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depth, node sizes, splitting criteria, and overall tre e structure. These hyperparame- ters are crucial for achieving accurate predictions in finit e-sample settings. However, the- oretical analyses of tree models often concentrate on their fundamental structural prop- erties, while overlooking finer details such as the ... | https://arxiv.org/abs/2503.14381v1 |
computational cost at this step is similar to that of trai ning a random forests model, as it introduces at most #S(b)≤2H−1additional splits. In summary, the overall computational 13 cost of training a RF+ S(b)model is comparable to training a random forests model plus o ur first step of optimizing oblique splits. 5 Sim... | https://arxiv.org/abs/2503.14381v1 |
(0.04) 0.01 (0.20) 0.07 (0.27) 0.44 (0.45) 0.90 (0.12) s0= 6 -0.03 (0.03) -0.03 (0.04) -0.03 (0.04) 0.00 (0.18) 0.09 (0.3 1) Table 1: Results for RF+ S(b). Each entry represents the average R2score over 100 indepen- dent trials, with standard deviations in parentheses. The r untime per iteration is approximately 0.0102... | https://arxiv.org/abs/2503.14381v1 |
0.84 (0.03) 0.95 (0.01) 0.93 (0.01) s0= 3 0.04 (0.02) 0.27 (0.05) 0.86 (0.04) 0.92 (0.01) s0= 4 -0.03 (0.01) 0.02 (0.03) 0.37 (0.06) 0.84 (0.04) s0= 5 -0.04 (0.01) -0.04 (0.02) 0.07 (0.04) 0.49 (0.09) s0= 6 -0.04 (0.01) -0.05 (0.02) -0.03 (0.03) 0.13 (0.06) Table 2: Results for Forest-RC. Each entry represents the av e... | https://arxiv.org/abs/2503.14381v1 |
-0.03 (0.03) / 139 sec - 0.13 (0.19) / 131 sec Table 3: Comparison with benchmarks using simulated data fr om (5) with n= 200 and s0= 2. Each entry shows the average R2over 100 independent experiments, with standard deviations in parentheses. The reported runtime r epresents the average across 100 experiments, includin... | https://arxiv.org/abs/2503.14381v1 |
dataset. The names of these datasets are given in Table 5, with other details ava ilable in (Grinsztajn et al., 2022). To assess prediction performance in high-dimensional sett ings, we incorporate feature interactions alongside the original dataset features. Eac h R2score (6) is computed by splitting the dataset into ... | https://arxiv.org/abs/2503.14381v1 |
prediction performance across different values of bis likely due to the inherent randomness in the sampling of split weights. Furthermore, the strong pe rformance of both RF+ S(b) and F-RC compared to RF suggests that the underlying data-ge nerating functions in these real-world prediction problems are complex and benefi... | https://arxiv.org/abs/2503.14381v1 |
should scale linearly with the actual number of iterations p erformed, i.e., (2l−⌊2l−1⌋)×1000, which is approximately observed in the house dataset in Tabl e 6. However, this trend is less evident for superconduct, where hyperparameter optimizat ion and its runtime variation dom- inate progressive tree training. Additi... | https://arxiv.org/abs/2503.14381v1 |
(0.926, 0.903, 0.855) (0.585, 0.502, 0.401) MORF (0.980, 0.964, 0.925) (0.901, 0.855, 0.781) (0.523, 0.455, 0.330) q/backslashbig d Ailerons (594) houses (44) house_16H (152) RF+S(1000)(0.845, 0.789, 0.719) (0.753, 0.716, 0.682) (0.601, 0.341, 0.190) F-RC (0.818, 0.784, 0.712) (0.742, 0.704, 0.626) (0.629, 0.361, 0.068... | https://arxiv.org/abs/2503.14381v1 |
Journal of Machine Learning Research , 17(1):126–151, 2016. Leo Breiman. Random forests. Machine learning , 45:5–32, 2001. Miguel A Carreira-Perpinán and Pooya Tavallali. Alternati ng optimization of decision trees, with application to learning sparse oblique trees. Advances in neural information process- ing systems ,... | https://arxiv.org/abs/2503.14381v1 |
Sameer K Deshpande. Obliqu e bayesian additive regression trees. arXiv preprint arXiv:2411.08849 , 2024. Lorien Y Pratt. Discriminability-based transfer between n eural networks. Advances in neural information processing systems , 5, 1992. Vasilis Syrgkanis and Manolis Zampetakis. Estimation and i nference with trees a... | https://arxiv.org/abs/2503.14381v1 |
klearn.ensemble.RandomForestClassifier.html 1 Parameter name Parameter type Parameter space n_estimators integer 100 min_samples_split integer {1,...,20} min_samples_leaf integer {2,...,20} min_impurity_decrease non-negative real number Uniform {0,0.01,0.02,0.05} max_depth non-negative integer Uniform {5,10,20,50,∞} cr... | https://arxiv.org/abs/2503.14381v1 |
.(B.8) Therefore, RHS of (B.1) ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleE(Y21X∈t′)−n−1/bracketleftBiggn/summationdisplay i=1(Yi1Xi∈t′)2/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsi... | https://arxiv.org/abs/2503.14381v1 |
almost surely. In the fourth equality, we rely on that E(Y|X∈t) =E(m(X)|X∈t). 7 By (B.17) and (B.18), it follows that on B∩D , E{[R(X)−m(X)]2|Xn} =/summationdisplay (t′,(# –w,c),t)∈/hatwideN(b) HE[Var(m(X)|1X∈t)1X∈t] =1 2×/summationdisplay (t′,(# –w,c),t)∈/hatwideN(b) HE[Var(m(X)|1X∈t′,1# –w⊤X>c)1X∈t′] ≤(ρn+CDmaxAn)2H ... | https://arxiv.org/abs/2503.14381v1 |
eachp≤nK01, eachH≥1, each (constant) s≥s0with√ 2n−1 2s/radicalbig H(1∨logs)logn≤ 1, eachB≥(s0logn)3/parenleftbigp s0/parenrightbig , and each #S×b≥D−1 min2H+s+1Blog(2HB), it holds that E{[/hatwideR(X)−m(X)]2}≤( sup # –x∈[0,1]p|m(# –x)|)2×(8n−1+4B−1+αH 0)+ρn2H×C,(B.27) which completes the desired proof of Theorem 4 for ... | https://arxiv.org/abs/2503.14381v1 |
–w,c),t)∈/hatwideN(b) hwith(# –w,c)∈E(ρn,t′,∪B q=1Λq)onB. Now, let one of the optimal split given the full available split be denoted b y (# –w⋆,c⋆)∈E(0,t′,Wp,s0). (B.35) Additionally, let# –u†∈P, in whichPis defined in (B.28), be given such that /vextenddouble/vextenddouble/vextenddouble# –u†−# –w⋆/vextenddouble/vexten... | https://arxiv.org/abs/2503.14381v1 |
–u−# –u†/vextenddouble/vextenddouble/vextenddouble 2√s}.(B.47) Similarly, we have that {|1# –u†⊤X>c⋆−1# –w⋆⊤X>c⋆|= 1}⊂{|# –w⋆⊤X−c⋆|≤4/vextenddouble/vextenddouble/vextenddouble# –u†−# –w⋆/vextenddouble/vextenddouble/vextenddouble 2√s}. (B.48) To bound the probabilities of the RHS of (B.47)–(B.48), we have to establish a... | https://arxiv.org/abs/2503.14381v1 |
deduce that (III)+(IV)+(I)+(II) ≤6( sup # –x∈[0,1]p|m(# –x)|)×P(|1# –u⊤X># –u⊤Xk⋆−1# –w⋆⊤X>c⋆|= 1).(B.59) By (B.52)–(B.53) and (B.40), the result of (B.42) is established. By (B.41) and (B.42), we have completed the proof of the third inequality of (B.37). 16 Optimizing obliques splits B.2.3 Proof of (B.21) We will pro... | https://arxiv.org/abs/2503.14381v1 |
below. Define E2=/intersectiondisplay J⊂{1,...,p},#J=s0/braceleftbig {a∈{0,1}s0: (Xij,j∈J) =afor some i∈{1,...,n}}={0,1}s0/bracerightbig . On the eventE2, for each# –w∈J(s0), splits other than those in {(# –w,# –w⊤Xi) :i∈ {1,...,n}}are equivalent splits to these splits. Specifically, for eac h# –w∈J(s0)and each c∈R, onE2... | https://arxiv.org/abs/2503.14381v1 |
nonzero elements of# –u, which should be s0. According to Condition 4 and (B.60), the probability of achieving this is s−1. (2) Sample a set of s0 coordinates, each corresponding to a nonzero element of the split weight vector# –u. This set must satisfy{j:|wj|>0}={j:|uj|>0}. The probability of achieving this is [/paren... | https://arxiv.org/abs/2503.14381v1 |
the second inequality results from that E(1Fh|1Θmh−1,h−1,Xn)≥ζhalmost surely for some ζh’s, and the third inequality follows from a recursive applic ation of similar reason- ing. Here, we define Θb,0= ΩwithP(Ω) = 1 , and that each ζhis aXn-measurable random variable to be defined shortly in (C.6) below. Also, recall th a... | https://arxiv.org/abs/2503.14381v1 |
We have completed the proof of the first assertion for the cont inuous features of Corollary 3. We now move on to proving the second assertion for the discret e features of Corollary 3. DefineO⋆={(J,# –v) :#J=s,J⊂{1,...,p},# –v∈{0,1}s}. It holds that #O⋆≤2s×/parenleftbiggp s/parenrightbigg . (C.11) DefineZn=/intersectiont... | https://arxiv.org/abs/2503.14381v1 |
–u,a),s)∈/hatwideN(b1) qsuch that n/summationdisplay i=1|1Xi∈t′−1Xi∈s′|= 0. Then, by the arguments in 1), it follows that on Θb1,h, for each (t′,(# –w,c),t)∈/hatwideN(b1+1) q, we haven/summationdisplay i=1|1Xi∈t′−1Xi∈s′|=n/summationdisplay i=1|1Xi∈t−1Xi∈s|= 0 for some (s′,(# –u,a),s)∈/hatwideN(b1) q. This implies that ... | https://arxiv.org/abs/2503.14381v1 |
To derive the desired reuslt, we first have to establish the lo wer bound of inft∈[t6,t7]h# –γ(t) in terms of supt∈RS(t)as follows. Denote t0= inf{t∈R:S(t)>0}, t8= sup{t∈R:S(t)>0}, tl=t0+l×t8−t0 8, δ=t8−t0. In the following, we examine two cases: Case 1 considersdS(t) dt≤0att=t7, while Case 2 considersdS(t) dt>0att=t7. ... | https://arxiv.org/abs/2503.14381v1 |
upper bounded as follows. E[Var(m(X)|1X∈t)1X∈t]≤E{[m(X)−E(m(X)|X∈t)]21X∈t} ≤δ2×P(X∈t).(C.30) 30 Optimizing obliques splits By (C.28)–(C.30), with the split (# –w,c) = (# –γ,E(m(X)|X∈t)), we deduce that inf (# –w′,c′)∈Wp,s0E[Var(m(X)|1X∈t,1# –w′⊤X>c′)1X∈t] ≤E[Var(m(X)|1X∈t,1# –w⊤X>c)1X∈t] ≤E[Var(m(X)|1X∈t)1X∈t] −P(X∈t,#... | https://arxiv.org/abs/2503.14381v1 |
in Example 5 are satisfied wit hK1≤2. Furthermore, the distributional assumptions on Xand other regularity conditions outlined in Example 5 are also fulfilled. Thus, we can apply the results from Exampl e 5 to establish the desired conclusion for Example 4. C.7 Proof that Example 5 satisfies Condition 3 Let us begin with ... | https://arxiv.org/abs/2503.14381v1 |
achieved by the split is proportional to the total variance conditional ontfor each of the following four cases. Specifically, we will show that E[Var(m(X)|1X∈t,1# –w⊤X>c)1X∈t]≤α0×E[Var(m(X)|1X∈t)1X∈t], thereby establishing that Condition 3 is satisfied. 35 Case 1 : #{k >0 :hk/\⌉}atio\slash=∅}= 0and#{k >0 :P(X∈t∩Ck)>0}>1... | https://arxiv.org/abs/2503.14381v1 |
deduce that P(X∈C(2) k∩t)≥D−1 maxDminρP(X∈ t)for each k∈{q:hq/\⌉}atio\slash=∅,1≤q≤K1}, based on the definition of C(2) k. Notably, in this case,tintersectsC0and the second highest and the highest platforms in the kth corner region for each k∈{q:hq/\⌉}atio\slash=∅,1≤q≤K1}. Letδ0=ρDmin 3×K1Dmaxbe given with ρ>0defined by (... | https://arxiv.org/abs/2503.14381v1 |
1X∈t, wheretis a leaf node formed by splits in T. We give a remark for Lemma 6. In this context, the number of fea turesp, the feature vectorX, the regression function m(X), and the model error εare assumed to satisfy the conditions stated in Lemma 6. It is important to note that whe np,X,m(X), andεare given and fixed, ... | https://arxiv.org/abs/2503.14381v1 |
that|# –w⊤X|≤/⌊ar⌈⌊l# –w/⌊ar⌈⌊l2√s=√s. Hence, P(# –w⊤X> c) =P(# –w⊤X≤−c) = 0for every c≥√s, which implies we only need to consider biases within [−√s,√s]. As a result, letC={−√s+t×dn}⌈2√s/dn⌉ t=1 . Moreover, let W(p,s,dn) ={(# –w,c) :# –w∈W,c∈C} (D.3) represent the set of all covering splits to be considered, wh erepis... | https://arxiv.org/abs/2503.14381v1 |
#T=h < H can be seen as a split set of Hsplits but H−hof them are trivial splits. Similarly, P(W13)≤[#W(p,s,dn)+1]H×2H+1×exp/parenleftBigg −nt2 13 2sup # –x∈[0,1]p|m(# –x)|/parenrightBigg ≤exp(−1 2s2(logn)2H(1∨logs)), P(W14)≤[#W(p,s,dn)+1]H×2H+1×exp/parenleftbigg −nt2 14 2Mǫ/parenrightbigg ≤exp(−1 2s2(logn)2H(1∨logs)),... | https://arxiv.org/abs/2503.14381v1 |
Lemma 6 in Section D.1, and recall that each of th ese sets of splits contains a split that is denoted by (# –w,c)with# –w∈J(s)andc∈R. Additionally, recall from the discussion at the beginning of the proof that we only need to consider splits with c∈[−√s,√s]. Now, let T†∈Q, defined in (D.9), be given such that #T†=h(rec... | https://arxiv.org/abs/2503.14381v1 |
most (2s+1)s+1≤2(s+1)2(D.29) 50 Optimizing obliques splits distinct ways to separate the vertices. Note that the expone ntial order of s2is similar to the one in (D.4). With (D.29), the number of distinct sets of sample-equivale nt splits in Wp,s, where two splits(# –w,c)and(# –u,a)are sample-equivalent splits if/summa... | https://arxiv.org/abs/2503.14381v1 |
Now, let us begin the formal proof of Lemma 9. Let S(t,# –w,t) =Area of{# –x∈t:# –w⊤# –x=t}. Here, area refers to the (p−1)-dimensional Lebesgue measure. In light of the definition of tin Lemma 9, it holds that for every# –w∈{# –x∈Rp:/⌊ar⌈⌊l# –x/⌊ar⌈⌊l2= 1}, S(t,# –w,t)is concave with respect to t. (D.33) To see the int... | https://arxiv.org/abs/2503.14381v1 |
lower bounded by Volume of C2∩t Volume of t=Volume of C2∩t Volume of C1∩t+Volume of C2∩t+Volume of C3∩t ≥Volume of C2∩t 2/parenleftBig 2s c0+√s/parenrightBig (a+b)+Volume of C2∩t ≥inf x≥c0 4(a+b)x 2/parenleftBig 2s c0+√s/parenrightBig (a+b)+x ≥c0 4(a+b) 2/parenleftBig 2s c0+√s/parenrightBig (a+b)+c0 4(a+b) =c2 0 16s+8c... | https://arxiv.org/abs/2503.14381v1 |
Robust tests for log-logistic models based on minimum density power divergence estimators Felipe, A., Jaenada, M., Miranda, P. and Pardo, L. Abstract The log-logistic distribution is a versatile parametric family widely used across various applied fields, including survival analysis, reliability engineering, and econom... | https://arxiv.org/abs/2503.14447v1 |
been widely applied across various fields, including Hydrology, Economics, Survival, and Quality Control, among many others. For instance, Shoukri et al. [1988] applied the log-logistic distribution to model precipitation data from some Canadian areas. In Ashkar and Mahdi [2003] the superior fit of the log-logistic dis... | https://arxiv.org/abs/2503.14447v1 |
et al. [2023] and Felipe et al.. The minimum density power divergence estimator (MDPDE) family generalizes the MLE by a tuning parameter τcontrolling the trade-off between efficiency and robustness on the estimation. Since the log-logistic distribution is widely used in different applied fields, defining statistical te... | https://arxiv.org/abs/2503.14447v1 |
Now, consider a set of data coming from an unknown distribution Gand corresponding density function g.To model the underlying distribution, a parametric family Fθ={fθ|θ∈Θ⊂Rk},is assumed. Therefore, we are interested in estimating the distribution parameter θsuch that the assumed parametric distribution is as closely as... | https://arxiv.org/abs/2503.14447v1 |
θ(x)dx−1 τ =1 nnX i=1 1 +1 τβτατβXτ(β−1) i Xβ i+αβ2τ−β ατ Bβτ+τ+β β,βτ−τ+β β −1 τ 4 forτ >0 and Hn,0(β, α) =1 nnX i=1( logβ+βlogXi α−2 log 1 +Xi αβ! +C) , forτ= 0,where Cis a constant that does not depend on the parameters βandα,andB(·,·) the beta function, B(a, b) =Z1 0xa−1(1−x)b−1dx. In the fo... | https://arxiv.org/abs/2503.14447v1 |
log-logistic distribution, X1, ....X n, the Wald test statistic under the null hypothesis (11) is given by Wn(bθ0) =n(bθ0−θ0)TI−1(θ0)(bθ0−θ0), wherebθ0is the MLE of the log-logistic model parameters θ= (α, β) and I(θ0) is the Fisher infor- mation matrix at θ0.If the null hypothesis holds, the Wald test statistic conver... | https://arxiv.org/abs/2503.14447v1 |
null hypotheses given in (14) and (15) should be rejected at significance level αif Wn(bατ)> χ2 1,αandWn bβτ > χ2 1,α respectively. Importantly, if we denote by βWn(α0,β0)(α∗, β∗) the power function of the Wald-type tests under two unknown parameters, we have that lim n→∞Wn(α0, β0) = 1 , i.e., the tests are consisten... | https://arxiv.org/abs/2503.14447v1 |
statistic for testing (11) is given by R(θ0) =nU(θ0)I−1(θ0)U(θ0), 9 with I(θ0) the fisher information matrix and U(θ0) =1 nnX i=1u(Xi,θ0) =1 nnX i=1∂logf θ(x) θ θ=θ0, the score function of the log-logistic model (first derivative of log-likelihood). Similar expressions can be defined for the tests (14) and (15). It can... | https://arxiv.org/abs/2503.14447v1 |
We only prove the result for the score of α,as the proof for βfollows similar steps. We consider the random variable uτ(X, α) where τis fixed. Firs, it is clear that, E[uτ(X, α)] = 0 and V[uτ(X, α)] = Eh (uτ(X, α))2i −E[uτ(X, α)]2 =E"∂logfα,β(X) ∂αfα,β(X)τ−Z∂logfα,β(x) ∂αfα,β(x)τ+1dx2# =E"∂logfα,β(X) ∂α2 fα,β(X)2τ#... | https://arxiv.org/abs/2503.14447v1 |
statistic based on the MDPDE with tuning parameter τfor test- ing the composite null hypothesis (19) is given by Rn eθτ =nUτ eθτT Qτ eθτ Qτ eθτT Kτ eθτ−1 Qτ eθτ−1 Qτ eθτT Uτ eθτ (25) where Uτ(α, β) = (Uτ(α), Uτ(β))TandKτ(α, β)is given in (9). (see Basu et al. [2022] for more details). Consequently, th... | https://arxiv.org/abs/2503.14447v1 |
On the other hand, the robust MDPDEs with moderate and high values of the tuning parameter, above τ= 0.3,keep competitive with low empirical level under contamination, thus demonstrating the advantage of the robust test in contaminated scenarios. (a) No contamination (b) 15% contamination Figure 1: Empirical level of t... | https://arxiv.org/abs/2503.14447v1 |
of the Wald-type test statistics with n= 50 (left) and n= 100 (rigth) under increasing contamination proportion with eα= 6 under different values of τ. scale parameter eα= 0.5 Figures 8 and 9 present the empirical power of the tests in the absence of contamination (left) and under a 15% of contamination (right) against... | https://arxiv.org/abs/2503.14447v1 |
Ghosh, A., Mart´ ın, N., and Pardo, L. (2022). A robust generalization of the Rao test. Journal ofBusiness andEconomic Statistics, 40(2):868–879. Basu, A., Harris, I., Hjort, N., and Jones, M. (1998). Robust and efficient estimation by minimizing a density power divergence. Biometrika, 85:549–559. Basu, A., Mandal, A.,... | https://arxiv.org/abs/2503.14447v1 |
Vetterling, W. (1986). Numerical recipes. Cambridge University Press, Cambridge (UK). Reath, J., Dong, J., and Wang, M. (2018). Improved parameter estimation of the log-logistic distribution with applications. Computational Statistics, 33(1):339–356. Rowinski, P., Strupczewski, W., and Singh, V. (2002). A note on the a... | https://arxiv.org/abs/2503.14447v1 |
arXiv:2503.14467v1 [math.ST] 18 Mar 2025Minimizers of U-processes and their domains of attraction Dietmar Ferger Fakult¨ at Mathematik, Technische Universit¨ at Dresden, Z ellescher Weg 12-14, Dresden, 01069, Germany. Contributing authors: dietmar.ferger@tu-dresden.de ; Abstract In this paper, we study the minimizers o... | https://arxiv.org/abs/2503.14467v1 |
∞as|t| → ∞for allx∈Sl. ThenUniscoercive, i.e.,Un(t)→ ∞as|t| → ∞. In this case the set Argmin( Un) of all minimizing points of Unis a non-empty compact interval. From Proposition 2.5 of Ferger [5] it follows that ∅ /\e}atio\slash= Argmin(Un) ={t∈R:V− n(t)≤0≤V+ n(t)}, (7) where V± n(t) =/parenleftbiggn l/parenrightbigg−1... | https://arxiv.org/abs/2503.14467v1 |
, x>0(c,α>0). (12) class 3:H(x) =/braceleftbigg Φσ(−c|x|α), x<0 Φσ(dxα), x>0(c,d,α> 0). (13) class 4:H(x) = 0, x<−c1 1 2,−c1<x<c 2 1, x>c 2(c1,c2≥0,max{c1,c2}>0).(14) 4 IfYis a real random variable with distribution function H, then the convergence in (9) is equivalent to the distributional convergence mn−m anD→Y. ... | https://arxiv.org/abs/2503.14467v1 |
0 by ( 5) and assumption (A1). Thus the first sum- mand in the decomposition ( 18) is a normalized and centered U-statistic with kernel D+h(x,m). Consequently, (A3) allows us to apply the Central Limit Theorem f or U-statistics, confer, e.g., Proposition 4.2.1 in Koroljuk and Borovsk ich [14], which yields: √n/parenleft... | https://arxiv.org/abs/2503.14467v1 |
Since Φ σis invertible, it follows that δn(x)→Φ−1 σ(H(x)) for every x∈CH. Thus (2) holds with δ(x) = Φ−1 σ(H(x)),x∈R,andD=CH, which is known to be dense in R. Conversely, assume that (2) is true. Then Hn(anx+m) =Rn(x) +Φσ(δn(x))→ Φσ(δ(x)) for allx∈Dby Theorem 2. Now, the functions Gngiven byGn(x) := Hn(anx+m),x∈R,are i... | https://arxiv.org/abs/2503.14467v1 |
δ(x)>0 for every x >0. In fact, assume that there is some x0>0 such that δ(x0)≤0. (28) Then actually δ(x)≤0 for allx>0. To see this fix some x>0 and put yl:=αl kx0, l∈N0. 9 Sinceαk>1 andx0is positive, it follows that yl↑ ∞asl→ ∞. In particular, yl≥xfor some sufficiently large l. Consequently, δ(x)≤δ(yl). (29) But δ(yl) =δ... | https://arxiv.org/abs/2503.14467v1 |
our assumption and thus max {c1,c2}>0. From(33)-(36)weimmediatelyobtainthefourpossibleclasses( 11)-(14)byanother application of Theorem 1(first part). This completes our proof of Theorem 1. 11 3 Domains of attraction of the limits H Now that we have specified all possible limit distributions, the next ste p is to develop... | https://arxiv.org/abs/2503.14467v1 |
ThenV(m+anx)<0 by (38) and√n≥1/V(m+an) by (41) and therefore √nV(m+anx)≤V(m+anx) V(m+an)=V(m+anx) V(m−anx)·V(m−anx) V(m+an).(47) 13 Here, on the rightside the first factoris negativefor all n∈Nand its inverseconverges to zero by condition ( 39) witht=−anx↓0. Thus the first factor converges to −∞, whereas the second facto... | https://arxiv.org/abs/2503.14467v1 |
δ(x)/\e}atio\slash= 0 for allx/\e}atio\slash= 0. Moreover,conditions (55) withA=−d/c<0 and (56) follow from ( 51) and (52). Theorem 7. Suppose that (A1)-(A4) hold with mbeing unique. Then (P,h)∈ D(H), whereH∈class 4 if and only if √nV(a∗ nx+m)→δ(x) = −∞, x<−c1 0,−c1<x<c 2 ∞, x>c 2.(57) 15 Here,c1,c2≥0withmax{c1,c2}... | https://arxiv.org/abs/2503.14467v1 |
require that ζ−>0. (Finiteness of ζ−follows from (B2).) (B4) There exist points s0<0<s1such that E[/parenleftbig D−h(X1,...,X l,m+t)−D−h(X1,...,X l,m)/parenrightbig2]<∞ fort∈ {s0,s1}. In complete analogy to the proof of Theorem 1 (part one), we obta in the following: Interim Conclusion Assume that mis a real number suc... | https://arxiv.org/abs/2503.14467v1 |
x√nx∀0/\e}atio\slash=x∈R.(71) Recall that δn(0) = 0. Therefore, δn(x)→V′(m)·x=δ(x) for allx∈Rand the assertions follow from Theorem 1. Indeed,H(x) = Φ σ(V′(m)x) is the distribution function of N(0,σ2/(V′(m))2), ifV′(m)>0. This shows ( 69). IfV′(m) = 0, then H(x) = 1/2 for allx∈R, which is the distribution function of Y... | https://arxiv.org/abs/2503.14467v1 |
n(˜mn) = 0, whence ( 76) is ful- filled. But, if for instance ψ(x,t) = 1{x≤t}−αwithx,t∈Randα∈(0,1), then V+ n(t) =Fn(t)−α, whereFnis the empirical distribution function pertaining to the 21 sampleX1,...,X n. Consequently, an estimator ˜ mnsatisfying (*) cannot exist, when- everα /∈ {k/n: 1≤k≤n−1}, because (*) reduces to... | https://arxiv.org/abs/2503.14467v1 |
necessarily at zero). Morover, assume that φis differentiable on R\{0}or equivalently that ϕis continuous onR\{0}. IfFis continuous, then there exists a unique m∈Rsatisfying condition 23 (A1). It fulfills the equation /integraldisplay R\{t}ϕ(t−x)R(dx) = 0, t∈R. (86) Forφdifferentiable on the entire real line the continuit... | https://arxiv.org/abs/2503.14467v1 |
is in accordance with the Central Limit Theorem for (non -degenerate) U-statistcs. Ifl= 1, then˜mnreduces to the arithmetic mean Xn=1 n/summationtextn i=1k(Xi)and if in addition S=Randkis equal to the identity, then ˜mnfurther simplifies to the arithmetic mean of the data X1,...,X n. It shouldbe mentioned that Example 1... | https://arxiv.org/abs/2503.14467v1 |
18]. Consequently 0≤1(m+t,∞)(x)ϕ(m+t−x)−ϕ(m−x) t≤ϕ(m+t−x)−ϕ(m−x) t ≤ϕ(m+t∗−x)−ϕ(m−x) t∗∀t∈(0,t∗], wheret∗is any fixed positive real number. Thus the Dominated Convergence Theorem yields that III→/integraldisplay (m,∞)ϕ′(m−x)R(dx)∈[0,∞), t↓0. This showsthat Visrightdifferentiableat m. Forthe derivationofleft differentiabil... | https://arxiv.org/abs/2503.14467v1 |
integral in the denominator. In case l= 1the variance τ2is equal to /integraltext |m−x|2(p−1)R(dx) (p−1)2/integraltext R\{m}|m−x|p−2R(dx). If in addition S=Randk(x) =x, then we obtain the result of Hjort and Pollard [ 11]. In our next example ϕis not continuous but has a jump at point zero. Recall that Fis the distribu... | https://arxiv.org/abs/2503.14467v1 |
Example 6isα=1 2resulting in the median of F. Here, the corresponding φcan be rewritten as φ(t) =1 2|t|. The factor1 2in front of |t|is obviously unneceessary and therefore will be ommitted in the seque l. So now we look ath(x,t) =|t−k(x)|and additionally consider S=R. Example 7. (Hodges-Lehmann estimator ) Here,k(x1,.... | https://arxiv.org/abs/2503.14467v1 |
the limit n→ ∞shows thatFhas a positive right derivative D+F(m) =¯δ(x)/(κρx)>0, because ¯δ(x)>0. Rearranging the last equality leads to ¯δ(x) =κρD+F(m)xfor allx >0. Consequently, δin (99) is given byδ(x) =ρ(κD+F(m) +V′ c(m))x, ifx >0and equal to −∞, ifx <0. Thus with Theorem 1and Slutsky’s Theorem we arrive at √n(˜mn−m... | https://arxiv.org/abs/2503.14467v1 |
and (B1), respectively. Infer from ( 101) of Lemma 3that (B2) is satisfied. To see the validity of (B3) we first simplify our notation: f±(x) :=D±h(x,m). A reformulation of ( 103) yields:Pl(N) = 0, where N={x∈Sl:f+(x)/\e}atio\slash=f−(x)}. An application of Fubini’s theorem yields that 0 =/integraltext SPl−1(Nx1)P(dx1) w... | https://arxiv.org/abs/2503.14467v1 |
K. L. Chung, A Course in Probability Theory , Second Edition, San Diego: Academic Press, Inc., 1974. [5] D. Ferger, Distributional Hyperspace Convergence of Argmin-Sets in C on- vex M-Estimation , Theor. Probability and Math. Statist. 109(2023), 3–35. https://doi.org/10.1090/tpms/1195. [6] D. Ferger, On semi-continuity... | https://arxiv.org/abs/2503.14467v1 |
arXiv:2503.14619v1 [math.ST] 18 Mar 2025The broken sample problem revisited: Proof of a conjecture b y Bai-Hsing and high-dimensional extensions Simiao Jiao Yihong Wu Jiaming Xu∗ March 20, 2025 Abstract We revisit the classical broken sample problem: Two samples of i.i.d. dat a points X= {X1,...,X n}andY={Y1,...,Y m}ar... | https://arxiv.org/abs/2503.14619v1 |
public da tasets (e.g., IMDb) to reveal the identities of anonymized users. Another modern applicatio n arises in e-commerce, where identity fragmentation occursasconsumersusemultipledeviceswit hdifferentidentifiers. Companies, facing fragmented views of user behavior, can use effective linking o f browsing logs across de... | https://arxiv.org/abs/2503.14619v1 |
Gaussian vectors with correlation ρ. The Gaussian case with equal dataset size m=nis considered [7,2] for low dimensions (fixed d) and more recently by [ 14,15,27,33,37] also for high dimensions (dgrowing with n). Example 2 (Bernoulli model) .LetX=Y={0,1}dandPX,Y=p⊗d, wherepdenotes the joint distribution of two Bernoull... | https://arxiv.org/abs/2503.14619v1 |
λ1→1,α= 1, and certain moment assumptions. The second test, leveraging the top- reigenfunctions and singular values (where ris appropriately chosen), succeeds in strong detection provided that/summationtextr k=1λ2 k→ ∞and certain additional moment conditions. Particularizing these general results to the Gaussian mode l... | https://arxiv.org/abs/2503.14619v1 |
results asser t that strong detection is impossible, provided that ρ2= 1−Ω(1) ford= 1 [2,15, Theorem 1] and ρ2< ρ∗(d) ford≥2, whereρ∗(d) the unique root of (1 −x)d=x[14, Theorem 4]. Despite these advances, the sharp threshold for strong dete ction remains open for both low and high dimensions, which we resolve in this ... | https://arxiv.org/abs/2503.14619v1 |
the rec overy problem under the planted model H1, where the goal is to estimate the latent permutation πbased on the XandYsamples. The Gaussian model has received the most attention , for which the maximum likelihood estimator (MLE) reduces to the following linear assignment problem: ˆπML∈argmin π∈Sn1 nn/summationdispl... | https://arxiv.org/abs/2503.14619v1 |
of X,YunderH1conditioned on the latent injection π∈Sm,n, so that P1=EπP1|π. Then L(X,Y) =dP1 dP0(X,Y) =1 |Sm,n|/summationdisplay π∈Sm,ndP1|π dP0(X,Y) =1 |Sm,n|/summationdisplay π∈Sm,nm/productdisplay i=1L(Xπ(i),Yi).(7) Directly computingthislikelihoodratiorequirestocalcu latethepermanent ofallm×msubmatrices of the matr... | https://arxiv.org/abs/2503.14619v1 |
asymptotic value of the cycle index polynomial using P´ olya’s Theorem and the Cauchy inte gral theorem, [ 27] obtains only loose upper bounds via Poisson approximation. 8 3.2 Test statistics and positive results Before presenting sufficient conditions on strong detection , we discuss the common rationale under- lying th... | https://arxiv.org/abs/2503.14619v1 |
covariance structure Σ0=/bracketleftbiggA0 0B/bracketrightbigg ,Σ1=/bracketleftbiggA√αC√αC⊤B/bracketrightbigg , (9) where the left (resp. right) singular vectors of Care given by those of A(resp.B). This reduces the problem of distinguishing two Gaussians with different co variance matrices and we can apply the optimal ... | https://arxiv.org/abs/2503.14619v1 |
were to hold for r=∞, then the sufficient condition ( m/n)I∞ χ2(X;Y)→ ∞would be tight, since I∞ χ2(X;Y) =Iχ2(X;Y) and strong detection is impossible if ( m/n)Iχ2(X;Y) =O(1) as established in Corollary 1. However, the moment condition may not hold for large r, preventing us from obtaining a tight sufficient condition in gen... | https://arxiv.org/abs/2503.14619v1 |
necessary for strong detection. 2In fact, instead of ( 16), it is sufficient to consider the simpler test statistic /bardbl¯X−¯Y/bardbl2for low dimensions ( ρclose to 1) and /an}bracketle{t¯X,¯Y/an}bracketri}htfor high dimensions ( ρclose to 0). The latter is the same test considered in [ 37, Theorem 1]. 3Since we can in... | https://arxiv.org/abs/2503.14619v1 |
discretized likelihood ratio kernel. 13 0.0 0.2 0.4 0.6 0.8 1.0 FPR0.00.20.40.60.81.0TPRα= 1, ρ = 0.9 Optimal likelihood ratio test Histogram test w = 50 Histogram test, w = 100 Histogram test w = 1000 0.0 0.2 0.4 0.6 0.8 1.0 FPR0.00.20.40.60.81.0TPRα= 0.5, ρ = 0.9 Optimal likelihood ratio test Histogram test w = 50 Hi... | https://arxiv.org/abs/2503.14619v1 |
threshold so that the Type-I error is fixed at 0.05 and plot the power (one minus Type-II err or) against the correlation level ρ∈[0.2,0.99]. Also included are the test ( 16) based on sample means (which is a special case of the spectral test for r= 1) and the trivial test (rejecting H0with probability 0 .05 independent... | https://arxiv.org/abs/2503.14619v1 |
panel for α= 0.5). 4 Proof of lower bounds Proof of Theorem 1.We first simplify the expression for E0L2=1 |Sm,n|2/summationdisplay π,π′∈Sm,nE0/bracketleftBiggm/productdisplay i=1L(Xπ(i),Yi)L(Xπ′(i),Yi)/bracketrightBigg =1 |Sm,n|/summationdisplay π∈Sm,nE0/bracketleftBiggm/productdisplay i=1L(Xπ(i),Yi)L(Xi,Yi)/bracketrigh... | https://arxiv.org/abs/2503.14619v1 |
Iis the left/right node set of the 2-core. Therefore, to specifyπ, it is equivalent to adding m−|I|edges between [ m]\Iand [n]\I. Additionally, for I being the 2-core, after adding those m−|I|edges, the induced subgraph of Gπon ([m]\I)×([n]\I) must be a vertex-disjoint collection of paths (cf. Fig. 3). We claim that th... | https://arxiv.org/abs/2503.14619v1 |
Tinner=1√mnn/summationdisplay i=1m/summationdisplay j=1Lr(Xi,Yj) =1√mnn/summationdisplay i=1m/summationdisplay j=1r/summationdisplay k=1λkφk(Xi)ψk(Yj), and the test ηr=/braceleftBigg 1, Tinner>√m n/summationtextr k=1λ2 k 2; 0,otherwise;. Suppose that for a given choice of r(which may depend on n), we have lim n→∞I(r) χ... | https://arxiv.org/abs/2503.14619v1 |
we fi rst project ( s,t) to the space spanned by eigenvectors of Σ 0corresponding to non-zero eigenvalues. Importantly, theleft (resp. right) singular vectors of the off-diagonal block Ccoincide withthose of the diagonal block A(resp.B). Indeed, one can verify that C⊤γ=0andCβ=0. Assume the 22 singular value decomposition... | https://arxiv.org/abs/2503.14619v1 |
Letting α=m/n→0, for fixed d, Corollary 1yields the impossibility condition of 1 −ρ2= Ω(α1/d) and our general results fail to yield any non-trivial upper bound. For growing d, there is a significant gap between the negative result of ρ2≤1 dlogC αand positive result ρ2≫1 dα. •Adapting to the joint distribution : The theor... | https://arxiv.org/abs/2503.14619v1 |
1−3 2q+o(q) = 1−3 2q+o(|ρ|q), where the last equality follows from our assumption|ρ| q→ ∞and high order terms have order at least 2. Similarly, (1−q+ρq)1 2= 1−1−ρ 2q+o(|ρ|q), hence/radicalbig (1−q)3(1−q+ρq) = 1−2q+ρq 2+o(|ρ|q). By the same argument, 2/radicalbig q2(1−q)2(1−ρ) = 2q(1−q)/parenleftbigg 1−1 2ρ+o(|ρ|)/paren... | https://arxiv.org/abs/2503.14619v1 |
that χ2(PG/bardblQG)≤χ2(PM/bardblQM)+ǫ. Note that Mmay not be induced by partitions on XandY. Thus, we define I=/braceleftBig ∩n k=1∩mk i=1C(k) i:C(k) i∈/braceleftBig A(k) i,/parenleftBig A(k) i/parenrightBigc/bracerightBig/bracerightBig , and relabel elements of Ito be{I1,I2,···,Iw}. Sets in Iare disjoint and ∪w k=1Ik=... | https://arxiv.org/abs/2503.14619v1 |
strong detection is impossible accordin g to Corollary 1. •Case 2: growing d. The goal is to show that strong detection is possible if and o nly if ρ2d→ ∞. For sufficiency, let r=d, then we havem nI(r) χ2(X;Y) =m n/summationtextd k=1λ2 k=m nρ2d→ ∞ andLr(x,y) =/summationtextd k=1ρφk(x)ψk(y) =ρ/a\}bracketle{tx,y/a\}bracket... | https://arxiv.org/abs/2503.14619v1 |
m/parenleftBig I(r) χ2(X;Y)/parenrightBig2=EPX,Y/bracketleftbigg/parenleftBig/summationtextd k=1(xk−q)(yk−q)/parenrightBig2/bracketrightbigg αnρ2d2q2(1−q)2 =(1−q)4q(q+ρ(1−q))+q4(1−q)(1−q+ρq)+2q3(1−q)3(1−ρ) αnρ2dq2(1−q)2+d−1 αnd =(1−q)2(q+ρ(1−q)) αnρ2dq+q2(1−q+ρq) αnρ2d(1−q)+2q(1−q)(1−ρ) αnρ2d+d−1 αnd. By condition ρ2d→... | https://arxiv.org/abs/2503.14619v1 |
planted matching pr oblem: Sharp threshold and infinite-order phase transition. Probability Theory and Related Fields , 187(1-2):1–71, 2023. [13] H. L. Dunn. Record linkage. American Journal of Public Health and the Nations Health , 36(12):1412–1416, 1946. [14] D. Elimelech and W. Huleihel. Phase transitions in the d et... | https://arxiv.org/abs/2503.14619v1 |
On the Precise Asymptotics of Universal Inference Kenta Takatsu Department of Statistics and Data Science, Carnegie Mellon University Abstract In statistical inference, confidence set procedures are typically evaluated based on their validity and width properties. Even when procedures achieve rate-optimal widths, confi... | https://arxiv.org/abs/2503.14717v1 |
(Cox, 1975) and Markov’s inequality to establish finite-sample validity. The procedure follows three steps. Suppose independent and identically distributed (IID) observations Z1, . . . , Z nfollow the distribution Pθ0, belonging to a collection of distributions {Pθ:θ∈Θ}, for an arbitrary set Θ. The procedure is as foll... | https://arxiv.org/abs/2503.14717v1 |
extreme conservativeness has been a pervasive limitation among universal inference and its derivatives. The two aforementioned issues of over-conservativeness and model misspecification can be addressed by slight changes in perspective. The conservativeness of universal inference might be mitigated if one is willing to... | https://arxiv.org/abs/2503.14717v1 |
. While the conservative property was mentioned in Wasserman et al. (2020) and Park et al. (2023), we present a formal explana- tion of this phenomenon and complete the story; (4) To mitigate conservative coverage, we develop a method based on studentization andbias correction . An interesting second-order bias conditi... | https://arxiv.org/abs/2503.14717v1 |
unless additional assumptions or modifications are imposed. Main claims We demonstrate that the nonasymptotic miscoverage probability of the con- fidence set under investigation can be expressed as follows: |PP(θ(P)̸∈CIn,α)− 1−Φ(zα+rn,P) | ≤∆n,P, (5) where Φ( ·) is the cumulative distribution function of the standard... | https://arxiv.org/abs/2503.14717v1 |
may not necessarily belong to the model PΘ. The target of inference is the parameter θG, indexing the distribution PθG∈ P Θwhich minimizes the Kullback–Leibler (KL) divergence between Gand the model PΘ. This parameter is defined by the following KL-projection functional such that θG≡θ(G) := arg min θ∈ΘKL(G∥Pθ). (6) 6 W... | https://arxiv.org/abs/2503.14717v1 |
following subsections will present the main results. 3.1 Extreme Conservativeness of Universal Inference To recall, the universal confidence set is defined as follows. Given IID observations Z1, . . . , Z N following the distribution G, (1) split the index set {1,2, . . . , N }into two non-overlapping setsD1andD2, with... | https://arxiv.org/abs/2503.14717v1 |
(2012). Assumption (A3) can also be dropped but (A5) will become more complicated (See Remark 4). Finally, assumptions (A1) —(A5) are sufficient conditions for uniformly controlling the Berry-Esseen bound over the class of distributions G, though it is not strictly necessary, as discussed in Takatsu and Kuchibhotla (20... | https://arxiv.org/abs/2503.14717v1 |
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