {1}{2}}\;{\bigr ]}&={\binom {80}{49}}({\tfrac {1}{2}})^{49}(1-{\tfrac {1}{2}})^{31}\approx 0.012,\\[6pt]\operatorname {\mathbb {p} } {\bigl [}\;\mathrm {h} =49\mid p={\tfrac {2}{3}}\;{\bigr ]}&={\binom {80}{49}}({\tfrac {2}{3}})^{49}(1-{\tfrac {2}{3}})^{31}\approx 0.054~.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/17e155bf64d121cf75477c514739f350b7df5111 height: height attribute not set width: width attribute not set description: {\displaystyle l(p)=f_{d}(\mathrm {h} =49\mid p)={\binom {80}{49}}p^{49}(1-p)^{31}~,} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/8d/mlfunctionbinomial-en.svg/200px-mlfunctionbinomial-en.svg.png height: 133 width: 200 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/11a8c9ed4cdd9536c6c56842e729a72bb71cdc3f height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}0&={\frac {\partial }{\partial p}}\left({\binom {80}{49}}p^{49}(1-p)^{31}\right)~,\\[8pt]0&=49p^{48}(1-p)^{31}-31p^{49}(1-p)^{30}\\[8pt]&=p^{48}(1-p)^{30}\left[49(1-p)-31p\right]\\[8pt]&=p^{48}(1-p)^{30}\left[49-80p\right]~.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {n}}(\mu ,\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b187aa258305f940d2de38a22c9135ffa574c766 height: height attribute not set width: width attribute not set description: {\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{{\sqrt {2\pi \sigma ^{2}}}\ }}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1b75a50d0a600772fcca460d04f1e3673f2a3c1f height: height attribute not set width: width attribute not set description: {\displaystyle f(x_{1},\ldots ,x_{n}\mid \mu ,\sigma ^{2})=\prod _{i=1}^{n}f(x_{i}\mid \mu ,\sigma ^{2})=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left(-{\frac {\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{2\sigma ^{2}}}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f81e77f9b7e8c4f6a0103452cc3e132ff40f957f height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {l}}(\mu ,\sigma ^{2})=f(x_{1},\ldots ,x_{n}\mid \mu ,\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc52ab503d9dfe69a8ffd168df952da18adb607 height: height attribute not set width: width attribute not set description: {\displaystyle \log {\bigl (}{\mathcal {l}}(\mu ,\sigma ^{2}){\bigr )}=-{\frac {\,n\,}{2}}\log(2\pi \sigma ^{2})-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(\,x_{i}-\mu \,)^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e000c425ac271525baf164ca92c860d038932ade height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}0&={\frac {\partial }{\partial \mu }}\log {\bigl (}{\mathcal {l}}(\mu ,\sigma ^{2}){\bigr )}=0-{\frac {\;-2n({\bar {x}}-\mu )\;}{2\sigma ^{2}}}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {x}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9c15f496b75fa38c43a35e101695314ab07b18f5 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\mu }}={\bar {x}}=\sum _{i=1}^{n}{\frac {\,x_{i}\,}{n}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/29f9c10198da22e4e62bfaf33c49fcb08fcf9640 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {\mathbb {e} } {\bigl [}\;{\widehat {\mu }}\;{\bigr ]}=\mu ,\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0771ade78a4446e1f7bf6900e637b788c0cbad92 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\mu }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/57eb67636ed6150576ffc86bd170be9cbd9ea67a height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}0&={\frac {\partial }{\partial \sigma }}\log {\bigl (}{\mathcal {l}}(\mu ,\sigma ^{2}){\bigr )}=-{\frac {\,n\,}{\sigma }}+{\frac {1}{\sigma ^{3}}}\sum _{i=1}^{n}(\,x_{i}-\mu \,)^{2}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1df96021e2250b3bd7ec3bbe19731bf156a1f093 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2021e311765713c0bddae20c7f72bab133627c28 height: height attribute not set width: width attribute not set description: {\displaystyle \mu ={\widehat {\mu }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/eb37cbe3319350d7505234ef2f387adfc6811e78 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}-{\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}x_{i}x_{j}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2e2a0bc5ad5ee9d6879b04c4c928665cb190fa59 height: height attribute not set width: width attribute not set description: {\displaystyle \delta _{i}\equiv \mu -x_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b14772ebcc7f56c75145115253ea2ad2430230aa height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(\mu -\delta _{i})^{2}-{\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}(\mu -\delta _{i})(\mu -\delta _{j}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d636e94e397dabd9c9247e3301e4cd6c26f3730a height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {\mathbb {e} } {\bigl [}\;\delta _{i}\;{\bigr ]}=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f659be5d63a212a73928224e9b2f71e369bc2f44 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {e} {\bigl [}\;\delta _{i}^{2}\;{\bigr ]}=\sigma ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b9b5471de022ca37fa914e6f48fa38f489206c height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {\mathbb {e} } {\bigl [}\;{\widehat {\sigma }}^{2}\;{\bigr ]}={\frac {\,n-1\,}{n}}\sigma ^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5afe69bbdc03070ef019f43fe451b847d2acf1ee height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\sigma }}^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3d821928565d31955f323900c86958a8a5219cae height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\sigma }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a76220bd34d7301e2c066e4cb89b1afaf88294 height: height attribute not set width: width attribute not set description: {\displaystyle \theta =(\mu ,\sigma ^{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fefa2a1d315e553b67a28535b148620bb76d85aa height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\theta \,}}=\left({\widehat {\mu }},{\widehat {\sigma }}^{2}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/019727a1303cc3d20999eba911a2291f9f588905 height: height attribute not set width: width attribute not set description: {\displaystyle \log {\bigl (}{\mathcal {l}}({\widehat {\mu }},{\widehat {\sigma }}){\bigr )}={\frac {\,-n\;\;}{2}}{\bigl (}\,\log(2\pi {\widehat {\sigma }}^{2})+1\,{\bigr )}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/eef4db76d658a98219aca14df06d9869d2b43c42 height: height attribute not set width: width attribute not set description: {\displaystyle y_{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7377c7399e662562cd420fa5c7ce49cfba574998 height: height attribute not set width: width attribute not set description: {\displaystyle y_{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/349b2b90fbc08d3759240c20f54656d22b4305e7 height: height attribute not set width: width attribute not set description: {\displaystyle f(y_{1},y_{2})=f(y_{1})f(y_{2})\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/44e34dc59edcc955dcaf524b13843536063ef6a0 height: height attribute not set width: width attribute not set description: {\displaystyle (y_{1},\ldots ,y_{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e45aca073b0619a226592fadba7f1fdc1fd14ca9 height: height attribute not set width: width attribute not set description: {\displaystyle (\mu _{1},\ldots ,\mu _{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa721f0ebb284d8fa648e04d8a4e4ee851621cb height: height attribute not set width: width attribute not set description: {\displaystyle {\mathit {\sigma }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/677eb0cae686c90d7da8e8f040a85d68f575d513 height: height attribute not set width: width attribute not set description: {\displaystyle f(y_{1},\ldots ,y_{n})={\frac {1}{(2\pi )^{n/2}{\sqrt {\det({\mathit {\sigma }})}}}}\exp \left(-{\frac {1}{2}}\left[y_{1}-\mu _{1},\ldots ,y_{n}-\mu _{n}\right]{\mathit {\sigma }}^{-1}\left[y_{1}-\mu _{1},\ldots ,y_{n}-\mu _{n}\right]^{\mathrm {t} }\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9251f77c3cfe253bdf5ef19bf6bb3a8b09266293 height: height attribute not set width: width attribute not set description: {\displaystyle f(y_{1},y_{2})={\frac {1}{2\pi \sigma _{1}\sigma _{2}{\sqrt {1-\rho ^{2}}}}}\exp \left[-{\frac {1}{2(1-\rho ^{2})}}\left({\frac {(y_{1}-\mu _{1})^{2}}{\sigma _{1}^{2}}}-{\frac {2\rho (y_{1}-\mu _{1})(y_{2}-\mu _{2})}{\sigma _{1}\sigma _{2}}}+{\frac {(y_{2}-\mu _{2})^{2}}{\sigma _{2}^{2}}}\right)\right]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/08fae9f72f9a081c0cf4e3347ad6460dbff7b9b2 height: height attribute not set width: width attribute not set description: {\displaystyle x_{1},\ x_{2},\ldots ,\ x_{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b height: height attribute not set width: width attribute not set description: {\displaystyle n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a49bf23b976e5613968296301d4b6e9e058b92da height: height attribute not set width: width attribute not set description: {\displaystyle x_{1}+x_{2}+\cdots +x_{m}=n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2 height: height attribute not set width: width attribute not set description: {\displaystyle p_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/46898b9c6f55fffd52781dd091b8827803114bd1 height: height attribute not set width: width attribute not set description: {\displaystyle p_{1}+p_{2}+\cdots +p_{m}=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d height: height attribute not set width: width attribute not set description: {\displaystyle x_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/237c0fd2860fa208d6420e53b05e4a71121fbe09 height: height attribute not set width: width attribute not set description: {\displaystyle x_{1},\ x_{2},\ldots ,x_{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/78559c4095eb3caca6a4bfbdf95a98eea25bff40 height: height attribute not set width: width attribute not set description: {\displaystyle f(x_{1},x_{2},\ldots ,x_{m}\mid p_{1},p_{2},\ldots ,p_{m})={\frac {n!}{\prod x_{i}!}}\prod p_{i}^{x_{i}}={\binom {n}{x_{1},x_{2},\ldots ,x_{m}}}p_{1}^{x_{1}}p_{2}^{x_{2}}\cdots p_{m}^{x_{m}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ee048a3b4d08b72ca2a58f8b6c2af7bdd850a387 height: height attribute not set width: width attribute not set description: {\displaystyle \ell (p_{1},p_{2},\ldots ,p_{m})=\log n!-\sum _{i=1}^{m}\log x_{i}!+\sum _{i=1}^{m}x_{i}\log p_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc0b320c0dfdc5e07a529c3531fc4928864d452 height: height attribute not set width: width attribute not set description: {\displaystyle l(p_{1},p_{2},\ldots ,p_{m},\lambda )=\ell (p_{1},p_{2},\ldots ,p_{m})+\lambda \left(1-\sum _{i=1}^{m}p_{i}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/adffd2b961b69cb8acead5a576f78bdfb8a149c4 height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {p}}_{i}={\frac {x_{i}}{n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/13cb54b8acf7c9e2a7f7e0183763b9e072ab4b15 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\partial \ell (\theta ;\mathbf {y} )}{\partial \theta }}=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2d972281ef7bab09767b172e3d854a1c935c7af0 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\theta }}={\widehat {\theta }}(\mathbf {y} )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/11959da426ce4ae531b2313bbfcd4c7f628291af height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\theta }}_{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/137dbfda0db89702cfb980f946f9631b05fd00f2 height: height attribute not set width: width attribute not set description: {\displaystyle \left\{{\widehat {\theta }}_{r}\right\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7a00fd84e6ca03a867837791011fa5376261c9dd height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\theta }}_{r+1}={\widehat {\theta }}_{r}+\eta _{r}\mathbf {d} _{r}\left({\widehat {\theta }}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a9195a6cc10b2ebd48d3864973f370cfeb57b644 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {d} _{r}\left({\widehat {\theta }}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/258e03a525beaf7ea2095a75a0978a5b53760823 height: height attribute not set width: width attribute not set description: {\displaystyle \eta _{r}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e9bd4d995e0365544884bac03269680f1ad096 height: height attribute not set width: width attribute not set description: {\displaystyle \eta _{r}\in \mathbb {r} ^{+}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d470ffbf73a01a8cfb5d6217de4e19eef5e6529e height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {d} _{r}\left({\widehat {\theta }}\right)=\nabla \ell \left({\widehat {\theta }}_{r};\mathbf {y} \right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/85522280f8cbd1ca9c6269873596ef3ab8db9913 height: height attribute not set width: width attribute not set description: {\displaystyle \eta _{r}=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/54bbc4e68d3f80518a09534f5e713ef83cd98aa3 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {d} _{r}\left({\widehat {\theta }}\right)=-\mathbf {h} _{r}^{-1}\left({\widehat {\theta }}\right)\mathbf {s} _{r}\left({\widehat {\theta }}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/207136b9cb61c9500e5590e6a39285b2d019e7fb height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {s} _{r}({\widehat {\theta }})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cd6cf7cc137c73c6a5531532108e241bed111b00 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {h} _{r}^{-1}\left({\widehat {\theta }}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/54504adfda94cd3accf328fc3a245468140acd35 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {d} _{r}\left({\widehat {\theta }}\right)=-\left[{\frac {1}{n}}\sum _{t=1}^{n}{\frac {\partial \ell (\theta ;\mathbf {y} )}{\partial \theta }}\left({\frac {\partial \ell (\theta ;\mathbf {y} )}{\partial \theta }}\right)^{\mathsf {t}}\right]^{-1}\mathbf {s} _{r}\left({\widehat {\theta }}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5189e84c5933650a98313d53e6f17397b7cd73f6 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {h} _{k+1}=\left(i-\gamma _{k}y_{k}s_{k}^{\mathsf {t}}\right)\mathbf {h} _{k}\left(i-\gamma _{k}s_{k}y_{k}^{\mathsf {t}}\right)+\gamma _{k}y_{k}y_{k}^{\mathsf {t}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e74bfbdba9572aa543e116cf3a71e6db9d92bb height: height attribute not set width: width attribute not set description: {\displaystyle y_{k}=\nabla \ell (x_{k}+s_{k})-\nabla \ell (x_{k}),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/64784fd63a3d6ddd03e498ad69b5f52b9f851d9c height: height attribute not set width: width attribute not set description: {\displaystyle \gamma _{k}={\frac {1}{y_{k}^{t}s_{k}}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d10f5859bb6deddd23400c17ccc2a58d7b0b6255 height: height attribute not set width: width attribute not set description: {\displaystyle s_{k}=x_{k+1}-x_{k}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/751ca32b4d3dbfc21d70530be9feb82aa8ed7ffb height: height attribute not set width: width attribute not set description: {\displaystyle b_{k+1}=b_{k}+{\frac {y_{k}y_{k}^{\mathsf {t}}}{y_{k}^{\mathsf {t}}s_{k}}}-{\frac {b_{k}s_{k}s_{k}^{\mathsf {t}}b_{k}^{\mathsf {t}}}{s_{k}^{\mathsf {t}}b_{k}s_{k}}}\ ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/79b549416c9863d8a2f90c6c50bbec905823ab21 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {i}}(\theta )=\operatorname {\mathbb {e} } \left[\mathbf {h} _{r}\left({\widehat {\theta }}\right)\right]} |
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