Converting SQL to LINQ With Sub Query

When building LINQ query, try do decompose parts for building final query.

var positions = 
    from dp in ctx.CCDailyPositions
    group by new { dp.masterline_id } into g
    select new 
    {
        ID = g.Max(x => x.ID),
        dp.masterline_id
    };

var purchases = 
    from pr in ctx.CCPurchase
    group by new { pr.masterline_id } into g
    select new 
    {
        ID = g.Max(x => x.ID),
        pr.masterline_id
    };

var query = 
    from m in ctx.CCMemberizationPII
    from dp in positions.Where(dp => dp.masterline_id == m.application_id)
        .DefaultIfEmpty()
    from pr in purchases.Where(pr => pr.masterline_id == m.application_id)
        .DefaultIfEmpty()
    orderby m.Id
    select new 
    {
        m.Id, 
        m.borrower_id, 
        m.application_id,  
        m.borrower_first_name,
        m.borrower_last_name, 
        DP_ID = dp.ID, 
        PR_ID = pr.ID, 
    };

Also your query can be written using OUTER APPLY. Note that EF Core can transform OUTER APPLY to LEFT JOIN with Window Function usage.

var query = 
    from m in ctx.CCMemberizationPII
    from dp in ctx.CCDailyPositions.Where(dp => dp.masterline_id == m.application_id)
        .OrderByDescending(dp => dp.ID)
        .Take(1)
        .DefaultIfEmpty()
    from pr in ctx.CCPurchase.Where(pr => pr.masterline_id == m.application_id)
        .OrderByDescending(pr => pr.ID)
        .Take(1)
        .DefaultIfEmpty()
    orderby m.Id
    select new 
    {
        m.Id, 
        m.borrower_id, 
        m.application_id,  
        m.borrower_first_name,
        m.borrower_last_name, 
        DP_ID = dp.ID, 
        PR_ID = pr.ID, 
    };

Which SQL will be better, depends on execution plan in your particular case.