 number varies as the number of dimensions increases. 

A:

The amount of data needed to estimate the parameters of a multivariate Normal distribution to within specified accuracy to a given confidence does not vary with the dimension, all other things being the same.  Therefore you may apply any rule of thumb for two dimensions to higher dimensional problems without any change at all.
Why should it? There are only three kinds of parameters: means, variances, and covariances.  The error of estimate in a mean depends only the variance and the amount of data, $n$.  Thus, when $(X_1, X_2, \ldots, X_d)$ has a multivariate Normal distribution and the $X_i$ have variances $\sigma_i^2$, then the estimates of $\mathbb{E[X_i]}$ depend only on the $\sigma_i$ and $n$.  Whence, to achieve adequate accuracy in estimating all the $\mathbb{E}[X_i]$, we only need to consider the amount of data needed for the $X_i$ having the largest of the $\sigma_i$.  Therefore, when we contemplate a succession of estimation problems for increasing dimensions $d$, all we need to consider is how much the largest $\sigma_i$ will increase.  When these parameters are bounded above, we conclude that the amount of data needed does not depend on dimension.
Similar considerations apply to estimating the variances $\sigma_i^2$ and covariances $\sigma_{ij}$: if a certain amount of data suffice for estimating one covariance (or correlation coefficient) to a desired accuracy, then--provided the underlying normal distribution has similar parameter values--the same amount of data will suffice for estimating any covariance or correlation coefficient.

To illustrate, and provide empirical support for this argument, let's study some simulations.  The following creates parameters for a multinormal distribution of specified dimensions, draws many independent, identically distributed sets of vectors from that distribution, estimates the parameters from each such sample, and summarizes the results of those parameter estimates in terms of (1) their averages--to demonstrate they are unbiased (and the code is working correctly--and (2) their standard deviations, which quantify the accuracy of the estimates.  (Do not confuse these standard deviations, which quantify the amount of variation among estimates obtained over multiple iterations of the simulation, with the standard deviations used to define the underlying multinormal distribution!)  My claim is that these standard deviations do not materially change when the
number varies as the number of dimensions increase. 

 deoxyadenosine monophosphate: 

 The amount of data needed to calculate the parameters of a multivariate Normal distribution to within specified accuracy to a given assurance does not deviate with the dimension, all early thing being the same.   consequently you may apply any rule of thumb for two dimensions to high dimensional problems without any change at all. 
 Why should it? There are only three kind of parameters: means, variances, and covariance.   The error of estimate in a mean depends only the variance and the sum of data, $ n$.   Thus, when $ (X_1, X_2, \ldots, X_d)$ has a multivariate Normal distribution and the $ X_i$ have variances $ \sigma_i^2 $, then the estimate of $ \mathbb{E[X_i]}$ depend only on the $ \sigma_i$ and $ n$.   Whence, to achieve adequate accuracy in estimating all the $ \mathbb{E}[X_i]$, we only need to regard the sum of data needed for the $ X_i$ having the largest of the $ \sigma_i$.   Therefore, when we contemplate a sequence of estimation trouble for increasing dimensions $ d$, all we need to think is how much the largest $ \sigma_i$ will increase.   When these parameters are restrict above, we conclude that the amount of data needed does not count on dimension. 
 Similar considerations apply to estimating the variances $ \sigma_i^2 $ and covariances $ \sigma_{ij}$: if a certain amount of data suffice for estimating one covariance (or correlation coefficient) to a desired accuracy, then -- provide the underlying normal distribution induce similar parameter values -- the same amount of datum will suffice for estimating any covariance or correlation coefficient. 

 To illustrate, and provide empirical support for this argument, let's study some pretense.   The following produce parameters for a multinormal distribution of specified dimension, draws many autonomous, identically distributed sets of vectors from that distribution, estimates the parameter from each such sample, and summarizes the results of those parameter estimates in term of (1) their averages -- to demonstrate they are unbiased (and the code is working correctly -- and (2) their standard deviations, which quantify the accuracy of the estimates.   (Do not confuse these standard deviations, which quantify the sum of variation among estimates obtain over multiple iterations of the simulation, with the standard deviations used to define the underlying multinormal distribution !)   My call is that these standard deviations do not materially change when the
 nulber varies as the numbev of dimensions nbcreasxs. 

A:

The amount uf data needed to estimate tie pqrameuvrs of a multivariate Normal dpstributiin ti within spxdified accuradn to c jiven confidencg does not vdry with the dhmdndion, all other things being the samq.  Therrflre you may apkly amr rumv if thumb for two dimensions tk higheg dimensional pronlems without any change ah alp.
Why should it? Thege are only thrqw kinds of pxrameters: means, variandes, and covariances.  The error uf escimate in a mwan gepends onlb the nariance and bne amognt of cata, $n$.  Thus, wmen $(X_1, X_2, \odots, X_d)$ has a multiveriate Normal distrifution ang che $X_i$ have variances $\sugma_i^2$, thet thd esgimztxs kf $\matjbb{X[X_i]}$ depend knly on the $\sigma_i$ and $n$.  Whenve, np achieve adsquate ascuracy in estimating all the $\mathbb{E}[X_i]$, we only need to consider rhe amount of data negded for tre $X_i$ having the largest of the $\sigma_i$.  Therefore, when de eintemooahe a succession of estimation problems for insdesspng dimensions $d$, cll we need to cpndicgr is how much the lcdgsst $\sigma_i$ will infrease.  When these pawameyers are bounded above, we cinclude that rhe amount of data needed doer noj depemd on dimension.
Similar eonsidsrations apoly to esfkmating the varixncvs $\shgma_i^2$ and covariances $\sigmw_{ij}$: if a rertann amoung of data fuffice fog estlkating one covariajce (ot corralation coffficient) to a desired accuracy, then--provided jhe ungerlying normsl distributijn has similar paramecer vauues--the saje amouit of data wyll suffice fmt estimating eny covaryancw or correlxgion coefficiemt.

To illustrate, and provide empirical sukpkrt for this arybmtnr, let's study spme siiupavions.  The followitg cfeagrs pafameters fov a mulyinormal distributiot of specified dimensipnf, draws nany indqpendent, idenyically distributef setv oh vectprs from that distribution, estimafes the pwraieters from qach such sample, and summarizes the results of those pacameter estimates in tgrms of (1) their averaggs--tp demonstrave ther are unbhased (and the code iw working correctky--and (2) their standard deviadions, which quantify the accuracy of the estimates.  (Do not confuse these standqrd detiwtions, whicg qusntifj tke cmount jf vermation among estikates obtained over multiple itxrations ox che simulation, with the standsra deviations osed to define the underlyjng multonormal distribution!)  My claim os that these standard dcviations do nkt matvtiauly change when thr
number varies as the number of dimensions The of data to estimate the distribution within specified accuracy a given confidence not vary with the dimension, all things being the same. Therefore you may apply any rule of thumb for dimensions to higher dimensional problems without any change at all. Why should it? are three of means, variances, and covariances. The error of estimate in a mean depends only the variance and amount of data, $n$. Thus, when $(X_1, X_2, X_d)$ has a multivariate distribution and the $X_i$ have $\sigma_i^2$, the estimates $\mathbb{E[X_i]}$ only the $\sigma_i$ and Whence, to achieve adequate accuracy in estimating all the $\mathbb{E}[X_i]$, we only need to consider the amount data needed $X_i$ having largest the Therefore, when we succession of estimation problems for increasing we need to consider is how much the $\sigma_i$ will When these parameters are bounded above, conclude that the amount of data needed does depend on dimension. Similar considerations apply to estimating the variances $\sigma_i^2$ and covariances $\sigma_{ij}$: if amount of data suffice estimating one covariance correlation to desired then--provided the normal distribution has similar parameter values--the same amount of data will for estimating any covariance or correlation coefficient. To illustrate, and support this argument, let's some simulations. The following parameters a multinormal distribution of draws independent, of from distribution, estimates the parameters each such sample, and summarizes results of those parameter their averages--to demonstrate they are unbiased (and the is working correctly--and (2) their standard deviations, quantify the accuracy of the estimates. (Do not confuse these standard deviations, quantify the variation among estimates obtained over multiple iterations of simulation, with the standard used to define the underlying multinormal distribution!) My claim that standard deviations not materially change the
 number varies as the number of Dimensions IncreAseS. 

A:

THe AmouNt of Data needed to esTImatE the parameters of a multiVariaTe nOrmaL DiStribUtion to WItHIN spEcIfIed AcCUrAcy to A giVen confIdence does Not VaRy with the dimENsIon, all otheR thIngs being the SamE.  ThereFoRe yOU may aPplY any rUle of tHUmb for Two dimensIoNS to higHEr dimenSIOnAl prOblems without any cHAnGE at all.
Why shoulD it? TheRe ARe ONLy tHreE kinds of paRaMeterS: Means, vaRIaNCES, anD Covariances.  ThE error of estIMatE in a meAn DepENds onlY the vArIAncE and the amouNt of Data, $n$.  Thus, When $(X_1, X_2, \LDots, X_d)$ hAS a multiVariatE NoRmaL disTRiBuTioN aND thE $x_i$ HavE VarIances $\siGmA_i^2$, Then tHe esTIMATes oF $\maThbb{e[X_i]}$ dePend only on the $\SigMa_i$ aND $n$.  WHence, To achIeve AdEquatE accurAcy in EsTimating all the $\mAthbB{E}[X_i]$, we onlY neEd To cOnSider THe amouNt oF daTa needeD for the $x_I$ haViNG THe Largest of the $\sigma_i$.  thEREfOre, when wE conteMPlAtE A successIoN of EstiMATion pRoblEMs For increAsing dIMeNsIons $d$, alL wE need tO cOnsIdeR is hoW Much The larGest $\sigmA_i$ wilL Increase.  When thESe parameters aRE bOUNdED aboVe, wE conclude thAt thE AmouNt of DAtA neEDed doEs not DePEnD On dimension.
Similar cOnSideraTions Apply to estimaTing the varIANCes $\sigma_I^2$ and COvARiances $\sigma_{ij}$: If a ceRtain amounT Of data suFfice For estimAting one cOVAriance (oR coRreLatIon COEfFicient) to a desIREd acCuRacy, theN--prOvided tHe uNdeRlyIng NoRmal distrIbution hAs SiMiLaR paRametER values--tHe SamE aMouNt of dATa will SuffiCe foR eStIMatIng any cOVaRIAnce Or CoRrelAtiOn CoeffIcieNT.

To IllustrAte, and proVidE EmpiRiCaL supporT for this argumEnT, let's study SoMe sImulatIONs.  The folLowing creates parameters FOr a multInoRmal dIstrIbution of SpeCified DimENsions, Draws mAny inDePenDENt, ideNTIcAllY dIstributed SETs oF vectOrS froM that diStribution, estimateS The Parameters froM eaCh suCH SaMplE, AnD SumMaRIzeS THe results of thosE parameter EsTImAtes in termS Of (1) tHeIr averaGes--to deMonstRAte they Are unbiasEd (and the cOdE is wORKinG correctly--And (2) their Standard dEViatiONs, Which QuaNtify tHe AccUracy Of the eSTimAtes.  (DO not coNfUse theSe staNdArd deviaTions, which quantify the amOunt of VariaTioN among estImaTEs oBtained ovEr muLtiple iterAtiOns Of the SimULatioN, witH ThE stANdard DeviATions used TO dEfiNE ThE underlying MULTinOrmal DisTRibutiOn!)  My Claim is that these sTAndard deviatioNs do NOT maTerIAlly ChAnge when the
 number varies as the numb er of dime nsion s i ncr ea ses.  

A :

The amounto f da ta needed to estimatethe p ar a mete r sof amultiva r ia t e  No rm al  di st r ib ution  to  within  specified  ac cu racy to a gi v en  confidenc e d oes not vary  wi th the  d ime n sion,  al l oth er thi n gs bei ng the sa me .   Ther e fore yo u ma y ap ply any rule of t h um b  for two dimen sionsto hi g h erdim ensional p ro blems without an y c han g e at all.
Why  should it? The re are  o nly threekinds  o f  pa rameters: m eans , varianc es, an d  covari a nces. The er ror  of  est i ma te  in  a mea n  d epe n dsonly the  v ar iance  and t h e  amo unt  ofdata,  $n$.  Thus,whe n $( X _1,  X_2,  \ldo ts,X_ d)$ h as a m ultiv ar iate Normal dis trib ution and  th e$X_ i$  have varian ces  $\ sigma_i ^2$, th e n t he e s ti mates of $\mathbb{ E[ X _ i] }$ depen d only on  t h e $\sigm a_ i$and$ n $.  W henc e ,to achie ve ade q ua te  accura cy  in es ti mat ing  allt he $ \mathb b{E}[X_i ]$, w e  only need toc onsider the a m ou n t  o f  dat a n eeded for t he $ X _i$havi n gthe large st of  t h e$ \sigma_i$.  Therefo re , when  we c ontemplate asuccession o f  estimat ionp ro b lems for incre asing  dimension s  $d$, al l weneed toconsideri s  how muc h t helar ges t $\ sigma_i$ will i ncre as e.  Whe n t hese pa ram ete rsare  b ounded ab ove, weco nc lu de  th at th e  amountof  da ta  ne ededd oes no t dep endon  d i men sion.
S i mi l a r co ns id erat ion sapply  toe sti matingthe varia nce s  $\s ig ma _i^2$ a nd covariance s$\sigma_{i j} $:if a c e r tain amo unt of data suffice for estimat ing  onecova riance (o r c orrela tio n  coeff icient ) toades i r ed ac c u ra cy,  t hen--provi d e d t he un de rlyi ng norm al distribution ha s  si milar paramet ervalu e s -- the sa m e a mo u nto f  data will suff ice for es ti m at ing any co v ari an ce or c orrelat ion c o efficie nt.

To i llustrate ,andp r ovi de empiric al suppo rt for th i s arg u me nt, l et' s stud ysom e sim ulatio n s. Thefollow in g crea tes p ar ametersfor a multinormal distr ibutio n ofspe cified di men s ion s, drawsmany  independe nt,  id entic all y  dist ribu t ed  se t s ofvect o rs from t h at  di s t ri bution, est i m a tes  thepar a meters  fro m each such sampl e , and summariz es t h e  re sul t s of  t hose parameter  es ti m a tes in t er ms of (1) t heir ave ra g es--t o demo nstrat e theya r eu nbiase d (a ndthe codeiswo r king co rr ec t ly--an d (2 )theirstanda r d de v i ations, which qu antif y the a c cur acy o fthe est i mate s.  (Do no t confuse t hese s tand ard d eviatio ns , whic h q ua ntify thea mount ofvaria tion am on g es tim ates o btai n e d ove r mu lt ipl e iterati o n so fth e  si mula tion,  w iththe stand a rd devia tio n s usedto  de f i ne the un d e rlying mul tin ormal d istributio n !) M yc laimis tha t thes e stand a rdde viation s d o not mater ially cha n gewh en t he
 number_varies as_the number of dimensions_increases. 

A:

The_amount_of data_needed_to estimate the_parameters of a_multivariate Normal distribution to_within specified accuracy_to_a given confidence does not vary with the dimension, all other things being the_same._ Therefore_you_may_apply any rule of thumb_for two dimensions to higher_dimensional problems_without any change at all.
Why should it? There_are_only three kinds_of parameters: means, variances, and covariances.  The error_of estimate in a mean depends_only the variance_and_the_amount of data, $n$._ Thus, when $(X_1, X_2, \ldots,_X_d)$ has a multivariate Normal distribution_and the $X_i$ have variances $\sigma_i^2$, then_the estimates of $\mathbb{E[X_i]}$ depend only_on the $\sigma_i$ and $n$._ Whence,_to achieve adequate accuracy in_estimating all the_$\mathbb{E}[X_i]$, we_only need to_consider the amount of data needed_for the $X_i$_having the largest of the $\sigma_i$.__Therefore, when we_contemplate_a_succession of_estimation problems for_increasing_dimensions $d$,_all_we need to consider is how_much_the largest $\sigma_i$ will increase.  When_these parameters are bounded_above,_we conclude that the_amount of data needed does_not depend on dimension.
Similar considerations apply_to estimating_the variances_$\sigma_i^2$ and covariances $\sigma_{ij}$: if a certain amount of data suffice_for estimating one covariance (or correlation_coefficient) to a desired_accuracy, then--provided_the_underlying normal distribution_has_similar parameter_values--the same amount of data will suffice_for estimating_any covariance or correlation coefficient.

To illustrate,_and provide empirical support_for_this argument, let's study some simulations._ The following creates parameters for_a multinormal distribution of specified_dimensions,_draws_many independent, identically distributed sets_of vectors from that distribution, estimates_the parameters from_each such sample, and summarizes the results_of_those parameter estimates in terms of_(1)_their averages--to demonstrate they are unbiased_(and_the_code is working correctly--and (2)_their standard deviations, which quantify the_accuracy of the estimates.  (Do not confuse these_standard deviations, which_quantify the amount of variation_among_estimates_obtained over multiple iterations of the simulation, with the standard_deviations used_to define the_underlying multinormal distribution!)  My claim is that these standard_deviations do not materially change when the