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astronomers commonly seek to estimate the _ space density _ of objects , and a sky survey such as the sloan digital sky survey ( sdss ) @xcite can yield a representative sample useful for this purpose , due to the assumed isotropy of the universe . figure [ rawdata ] depicts redshift and absolute magnitude measurements for a sample of quasars given in @xcite . these are a subset of the sdss quasar sample ( data release 3 ) , chosen to be statistically valid for purposes such as exploring the evolution with redshift of the luminosity function , i.e. the space density of quasars as a function of absolute magnitude . this paper describes a new method for estimating these luminosity functions , and presents results from the analysis of this quasar sample . for the purposes of the statistical inference problem , imagine the dots in figure [ rawdata ] as observations of bivariate data @xmath7 from some distribution with probability density @xmath2 , i.e. the probability that a randomly chosen quasar falls in a region @xmath8 is @xmath9 . ( equivalently , in a sample of size @xmath10 , one expects that @xmath11 will fall in the region @xmath8 . ) hence , the luminosity function at redshift @xmath12 is , up to a multiplicative constant , the cross - section of the bivariate density at @xmath12 , denoted @xmath13 . the main challenge is estimation of this bivariate density given truncated data . only objects with apparent magnitude within some range are observable . when this bound on apparent magnitude is transformed into a bound on absolute magnitude , @xmath14 , @xmath15 is assumed when making this transformation . ] , the truncation bound takes an irregular shape , varying with redshift . @xmath16-corrections further complicate this boundary , leading to the dashed region in figure [ rawdata ] . also , the sample is not assumed to be complete within this region , and the probability of observing an object will vary with position on the sky , along with other factors . incorporating this _ selection function _ into the analysis is a secondary challenge . nonparametric estimators are advantageous in cases where either there does not exist a commonly agreed upon parametric physical model , or there is a desire to validate a parametric model . see @xcite for an overview of the potential of nonparametric methods in astronomy and cosmology . a fully nonparametric approach is not possible here , since some assumptions must be placed on the form of the density in order to infer its shape over the unobservable region . under such conditions , one approach would be to fit a sequence of increasingly complex parametric models in an attempt to obtain a good fit to the data . a less subjective alternative is a _ semiparametric _ approach which merges a nonparametric method with sufficient structure from a parametric form to obtain useful results . this work describes a semiparametric approach to estimating the bivariate density , and hence the luminosity functions , under irregular truncation . this is a long - standing challenge in astronomical data analysis , with a variety of proposed methods . interesting qualitative and simulations - based comparisons between different approaches can be found in @xcite and @xcite . a parametric model fit using maximum likelihood is a common choice , since it addresses the truncation bias in a natural manner ; see , for instance , @xcite , @xcite and the parametric models fit and referenced in  6 of @xcite . these models have the drawback of imposing a tight constraint on the luminosity function in a case where there is not a consensus parametric form . some proposed methods are nonparametric , but assume that redshift and absolute magnitude are independent , and hence assume that there is no evolution of the luminosity function with redshift . these include the nonparametric maximum likelihood method described in @xcite and @xcite and adapted for double truncation in @xcite , along with the methods in @xcite , @xcite , the @xmath17 estimator of @xcite and @xcite . the semiparametric method of @xcite also assumes independence . @xcite apply a nonparametric technique which assumes independence after having transformed the bivariate data using a parametric form . any method which assumes independence can be applied over small redshift ranges ( usually called bins ) . @xcite and @xcite describe other binning approaches . binning forces the difficult choices of bin centers and widths , and independence is still assumed over the width of the bin . this work was motivated by the goal of developing a statistically rigorous method which ( 1 ) does not assume a strict parametric form for the bivariate density ; ( 2 ) does not assume independence between redshift and absolute magnitude ; ( 3 ) does not require dividing the data into arbitrary bins ; and ( 4 ) naturally incorporates a varying selection function . this was accomplished by decomposing the bivariate density @xmath2 into @xmath3 where @xmath18 will take an assumed parametric form ; it is intended to model the dependence between the two random variables . for example , there may be a physical , parametric model for the evolution of the luminosity function which could be incorporated into @xmath18 . alternatively , one could use @xmath19 as a first - order approximation to the dependence . the functions @xmath4 and @xmath5 are estimated nonparametrically , with _ bandwidth _ parameters to control the amount of smoothness in the estimate . using the quasar sample of figure [ rawdata ] , the estimates obtained here are quite consistent , if not a bit smoother , than those found in @xcite . this analysis confirms the finding of the flattening of the slope of the luminosity function at higher redshift . the paper is organized as follows .  [ data ] briefly describes the quasar sample used here . [ ourmodel ] gives an overview of the idea of local maximum likelihood , a nonparametric extension of maximum likelihood , and describes in detail the semiparametric approach taken here . [ bandselect ] describes how the integrated mean squared error can be approximated using cross - validation ; the bandwidths can then be chosen to minimize this quantity . [ results ] presents some results from the analysis of the @xcite quasar sample , along with the results from some simulations . more detailed derivations , along with theory for approximating the distribution of the estimator , can be found in @xcite . the approach was implemented as a fortran subroutine with r wrapper ] ] . the full @xcite sample , shown in figure [ rawdata ] , consists of 15,343 quasars . from these , any quasar is removed if it has @xmath20 , @xmath21 , @xmath22 , or @xmath23 . in addition , for quasars of redshift less than 3.0 , only those with apparent magnitude between 15.0 and 19.1 , inclusive , ( after the application of @xmath16-corrections ) are kept ; for quasars of redshift greater than or equal to 3.0 , only those with apparent magnitude between 15.0 and 20.2 are retained . these boundaries combine to create the irregular shape shown by the dashed line in figure [ rawdata ] . this truncation removes two groups of quasars from the @xcite sample . first , there are 62 quasars removed with @xmath22 . this was done to mitigate the effect of the irregularly - shaped , very narrow region in the lower left corner of figure [ rawdata ] . second , there are 224 additional quasars with @xmath24 and apparent magnitude larger than 19.1 ; these fall in an extremely poorly sampled region , which can also be noted from figure [ rawdata ] . hence there are 15,057 quasars remaining after this truncation . the sample is not assumed to be complete within this region . associated with each sampled quasar is a value for the _ selection function _ , which can be interpreted as the probability that a quasar at this location , and of these characteristics would be captured by the sample . details regarding how the selection function was approximated via simulations , along with many other details regarding the sample , can be found in @xcite . the approach taken here is built upon a nonparametric extension of maximum likelihood called _ local likelihood modeling_. this section begins by describing local likelihood density estimation in the general case . this is then adapted to the problem at hand , initially for the case assuming the random variables are independent . the case where dependence is allowed is then described as a simple extension .      to contrast the standard _ global _ approach to estimation with the local approach employed here , consider the following . assume the data @xmath25 are realizations ( observations ) of independent , identically distributed random variables from a distribution with density @xmath26 . with classic maximum likelihood estimation , one chooses a single estimate from among a class of candidates for @xmath26 ; let @xmath27 denote this class . specifically , the maximum likelihood estimator ( @xmath28 ) for @xmath26 is defined as the @xmath29 which maximizes @xmath30     \label{usuallike}\ ] ] or , equivalently , the @xmath29 which maximizes @xmath31 ( the notation @xmath32 simultaneously indicates a random variable with unknown density @xmath26 , and the observed realization of that random variable . ) although written here like a density estimation problem , one could imagine the class @xmath27 being indexed by a parameter @xmath33 ; hence this also captures the usual maximum likelihood estimator for parametric problems . for example , one could define @xmath27 to consist of all gaussian densities as mean @xmath34 and variance @xmath35 vary . in cases where each @xmath29 is a density ( e.g. , the aforementioned gaussian case ) , the expression in brackets of equation ( [ usuallike ] ) is always zero , and thus unnecessary . however , it is often advantageous to let @xmath27 be a wider class of smooth , nonnegative functions ; then the bracketed term forces @xmath28 to be a probability density . with local modeling , instead of seeking the single member of the class @xmath27 to be the estimate of @xmath36 , the goal is to approximate @xmath37 for @xmath38 near @xmath39 , yielding the _ local estimate _ @xmath40 . typically , @xmath41 can be approximated locally by a polynomial ; in fact , a linear form for @xmath42 usually suffices . see figure [ explainloclike ] . on the left plot , the dashed line gives the logarithm of the gaussian density with mean zero and variance one . local linear estimates @xmath42 are shown for each of @xmath43 . it is unimportant that @xmath44 is not a good estimate of @xmath37 for @xmath38 far from @xmath39 , since many such local estimates will be found and then smoothed together . these local estimates were calculated with a simulated data set consisting of 10,000 values . the method for finding these local estimates is outlined next .    in independent work , @xcite and @xcite localized the likelihood criterion of equation ( [ shortlike ] ) near @xmath45 by writing @xmath46 where @xmath47 is a kernel function parametrized by @xmath48 . a standard choice would be @xmath49 where @xmath16 is a probability density , but more specific forms will be considered ( and required ) below . the choice of @xmath50 typically has much more influence on the estimator than does the choice of the kernel function . the local estimate @xmath40 is found by maximizing @xmath51 over @xmath52 belonging to some simple class , usually degree @xmath53 polynomials expanded around @xmath39 : @xmath54 thus , the model is locally parametric with parameters @xmath55 . one imagines repeating this procedure at a grid of @xmath39-values , call this grid @xmath56 , and hence obtaining a family of local estimates @xmath57 . as a result , @xmath58 is the family @xmath4 of local estimates which maximizes @xmath59 the final local likelihood estimator @xmath60 is constructed by smoothing together the local estimates : @xmath61 thus making dual use of @xmath50 . returning to figure [ explainloclike ] , the plot on the right shows @xmath60 , the result of smoothing together 101 local linear estimates ( @xmath56 consists of 101 values between -3 and 3 ) . in this case , @xmath62 . it is clear that the estimate comes very close to the true density .    in what follows , simply assume that @xmath63 is chosen so that @xmath64 for all @xmath38 and hence @xmath65 this is a departure from the original approach of @xcite and @xcite , who instead used @xmath66 . the criterion @xmath67 appears awkward upon first sight , but it possesses the following property : considering @xmath68 again as random variables with unknown density @xmath26 , then @xmath69 is maximized by choosing the family @xmath4 which sets @xmath70 for all @xmath39 and all @xmath38 . if that choice were made , the estimate would be @xmath71 . thus , since @xmath72 , the local estimate @xmath42 will approximate the degree @xmath53 taylor expansion of @xmath73 for @xmath38 around @xmath39 . the expected value of the standard likelihood criterion is also maximized by setting the density equal to the truth , but this localized version has the advantage of allowing the choice of @xmath50 to adjust the amount of smoothness in the estimator . in  [ bandselect ] , an objective method for bandwidth selection is described . there is an apparent conflict between the choice of @xmath50 and the choice of the number of local models ( the cardinality of @xmath56 ) since small @xmath56 will lead to smooth estimates . in the applications here , @xmath56 is chosen large , so that the amount of smoothing is completely dictated by @xmath50 . now return to the bivariate density estimation problem using truncated astronomical data . the available data are denoted @xmath74 and @xmath75 , the vectors of redshifts and absolute magnitudes , respectively . let @xmath76 denote the region outside of which the data are truncated and let @xmath77 denote the cross - section of @xmath76 at @xmath12 ; @xmath78 is defined similarly . let @xmath2 denote the unknown joint density of random variables @xmath12 and @xmath79 . the approach taken here originates in the following naive method . for the moment assume @xmath12 and @xmath79 are independent so that @xmath80 where @xmath81 is the density for redshift and @xmath82 is the density for absolute magnitude . clearly , the available data allow estimation of the redshift density for observable quasars , denote this density @xmath83 . this is related to @xmath81 by @xmath84 where @xmath85 is a normalizing constant which forces @xmath83 to integrate to one . assuming for the moment that @xmath82 were known , it is possible to turn an estimator for @xmath83 into an estimator for @xmath81 by solving equation ( [ xmarg ] ) for @xmath81 : @xmath86 starting with an initial guess at @xmath82 , we could iterate between assuming @xmath82 is known , and estimating @xmath81 , and vice versa . this procedure is portrayed in figure [ explainmethod ] . using the quasar data set described in  [ data ] , the upper left plot depicts @xmath87 along the vertical axis , with absolute magnitudes ranging from -29.9 to -25.85 . an ( arbitrary ) assumption is made regarding the density for absolute magnitude @xmath88 , shown as the solid curve in the upper right plot . for example , one can find that @xmath89 and thus conclude that the observed sample catches 24% of the quasars at @xmath90 . ( the fact that some quasars are missed within @xmath76 is considered later when the selection function is incorporated into the analysis . ) the lower left plot shows how the proportion of quasars observed varies with redshift , i.e. it is a graph of @xmath91 versus @xmath92 . the dashed line in the lower right plot is @xmath93 , the estimated redshift density for observable quasars . the solid curve is @xmath94 , as defined above , found by dividing @xmath93 by the proportion of quasars observed at redshift @xmath12 , and then normalizing to force the estimate to be a density . figure [ explainmethod ] also illustrates problems with this approach . first , the sharp corner of @xmath76 at @xmath95 leads to a sharp feature in the estimate @xmath94 . in other words , smooth @xmath83 does not produce a smooth @xmath94 . second , consider the behavior of @xmath96 for @xmath12 where @xmath97 is small , for instance @xmath98 : even a small error in the estimate of @xmath97 will lead to a large error in @xmath96 . the fundamental challenge is that a well - chosen estimator ( i.e. , well - chosen smoothing parameters ) for @xmath83 does not necessarily lead to @xmath94 being a good estimator for @xmath81 . in addition , it is possible to construct examples where this iterative approach will converge to different estimates starting from different initial values for @xmath82 . despite the aforementioned problems with the use of @xmath94 , that approach can be improved using the local likelihood methods of  [ locliketut ] . in what follows , @xmath83 is estimated using local polynomial models which include an additive _ offset _ term . this offset is chosen so that when subtracted off , what remains is a good estimator for @xmath81 . the procedure is fundamentally the same as that for constructing @xmath94 : starting with an initial guess as to the value of the density for absolute magnitude ( @xmath82 ) , the relationship between @xmath81 , @xmath82 , and @xmath83 ( shown in equation ( [ xmarg ] ) ) is exploited to construct an estimator for @xmath81 . ( here it is assumed that @xmath2 is normalized so that @xmath99 , but this choice is arbitrary since the estimate can be extended outside of @xmath76 and then renormalized as appropriate . )    to start , rewrite equation ( [ xmarg ] ) as @xmath100 where @xmath85 is the constant required to force @xmath99 . consider the goal of estimating @xmath101 for @xmath38 near @xmath39 . ideally , it would be possible to fit a local model @xmath102 to obtain both the local estimate @xmath40 and the needed normalizing constant @xmath85 , but truncation does not allow for direct estimation of @xmath81 . instead , write a local version of equation ( [ withoff ] ) as @xmath103 and then substitute in the expression for @xmath104 from equation ( [ ideal ] ) into equation ( [ locverswithoff ] ) to get @xmath105 of course , it is possible to estimate @xmath83 with the available data and equation ( [ offsetexp ] ) makes it clear that a good way of doing this would be to fit a local polynomial model with @xmath106 included as an offset . ( recall that , for the moment , @xmath82 is assumed known . ) in other words , instead of maximizing the local likelihood criterion @xmath107 over @xmath108 that are polynomials expanded around @xmath39 ( as in equation ( [ polyexp ] ) ) , maximize over functions of the form @xmath109 write @xmath110 as the local likelihood at @xmath39 when the offset is included . label the parameters which maximize @xmath110 as @xmath111 . comparing equations ( [ ideal ] ) and ( [ offsetexp ] ) , note that @xmath112 is an estimate of @xmath113 and hence @xmath114 is the local ( near @xmath39 ) estimate of @xmath115 . as before , this is repeated for a grid of values @xmath116 and the result is the family @xmath58 which maximizes @xmath117 and the estimate of @xmath118 is found by smoothing together these local estimates : @xmath119 here , it is stressed that estimates of @xmath118 are smoothed together , instead of estimates of @xmath83 . this is important because now @xmath50 can be chosen to obtain the optimal amount of smoothing for the best estimate of @xmath118 . this avoids the problems which were evident at @xmath95 in figure [ explainmethod ] . a method for choosing @xmath50 is described in  [ bandselect ] . also , the constant @xmath85 is present in all of these estimates , but it will turn out in the next step that this is exactly what we need : there is no need to renormalize and get separate estimates of @xmath81 and @xmath85 .    in this next step , @xmath82 will be estimated holding @xmath118 fixed at its current estimate . to ease notation , define @xmath120 with an estimate of @xmath118 in hand , now let @xmath121 denote the density for the observable @xmath79 so that since @xmath122 it follows that @xmath123 now consider local models of the form @xmath124 where @xmath125 and now @xmath126 is the logarithm of the offset ; note that an estimator for this was found above in equation ( [ estoff ] ) : @xmath127d{z}.\ ] ] this leaves @xmath128 as an estimator for @xmath82 . this is then used to reestimate the offset term used in equation ( [ offsetexp ] ) , and the process repeats . this is conceptually the same procedure as was used to create @xmath94 above , since the estimate of @xmath129 is found by alternating estimating @xmath81 and @xmath82 . this section will tie together the ideas of the previous . the iterative procedure described above is computationally tractable , and has intuitive appeal . remarkably , it is also possible to pose the estimation problem in another manner which is not as computationally useful , but will lead to analytical results . define @xmath130     \left [     \sum_{u \in { \cal g } }      k^*{\hspace{-.04in}}\left({z},u,\lambda\right )     \exp\!\left (     { \bf a}_u\!\left({z}\right)\right )     \right ]     d{m}\:d{z}\right\}\label{critind } \ ] ] as the global criterion . it is a function of both families of local models , @xmath4 and @xmath5 . the key is to notice that if @xmath5 is held fixed at its current estimate @xmath131 , maximizing @xmath132 over local models @xmath4 is identical to maximizing @xmath133 with fixed estimate of the offset term . to see this , recall equation ( [ foffset ] ) and note that an estimator for @xmath134 is @xmath135 and from equations ( [ callf ] ) and ( [ loclike ] ) , @xmath136     \right ] d{z}\\     & & \hspace{-1.2 in } = { k'}+ \sum_{j=1}^n     \left\ {     \sum_{u \in { \cal g } }     k^*{\hspace{-.04in}}\left({z}_j , u,\lambda\right ) { \bf a}_u\!\left({z}_j\right ) \right.\\     & & \hspace{-1.0in}\left.- \!\int_{\cal a }      \left[\sum_{u \in { \cal g}}k^*{\hspace{-.04in}}\left({z},u,\lambda\right )     \exp\!\left({\bf a}_u\!\left({z}\right)\right)\right ]     \left [     \sum_{v \in { \cal g } } k^*\!\left({m},v,\lambda\right )      \exp(\widehat { \bf b}_v\!\left({m}\right))\right ] d{z}\:d{m}\right\ } \\     & & \hspace{-1.2 in } = { k''}+ { \cal l}^*\!\left({\bf f } , \widehat { \bf g } , { \bf { z}},{\bf { m}}\right)\end{aligned}\ ] ] where @xmath137 and @xmath138 are constants which do not depend on @xmath4 , and @xmath139 and @xmath140 are the lower and upper bounds on redshift , respectively . an analogous statement could be made for finding @xmath5 when @xmath58 is held fixed . thus , the iterative search method described in  [ itermethod ] is equivalent to maximizing this global criterion .      until now , the derivation of the approach has assumed that random variables @xmath12 and @xmath79 are independent . dependence will be incorporated by including a parametric portion @xmath18 so that the assumption becomes that @xmath141 a restriction placed on @xmath6 is that it must be linear in the real - valued parameters @xmath33 . in the absence of a physically - motivated model , a useful first - order approximation is @xmath19 . the global criterion of equation ( [ critind ] ) is naturally updated to @xmath142     \!\left [     \sum_{u \in { \cal g } }      k^*{\hspace{-.04in}}\left({z},u,\lambda\right )     \exp\!\left (     { \bf a}_u\!\left({z}\right)\right )     \right ]     \!d{m}\:d{z}\right\}.     \label{globcrit}\end{aligned}\ ] ] note that with this form , when @xmath4 and @xmath5 are held constant , maximizing @xmath143 over @xmath33 is equivalent to finding the maximum likelihood estimate of @xmath33 . note also that the sum over the @xmath10 data pairs has also been updated to allow specification of a weight @xmath144 . in this case , the natural choice for the weight is the inverse of the selection function for that data pair . the intuition is that a pair with selection function of 0.5 is `` like '' two observations at that location .    finally , with a criterion of this form , this estimator can be fit into a general class of statistical procedures called _ m - estimators_. see the appendix (  [ mest ] ) for an overview of m - estimators . the described procedure returns an estimate normalized to be a probability density over the observable region @xmath76 . of course , it could be renormalized to meet the goals of the analysis , but care should be taken if the renormalization involves multiplying by a constant which is itself estimated from the data . in certain cases , namely when there is a small sample , this could result in significantly understated standard errors . luminosity curves are usually stated in units of @xmath145 , and are obtained by multiplying the bivariate density ( normalized to be a probability density over @xmath76 ) by a redshift - dependent constant ; thus no adjustment of the standard errors is needed in this case . the choice of the bandwidth @xmath50 ( the smoothing parameter ) is critical . choosing @xmath50 too large results in an oversmoothed , highly biased estimator ; choosing @xmath50 too small leads to a rough , highly variable estimator . this is the _ bias / variance tradeoff_. fortunately , it is possible to select @xmath50 to balance these two in a meaningful , objective manner . although this discussion applies in general to the problem of density estimation , here it will be described in terms of estimating the bivariate density @xmath146 over @xmath76 . let @xmath147 denote a general estimator for @xmath146 which is a function of a smoothing parameter @xmath50 . then , @xmath148     \:d{z}\:d{m}\nonumber \\      & = &     \int_{\cal a }      \left [     \mbox{variance}\!\left(\widehat { \phi}_{\lambda}\!\left({z},{m}\right)\right )     +     \mbox{bias}^2\!\left(\widehat { \phi}_{\lambda}\!\left({z},{m}\right)\right )     \right ]     \:d{z}\:d{m}\label{balance}\end{aligned}\ ] ] is the _ integrated mean squared error _ for @xmath147 . imse is a natural measure of the error in the estimator , and it is apparent from equation ( [ balance ] ) how it balances the bias and variance of the estimator . although imse can not be calculated , there is an unbiased estimator . it holds that @xmath149 where @xmath85 is a constant which does not depend on @xmath50 , so it can be ignored . let @xmath150 denote the estimate of the density at @xmath151 found using the data set with this @xmath152 data pair removed . following @xcite , @xmath153 so that the _ least - squares cross - validation score _ ( lscv ) , @xmath154 is an unbiased estimator for imse@xmath155 , and hence minimizing it over @xmath50 approximates minimizing the imse . see @xcite and @xcite for theoretical results showing the large - sample optimality of choosing smoothing parameters to minimize this criterion . figure [ explainlscv ] gives an example of bandwidth selection by minimizing lscv . here , 100 simulated values are taken from the gaussian distribution with mean zero and variance one . the left plot shows how lscv varies with the choice of bandwidth , and leads to a choice of @xmath156 . the right plot compares the density estimate using three bandwidths @xmath157 with the true density . with the bandwidth too small , there are nonsmooth features , and the bias is low but the variance is high . with the bandwidth too large , the estimate is smoothing out the prominent peak in the center . here , the variance of the estimate is low , but the bias is high . the optimal choice gives an estimate close to the truth , and is found using a bandwidth which balances estimates of the bias and variance . the weighting due to the selection function needs to be taken into account in the previous discussion . recall that the weight @xmath158 is conceptualized as the number of equivalent observations represented by this data pair . thus `` leaving out '' observation @xmath159 is achieved by reducing its weight from @xmath158 to @xmath160 in the criterion ( equation [ globcrit ] ) . but one must imagine repeating this @xmath158 times ( for each equivalent observation which observation @xmath159 represents ) . let @xmath161 denote the _ effective sample size_. the new relationship is @xmath162 where @xmath150 now indicates the estimator evaluated at @xmath151 when the weight on observation @xmath159 is reduced from @xmath158 to @xmath163 . direct calculation of the leave - one - out estimates would be computationally intractable . @xcite describes an approximation based on the second - order taylor expansion of the criterion function . this approximation proves to be highly accurate and computationally simple . the method described in  [ itermethod ] involves fitting local polynomial models at each of a grid of values @xmath116 , for both the @xmath12 and @xmath79 directions . these derivations were all performed assuming fixed bandwidth @xmath50 used for each of these models . this was merely for notational convenience ; there is no reason that different bandwidths could not be chosen for each of these local models . in fact , given that the variables are on different scales , it would be unreasonable to assume the same bandwidth would be a good choice for each . in the results given in the next section , a stated bandwidth is assumed to be on the scale of the variables after they have been transformed to lie in the unit interval , and bandwidths are given as @xmath164 pairs . allowing the bandwidth to further vary over the different local models gives the overall model fit much flexibility , and lscv can be minimized as before . a full search over this high - dimensional space is not feasible in practice , however . this section describes the results of the application of this method to some real and simulated data sets . in all cases , linear models are fit when doing the local likelihood modeling ( @xmath165 ) , and @xmath56 is a grid of 100 evenly spaced values in both the @xmath92 and @xmath166 dimensions . the parametric portion is set as @xmath19 . bandwidths @xmath164 are stated as proportions of the range for that variable , e.g. @xmath167 means that the bandwidth for the local models for redshift cover 5% of the range @xmath168 . this method was applied to the sample of quasars described in  [ data ] . as stated above , the method is capable of incorporating the selection function via differential weighting in the criterion ( equation ( [ globcrit ] ) ) , but the selection function does present some challenges in this case . for quasars with @xmath169 the selection function drops as low as 0.04 due to difficulty in distinguishing quasars from stars of spectral type a and f. this gives a weight of 25 to these quasars , which would be fine if it were exact , but these weights are calculated based on simulations and @xcite states that the selection function in this region `` is quite sensitive to such uncertain details of the simulation . '' they limit the weight on any observation to 3.0 to account for this . this limit was also imposed in the analysis here . figure [ lscvplot ] shows how lscv varies with @xmath170 and @xmath171 . the criterion is minimized when @xmath172 and @xmath173 . the grid of values at which lscv is calculated is spaced by 0.01 because , as will be seen below , fluctuations of the bandwidths on this scale lead to very little change in the estimates . the minimum value is -0.0078262 , but no significance can be attached to this value , since lscv is not an unbiased estimate of imse , but instead of imse plus an unknown additive constant . figure [ bivest ] shows , using the solid contours , the estimate of the quasar density ( two - dimensional luminosity function ) as a function of @xmath92 and @xmath166 , when @xmath167 and @xmath174 . this estimate is normalized to integrate to one over the entire dashed ( observable ) region . recall from  [ loclikewithoff ] that this is the form which the algorithm provides . fortunately , this is the ideal form for the estimate . the ( effective ) count of quasars in the surveyed region is @xmath175 and the survey covers @xmath176 . thus , the quasar count in a region @xmath177 of @xmath178 space can be estimated using @xmath179.\ ] ] the estimate of @xmath33 is @xmath180 , with a standard error of 0.03 . although it is not possible to assign physical significance to this value for @xmath33 , it is clear that the possibility that @xmath181 is ruled out , and hence there is very strong evidence for evolution of the luminosity function with redshift . this estimate has an apparent irregularity in the shape of the density estimate for @xmath182 . ( note the `` bumps '' in the solid contours for all values of @xmath166 at @xmath182 . ) quasars of this redshift are given larger weight due to interference from stars of spectral type g and k. although it is not possible to be certain , it appears that the weighting may not be sufficiently accurate for the quasars . the weights may be underestimated leading to a corresponding dip in the density estimate . the bandwidth ( @xmath170 ) is sufficiently small to pick up this artifact . in fact , lscv forces the bandwidth to be small enough so it can model this feature . it is hoped that in future work the uncertainty in the weights can be incorporated into lscv . for comparison , another estimate was constructed using @xmath183 for local models centered on redshift values larger than 2.0 , while still using @xmath167 for @xmath184 . this estimate is shown as the dotted contours . the increased smoothing removes the artifact . figure [ margs ] shows the estimated count of quasars with @xmath185 as a function of redshift . as in figure [ bivest ] , the solid curve is the estimate with the lscv - optimal bandwidths , and the dashed estimate is found using the increased smoothing . figure [ allconds ] shows quasar counts as a function of absolute magnitude at a collection of redshift values . comparisons are made with the estimates given in @xcite which were found using the bin - based method of @xcite . the error bars in both figures [ margs ] and [ allconds ] are one standard error , but represent statistical errors only . the error bars do not account for incorrect specification for the parametric form @xmath6 . but , if there is bias from the incorrect specification of @xmath6 , the binned estimates must share these biases . this would be surprising since , while having higher variance , estimates constructed from binning do not make assumptions regarding the evolution of the luminosity function , and hence a well - constructed estimate should be approximately unbiased .    figure [ allconds ] also provides insight into the sensitivity of the estimate to the bandwidth choice . it would be of great concern if small changes in bandwidth led to significant changes in the estimate . to explore this , eight additional estimates were constructed using every possible combination of @xmath186 and @xmath187 . the results are shown as gray curves in each plot of figure [ allconds ] , but are only visible at @xmath188 and @xmath189 . the fluctuations are small relative to the size of the error bars . clearly , the estimates are insensitive to these perturbations . simulations were performed to further explore the behavior of the estimator . for these , the estimate shown in the dotted contours in figure [ bivest ] is taken to be the true bivariate density ; the truncation region is unchanged . the idea is to ask the following : if the truth were , in fact , the estimate found here , would this method be able to reach a good estimate of the density under identical conditions ( same sample size and truncation region ) ? hence , the new data sets were simulated consisting of 16,589 @xmath178 pairs within the observable region . the first of these data sets was utilized to find the optimal smoothing parameters ; these were found to be @xmath190 and @xmath191 . each of the other 19 data sets was analyzed using these values , so that these simulations also provide insight into the adequacy of this approach to bandwidth selection . figure [ simconds ] shows the results from the simulations by comparing estimates of the cross - sections of the estimates @xmath192 at four different redshifts . each dashed curve is an estimate from one of the 20 data sets . the solid curve is the truth . these results show strong agreement between the estimates and the truth over the regions where data are observed . there is some bias in the tails , but this is in regions far from any observable data . in addition , these simulations provide strong evidence that the estimates of the standard errors are accurate : the variability in the estimates is comparable to the size of the error bars . the semiparametric method described here is a strong alternative to previous approaches to estimating luminosity functions . the primary advantage is that it allows one to estimate the evolution of the luminosity function with redshift without assuming a strict parametric form for the bivariate density . instead , one only needs to specify the parametric form for a term which models the dependence between redshift and absolute magnitude . future work will focus on specifying a physically - motivated form for this parametric portion , but the results from the analysis of a sample of quasars reproduce well those from @xcite while only assuming a simple , first - order approximation to the dependence . other portions of the bivariate density are modeled nonparametrically , and are functions of smoothing parameters . using least - squares cross - validation , these smoothing parameters can be chosen in an objective manner , by minimizing a quantity which is a good approximation to the integrated mean squared error . results from simulations show that , with a data set of this size , the method is indeed capable of recapturing the true luminosity curves under the truncation observed in these cosmological data sets . the author gratefully acknowledges the comments of the referee , which greatly improved this paper , and the contributions of peter freeman , chris genovese , and larry wasserman of the department of statistics at carnegie mellon university . the author s work is supported by nsf grants # 0434343 and # 0240019 . funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s . department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium for the participating institutions . the participating institutions are the american museum of natural history , astrophysical institute potsdam , university of basel , cambridge university , case western reserve university , university of chicago , drexel university , fermilab , the institute for advanced study , the japan participation group , johns hopkins university , the joint institute for nuclear astrophysics , the kavli institute for particle astrophysics and cosmology , the korean scientist group , the chinese academy of sciences ( lamost ) , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , ohio state university , university of pittsburgh , university of portsmouth , princeton university , the united states naval observatory , and the university of washington . 15    natexlab#1#1    boyle , b. , shanks , t. , croom , s. , smith , r. , miller , l. , loaring , n. , and heymans , c. 2000 , mnras , 317 , 1014    choloniewski , j. 1986 , mnras , 223 , 1    efron , b. & petrosian , v. 1999 , j. am . assoc . , 94 , 824    efstathiou , g. , ellis , r. , peterson , b. 1988 , mnras , 232 , 431    felten , j. 1976 , apj , 207 , 700    hall , p. 1983 , ann . stat . , 11 , 1156    hjort , n. & jones , m. 1996 , ann . stat . , 24 , 1619    jackson , j. 1974 , mnras , 166 , 281    loader , c. 1996 , ann . stat . , 24 , 1602    lynden - bell , d. 1971 , mnras , 155 , 95    maloney , a. , & petrosian , v. 1999 , apj , 518 , 32    nicoll , j. & segal , i. 1983 , a&a , 118 , 180    page , m. and carrera , f. 2000 , mnras , 433    qin , y .- p . , & xie , g .- z . 1999 , a&a , 341 , 693    richards , g. , et al . 2006 , apj , 131 , 2766    rudemo , m. 1982 , scan . j. of stat . , 9 , 65    sandage , a. , tammann , g. , & yahil , a. 1979 , apj , 232 , 352    schafer , c. ( 2006 ) submitted . available as cmu dept . of stat . tech report # 842 http://www.stat.cmu.edu/tr/tr842/tr842.html    schmidt , m. 1968 , apj , 151 , 393    stone , c. 1983 , ann . , 12 , 1285    takeuchi , t. , yoshikawa , k. , & ishi , t. 2000 , apjs , 129 , 1    willmer , c. 1997 aj , 114 , 898    wang , m .- c . 1989 , j. of amer . 84 , 742    wasserman , l. , miller , c. , nicol , r. , genovese , c. jang , w. , connolly , a. , moore , a. , & schneider , j. 2001 , astro - ph/0112050    woodroofe , m. 1985 , ann . stat . , 13 , 163    york , d. 2000 , aj , 120 , 1579 the procedure described in  [ ourmodel ] can be fit into a general class of statistical estimators called _ m - estimators_. in the simplest case , a m - estimator for a parameter is constructed by maximizing a criterion of the form @xmath193\ ] ] where @xmath194 are the observed data , assumed to be realizations of independent , identically distributed random variables and @xmath195 is the parameter to be estimated . the function @xmath196 is some criterion . for example , in the case of finding the maximum likelihood estimate of @xmath195 , the function @xmath197 , where @xmath198 is the density corresponding to parameter @xmath195 . most least squares problems can be stated as m - estimators . standard theory for m - estimators can be applied to obtain an approximation to the distribution of @xmath199 , which can then be used to find standard errors and form confidence intervals .    in the case at hand , @xmath32 is the pair @xmath200 , @xmath201 , and @xmath202     \left [     \sum_{u \in { \cal g } }      k^*{\hspace{-.04in}}\left({z},u,\lambda\right )     \exp\!\left (     { \bf a}_u\!\left({z}\right)\right )     \right ]     d{m}\:d{z}. \ ] ] see schafer ( 2006 ) to see the derivations of the approximate distribution for the estimator in this case . the m - estimator could be generalized to the following : @xmath203\ ] ] where @xmath204 is the weight given to the @xmath152 observation . this allows for easy incorporation of the selection function into the analysis . the statistical theory for this _ weighted m - estimator _ is a simple extension of that for the standard m - estimator .
the observational limitations of astronomical surveys lead to significant statistical inference challenges . 
 one such challenge is the estimation of luminosity functions given redshift @xmath0 and absolute magnitude @xmath1 measurements from an irregularly truncated sample of objects . 
 this is a bivariate density estimation problem ; we develop here a statistically rigorous method which ( 1 ) does not assume a strict parametric form for the bivariate density ; ( 2 ) does not assume independence between redshift and absolute magnitude ( and hence allows evolution of the luminosity function with redshift ) ; ( 3 ) does not require dividing the data into arbitrary bins ; and ( 4 ) naturally incorporates a varying selection function . 
 we accomplish this by decomposing the bivariate density @xmath2 via @xmath3 where @xmath4 and @xmath5 are estimated nonparametrically , and @xmath6 takes an assumed parametric form . 
 there is a simple way of estimating the integrated mean squared error of the estimator ; smoothing parameters are selected to minimize this quantity . 
 results are presented from the analysis of a sample of quasars .