title: Error using integral (line 85): A and B must be floating-point scalars
body: <p>I have the following code that I wish to estimate the parameters of a custom distribution using MATLAB's function <code>mle()</code>. <a href="https://stackoverflow.com/q/56522903/4930944">For more details on the distribution</a>. </p>
  
  <p>The main code is:</p>
  
  <pre><code>x           =  [0   0   0   0   0.000649967501624919    0.00569971501424929 0.0251487425628719  0.0693465326733663  0.155342232888356   0.284835758212089   0.458277086145693   0.658567071646418   0.908404579771011   1.17284135793210    1.43977801109945    1.71951402429879    1.98925053747313    2.27553622318884    2.57147142642868    2.80390980450977    3.03829808509575    3.26583670816459    3.45642717864107    3.65106744662767    3.81950902454877    3.98275086245688    4.11259437028149    4.24683765811709    4.35043247837608    4.43832808359582    4.58427078646068    4.62286885655717    4.68361581920904    4.75686215689216    4.80245987700615    4.84005799710015    4.86280685965702    4.91675416229189    4.92725363731813    4.90890455477226    4.96570171491425    4.92315384230789    4.95355232238388    4.92790360481976    4.93135343232838    4.90310484475776    4.90885455727214    4.86765661716914    4.85490725463727    4.81940902954852    4.81450927453627    4.78621068946553    4.74206289685516    4.71791410429479    4.69961501924904    4.65706714664267    4.63611819409030    4.60176991150443    4.57512124393780    4.53507324633768    4.48252587370631    4.47062646867657    4.43127843607820    4.39963001849908    4.37598120093995    4.29548522573871    4.31033448327584    4.21708914554272    4.21913904304785    4.18669066546673    4.16719164041798    4.09774511274436    4.07989600519974    4.02869856507175    3.98485075746213    3.95785210739463    3.93945302734863    3.90240487975601    3.87025648717564    3.81185940702965    3.78461076946153    3.74091295435228    3.71666416679166    3.67276636168192    3.65846707664617    3.61361931903405    3.58712064396780    3.55452227388631    3.53082345882706    3.49197540122994    3.48582570871456    3.46512674366282    3.41227938603070    3.36278186090695    3.35528223588821    3.31238438078096    3.27213639318034    3.23863806809660    3.24173791310434    3.19339033048348    3.20118994050298    3.16489175541223    3.10739463026849    3.09484525773711    3.08094595270237    3.02129893505325    3.02309884505775    2.99375031248438    2.95765211739413    2.93230338483076    2.89560521973901    2.87805609719514    2.85440727963602    2.82285885705715    2.80175991200440    2.79091045447728    2.73901304934753    2.72701364931753    2.73441327933603    2.71646417679116    2.68236588170592    2.65551722413879    2.63356832158392    2.60361981900905    2.58147092645368    2.57697115144243    2.54287285635718    2.53502324883756    2.47702614869257    2.50387480625969    2.46487675616219    2.45722713864307    2.42707864606770    2.41762911854407    2.39823008849558    2.38708064596770    2.34058297085146    2.35613219339033    2.32123393830309    2.30503474826259    2.27613619319034    2.27248637568122    2.25113744312784    2.24908754562272    2.22703864806760    2.20583970801460    2.17244137793110    2.15709214539273    2.16469176541173    2.12139393030348    2.12809359532023    2.11389430528474    2.09774511274436    2.07629618519074    2.07459627018649    2.05394730263487    2.04724763761812    2.01684915754212    2.01684915754212    2.00409979501025    1.98955052247388    1.96540172991350    1.95890205489726    1.93035348232588    1.92295385230738    1.90605469726514    1.89785510724464    1.87070646467677    1.88000599970002    1.86295685215739    1.84420778961052    1.82510874456277    1.80480975951202    1.80785960701965    1.80870956452177    1.77581120943953    1.76771161441928    1.77131143442828    1.76636168191590    1.75081245937703    1.73156342182891    1.69876506174691    1.70836458177091    1.70376481175941    1.67196640167992    1.68101594920254    1.66586670666467    1.66061696915154    1.64296785160742    1.63291835408230    1.62506874656267    1.62516874156292    1.60556972151392    1.59007049647518    1.59187040647968    1.57947102644868    1.57577121143943    1.54527273636318    1.57237138143093    1.54637268136593    1.54802259887006    1.50492475376231    1.52077396130193    1.50417479126044    1.50162491875406    1.50062496875156    1.48957552122394    1.47997600119994    1.47027648617569    1.44452777361132    1.45407729613519    1.44272786360682    1.43247837608120    1.41657917104145    1.40787960601970    1.39323033848308    1.40282985850707    1.39403029848508    1.38233088345583    1.37888105594720    1.37943102844858    1.36183190840458    1.34808259587021    1.34503274836258    1.33703314834258    1.33308334583271    1.32253387330633    1.32698365081746    1.29963501824909    1.30758462076896    1.29103544822759    1.29473526323684    1.27413629318534    1.26858657067147    1.27888605569722    1.26063696815159    1.27863606819659    1.25168741562922    1.23913804309785    1.24788760561972    1.22308884555772    1.24198790060497    1.22133893305335    1.20678966051697    1.20098995050247    1.20343982800860    1.18779061046948    1.19024048797560    1.17194140292985    1.17369131543423    1.16869156542173    1.15814209289536    1.15429228538573    1.15904204789761    1.12774361281936    1.15344232788361    1.13744312784361    1.12909354532273    1.12479376031198    1.11099445027749    1.11469426528674    1.11064446777661    1.10464476776161    1.10309484525774    1.10689465526724    1.07654617269137    1.07884605769712    1.07359632018399    1.06864656767162    1.07544622768862    1.06689665516724    1.04884755762212    1.06164691765412    1.04979751012449    1.04529773511324    1.02839858007100    1.03634818259087    1.01709914504275    1.02089895505225    1.01024948752562    1.01549922503875    1.01319934003300    1.01404929753512    1.00839958002100    0.995400229988501   0.989850507474626   0.978801059947003   0.977551122443878   0.980450977451127   0.975451227438628   0.969201539923004   0.964151792410380   0.964601769911504   0.958802059897005   0.955702214889256   0.948602569871506   0.960751962401880   0.941352932353382   0.928653567321634   0.949002549872506   0.937053147342633   0.913854307284636   0.916204189790510   0.915454227288636   0.902604869756512   0.909454527273636   0.895505224738763   0.898355082245888   0.894455277236138   0.902454877256137   0.883705814709265   0.888405579721014   0.876356182190891   0.881555922203890   0.878156092195390   0.868456577171141   0.870406479676016   0.863906804659767   0.862456877156142   0.858757062146893   0.851307434628269   0.851107444627769   0.833908304584771   0.843507824608770   0.831708414579271   0.836858157092145   0.829058547072646   0.828508574571272   0.822908854557272   0.820508974551273   0.815559222038898   0.819709014549273   0.809609519524024   0.813409329533523   0.800759962001900   0.806609669516524   0.806959652017399   0.792260386980651   0.787660616969152   0.783810809459527   0.794960251987401   0.771061446927654   0.788910554472276   0.789510524473776   0.763061846907655   0.776761161941903   0.767561621918904   0.773611319434028   0.750262486875656   0.765811709414529   0.765911704414779   0.748012599370032   0.741612919354032   0.757312134393280   0.752612369381531   0.741362931853407   0.742212889355532   0.741912904354782   0.743162841857907   0.732963351832408   0.732813359332033   0.733363331833408   0.721913904304785   0.716664166791661   0.726713664316784   0.709764511774411   0.700064996750163   0.710764461776911   0.717664116794160   0.707314634268287   0.707114644267787   0.705614719264037   0.709164541772911   0.696665166741663   0.680765961701915   0.686715664216789   0.694465276736163   0.683015849207540   0.681715914204290   0.694465276736163   0.688615569221539   0.680665966701665   0.672316384180791   0.672866356682166   0.656517174141293   0.665316734163292   0.671566421678916   0.666266686665667   0.652917354132293   0.663366831658417   0.651917404129794   0.663816809159542   0.661366931653417   0.647017649117544   0.655167241637918   0.647867606619669   0.636918154092295   0.645467726613669   0.633118344082796   0.640217989100545   0.634668266586671   0.618669066546673   0.635068246587671   0.632568371581421   0.623118844057797   0.623868806559672   0.623718814059297   0.621368931553422   0.623768811559422   0.608419579021049   0.616019199040048   0.609869506524674   0.606569671516424   0.614019299035048   0.610269486525674   0.596520173991300   0.595570221488926   0.593270336483176   0.596670166491675   0.598470076496175   0.597770111494425   0.593720313984301   0.592770361481926   0.585420728963552   0.580870956452177   0.584120793960302   0.580270986450677   0.577971101444928   0.579021048947553   0.572821358932053   0.585970701464927   0.572921353932303   0.567071646417679   0.569971501424929   0.571271436428179   0.568421578921054   0.567421628918554   0.569521523923804   0.563721813909305   0.558772061396930   0.562171891405430   0.557872106394680   0.549072546372681   0.558722063896805   0.536973151342433   0.561021948902555   0.544172791360432   0.552122393880306   0.553072346382681   0.546222688865557   0.551472426378681   0.540772961351932   0.541122943852807   0.542772861356932   0.530323483825809   0.526023698815059   0.529273536323184   0.524573771311435   0.525923703814809   0.524923753812309   0.516474176291185   0.527273636318184   0.527723613819309   0.518424078796060   0.517874106294685   0.516074196290186   0.517924103794810   0.523173841307935   0.514474276286186   0.513174341282936   0.498875056247188   0.518024098795060   0.507924603769812   0.505524723763812   0.507174641267937   0.502874856257187   0.502624868756562   0.500624968751562   0.510824458777061   0.490925453727314   0.492675366231688   0.489925503724814   0.478126093695315   0.485775711214439   0.491775411229439   0.489925503724814   0.491325433728314   0.487225638718064   0.485725713714314   0.485675716214189   0.477676116194190   0.483875806209690   0.478026098695065   0.470176491175441   0.471926403679816   0.483625818709065   0.469376531173441   0.474026298685066   0.467826608669567   0.462426878656067];
  
  Censored    =  ones(1,size(x,2));% 
  custpdf     =  @eval_custpdf;
  custcdf     =  @eval_custcdf;
  phat        =  mle(x,'pdf', custpdf,'cdf', custcdf,'start',[1 0.1 0.3 0.1 0.01 -0.3],...
                 'lowerbound',[0 0 0 0 0 -inf],'upperbound',[inf inf inf inf inf inf],'Censoring',Censored);
  
  % Cheking how close the estimated PDF and CDF match with those from the data x
  t   =  0.001:0.001:0.5;
  figure();
  plot(t,x);hold on
  plot(t,custpdf(t, phat(1), phat(2), phat(3), phat(4), phat(5), phat(6))) 
  
  figure();
  plot(t,cumsum(x)./sum(x));hold on
  plot(t,custcdf(t, phat(1), phat(2), phat(3), phat(4), phat(5), phat(6)))
  </code></pre>
  
  <p>The functions are:</p>
  
  <pre><code>function out = eval_custpdf(x,myalpha,mybeta,mytheta,a,b,c)
    first_integral      =  integral(@(x) eval_K(x,a,b,c),0,1).^-1;
    theta_t_ratio       =  (mytheta./x);
    incomplete_gamma    =  igamma(myalpha,theta_t_ratio.^mybeta);
    n_gamma             =  gamma(myalpha);
    exponent_term       =  exp(-theta_t_ratio.^mybeta-(c.*(incomplete_gamma./n_gamma)));
  
  
    numerator   =  first_integral.* mybeta.*incomplete_gamma.^(a-1).*...
                 theta_t_ratio.^(myalpha*mybeta+1).*exponent_term;
    denominator =  mytheta.* n_gamma.^(a+b-1).* (n_gamma-incomplete_gamma.^mybeta).^(1-b);
  
    out         =  numerator./denominator;
  end
  
  function out = eval_custcdf(x,myalpha,mybeta,mytheta,a,b,c)
    first_integral      =  integral(@(x) eval_K(x,a,b,c),0,1).^-1;
    theta_t_ratio       =  (mytheta./x);
    incomplete_gamma    =  igamma(myalpha,theta_t_ratio.^mybeta);
    n_gamma             =  gamma(myalpha);
    second_integral     =  integral(@(x) eval_K(x,a,b,c),0, incomplete_gamma.^mybeta./n_gamma);
                                                           % |&lt;----- PROBLEMATIC LINE -----&gt;|
  
    out     =  first_integral*second_integral;
  
  end    
  
  function out = eval_K(x,a,b,c)
  
    out =  x.^(a-1).*(1-x).^(b-1).*exp(-c.*x);
  
  end
  </code></pre>
  
  <p>The integral that is causing the problem is the second intergral in the function <code>eval_custcdf()</code> as its upper limit is an array (denoted by <em>PROBLEMATIC LINE</em>).</p>
  
  <p>Is there a way to take a single value from the array <code>x</code> such that the upper limit remains a scalar? And then calculate the cdf such that the output of the cdf is an array? Using a <code>forloop</code>, maybe? But I cannot seem to figure how to implement that?</p>
  
  <p>How can I work around this problem?</p>
  
  <p>Any help would be appreciated.</p>
  
  <p>Thanks in advance.</p>